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SUPERGRAVEDAD CUÁNTICA RELATIVISTA.
TEORIZACIÓN INICIAL
RELATIVISTIC QUANTUM SUPERGRAVITY. INITIAL
THEORIZATION
Manuel Ignacio Albuja Bustamante
Investigador independiente
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DOI: https://doi.org/10.37811/cl_rcm.v9i4.18474
Supergravedad cuántica relativista. Teorización inicial
Manuel Ignacio Albuja Bustamante1
ignaciomanuelalbujabustamante@gmail.com
Investigador independiente
RESUMEN
La supergravedad cuántica, en contraposición a la gravedad cuántica, es un estado del espacio – tiempo
cuántico, en el que, una partícula oscura o una partícula estrella, deforman hipergeométricamente el
referido espacio, provocando agujeros negros cuánticos, a razón de su aniquilación o colapso
gravitacional, o en su defecto o simultáneamente, provocando D – dimensiones, en los que es
perfectamente posible la transformación y desplazamiento de materia y energía, de un punto a otro, en
dimensiones distintas, es decir, por fuera de la dimensión en ℝ4. En este punto, la dimensión del tiempo
es maleable a razón de la supergravedad. La dualidad holográfica, es una de las principales
características de este fenómeno, en la medida en que, si bien se tratan de desdoblamientos espaciales,
las leyes de la física no son las mismas, lo que al contrario ocurre con la gravedad cuántica. En este
artículo, me propongo sentar bases formales de la supergravedad cuántica, la misma que, puede ser
endógena o exógena, al tenor de las premisas fundamentales contenidas en la Teoría Cuántica de
Campos Relativistas o Curvos, formulada por este autor en trabajos anteriores. Este trabajo, al igual que
los anteriores y paralelos, se conciben como un intento de unificación.
Palabras clave: supergravedad cuántica, partícula estrella, partícula oscura, supermembranas,
supersimetría de gauge, supersimetría de Yang – Mills
1 Autor principal
Correspondencia: ignaciomanuelalbujabustamante@gmail.com
DOI: https://doi.org/10.37811/cl_rcm.v9i4.18474
Supergravedad cuántica relativista. Teorización inicial
Manuel Ignacio Albuja Bustamante1
ignaciomanuelalbujabustamante@gmail.com
Investigador independiente
RESUMEN
La supergravedad cuántica, en contraposición a la gravedad cuántica, es un estado del espacio – tiempo
cuántico, en el que, una partícula oscura o una partícula estrella, deforman hipergeométricamente el
referido espacio, provocando agujeros negros cuánticos, a razón de su aniquilación o colapso
gravitacional, o en su defecto o simultáneamente, provocando D – dimensiones, en los que es
perfectamente posible la transformación y desplazamiento de materia y energía, de un punto a otro, en
dimensiones distintas, es decir, por fuera de la dimensión en ℝ4. En este punto, la dimensión del tiempo
es maleable a razón de la supergravedad. La dualidad holográfica, es una de las principales
características de este fenómeno, en la medida en que, si bien se tratan de desdoblamientos espaciales,
las leyes de la física no son las mismas, lo que al contrario ocurre con la gravedad cuántica. En este
artículo, me propongo sentar bases formales de la supergravedad cuántica, la misma que, puede ser
endógena o exógena, al tenor de las premisas fundamentales contenidas en la Teoría Cuántica de
Campos Relativistas o Curvos, formulada por este autor en trabajos anteriores. Este trabajo, al igual que
los anteriores y paralelos, se conciben como un intento de unificación.
Palabras clave: supergravedad cuántica, partícula estrella, partícula oscura, supermembranas,
supersimetría de gauge, supersimetría de Yang – Mills
1 Autor principal
Correspondencia: ignaciomanuelalbujabustamante@gmail.com
pág. 2
Relativistic quantum supergravity. Initial theorization
ABSTRACT
Quantum supergravity, as opposed to quantum gravity, is a state of quantum space-time, in which a dark
particle or a star particle hypergeometrically deforms the aforementioned space, causing quantum black
holes, due to their annihilation or gravitational collapse, or failing that or simultaneously, causing D-
dimensions, in which the transformation and displacement of matter and energy is perfectly possible.
from one point to another, in different dimensions, that is, outside the dimension in ℝ4. At this point,
the dimension of time is malleable due to supergravity. Holographic duality is one of the main
characteristics of this phenomenon, insofar as, although they are spatial splittings, the laws of physics
are not the same, which is the opposite of quantum gravity. In this article, I propose to lay the formal
foundations of quantum supergravity, which can be endogenous or exogenous, according to the
fundamental premises contained in the Quantum Theory of Relativistic or Curved Fields, formulated by
this author in previous works. This work, like the previous and parallel ones, is conceived as an attempt
at unification.
Keywords: quantum supergravity, star particle, dark particle, supermembranes, gauge supersymmetry,
yang-mills supersymmetry.
Artículo recibido 07 mayo 2025
Aceptado para publicación: 11 junio 2025
Relativistic quantum supergravity. Initial theorization
ABSTRACT
Quantum supergravity, as opposed to quantum gravity, is a state of quantum space-time, in which a dark
particle or a star particle hypergeometrically deforms the aforementioned space, causing quantum black
holes, due to their annihilation or gravitational collapse, or failing that or simultaneously, causing D-
dimensions, in which the transformation and displacement of matter and energy is perfectly possible.
from one point to another, in different dimensions, that is, outside the dimension in ℝ4. At this point,
the dimension of time is malleable due to supergravity. Holographic duality is one of the main
characteristics of this phenomenon, insofar as, although they are spatial splittings, the laws of physics
are not the same, which is the opposite of quantum gravity. In this article, I propose to lay the formal
foundations of quantum supergravity, which can be endogenous or exogenous, according to the
fundamental premises contained in the Quantum Theory of Relativistic or Curved Fields, formulated by
this author in previous works. This work, like the previous and parallel ones, is conceived as an attempt
at unification.
Keywords: quantum supergravity, star particle, dark particle, supermembranes, gauge supersymmetry,
yang-mills supersymmetry.
Artículo recibido 07 mayo 2025
Aceptado para publicación: 11 junio 2025
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INTRODUCCIÓN.
Existe una diferencia sustancial entre gravedad cuántica y supergravedad cuántica, pues la primera (GC)
consiste esencialmente en la deformación del espacio – tiempo cuántico, a propósito de las interacciones
y simetría fija o corregida de una partícula supermasiva que a razón de su masa y energía, esto es, de su
centro de materia – energía, curva geométricamente el plano cuántico en el que se propaga
vectorialmente y lo torsiona o tuerce, cuya matematización es puramente tensorial y geométrica, esto, a
raíz de un espacio de Hilbert – Einstein y una superficie de Riemann, sin que por esto, se produzcan
pluridimensiones de gauge. Asimismo, la gravedad cuántica (GC), se desarrolla o cumple los parámetros
anteriores, cuando una hiperpartícula, con o sin masa, se aproxima, iguala o supera la velocidad de la
luz (momentum), por lo que, en cualquiera de los casos anteriores, el resultado es la formación de la
curvatura de Dirac del espacio – tiempo cuántico en el que interactúa, lo que sería el caso del taquión,
partícula hipotética que cumple esta descripción. En cuanto a la gravedad cuántica se refiere, la dualidad
holográfica se tiene por inexistente, lo que implica, que para el espacio – tiempo cuántico deformado
geométricamente, operan las mismas leyes de la física cuántica, pues se trata de la misma dimensión en
ℝ4. Ahora bien, la supergravedad cuántica, en contrario a la gravedad cuántica, comporta las siguientes
características: 1) el espacio – tiempo cuántico se deforma hipergeométricamente; 2) comporta la
existencia de D – dimensiones, esto es, supermembranas y por ende, superespacios; 3) comporta la
existencia de agujeros negros cuánticos masivos o supermasivos, según sea el caso; provocados por
aniquilación entre dos partículas supermasivas, una partícula supermasiva y otra partícula repercutida o
por colisión, según las categorías antes referidas; 4) comporta la existencia de supersimetrías de gauge
en invariancia o covariancia; 5) comporta la existencia de dualidad holográfica; 6) la deformación del
espacio – tiempo cuántico opera por una partícula oscura o una partícula estrella, siendo la primera,
aquella que es extremadamente densa pero su energía – momentum es inferior, en tanto que la segunda,
es aquella cuya masa es extremadamente densa al igual que su energía – momentum, por lo que, su
colapso o aniquilación, genera radiación. Sin perjuicio de lo anterior, ambos tipos de gravedad, surgen
de forma endógena o exógena, según corresponda. En este artículo, abordaremos específicamente la
supergravedad cuántica.
INTRODUCCIÓN.
Existe una diferencia sustancial entre gravedad cuántica y supergravedad cuántica, pues la primera (GC)
consiste esencialmente en la deformación del espacio – tiempo cuántico, a propósito de las interacciones
y simetría fija o corregida de una partícula supermasiva que a razón de su masa y energía, esto es, de su
centro de materia – energía, curva geométricamente el plano cuántico en el que se propaga
vectorialmente y lo torsiona o tuerce, cuya matematización es puramente tensorial y geométrica, esto, a
raíz de un espacio de Hilbert – Einstein y una superficie de Riemann, sin que por esto, se produzcan
pluridimensiones de gauge. Asimismo, la gravedad cuántica (GC), se desarrolla o cumple los parámetros
anteriores, cuando una hiperpartícula, con o sin masa, se aproxima, iguala o supera la velocidad de la
luz (momentum), por lo que, en cualquiera de los casos anteriores, el resultado es la formación de la
curvatura de Dirac del espacio – tiempo cuántico en el que interactúa, lo que sería el caso del taquión,
partícula hipotética que cumple esta descripción. En cuanto a la gravedad cuántica se refiere, la dualidad
holográfica se tiene por inexistente, lo que implica, que para el espacio – tiempo cuántico deformado
geométricamente, operan las mismas leyes de la física cuántica, pues se trata de la misma dimensión en
ℝ4. Ahora bien, la supergravedad cuántica, en contrario a la gravedad cuántica, comporta las siguientes
características: 1) el espacio – tiempo cuántico se deforma hipergeométricamente; 2) comporta la
existencia de D – dimensiones, esto es, supermembranas y por ende, superespacios; 3) comporta la
existencia de agujeros negros cuánticos masivos o supermasivos, según sea el caso; provocados por
aniquilación entre dos partículas supermasivas, una partícula supermasiva y otra partícula repercutida o
por colisión, según las categorías antes referidas; 4) comporta la existencia de supersimetrías de gauge
en invariancia o covariancia; 5) comporta la existencia de dualidad holográfica; 6) la deformación del
espacio – tiempo cuántico opera por una partícula oscura o una partícula estrella, siendo la primera,
aquella que es extremadamente densa pero su energía – momentum es inferior, en tanto que la segunda,
es aquella cuya masa es extremadamente densa al igual que su energía – momentum, por lo que, su
colapso o aniquilación, genera radiación. Sin perjuicio de lo anterior, ambos tipos de gravedad, surgen
de forma endógena o exógena, según corresponda. En este artículo, abordaremos específicamente la
supergravedad cuántica.
pág. 4
RESULTADOS Y DISCUSIÓN.
Supergravedad cuántica. Modelo Matemático.
Cálculos preliminares para campos superplanckianos.
𝑚2 = 𝐽
𝛼′
𝑠 = −(𝑝1 + 𝑝2)2 , 𝑡 = −(𝑝2 + 𝑝3)2 , 𝑢 = −(𝑝1 + 𝑝3)2
𝐴𝐽(𝑠, 𝑡) ∼ (−𝑠)𝐽
𝑡 − 𝑀2
𝐴(𝑠, 𝑡) = Γ(−𝛼(𝑠))Γ(−𝛼(𝑡))
Γ(−𝛼(𝑠) − 𝛼(𝑡))
𝛼(𝑠) = 𝛼(0) + 𝛼′𝑠
𝐴(𝑠, 𝑡) = − ∑
∞
𝑛=0
(𝛼(𝑠) + 1) … (𝛼(𝑠) + 𝑛)
𝑛!
1
𝛼(𝑡) − 𝑛
1
𝑀Planck
4 ∫
Λ
0
𝑑𝐸𝐸3 ∼ Λ4
𝑀Planck
4
Partícula Supermasiva. Comportamiento de campo.
𝑆 = 𝑚 ∫
𝑠𝑓
𝑠𝑖
𝑑𝑠 = 𝑚 ∫
𝜏1
𝜏0
𝑑𝜏√−𝜂𝜇𝜈𝑥˙ 𝜇𝑥˙ 𝜈
𝑝𝜇 = − 𝛿𝐿
𝛿𝑥˙ 𝜇 = 𝑚𝑥˙𝜇
√−𝑥˙ 2
𝜕𝜏 ( 𝑚𝑥˙𝜇
√−𝑥˙ 2) = 0
𝑝2 + 𝑚2 = 0
𝐻𝑐𝑎𝑛 = 𝜕𝐿
𝜕𝑥˙ 𝜇 𝑥˙ 𝜇 − 𝐿
𝐻 = 𝑁
2𝑚 (𝑝2 + 𝑚2)
𝑥˙ 𝜇 = {𝑥𝜇, 𝐻} = 𝑁
𝑚 𝑝𝜇 = 𝑁𝑥˙ 𝜇
√−𝑥˙ 2
𝑥˙ 2 = −𝑁2
RESULTADOS Y DISCUSIÓN.
Supergravedad cuántica. Modelo Matemático.
Cálculos preliminares para campos superplanckianos.
𝑚2 = 𝐽
𝛼′
𝑠 = −(𝑝1 + 𝑝2)2 , 𝑡 = −(𝑝2 + 𝑝3)2 , 𝑢 = −(𝑝1 + 𝑝3)2
𝐴𝐽(𝑠, 𝑡) ∼ (−𝑠)𝐽
𝑡 − 𝑀2
𝐴(𝑠, 𝑡) = Γ(−𝛼(𝑠))Γ(−𝛼(𝑡))
Γ(−𝛼(𝑠) − 𝛼(𝑡))
𝛼(𝑠) = 𝛼(0) + 𝛼′𝑠
𝐴(𝑠, 𝑡) = − ∑
∞
𝑛=0
(𝛼(𝑠) + 1) … (𝛼(𝑠) + 𝑛)
𝑛!
1
𝛼(𝑡) − 𝑛
1
𝑀Planck
4 ∫
Λ
0
𝑑𝐸𝐸3 ∼ Λ4
𝑀Planck
4
Partícula Supermasiva. Comportamiento de campo.
𝑆 = 𝑚 ∫
𝑠𝑓
𝑠𝑖
𝑑𝑠 = 𝑚 ∫
𝜏1
𝜏0
𝑑𝜏√−𝜂𝜇𝜈𝑥˙ 𝜇𝑥˙ 𝜈
𝑝𝜇 = − 𝛿𝐿
𝛿𝑥˙ 𝜇 = 𝑚𝑥˙𝜇
√−𝑥˙ 2
𝜕𝜏 ( 𝑚𝑥˙𝜇
√−𝑥˙ 2) = 0
𝑝2 + 𝑚2 = 0
𝐻𝑐𝑎𝑛 = 𝜕𝐿
𝜕𝑥˙ 𝜇 𝑥˙ 𝜇 − 𝐿
𝐻 = 𝑁
2𝑚 (𝑝2 + 𝑚2)
𝑥˙ 𝜇 = {𝑥𝜇, 𝐻} = 𝑁
𝑚 𝑝𝜇 = 𝑁𝑥˙ 𝜇
√−𝑥˙ 2
𝑥˙ 2 = −𝑁2
pág. 5
𝑆 = − 1
2 ∫ 𝑑𝜏𝑒(𝜏)(𝑒−2(𝜏)(𝑥˙ 𝜇)2 − 𝑚2)
𝑆 = − 1
2 ∫ 𝑑𝜏√det𝑔𝜏𝜏(𝑔𝜏𝜏𝜕𝜏𝑥 ⋅ 𝜕𝜏𝑥 − 𝑚2)
𝛿𝑥𝜇(𝜏) = 𝑥𝜇(𝜏 + 𝜉(𝜏)) − 𝑥𝜇(𝜏) = 𝜉(𝜏)𝑥˙ 𝜇 + 𝒪(𝜉2)
𝛿𝑆 = 1
2 ∫ 𝑑𝜏 ( 1
𝑒2(𝜏) (𝑥˙ 𝜇)2 + 𝑚2) 𝛿𝑒(𝜏)
𝑒−2𝑥2 + 𝑚2 = 0 → 𝑒 = 1
𝑚 √−𝑥˙ 2
𝛿𝑆 = 1
2 ∫ 𝑑𝜏𝑒(𝜏)(𝑒−2(𝜏)2𝑥˙ 𝜇)𝜕𝜏𝛿𝑥𝜇
𝜕𝜏(𝑒−1𝑥˙ 𝜇) = 0
⟨𝑥 ∣ 𝑥′⟩ = 𝑁 ∫
𝑥(1)=𝑥′
𝑥(0)=𝑥
𝐷𝑒𝐷𝑥𝜇exp (1
2 ∫
1
0
(1
𝑒 (𝑥˙ 𝜇)2 − 𝑒𝑚2) 𝑑𝜏)
𝛿𝑒 = 𝜕𝜏(𝜉𝑒)
𝐿 = ∫
1
0
𝑑𝜏√det𝑔𝜏𝜏 = ∫
1
0
𝑑𝜏𝑒
⟨𝑥 ∣ 𝑥′⟩ = 𝑁 ∫
∞
0
𝑑𝐿 ∫
𝑥(1)=𝑥′
𝑥(0)=𝑥
𝐷𝑥𝜇exp (− 1
2 ∫
1
0
(1
𝐿 𝑥˙ 2 + 𝐿𝑚2) 𝑑𝜏)
𝑥𝜇(𝜏) = 𝑥𝜇 + (𝑥′𝜇 − 𝑥𝜇)𝜏 + 𝛿𝑥𝜇(𝜏),
‖𝛿𝑥‖2 = ∫
1
0
𝑑𝜏𝑒(𝛿𝑥𝜇)2 = 𝐿 ∫
1
0
𝑑𝜏(𝛿𝑥𝜇)2
𝐷𝑥𝜇 ∼ ∏
𝜏
√𝐿𝑑𝛿𝑥𝜇(𝜏)
⟨𝑥 ∣ 𝑥′⟩ = 𝑁 ∫
∞
0
𝑑𝐿 ∫ ∏
𝜏
√𝐿𝑑𝛿𝑥𝜇(𝜏)𝑒−(𝑥′−𝑥)2
2𝐿 −𝑚2𝐿/2𝑒− 1
2𝐿 ∫1
0 (𝛿𝑥𝜇)2
∫ ∏
𝜏
√𝐿𝑑𝛿𝑥𝜇(𝜏)𝑒−1
𝐿 ∫1
0 (𝛿𝑥˙ 𝜇)2
∼ (det (− 1
𝐿 𝜕𝜏
2))
−𝐷
2
− 1
𝐿 𝜕𝜏
2𝜓(𝜏) = 𝜆𝜓(𝜏)
𝑆 = − 1
2 ∫ 𝑑𝜏𝑒(𝜏)(𝑒−2(𝜏)(𝑥˙ 𝜇)2 − 𝑚2)
𝑆 = − 1
2 ∫ 𝑑𝜏√det𝑔𝜏𝜏(𝑔𝜏𝜏𝜕𝜏𝑥 ⋅ 𝜕𝜏𝑥 − 𝑚2)
𝛿𝑥𝜇(𝜏) = 𝑥𝜇(𝜏 + 𝜉(𝜏)) − 𝑥𝜇(𝜏) = 𝜉(𝜏)𝑥˙ 𝜇 + 𝒪(𝜉2)
𝛿𝑆 = 1
2 ∫ 𝑑𝜏 ( 1
𝑒2(𝜏) (𝑥˙ 𝜇)2 + 𝑚2) 𝛿𝑒(𝜏)
𝑒−2𝑥2 + 𝑚2 = 0 → 𝑒 = 1
𝑚 √−𝑥˙ 2
𝛿𝑆 = 1
2 ∫ 𝑑𝜏𝑒(𝜏)(𝑒−2(𝜏)2𝑥˙ 𝜇)𝜕𝜏𝛿𝑥𝜇
𝜕𝜏(𝑒−1𝑥˙ 𝜇) = 0
⟨𝑥 ∣ 𝑥′⟩ = 𝑁 ∫
𝑥(1)=𝑥′
𝑥(0)=𝑥
𝐷𝑒𝐷𝑥𝜇exp (1
2 ∫
1
0
(1
𝑒 (𝑥˙ 𝜇)2 − 𝑒𝑚2) 𝑑𝜏)
𝛿𝑒 = 𝜕𝜏(𝜉𝑒)
𝐿 = ∫
1
0
𝑑𝜏√det𝑔𝜏𝜏 = ∫
1
0
𝑑𝜏𝑒
⟨𝑥 ∣ 𝑥′⟩ = 𝑁 ∫
∞
0
𝑑𝐿 ∫
𝑥(1)=𝑥′
𝑥(0)=𝑥
𝐷𝑥𝜇exp (− 1
2 ∫
1
0
(1
𝐿 𝑥˙ 2 + 𝐿𝑚2) 𝑑𝜏)
𝑥𝜇(𝜏) = 𝑥𝜇 + (𝑥′𝜇 − 𝑥𝜇)𝜏 + 𝛿𝑥𝜇(𝜏),
‖𝛿𝑥‖2 = ∫
1
0
𝑑𝜏𝑒(𝛿𝑥𝜇)2 = 𝐿 ∫
1
0
𝑑𝜏(𝛿𝑥𝜇)2
𝐷𝑥𝜇 ∼ ∏
𝜏
√𝐿𝑑𝛿𝑥𝜇(𝜏)
⟨𝑥 ∣ 𝑥′⟩ = 𝑁 ∫
∞
0
𝑑𝐿 ∫ ∏
𝜏
√𝐿𝑑𝛿𝑥𝜇(𝜏)𝑒−(𝑥′−𝑥)2
2𝐿 −𝑚2𝐿/2𝑒− 1
2𝐿 ∫1
0 (𝛿𝑥𝜇)2
∫ ∏
𝜏
√𝐿𝑑𝛿𝑥𝜇(𝜏)𝑒−1
𝐿 ∫1
0 (𝛿𝑥˙ 𝜇)2
∼ (det (− 1
𝐿 𝜕𝜏
2))
−𝐷
2
− 1
𝐿 𝜕𝜏
2𝜓(𝜏) = 𝜆𝜓(𝜏)
pág. 6
𝜓𝑛(𝜏) = 𝐶𝑛sin (𝑛𝜋𝜏) , 𝜆𝑛 = 𝑛2
𝐿 , 𝑛 = 1,2, …
det (− 1
𝐿 𝜕𝜏
2) = ∏
∞
𝑛=1
𝑛2
𝐿
∏
∞
𝑛=1
𝐿−1 = 𝐿−𝜁(0) = 𝐿1/2 , ∏
∞
𝑛=1
𝑛𝑎 = 𝑒−𝑎𝜁′(0) = (2𝜋)𝑎/2
⟨𝑥 ∣ 𝑥′⟩ = 1
2(2𝜋)𝐷/2 ∫
∞
0
𝑑𝐿𝐿−𝐷
2 𝑒−(𝑥′−𝑥)2
2𝐿 −𝑚2𝐿/2 =
= 1
(2𝜋)𝐷/2 (|𝑥 − 𝑥′|
𝑚 )
(2−𝐷)/2
𝐾(𝐷−2)/2(𝑚|𝑥 − 𝑥′|)
|𝑝⟩ = ∫ 𝑑𝐷𝑥𝑒𝑖𝑝⋅𝑥|𝑥⟩
⟨𝑝 ∣ 𝑝′⟩ = ∫ 𝑑𝐷𝑥𝑒−𝑖𝑝⋅𝑥 ∫ 𝑑𝐷𝑥′𝑒𝑖𝑝′⋅𝑥′
⟨𝑥 ∣ 𝑥′⟩
= 1
2 ∫ 𝑑𝐷𝑥′𝑒𝑖(𝑝′−𝑝)⋅𝑥′
∫
∞
0
𝑑𝐿𝑒−𝐿
2(𝑝2+𝑚2)
= (2𝜋)𝐷𝛿(𝑝 − 𝑝′) 1
𝑝2 + 𝑚2
Campos cuánticos relativistas o curvos. Propagadores, operadores, números fantasma,
supersimetrías, antisimetrías, gravedad supermembranas y superespacios y pluridimensiones en
un campo de Hilbert – Einstein. Métrica de Nambu – Goto.
𝑆𝑁𝐺 = −𝑇 ∫ 𝑑𝐴
𝑑𝑠2 = 𝐺𝜇𝜈(𝑋)𝑑𝑋𝜇𝑑𝑋𝜈 = 𝐺𝜇𝜈
𝜕𝑋𝜇
𝜕𝜉𝑖
𝜕𝑋𝜈
𝜕𝜉𝑗 𝑑𝜉𝑖𝑑𝜉𝑗 = 𝐺𝑖𝑗𝑑𝜉𝑖𝑑𝜉𝑗
𝐺𝑖𝑗 = 𝐺𝜇𝜈𝜕𝑖𝑋𝜇𝜕𝑗𝑋𝜈
𝑆𝑁𝐺 = −𝑇 ∫ √−det𝐺𝑖𝑗𝑑2𝜉 = −𝑇 ∫ √(𝑋˙ ⋅ 𝑋′)2 − (𝑋˙ 2)(𝑋′2)𝑑2𝜉
𝜕𝜏 ( 𝛿𝐿
𝛿𝑋˙𝜇) + 𝜕𝜎 ( 𝛿𝐿
𝛿𝑋𝜇) = 0
𝑋𝜇(𝜎 + 𝜎‾) = 𝑋𝜇(𝜎)
• Neumann :
𝛿𝐿
𝛿𝑋′𝜇|𝜎=0,𝜎‾
= 0
𝜓𝑛(𝜏) = 𝐶𝑛sin (𝑛𝜋𝜏) , 𝜆𝑛 = 𝑛2
𝐿 , 𝑛 = 1,2, …
det (− 1
𝐿 𝜕𝜏
2) = ∏
∞
𝑛=1
𝑛2
𝐿
∏
∞
𝑛=1
𝐿−1 = 𝐿−𝜁(0) = 𝐿1/2 , ∏
∞
𝑛=1
𝑛𝑎 = 𝑒−𝑎𝜁′(0) = (2𝜋)𝑎/2
⟨𝑥 ∣ 𝑥′⟩ = 1
2(2𝜋)𝐷/2 ∫
∞
0
𝑑𝐿𝐿−𝐷
2 𝑒−(𝑥′−𝑥)2
2𝐿 −𝑚2𝐿/2 =
= 1
(2𝜋)𝐷/2 (|𝑥 − 𝑥′|
𝑚 )
(2−𝐷)/2
𝐾(𝐷−2)/2(𝑚|𝑥 − 𝑥′|)
|𝑝⟩ = ∫ 𝑑𝐷𝑥𝑒𝑖𝑝⋅𝑥|𝑥⟩
⟨𝑝 ∣ 𝑝′⟩ = ∫ 𝑑𝐷𝑥𝑒−𝑖𝑝⋅𝑥 ∫ 𝑑𝐷𝑥′𝑒𝑖𝑝′⋅𝑥′
⟨𝑥 ∣ 𝑥′⟩
= 1
2 ∫ 𝑑𝐷𝑥′𝑒𝑖(𝑝′−𝑝)⋅𝑥′
∫
∞
0
𝑑𝐿𝑒−𝐿
2(𝑝2+𝑚2)
= (2𝜋)𝐷𝛿(𝑝 − 𝑝′) 1
𝑝2 + 𝑚2
Campos cuánticos relativistas o curvos. Propagadores, operadores, números fantasma,
supersimetrías, antisimetrías, gravedad supermembranas y superespacios y pluridimensiones en
un campo de Hilbert – Einstein. Métrica de Nambu – Goto.
𝑆𝑁𝐺 = −𝑇 ∫ 𝑑𝐴
𝑑𝑠2 = 𝐺𝜇𝜈(𝑋)𝑑𝑋𝜇𝑑𝑋𝜈 = 𝐺𝜇𝜈
𝜕𝑋𝜇
𝜕𝜉𝑖
𝜕𝑋𝜈
𝜕𝜉𝑗 𝑑𝜉𝑖𝑑𝜉𝑗 = 𝐺𝑖𝑗𝑑𝜉𝑖𝑑𝜉𝑗
𝐺𝑖𝑗 = 𝐺𝜇𝜈𝜕𝑖𝑋𝜇𝜕𝑗𝑋𝜈
𝑆𝑁𝐺 = −𝑇 ∫ √−det𝐺𝑖𝑗𝑑2𝜉 = −𝑇 ∫ √(𝑋˙ ⋅ 𝑋′)2 − (𝑋˙ 2)(𝑋′2)𝑑2𝜉
𝜕𝜏 ( 𝛿𝐿
𝛿𝑋˙𝜇) + 𝜕𝜎 ( 𝛿𝐿
𝛿𝑋𝜇) = 0
𝑋𝜇(𝜎 + 𝜎‾) = 𝑋𝜇(𝜎)
• Neumann :
𝛿𝐿
𝛿𝑋′𝜇|𝜎=0,𝜎‾
= 0
pág. 7
• Dirichlet :
𝛿𝐿
𝛿𝑋˙𝜇|𝜎=0,𝜎‾
= 0
Π𝜇 = 𝛿𝐿
𝛿𝑋˙𝜇 = −𝑇 (𝑋˙ ⋅ 𝑋′)𝑋′𝜇 − (𝑋′)2𝑋˙𝜇
[(𝑋′ ⋅ 𝑋˙ )2 − (𝑋˙ )2(𝑋′)2]1/2
Π ⋅ 𝑋′ = 0 , Π2 + 𝑇2𝑋′2 = 0
𝐻 = ∫
𝜎‾
0
𝑑𝜎(𝑋˙ ⋅ Π − 𝐿)
𝑆𝑃 = − 𝑇
2 ∫ 𝑑2𝜉√−det𝑔𝑔𝛼𝛽𝜕𝛼𝑋𝜇𝜕𝛽𝑋𝜈𝜂𝜇𝜈
𝑇𝛼𝛽 ≡ − 2
𝑇
1
√−det𝑔
𝛿𝑆𝑃
𝛿𝑔𝛼𝛽 = 𝜕𝛼𝑋 ⋅ 𝜕𝛽𝑋 − 1
2 𝑔𝛼𝛽𝑔𝛾𝛿𝜕𝛾𝑋 ⋅ 𝜕𝛿 𝑋
𝑔𝛼𝛽 = 𝜕𝛼𝑋 ⋅ 𝜕𝛽𝑋
1
√−det𝑔 𝜕𝛼(√−det𝑔𝑔𝛼𝛽𝜕𝛽𝑋𝜇) = 0
𝜆1 ∫ √−det𝑔
𝜆2 ∫ √−det𝑔𝑅(2)
• Poincaré:
𝛿𝑋𝜇 = 𝜔𝜈
𝜇𝑋𝜈 + 𝛼𝜇 , 𝛿𝑔𝛼𝛽 = 0
𝛿𝑔𝛼𝛽 = 𝜉𝛾𝜕𝛾𝑔𝛼𝛽 + 𝜕𝛼𝜉𝛾𝑔𝛽𝛾 + 𝜕𝛽𝜉𝛾𝑔𝛼𝛾 = ∇𝛼𝜉𝛽 + ∇𝛽𝜉𝛼
𝛿𝑋𝜇 = 𝜉𝛼𝜕𝛼𝑋𝜇
𝛿(√−det𝑔) = 𝜕𝛼(𝜉𝛼√−det𝑔)
• Weyl:
𝛿𝑋𝜇 = 0 , 𝛿𝑔𝛼𝛽 = 2Λ𝑔𝛼𝛽
𝛿𝑔𝛼𝛽 = 2Λ(𝑥)𝑔𝛼𝛽 , 𝛿𝜙𝑖 = 𝑑𝑖Λ(𝑥)𝜙𝑖
0 = 𝛿𝑆 = ∫ 𝑑2𝜉 [2 𝛿𝑆
𝛿𝑔𝛼𝛽 𝑔𝛼𝛽 + ∑
𝑖
𝑑𝑖
𝛿𝑆
𝛿𝜙𝑖
𝜙𝑖] Λ
• Dirichlet :
𝛿𝐿
𝛿𝑋˙𝜇|𝜎=0,𝜎‾
= 0
Π𝜇 = 𝛿𝐿
𝛿𝑋˙𝜇 = −𝑇 (𝑋˙ ⋅ 𝑋′)𝑋′𝜇 − (𝑋′)2𝑋˙𝜇
[(𝑋′ ⋅ 𝑋˙ )2 − (𝑋˙ )2(𝑋′)2]1/2
Π ⋅ 𝑋′ = 0 , Π2 + 𝑇2𝑋′2 = 0
𝐻 = ∫
𝜎‾
0
𝑑𝜎(𝑋˙ ⋅ Π − 𝐿)
𝑆𝑃 = − 𝑇
2 ∫ 𝑑2𝜉√−det𝑔𝑔𝛼𝛽𝜕𝛼𝑋𝜇𝜕𝛽𝑋𝜈𝜂𝜇𝜈
𝑇𝛼𝛽 ≡ − 2
𝑇
1
√−det𝑔
𝛿𝑆𝑃
𝛿𝑔𝛼𝛽 = 𝜕𝛼𝑋 ⋅ 𝜕𝛽𝑋 − 1
2 𝑔𝛼𝛽𝑔𝛾𝛿𝜕𝛾𝑋 ⋅ 𝜕𝛿 𝑋
𝑔𝛼𝛽 = 𝜕𝛼𝑋 ⋅ 𝜕𝛽𝑋
1
√−det𝑔 𝜕𝛼(√−det𝑔𝑔𝛼𝛽𝜕𝛽𝑋𝜇) = 0
𝜆1 ∫ √−det𝑔
𝜆2 ∫ √−det𝑔𝑅(2)
• Poincaré:
𝛿𝑋𝜇 = 𝜔𝜈
𝜇𝑋𝜈 + 𝛼𝜇 , 𝛿𝑔𝛼𝛽 = 0
𝛿𝑔𝛼𝛽 = 𝜉𝛾𝜕𝛾𝑔𝛼𝛽 + 𝜕𝛼𝜉𝛾𝑔𝛽𝛾 + 𝜕𝛽𝜉𝛾𝑔𝛼𝛾 = ∇𝛼𝜉𝛽 + ∇𝛽𝜉𝛼
𝛿𝑋𝜇 = 𝜉𝛼𝜕𝛼𝑋𝜇
𝛿(√−det𝑔) = 𝜕𝛼(𝜉𝛼√−det𝑔)
• Weyl:
𝛿𝑋𝜇 = 0 , 𝛿𝑔𝛼𝛽 = 2Λ𝑔𝛼𝛽
𝛿𝑔𝛼𝛽 = 2Λ(𝑥)𝑔𝛼𝛽 , 𝛿𝜙𝑖 = 𝑑𝑖Λ(𝑥)𝜙𝑖
0 = 𝛿𝑆 = ∫ 𝑑2𝜉 [2 𝛿𝑆
𝛿𝑔𝛼𝛽 𝑔𝛼𝛽 + ∑
𝑖
𝑑𝑖
𝛿𝑆
𝛿𝜙𝑖
𝜙𝑖] Λ
pág. 8
𝑇𝛼
𝛼 ∼ 𝛿𝑆
𝛿𝑔𝛼𝛽 𝑔𝛼𝛽 = 0
𝑔𝛼𝛽 = 𝑒2Λ(𝜉)𝜂𝛼𝛽
𝜉+ = 𝜏 + 𝜎 , 𝜉− = 𝜏 − 𝜎
𝑑𝑠2 = −𝑑𝜉+𝑑𝜉−
𝑔++ = 𝑔−− = 0 , 𝑔+− = 𝑔−+ = − 1
2
𝜕± = 1
2 (𝜕𝜏 ± 𝜕𝜎)
• Polyakov:
𝑆𝑃 ∼ 𝑇 ∫ 𝑑2𝜉𝜕+𝑋𝜇𝜕−𝑋𝜈𝜂𝜇𝜈
𝜉+ ⟶ 𝑓(𝜉+) , 𝜉− ⟶ 𝑔(𝜉−)
𝑑(𝑑 + 1)
2 − 𝑑 − 1
𝛿𝑆 = 𝑇 ∫ 𝑑2𝜉(𝛿𝑋𝜇𝜕+𝜕−𝑋𝜇) − 𝑇 ∫
𝜏1
𝜏0
𝑑𝜏𝑋𝜇
′ 𝛿𝑋𝜇
𝑋′𝜇|𝜎=0,𝜎‾ = 0
𝜕+𝜕−𝑋𝜇 = 0
𝑇𝛼𝛽 = 0
𝑇10 = 𝑇01 = 1
2 𝑋˙ ⋅ 𝑋′ = 0 , 𝑇00 = 𝑇11 = 1
4 (𝑋˙ 2 + 𝑋′2) = 0
(𝑋˙ ± 𝑋′)2 = 0
𝑇++ = 1
2 𝜕+𝑋 ⋅ 𝜕+𝑋 , 𝑇−− = 1
2 𝜕−𝑋 ⋅ 𝜕−𝑋 , 𝑇+− = 𝑇−+ = 0
𝜕−𝑇++ + 𝜕+𝑇−+ = 𝜕+𝑇−− + 𝜕−𝑇+− = 0
𝜕−𝑇++ = 𝜕+𝑇−− = 0
𝑄𝑓 = ∫
𝜎‾
0
𝑓(𝜉+)𝑇++(𝜉+)
0 = ∫ 𝑑𝜎𝜕−(𝑓(𝜉+)𝑇++) = 𝜕𝜏𝑄𝑓 + 𝑓(𝜉+)𝑇++|0
𝜎‾
𝑇𝛼
𝛼 ∼ 𝛿𝑆
𝛿𝑔𝛼𝛽 𝑔𝛼𝛽 = 0
𝑔𝛼𝛽 = 𝑒2Λ(𝜉)𝜂𝛼𝛽
𝜉+ = 𝜏 + 𝜎 , 𝜉− = 𝜏 − 𝜎
𝑑𝑠2 = −𝑑𝜉+𝑑𝜉−
𝑔++ = 𝑔−− = 0 , 𝑔+− = 𝑔−+ = − 1
2
𝜕± = 1
2 (𝜕𝜏 ± 𝜕𝜎)
• Polyakov:
𝑆𝑃 ∼ 𝑇 ∫ 𝑑2𝜉𝜕+𝑋𝜇𝜕−𝑋𝜈𝜂𝜇𝜈
𝜉+ ⟶ 𝑓(𝜉+) , 𝜉− ⟶ 𝑔(𝜉−)
𝑑(𝑑 + 1)
2 − 𝑑 − 1
𝛿𝑆 = 𝑇 ∫ 𝑑2𝜉(𝛿𝑋𝜇𝜕+𝜕−𝑋𝜇) − 𝑇 ∫
𝜏1
𝜏0
𝑑𝜏𝑋𝜇
′ 𝛿𝑋𝜇
𝑋′𝜇|𝜎=0,𝜎‾ = 0
𝜕+𝜕−𝑋𝜇 = 0
𝑇𝛼𝛽 = 0
𝑇10 = 𝑇01 = 1
2 𝑋˙ ⋅ 𝑋′ = 0 , 𝑇00 = 𝑇11 = 1
4 (𝑋˙ 2 + 𝑋′2) = 0
(𝑋˙ ± 𝑋′)2 = 0
𝑇++ = 1
2 𝜕+𝑋 ⋅ 𝜕+𝑋 , 𝑇−− = 1
2 𝜕−𝑋 ⋅ 𝜕−𝑋 , 𝑇+− = 𝑇−+ = 0
𝜕−𝑇++ + 𝜕+𝑇−+ = 𝜕+𝑇−− + 𝜕−𝑇+− = 0
𝜕−𝑇++ = 𝜕+𝑇−− = 0
𝑄𝑓 = ∫
𝜎‾
0
𝑓(𝜉+)𝑇++(𝜉+)
0 = ∫ 𝑑𝜎𝜕−(𝑓(𝜉+)𝑇++) = 𝜕𝜏𝑄𝑓 + 𝑓(𝜉+)𝑇++|0
𝜎‾
pág. 9
𝑃𝜇
𝛼 = −𝑇√det𝑔𝑔𝛼𝛽𝜕𝛽𝑋𝜇
𝐽𝜇𝜈
𝛼 = −𝑇√det𝑔𝑔𝛼𝛽(𝑋𝜇𝜕𝛽𝑋𝜈 − 𝑋𝜈𝜕𝛽𝑋𝜇)
𝑃𝜇 = ∫
𝜎‾
0
𝑑𝜎𝑃𝜇
𝜏 , 𝐽𝜇𝜈 = ∫
𝜎‾
0
𝑑𝜎𝐽𝜇𝜈
𝜏
𝜕𝑃𝜇
𝜕𝜏 = 𝑇 ∫
𝜎‾
0
𝑑𝜎𝜕𝜏
2𝑋𝜇 = 𝑇 ∫
𝜎‾
0
𝑑𝜎𝜕𝜎
2𝑋𝜇 = 𝑇(𝜕𝜎𝑋𝜇(𝜎 = 𝜎‾) − 𝜕𝜎𝑋𝜇(𝜎 = 0))
Expansiones oscilatorias.
𝜕+𝜕−𝑋𝜇 = 0
𝑋𝜇(𝜏, 𝜎 + 2𝜋) = 𝑋𝜇(𝜏, 𝜎)
𝑋𝜇(𝜏, 𝜎) = 𝑋𝐿
𝜇(𝜏 + 𝜎) + 𝑋𝑅
𝜇(𝜏 − 𝜎)
𝑋𝐿
𝜇(𝜏 + 𝜎) = 𝑥𝜇
2 + 𝑝𝜇
4𝜋𝑇 (𝜏 + 𝜎) + 𝑖
√4𝜋𝑇 ∑
𝑘≠0
𝛼‾𝑘
𝜇
𝑘 𝑒−𝑖𝑘(𝜏+𝜎)
𝑋𝑅
𝜇(𝜏 − 𝜎) = 𝑥𝜇
2 + 𝑝𝜇
4𝜋𝑇 (𝜏 − 𝜎) + 𝑖
√4𝜋𝑇 ∑
𝑘≠0
𝛼𝑘
𝜇
𝑘 𝑒−𝑖𝑘(𝜏−𝜎)
(𝛼𝑘
𝜇)∗ = 𝛼−𝑘
𝜇 and (𝛼‾𝑘
𝜇)∗ = 𝛼‾−𝑘
𝜇
𝜕−𝑋𝑅
𝜇 = 1
√4𝜋𝑇 ∑
𝑘∈ℤ
𝛼𝑘
𝜇𝑒−𝑖𝑘(𝜏−𝜎),
𝜕+𝑋𝐿
𝜇 = 1
√4𝜋𝑇 ∑
𝑘∈ℤ
𝛼‾𝑘
𝜇𝑒−𝑖𝑘(𝜏+𝜎).
𝑋′𝜇(𝜏, 𝜎)|𝜎=0,𝜋 = 0
𝑋′𝜇|𝜎=0 = 𝑝𝜇 − 𝑝‾𝜇
√4𝜋𝑇 + 1
√4𝜋𝑇 ∑
𝑘≠0
𝑒𝑖𝑘𝜏(𝛼‾𝑘
𝜇 − 𝛼𝑘
𝜇)
𝑝𝜇 = 𝑝‾𝜇 and 𝛼𝑘
𝜇 = 𝛼‾𝑘
𝜇
𝑋𝜇(𝜏, 𝜎) = 𝑥𝜇 + 𝑝𝜇𝜏
𝜋𝑇 + 𝑖
√𝜋𝑇 ∑
𝑘≠0
𝛼𝑘
𝜇
𝑘 𝑒−𝑖𝑘𝜏cos (𝑘𝜎)
𝜕±𝑋𝜇 = 1
√4𝜋𝑇 ∑
𝑘∈ℤ
𝛼𝑘
𝜇𝑒−𝑖𝑘(𝜏±𝜎)
𝑋𝐶𝑀
𝜇 ≡ 1
𝜎‾ ∫
𝜎‾
0
𝑑𝜎𝑋𝜇(𝜏, 𝜎) = 𝑥𝜇 + 𝑝𝜇𝜏
𝜋𝑇
𝑃𝜇
𝛼 = −𝑇√det𝑔𝑔𝛼𝛽𝜕𝛽𝑋𝜇
𝐽𝜇𝜈
𝛼 = −𝑇√det𝑔𝑔𝛼𝛽(𝑋𝜇𝜕𝛽𝑋𝜈 − 𝑋𝜈𝜕𝛽𝑋𝜇)
𝑃𝜇 = ∫
𝜎‾
0
𝑑𝜎𝑃𝜇
𝜏 , 𝐽𝜇𝜈 = ∫
𝜎‾
0
𝑑𝜎𝐽𝜇𝜈
𝜏
𝜕𝑃𝜇
𝜕𝜏 = 𝑇 ∫
𝜎‾
0
𝑑𝜎𝜕𝜏
2𝑋𝜇 = 𝑇 ∫
𝜎‾
0
𝑑𝜎𝜕𝜎
2𝑋𝜇 = 𝑇(𝜕𝜎𝑋𝜇(𝜎 = 𝜎‾) − 𝜕𝜎𝑋𝜇(𝜎 = 0))
Expansiones oscilatorias.
𝜕+𝜕−𝑋𝜇 = 0
𝑋𝜇(𝜏, 𝜎 + 2𝜋) = 𝑋𝜇(𝜏, 𝜎)
𝑋𝜇(𝜏, 𝜎) = 𝑋𝐿
𝜇(𝜏 + 𝜎) + 𝑋𝑅
𝜇(𝜏 − 𝜎)
𝑋𝐿
𝜇(𝜏 + 𝜎) = 𝑥𝜇
2 + 𝑝𝜇
4𝜋𝑇 (𝜏 + 𝜎) + 𝑖
√4𝜋𝑇 ∑
𝑘≠0
𝛼‾𝑘
𝜇
𝑘 𝑒−𝑖𝑘(𝜏+𝜎)
𝑋𝑅
𝜇(𝜏 − 𝜎) = 𝑥𝜇
2 + 𝑝𝜇
4𝜋𝑇 (𝜏 − 𝜎) + 𝑖
√4𝜋𝑇 ∑
𝑘≠0
𝛼𝑘
𝜇
𝑘 𝑒−𝑖𝑘(𝜏−𝜎)
(𝛼𝑘
𝜇)∗ = 𝛼−𝑘
𝜇 and (𝛼‾𝑘
𝜇)∗ = 𝛼‾−𝑘
𝜇
𝜕−𝑋𝑅
𝜇 = 1
√4𝜋𝑇 ∑
𝑘∈ℤ
𝛼𝑘
𝜇𝑒−𝑖𝑘(𝜏−𝜎),
𝜕+𝑋𝐿
𝜇 = 1
√4𝜋𝑇 ∑
𝑘∈ℤ
𝛼‾𝑘
𝜇𝑒−𝑖𝑘(𝜏+𝜎).
𝑋′𝜇(𝜏, 𝜎)|𝜎=0,𝜋 = 0
𝑋′𝜇|𝜎=0 = 𝑝𝜇 − 𝑝‾𝜇
√4𝜋𝑇 + 1
√4𝜋𝑇 ∑
𝑘≠0
𝑒𝑖𝑘𝜏(𝛼‾𝑘
𝜇 − 𝛼𝑘
𝜇)
𝑝𝜇 = 𝑝‾𝜇 and 𝛼𝑘
𝜇 = 𝛼‾𝑘
𝜇
𝑋𝜇(𝜏, 𝜎) = 𝑥𝜇 + 𝑝𝜇𝜏
𝜋𝑇 + 𝑖
√𝜋𝑇 ∑
𝑘≠0
𝛼𝑘
𝜇
𝑘 𝑒−𝑖𝑘𝜏cos (𝑘𝜎)
𝜕±𝑋𝜇 = 1
√4𝜋𝑇 ∑
𝑘∈ℤ
𝛼𝑘
𝜇𝑒−𝑖𝑘(𝜏±𝜎)
𝑋𝐶𝑀
𝜇 ≡ 1
𝜎‾ ∫
𝜎‾
0
𝑑𝜎𝑋𝜇(𝜏, 𝜎) = 𝑥𝜇 + 𝑝𝜇𝜏
𝜋𝑇
pág. 10
𝑝𝐶𝑀
𝜇 = 𝑇 ∫
𝜎‾
0
𝑑𝜎𝑋˙𝜇 = 𝑇
√4𝜋𝑇 ∫ 𝑑𝜎 ∑
𝑘
(𝛼𝑘
𝜇 + 𝛼‾𝑘
𝜇)𝑒−𝑖𝑘(𝜏±𝜎) = 2𝜋𝑇
√4𝜋𝑇 (𝛼0
𝜇 + 𝛼‾0
𝜇) = 𝑝𝜇
𝐽𝜇𝜈 = 𝑇 ∫
𝜎‾
0
𝑑𝜎(𝑋𝜇𝑋˙ 𝜈 − 𝑋𝜈𝑋˙𝜇) = 𝑙𝜇𝜈 + 𝐸𝜇𝜈 + 𝐸‾𝜇𝜈
𝑙𝜇𝜈 = 𝑥𝜇𝑝𝜈 − 𝑥𝜈𝑝𝜇
𝐸𝜇𝜈 = −𝑖 ∑
∞
𝑛=1
1
𝑛 (𝛼−𝑛
𝜇 𝛼𝑛
𝜈 − 𝛼−𝑛
𝜈 𝛼𝑛
𝜇)
𝐸‾𝜇𝜈 = −𝑖 ∑
∞
𝑛=1
1
𝑛 (𝛼‾−𝑛
𝜇 𝛼‾𝑛
𝜈 − 𝛼‾−𝑛
𝜈 𝛼‾𝑛
𝜇)
{𝑋𝜇(𝜎, 𝜏), 𝑋˙ 𝜈(𝜎′, 𝜏)}𝑃𝐵 = 1
𝑇 𝛿(𝜎 − 𝜎′)𝜂𝜇𝜈
{𝛼𝑚
𝜇 , 𝛼𝑛
𝜈} = {𝛼‾𝑚
𝜇 , 𝛼‾𝑛
𝜈} = −𝑖𝑚𝛿𝑚+𝑛,0𝜂𝜇𝜈
{𝛼‾𝑚
𝜇 , 𝛼𝑛
𝜈} = 0, {𝑥𝜇, 𝑝𝜈} = 𝜂𝜇𝜈.
𝐻 = ∫ 𝑑𝜎(𝑋˙ Π − 𝐿) = 𝑇
2 ∫ 𝑑𝜎(𝑋˙ 2 + 𝑋′2)
𝐻 = 1
2 ∑
𝑛∈𝑍
(𝛼−𝑛𝛼𝑛 + 𝛼‾−𝑛𝛼‾𝑛)
𝐻 = 1
2 ∑
𝑛∈𝑍
𝛼−𝑛𝛼𝑛
𝐿𝑚 = 2𝑇 ∫
2𝜋
0
𝑑𝜎𝑇−−𝑒𝑖𝑚(𝜏−𝜎) , 𝐿‾𝑚 = 2𝑇 ∫
2𝜋
0
𝑑𝜎𝑇++𝑒𝑖𝑚(𝜎+𝜏)
𝐿𝑚 = 1
2 ∑
𝑛
𝛼𝑚−𝑛𝛼𝑛 , 𝐿‾𝑚 = 1
2 ∑
𝑛
𝛼‾𝑚−𝑛𝛼‾𝑛
𝐿𝑚
∗ = 𝐿−𝑚 and 𝐿‾𝑚
∗ = 𝐿‾−𝑚
𝐻 = 𝐿0 + 𝐿‾0.
𝐿𝑚 = 2𝑇 ∫
𝜋
0
𝑑𝜎{𝑇−−𝑒𝑖𝑚(𝜏−𝜎) + 𝑇++𝑒𝑖𝑚(𝜎+𝜏)}
𝐿𝑚 = 1
2 ∑
𝑛
𝛼𝑚−𝑛𝛼𝑛
𝐻 = 𝐿0
𝑝𝐶𝑀
𝜇 = 𝑇 ∫
𝜎‾
0
𝑑𝜎𝑋˙𝜇 = 𝑇
√4𝜋𝑇 ∫ 𝑑𝜎 ∑
𝑘
(𝛼𝑘
𝜇 + 𝛼‾𝑘
𝜇)𝑒−𝑖𝑘(𝜏±𝜎) = 2𝜋𝑇
√4𝜋𝑇 (𝛼0
𝜇 + 𝛼‾0
𝜇) = 𝑝𝜇
𝐽𝜇𝜈 = 𝑇 ∫
𝜎‾
0
𝑑𝜎(𝑋𝜇𝑋˙ 𝜈 − 𝑋𝜈𝑋˙𝜇) = 𝑙𝜇𝜈 + 𝐸𝜇𝜈 + 𝐸‾𝜇𝜈
𝑙𝜇𝜈 = 𝑥𝜇𝑝𝜈 − 𝑥𝜈𝑝𝜇
𝐸𝜇𝜈 = −𝑖 ∑
∞
𝑛=1
1
𝑛 (𝛼−𝑛
𝜇 𝛼𝑛
𝜈 − 𝛼−𝑛
𝜈 𝛼𝑛
𝜇)
𝐸‾𝜇𝜈 = −𝑖 ∑
∞
𝑛=1
1
𝑛 (𝛼‾−𝑛
𝜇 𝛼‾𝑛
𝜈 − 𝛼‾−𝑛
𝜈 𝛼‾𝑛
𝜇)
{𝑋𝜇(𝜎, 𝜏), 𝑋˙ 𝜈(𝜎′, 𝜏)}𝑃𝐵 = 1
𝑇 𝛿(𝜎 − 𝜎′)𝜂𝜇𝜈
{𝛼𝑚
𝜇 , 𝛼𝑛
𝜈} = {𝛼‾𝑚
𝜇 , 𝛼‾𝑛
𝜈} = −𝑖𝑚𝛿𝑚+𝑛,0𝜂𝜇𝜈
{𝛼‾𝑚
𝜇 , 𝛼𝑛
𝜈} = 0, {𝑥𝜇, 𝑝𝜈} = 𝜂𝜇𝜈.
𝐻 = ∫ 𝑑𝜎(𝑋˙ Π − 𝐿) = 𝑇
2 ∫ 𝑑𝜎(𝑋˙ 2 + 𝑋′2)
𝐻 = 1
2 ∑
𝑛∈𝑍
(𝛼−𝑛𝛼𝑛 + 𝛼‾−𝑛𝛼‾𝑛)
𝐻 = 1
2 ∑
𝑛∈𝑍
𝛼−𝑛𝛼𝑛
𝐿𝑚 = 2𝑇 ∫
2𝜋
0
𝑑𝜎𝑇−−𝑒𝑖𝑚(𝜏−𝜎) , 𝐿‾𝑚 = 2𝑇 ∫
2𝜋
0
𝑑𝜎𝑇++𝑒𝑖𝑚(𝜎+𝜏)
𝐿𝑚 = 1
2 ∑
𝑛
𝛼𝑚−𝑛𝛼𝑛 , 𝐿‾𝑚 = 1
2 ∑
𝑛
𝛼‾𝑚−𝑛𝛼‾𝑛
𝐿𝑚
∗ = 𝐿−𝑚 and 𝐿‾𝑚
∗ = 𝐿‾−𝑚
𝐻 = 𝐿0 + 𝐿‾0.
𝐿𝑚 = 2𝑇 ∫
𝜋
0
𝑑𝜎{𝑇−−𝑒𝑖𝑚(𝜏−𝜎) + 𝑇++𝑒𝑖𝑚(𝜎+𝜏)}
𝐿𝑚 = 1
2 ∑
𝑛
𝛼𝑚−𝑛𝛼𝑛
𝐻 = 𝐿0
pág. 11
{𝐿𝑚, 𝐿𝑛}𝑃𝐵 = −𝑖(𝑚 − 𝑛)𝐿𝑚+𝑛
{𝐿‾𝑚, 𝐿‾𝑛}𝑃𝐵 = −𝑖(𝑚 − 𝑛)𝐿‾𝑚+𝑛
{𝐿𝑚, 𝐿‾𝑛}𝑃𝐵 = 0
{ , }𝑃𝐵 ⟶ −𝑖[, ].
Cuantización canonical- Bosonificación.
[𝑥𝜇, 𝑝𝜈] = 𝑖𝜂𝜇𝜈
[𝛼𝑚
𝜇 , 𝛼𝑛
𝜈] = 𝑚𝛿𝑚+𝑛,0𝜂𝜇𝜈
[𝑎𝑚
𝜇 , 𝑎𝑛
𝜈†] = 𝛿𝑚,𝑛𝜂𝜇𝜈
𝛼𝑚|𝑝⟩ = 0 ∀𝑚 > 0.
|𝑝⟩, 𝛼−1
𝜇 |𝑝𝜇⟩, 𝛼−1
𝜇 𝛼−1
𝜈 𝛼−2
𝜈 |𝑝𝜇⟩, etc.
|𝛼−1
0 |𝑝⟩ |2
= ⟨𝑝|𝛼1
0𝛼−1
0 |𝑝⟩ = −1,
𝐿𝑚 = 1
2 ∑
𝑛∈ℤ
: 𝛼𝑚−𝑛 ⋅ 𝛼𝑛: .
𝐿0 = 1
2 𝛼0
2 + ∑
∞
𝑛=1
𝛼−𝑛 ⋅ 𝛼𝑛
[𝐿𝑚, 𝐿𝑛] = (𝑚 − 𝑛)𝐿𝑚+𝑛 + 𝑐
12 𝑚(𝑚2 − 1)𝛿𝑚+𝑛,0
0 = ⟨𝜙|[𝐿𝑚, 𝐿−𝑚]|𝜙⟩ = 2𝑚⟨𝜙|𝐿0|𝜙⟩ + 𝑑
12 𝑚(𝑚2 − 1)⟨𝜙 ∣ 𝜙⟩ ≠ 0
𝐿𝑚>0 ∣ phys ⟩ = 0, (𝐿0 − 𝑎) ∣ phys ⟩ = 0
𝛼′𝑚2 = 4(𝑁 − 𝑎)
𝑁 = ∑
∞
𝑚=1
𝛼−𝑚 ⋅ 𝛼𝑚
𝜉+
′ = 𝑓(𝜉+), 𝜉−
′ = 𝑔(𝜉−)
𝑋+ = 𝑥+ + 𝛼′𝑝+𝜏
𝑋± = 𝑋0 ± 𝑋1
𝛼𝑛
+ = 𝛼‾𝑛
+ = √𝛼′
2 𝑝+𝛿𝑛,0
𝛼𝑛
− = 1
√2𝛼′𝑝+ { ∑
𝑚∈ℤ
: 𝛼𝑛−𝑚
𝑖 𝛼𝑚
𝑖 : −2𝑎𝛿𝑛,0}
{𝐿𝑚, 𝐿𝑛}𝑃𝐵 = −𝑖(𝑚 − 𝑛)𝐿𝑚+𝑛
{𝐿‾𝑚, 𝐿‾𝑛}𝑃𝐵 = −𝑖(𝑚 − 𝑛)𝐿‾𝑚+𝑛
{𝐿𝑚, 𝐿‾𝑛}𝑃𝐵 = 0
{ , }𝑃𝐵 ⟶ −𝑖[, ].
Cuantización canonical- Bosonificación.
[𝑥𝜇, 𝑝𝜈] = 𝑖𝜂𝜇𝜈
[𝛼𝑚
𝜇 , 𝛼𝑛
𝜈] = 𝑚𝛿𝑚+𝑛,0𝜂𝜇𝜈
[𝑎𝑚
𝜇 , 𝑎𝑛
𝜈†] = 𝛿𝑚,𝑛𝜂𝜇𝜈
𝛼𝑚|𝑝⟩ = 0 ∀𝑚 > 0.
|𝑝⟩, 𝛼−1
𝜇 |𝑝𝜇⟩, 𝛼−1
𝜇 𝛼−1
𝜈 𝛼−2
𝜈 |𝑝𝜇⟩, etc.
|𝛼−1
0 |𝑝⟩ |2
= ⟨𝑝|𝛼1
0𝛼−1
0 |𝑝⟩ = −1,
𝐿𝑚 = 1
2 ∑
𝑛∈ℤ
: 𝛼𝑚−𝑛 ⋅ 𝛼𝑛: .
𝐿0 = 1
2 𝛼0
2 + ∑
∞
𝑛=1
𝛼−𝑛 ⋅ 𝛼𝑛
[𝐿𝑚, 𝐿𝑛] = (𝑚 − 𝑛)𝐿𝑚+𝑛 + 𝑐
12 𝑚(𝑚2 − 1)𝛿𝑚+𝑛,0
0 = ⟨𝜙|[𝐿𝑚, 𝐿−𝑚]|𝜙⟩ = 2𝑚⟨𝜙|𝐿0|𝜙⟩ + 𝑑
12 𝑚(𝑚2 − 1)⟨𝜙 ∣ 𝜙⟩ ≠ 0
𝐿𝑚>0 ∣ phys ⟩ = 0, (𝐿0 − 𝑎) ∣ phys ⟩ = 0
𝛼′𝑚2 = 4(𝑁 − 𝑎)
𝑁 = ∑
∞
𝑚=1
𝛼−𝑚 ⋅ 𝛼𝑚
𝜉+
′ = 𝑓(𝜉+), 𝜉−
′ = 𝑔(𝜉−)
𝑋+ = 𝑥+ + 𝛼′𝑝+𝜏
𝑋± = 𝑋0 ± 𝑋1
𝛼𝑛
+ = 𝛼‾𝑛
+ = √𝛼′
2 𝑝+𝛿𝑛,0
𝛼𝑛
− = 1
√2𝛼′𝑝+ { ∑
𝑚∈ℤ
: 𝛼𝑛−𝑚
𝑖 𝛼𝑚
𝑖 : −2𝑎𝛿𝑛,0}
pág. 12
𝛼−1
𝑖 𝛼‾−1
𝑗 |𝑝⟩
𝛼−1
𝑖 𝛼‾−1
𝑗 |𝑝⟩ = 𝛼−1
[𝑖 𝛼‾−1
𝑗] |𝑝⟩ + [𝛼−1
{𝑖 𝛼‾−1
𝑗} − 1
𝑑 − 2 𝛿𝑖𝑗𝛼−1
𝑘 𝛼‾−1
𝑘 ] |𝑝⟩ +
+ 1
𝑑 − 2 𝛿𝑖𝑗𝛼−1
𝑘 𝛼‾−1
𝑘 |𝑝⟩
𝛼′𝑚2 = 4(1 − 𝑎)
𝛼−1
𝑖 |𝑝⟩,
𝛼−2
𝑖 |𝑝⟩, 𝛼−1
𝑖 𝛼−1
𝑗 |𝑝⟩,
𝑗max = 𝛼′𝑚2 + 1
Ω|𝑝, 𝑖, 𝑗⟩ = 𝜖|𝑝, 𝑗, 𝑖⟩
𝑍 = ∫ 𝒟𝑔𝒟𝑋𝜇
𝑉gauge
𝑒𝑖𝑆𝑝(𝑔,𝑋𝜇)
‖𝛿𝑔‖ = ∫ 𝑑2𝜎√𝑔𝑔𝛼𝛽𝑔𝛿𝛾𝛿𝑔𝛼𝛾𝛿𝑔𝛽𝛿
‖𝛿𝑋𝜇‖ = ∫ 𝑑𝜎√𝑔𝛿𝑋𝜇𝛿𝑋𝜈𝜂𝜇𝜈
𝑔𝛼𝛽 = 𝑒2𝜙ℎ𝛼𝛽
𝛿𝑔𝛼𝛽 = ∇𝛼𝜉𝛽 + ∇𝛽𝜉𝛼 + 2Λ𝑔𝛼𝛽 = (𝑃ˆ 𝜉)𝛼𝛽 + 2Λ˜ 𝑔𝛼𝛽
𝒟𝑔 = 𝒟(𝑃ˆ 𝜉)𝒟(Λ˜ ) = 𝒟𝜉𝒟Λ |𝜕(𝑃𝜉, Λ˜ )
𝜕(𝜉, Λ) | ,
|𝜕(𝑃𝜉, Λ˜ )
𝜕(𝜉, Λ) | = |det (𝑃ˆ 0
∗ 1)| = |det𝑃| = √det𝑃ˆ 𝑃ˆ †
𝑍 = ∫ 𝒟𝑋𝜇√det𝑃𝑃†𝑒𝑖𝑆𝑝(ℎˆ 𝛼𝛽,𝑋𝜇)
• Faddeev-Popov:
√det𝑃𝑃† = ∫ 𝒟𝑐𝒟𝑏𝑒𝑖 ∫ 𝑑2𝜎√𝑔𝑔𝛼𝛽𝑏𝛼𝛾∇𝛽𝑐𝛼
𝑍 = ∫ 𝒟𝑋𝒟𝑐𝒟𝑏𝑒𝑖(𝑆𝑝[𝑋]+𝑆𝑔ℎ[𝑐,𝑏])
𝑆𝑝[𝑋] = 𝑇 ∫ 𝑑2𝜎𝜕+𝑋𝜇𝜕−𝑋𝜇
𝑆𝑔ℎ[𝑏, 𝑐] = ∫ 𝑏++𝜕−𝑐+ + 𝑏−−𝜕+𝑐−
𝛼−1
𝑖 𝛼‾−1
𝑗 |𝑝⟩
𝛼−1
𝑖 𝛼‾−1
𝑗 |𝑝⟩ = 𝛼−1
[𝑖 𝛼‾−1
𝑗] |𝑝⟩ + [𝛼−1
{𝑖 𝛼‾−1
𝑗} − 1
𝑑 − 2 𝛿𝑖𝑗𝛼−1
𝑘 𝛼‾−1
𝑘 ] |𝑝⟩ +
+ 1
𝑑 − 2 𝛿𝑖𝑗𝛼−1
𝑘 𝛼‾−1
𝑘 |𝑝⟩
𝛼′𝑚2 = 4(1 − 𝑎)
𝛼−1
𝑖 |𝑝⟩,
𝛼−2
𝑖 |𝑝⟩, 𝛼−1
𝑖 𝛼−1
𝑗 |𝑝⟩,
𝑗max = 𝛼′𝑚2 + 1
Ω|𝑝, 𝑖, 𝑗⟩ = 𝜖|𝑝, 𝑗, 𝑖⟩
𝑍 = ∫ 𝒟𝑔𝒟𝑋𝜇
𝑉gauge
𝑒𝑖𝑆𝑝(𝑔,𝑋𝜇)
‖𝛿𝑔‖ = ∫ 𝑑2𝜎√𝑔𝑔𝛼𝛽𝑔𝛿𝛾𝛿𝑔𝛼𝛾𝛿𝑔𝛽𝛿
‖𝛿𝑋𝜇‖ = ∫ 𝑑𝜎√𝑔𝛿𝑋𝜇𝛿𝑋𝜈𝜂𝜇𝜈
𝑔𝛼𝛽 = 𝑒2𝜙ℎ𝛼𝛽
𝛿𝑔𝛼𝛽 = ∇𝛼𝜉𝛽 + ∇𝛽𝜉𝛼 + 2Λ𝑔𝛼𝛽 = (𝑃ˆ 𝜉)𝛼𝛽 + 2Λ˜ 𝑔𝛼𝛽
𝒟𝑔 = 𝒟(𝑃ˆ 𝜉)𝒟(Λ˜ ) = 𝒟𝜉𝒟Λ |𝜕(𝑃𝜉, Λ˜ )
𝜕(𝜉, Λ) | ,
|𝜕(𝑃𝜉, Λ˜ )
𝜕(𝜉, Λ) | = |det (𝑃ˆ 0
∗ 1)| = |det𝑃| = √det𝑃ˆ 𝑃ˆ †
𝑍 = ∫ 𝒟𝑋𝜇√det𝑃𝑃†𝑒𝑖𝑆𝑝(ℎˆ 𝛼𝛽,𝑋𝜇)
• Faddeev-Popov:
√det𝑃𝑃† = ∫ 𝒟𝑐𝒟𝑏𝑒𝑖 ∫ 𝑑2𝜎√𝑔𝑔𝛼𝛽𝑏𝛼𝛾∇𝛽𝑐𝛼
𝑍 = ∫ 𝒟𝑋𝒟𝑐𝒟𝑏𝑒𝑖(𝑆𝑝[𝑋]+𝑆𝑔ℎ[𝑐,𝑏])
𝑆𝑝[𝑋] = 𝑇 ∫ 𝑑2𝜎𝜕+𝑋𝜇𝜕−𝑋𝜇
𝑆𝑔ℎ[𝑏, 𝑐] = ∫ 𝑏++𝜕−𝑐+ + 𝑏−−𝜕+𝑐−
pág. 13
Geometría topológica de un campo de gauge. Supergeometría e hipergeometrización tensorial.
𝛿𝑔𝛼𝛽 = ∇𝛼𝜉𝛽 + ∇𝛽𝜉𝛼 + 2Λ𝑔𝛼𝛽 = (𝑃ˆ 𝜉)𝛼𝛽 + 2Λ˜ 𝑔𝛼𝛽
𝑃ˆ 𝜉∗ = 0
(𝑉𝛼, 𝑊𝛼) = ∫ 𝑑2𝜉√det𝑔𝑔𝛼𝛽𝑉𝛼𝑊𝛽
(𝑇𝛼𝛽, 𝑆𝛼𝛽) = ∫ 𝑑2𝜉√det𝑔𝑔𝛼𝛾𝑔𝛽𝛿 𝑇𝛼𝛽𝑆𝛾𝛿
(𝑃ˆ †𝑡)𝛼 = −2∇𝛽𝑡𝛼𝛽.
𝑃ˆ †𝑡∗ = 0
[𝛿𝛼, 𝛿𝛽 ] = 𝑓𝛼𝛽
𝛾 𝛿𝛾
𝐹𝐴(𝜙𝑖) = 0
∫ 𝒟𝜙
𝑉gauge
𝑒−𝑆0 ∼ ∫ 𝒟𝜙𝛿(𝐹𝐴(𝜙) = 0)𝒟𝑏𝐴𝒟𝑐𝛼𝑒−𝑆0−∫ 𝑏𝐴(𝛿𝛼𝐹𝐴)𝑐𝛼
∼ ∫ 𝒟𝜙𝒟𝐵𝐴𝒟𝑏𝐴𝒟𝑐𝛼𝑒−𝑆0−𝑖 ∫ 𝐵𝐴𝐹𝐴(𝜙)−∫ 𝑏𝐴(𝛿𝛼𝐹𝐴)𝑐𝛼
= ∫ 𝒟𝜙𝒟𝐵𝐴𝒟𝑏𝐴𝒟𝑐𝛼𝑒−𝑆
𝑆 = 𝑆0 + 𝑆1 + 𝑆2 , 𝑆1 = 𝑖 ∫ 𝐵𝐴𝐹𝐴(𝜙) , 𝑆2 = ∫ 𝑏𝐴(𝛿𝛼𝐹𝐴)𝑐𝛼
𝛿𝐵𝑅𝑆𝑇𝜙𝑖 = −𝑖𝜖𝑐𝛼𝛿𝛼𝜙𝑖
𝛿𝐵𝑅𝑆𝑇𝑏𝐴 = −𝜖𝐵𝐴
𝛿𝐵𝑅𝑆𝑇𝑐𝛼 = − 1
2 𝜖𝑐𝛽𝑐𝛾𝑓𝛽𝛾
𝛼
𝛿𝐵𝑅𝑆𝑇𝐵𝐴 = 0
𝛿𝐵𝑅𝑆𝑇(𝑏𝐴𝐹𝐴) = 𝜖[𝐵𝐴𝐹𝐴(𝜙) + 𝑏𝐴𝑐𝛼𝛿𝛼𝐹𝐴(𝜙)].
𝜖𝛿𝐹⟨𝜓 ∣ 𝜓′⟩ = −𝑖⟨𝜓|𝛿𝐵𝑅𝑆𝑇(𝑏𝐴𝛿𝐹𝐴)|𝜓′⟩ = ⟨𝜓|{𝑄𝐵, 𝑏𝐴𝛿𝐹𝐴}|𝜓′⟩,
𝑄𝐵 ∣ phys ⟩ = 0
0 = [𝑄𝐵, 𝛿𝐻] = [𝑄𝐵, 𝛿𝐵(𝑏𝐴𝛿𝐹𝐴)]
= [𝑄𝐵, {𝑄𝐵, 𝑏𝐴𝛿𝐹𝐴}] = [𝑄𝐵
2, 𝑏𝐴𝛿𝐹𝐴]
𝑄𝐵
2 = 0
|𝜓′⟩ = |𝜓⟩ + 𝑄𝐵|𝜒⟩
𝑄𝐵 ∣ phys ⟩ = 0
and ∣ phys ⟩≠ 𝑄𝐵 ∣ something ⟩.
Geometría topológica de un campo de gauge. Supergeometría e hipergeometrización tensorial.
𝛿𝑔𝛼𝛽 = ∇𝛼𝜉𝛽 + ∇𝛽𝜉𝛼 + 2Λ𝑔𝛼𝛽 = (𝑃ˆ 𝜉)𝛼𝛽 + 2Λ˜ 𝑔𝛼𝛽
𝑃ˆ 𝜉∗ = 0
(𝑉𝛼, 𝑊𝛼) = ∫ 𝑑2𝜉√det𝑔𝑔𝛼𝛽𝑉𝛼𝑊𝛽
(𝑇𝛼𝛽, 𝑆𝛼𝛽) = ∫ 𝑑2𝜉√det𝑔𝑔𝛼𝛾𝑔𝛽𝛿 𝑇𝛼𝛽𝑆𝛾𝛿
(𝑃ˆ †𝑡)𝛼 = −2∇𝛽𝑡𝛼𝛽.
𝑃ˆ †𝑡∗ = 0
[𝛿𝛼, 𝛿𝛽 ] = 𝑓𝛼𝛽
𝛾 𝛿𝛾
𝐹𝐴(𝜙𝑖) = 0
∫ 𝒟𝜙
𝑉gauge
𝑒−𝑆0 ∼ ∫ 𝒟𝜙𝛿(𝐹𝐴(𝜙) = 0)𝒟𝑏𝐴𝒟𝑐𝛼𝑒−𝑆0−∫ 𝑏𝐴(𝛿𝛼𝐹𝐴)𝑐𝛼
∼ ∫ 𝒟𝜙𝒟𝐵𝐴𝒟𝑏𝐴𝒟𝑐𝛼𝑒−𝑆0−𝑖 ∫ 𝐵𝐴𝐹𝐴(𝜙)−∫ 𝑏𝐴(𝛿𝛼𝐹𝐴)𝑐𝛼
= ∫ 𝒟𝜙𝒟𝐵𝐴𝒟𝑏𝐴𝒟𝑐𝛼𝑒−𝑆
𝑆 = 𝑆0 + 𝑆1 + 𝑆2 , 𝑆1 = 𝑖 ∫ 𝐵𝐴𝐹𝐴(𝜙) , 𝑆2 = ∫ 𝑏𝐴(𝛿𝛼𝐹𝐴)𝑐𝛼
𝛿𝐵𝑅𝑆𝑇𝜙𝑖 = −𝑖𝜖𝑐𝛼𝛿𝛼𝜙𝑖
𝛿𝐵𝑅𝑆𝑇𝑏𝐴 = −𝜖𝐵𝐴
𝛿𝐵𝑅𝑆𝑇𝑐𝛼 = − 1
2 𝜖𝑐𝛽𝑐𝛾𝑓𝛽𝛾
𝛼
𝛿𝐵𝑅𝑆𝑇𝐵𝐴 = 0
𝛿𝐵𝑅𝑆𝑇(𝑏𝐴𝐹𝐴) = 𝜖[𝐵𝐴𝐹𝐴(𝜙) + 𝑏𝐴𝑐𝛼𝛿𝛼𝐹𝐴(𝜙)].
𝜖𝛿𝐹⟨𝜓 ∣ 𝜓′⟩ = −𝑖⟨𝜓|𝛿𝐵𝑅𝑆𝑇(𝑏𝐴𝛿𝐹𝐴)|𝜓′⟩ = ⟨𝜓|{𝑄𝐵, 𝑏𝐴𝛿𝐹𝐴}|𝜓′⟩,
𝑄𝐵 ∣ phys ⟩ = 0
0 = [𝑄𝐵, 𝛿𝐻] = [𝑄𝐵, 𝛿𝐵(𝑏𝐴𝛿𝐹𝐴)]
= [𝑄𝐵, {𝑄𝐵, 𝑏𝐴𝛿𝐹𝐴}] = [𝑄𝐵
2, 𝑏𝐴𝛿𝐹𝐴]
𝑄𝐵
2 = 0
|𝜓′⟩ = |𝜓⟩ + 𝑄𝐵|𝜒⟩
𝑄𝐵 ∣ phys ⟩ = 0
and ∣ phys ⟩≠ 𝑄𝐵 ∣ something ⟩.
pág. 14
𝛿𝐵𝑋𝜇 = 𝑖𝜖(𝑐+𝜕+ + 𝑐−𝜕−)𝑋𝜇
𝛿𝐵𝑐± = ±𝑖𝜖(𝑐+𝜕+ + 𝑐−𝜕−)𝑐±
𝛿𝐵𝑏± = ±𝑖𝜖(𝑇±
𝑋 + 𝑇±
𝑔ℎ)
𝑆𝑔ℎ = ∫ 𝑑2𝜎(𝑏++𝜕−𝑐+ + 𝑏−−𝜕+𝑐−)
𝑇++
𝑔ℎ = 𝑖(2𝑏++𝜕+𝑐+ + 𝜕+𝑏++𝑐+)
𝑇−−
𝑔ℎ = 𝑖(2𝑏−−𝜕−𝑐− + 𝜕−𝑏−−𝑐−)
𝜕−𝑇++
𝑔ℎ = 𝜕+𝑇−−
𝑔ℎ = 0
𝜕−𝑏++ = 𝜕+𝑏−− = 𝜕−𝑐+ = 𝜕+𝑐− = 0
𝑐+ = ∑ 𝑐‾𝑛𝑒−𝑖𝑛(𝜏+𝜎), 𝑐− = ∑ 𝑐𝑛𝑒−𝑖𝑛(𝜏−𝜎)
𝑏++ = ∑ 𝑏‾𝑛𝑒−𝑖𝑛(𝜏+𝜎), 𝑏−− = ∑ 𝑐𝑛𝑒−𝑖𝑛(𝜏−𝜎)
{𝑏𝑚, 𝑐𝑛} = 𝛿𝑚+𝑛,0 , {𝑏𝑚, 𝑏𝑛} = {𝑐𝑚, 𝑐𝑛} = 0
𝐿𝑚
𝑔ℎ = ∑
𝑛
(𝑚 − 𝑛): 𝑏𝑚+𝑛𝑐−𝑛: , 𝐿‾𝑚
𝑔ℎ = ∑
𝑛
(𝑚 − 𝑛): 𝑏‾𝑚+𝑛𝑐‾−𝑛:
• Virasoro: Campos Fantasma.
[𝐿𝑚
𝑔ℎ, 𝐿𝑛
𝑔ℎ] = (𝑚 − 𝑛)𝐿𝑚+𝑛
𝑔ℎ + 1
6 (𝑚 − 13𝑚3)𝛿𝑚+𝑛,0
𝐿𝑚 = 𝐿𝑚
𝑋 + 𝐿𝑚
𝑔ℎ − 𝑎𝛿𝑚
[𝐿𝑚, 𝐿𝑛] = (𝑚 − 𝑛)𝐿𝑚+𝑛 + 𝐴(𝑚)𝛿𝑚+𝑛
𝐴(𝑚) = 𝑑
12 𝑚(𝑚2 − 1) + 1
6 (𝑚 − 13𝑚3) + 2𝑎𝑚
𝑗𝐵 = 𝑐𝑇𝑋 + 1
2 : 𝑐𝑇𝑔ℎ: = 𝑐𝑇𝑋+: 𝑏𝑐𝜕𝑐: ,
𝑄𝐵 = ∫ 𝑑𝜎𝑗𝐵
𝑄𝐵 = ∑
𝑛
𝑐𝑛𝐿−𝑛
𝑋 + ∑
𝑚,𝑛
𝑚 − 𝑛
2 : 𝑐𝑚𝑐𝑛𝑏−𝑚−𝑛: −𝑐0
𝑏0|phys⟩ = 0
𝑏𝑛>0 ∣ ghost vacuum ⟩ = 𝑐𝑛>0 ∣ ghost vacuum ⟩ = 0.
𝑏0| ↓⟩ = 0, 𝑏0| ↑⟩ = | ↓⟩,
𝑐0| ↑⟩ = 0, 𝑐0| ↓⟩ = | ↑⟩.
𝛿𝐵𝑋𝜇 = 𝑖𝜖(𝑐+𝜕+ + 𝑐−𝜕−)𝑋𝜇
𝛿𝐵𝑐± = ±𝑖𝜖(𝑐+𝜕+ + 𝑐−𝜕−)𝑐±
𝛿𝐵𝑏± = ±𝑖𝜖(𝑇±
𝑋 + 𝑇±
𝑔ℎ)
𝑆𝑔ℎ = ∫ 𝑑2𝜎(𝑏++𝜕−𝑐+ + 𝑏−−𝜕+𝑐−)
𝑇++
𝑔ℎ = 𝑖(2𝑏++𝜕+𝑐+ + 𝜕+𝑏++𝑐+)
𝑇−−
𝑔ℎ = 𝑖(2𝑏−−𝜕−𝑐− + 𝜕−𝑏−−𝑐−)
𝜕−𝑇++
𝑔ℎ = 𝜕+𝑇−−
𝑔ℎ = 0
𝜕−𝑏++ = 𝜕+𝑏−− = 𝜕−𝑐+ = 𝜕+𝑐− = 0
𝑐+ = ∑ 𝑐‾𝑛𝑒−𝑖𝑛(𝜏+𝜎), 𝑐− = ∑ 𝑐𝑛𝑒−𝑖𝑛(𝜏−𝜎)
𝑏++ = ∑ 𝑏‾𝑛𝑒−𝑖𝑛(𝜏+𝜎), 𝑏−− = ∑ 𝑐𝑛𝑒−𝑖𝑛(𝜏−𝜎)
{𝑏𝑚, 𝑐𝑛} = 𝛿𝑚+𝑛,0 , {𝑏𝑚, 𝑏𝑛} = {𝑐𝑚, 𝑐𝑛} = 0
𝐿𝑚
𝑔ℎ = ∑
𝑛
(𝑚 − 𝑛): 𝑏𝑚+𝑛𝑐−𝑛: , 𝐿‾𝑚
𝑔ℎ = ∑
𝑛
(𝑚 − 𝑛): 𝑏‾𝑚+𝑛𝑐‾−𝑛:
• Virasoro: Campos Fantasma.
[𝐿𝑚
𝑔ℎ, 𝐿𝑛
𝑔ℎ] = (𝑚 − 𝑛)𝐿𝑚+𝑛
𝑔ℎ + 1
6 (𝑚 − 13𝑚3)𝛿𝑚+𝑛,0
𝐿𝑚 = 𝐿𝑚
𝑋 + 𝐿𝑚
𝑔ℎ − 𝑎𝛿𝑚
[𝐿𝑚, 𝐿𝑛] = (𝑚 − 𝑛)𝐿𝑚+𝑛 + 𝐴(𝑚)𝛿𝑚+𝑛
𝐴(𝑚) = 𝑑
12 𝑚(𝑚2 − 1) + 1
6 (𝑚 − 13𝑚3) + 2𝑎𝑚
𝑗𝐵 = 𝑐𝑇𝑋 + 1
2 : 𝑐𝑇𝑔ℎ: = 𝑐𝑇𝑋+: 𝑏𝑐𝜕𝑐: ,
𝑄𝐵 = ∫ 𝑑𝜎𝑗𝐵
𝑄𝐵 = ∑
𝑛
𝑐𝑛𝐿−𝑛
𝑋 + ∑
𝑚,𝑛
𝑚 − 𝑛
2 : 𝑐𝑚𝑐𝑛𝑏−𝑚−𝑛: −𝑐0
𝑏0|phys⟩ = 0
𝑏𝑛>0 ∣ ghost vacuum ⟩ = 𝑐𝑛>0 ∣ ghost vacuum ⟩ = 0.
𝑏0| ↓⟩ = 0, 𝑏0| ↑⟩ = | ↓⟩,
𝑐0| ↑⟩ = 0, 𝑐0| ↓⟩ = | ↑⟩.
pág. 15
0 = 𝑄𝐵| ↓, 𝑝⟩ = (𝐿0
𝑋 − 1)𝑐0| ↓, 𝑝⟩.
|𝜓⟩ = (𝜁 ⋅ 𝛼−1 + 𝜉1𝑐−1 + 𝜉2𝑏−1)| ↓, 𝑝⟩,
0 = 𝑄𝐵|𝜓⟩ = 2(𝑝2𝑐0 + (𝑝 ⋅ 𝜁)𝑐−1 + 𝜉1𝑝 ⋅ 𝛼−1)| ↓, 𝑝⟩.
𝑄𝐵|𝜒⟩ = 2(𝑝 ⋅ 𝜁′𝑐−1 + 𝜉1
′ 𝑝 ⋅ 𝛼−1)| ↓, 𝑝⟩.
Interacciones y amplitudes.
𝑔𝜇𝜈 → 𝑔𝜇𝜈
′ (𝑥′) = 𝜕𝑥𝛼
𝜕𝑥′𝜇
𝜕𝑥𝛽
𝜕𝑥′𝜈 𝑔𝛼𝛽(𝑥)
𝑔𝜇𝜈(𝑥) → 𝑔𝜇𝜈
′ (𝑥′) = Ω(𝑥)𝑔𝜇𝜈(𝑥)
𝑑𝑠′2 = 𝑑𝑠2 − (𝜕𝜇𝜖𝜈 + 𝜕𝜈𝜖𝜇)𝑑𝑥𝜇𝑑𝑥𝜈
𝜕𝜇𝜖𝜈 + 𝜕𝜈𝜖𝜇 = 2
𝑑 (𝜕 ⋅ 𝜖)𝜂𝜇𝜈
◻ 𝜖𝜈 + (1 − 2
𝑑) 𝜕𝜈(𝜕 ⋅ 𝜖) = 0
𝜕𝜇 ◻ 𝜖𝜈 + 𝜕𝜈 ◻ 𝜖𝜇 = 2
𝑑 𝜂𝜇𝜈 ◻ (𝜕 ⋅ 𝜖)
[𝜂𝜇𝜈 ◻ +(𝑑 − 2)𝜕𝜇𝜕𝜈]𝜕 ⋅ 𝜖 = 0
𝜖𝜇 = 𝑎𝜇translationes
𝜖𝜇 = 𝜔𝜈
𝜇𝑥𝜈 rotationes (𝜔𝜇𝜈 = −𝜔𝜈𝜇),
𝜖𝜇 = 𝜆𝑥𝜇 transformationes escalares
𝜖𝜇 = 𝑏𝜇𝑥2 − 2𝑥𝜇(𝑏 ⋅ 𝑥)
𝑑 + 1
2 𝑑(𝑑 − 1) + 1 + 𝑑 = 1
2 (𝑑 + 2)(𝑑 + 1)
𝜕1𝜖1 = 𝜕2𝜖2, 𝜕1𝜖2 = −𝜕2𝜖1
𝜕𝜖‾ = 0, 𝜕‾𝜖 = 0
𝑧 → 𝑓(𝑧) and 𝑧‾ → 𝑓‾(𝑧‾)
𝜖(𝑧) = − ∑ 𝑎𝑛𝑧𝑛+1
ℓ𝑛 = −𝑧𝑛+1𝜕𝑧
[ℓ𝑚, ℓ𝑛] = (𝑚 − 𝑛)ℓ𝑚+𝑛, [ℓ‾𝑚, ℓ‾𝑛] = (𝑚 − 𝑛)ℓ‾𝑚+𝑛
𝑧 → 𝑎𝑧 + 𝑏
𝑐𝑧 + 𝑑
0 = 𝑄𝐵| ↓, 𝑝⟩ = (𝐿0
𝑋 − 1)𝑐0| ↓, 𝑝⟩.
|𝜓⟩ = (𝜁 ⋅ 𝛼−1 + 𝜉1𝑐−1 + 𝜉2𝑏−1)| ↓, 𝑝⟩,
0 = 𝑄𝐵|𝜓⟩ = 2(𝑝2𝑐0 + (𝑝 ⋅ 𝜁)𝑐−1 + 𝜉1𝑝 ⋅ 𝛼−1)| ↓, 𝑝⟩.
𝑄𝐵|𝜒⟩ = 2(𝑝 ⋅ 𝜁′𝑐−1 + 𝜉1
′ 𝑝 ⋅ 𝛼−1)| ↓, 𝑝⟩.
Interacciones y amplitudes.
𝑔𝜇𝜈 → 𝑔𝜇𝜈
′ (𝑥′) = 𝜕𝑥𝛼
𝜕𝑥′𝜇
𝜕𝑥𝛽
𝜕𝑥′𝜈 𝑔𝛼𝛽(𝑥)
𝑔𝜇𝜈(𝑥) → 𝑔𝜇𝜈
′ (𝑥′) = Ω(𝑥)𝑔𝜇𝜈(𝑥)
𝑑𝑠′2 = 𝑑𝑠2 − (𝜕𝜇𝜖𝜈 + 𝜕𝜈𝜖𝜇)𝑑𝑥𝜇𝑑𝑥𝜈
𝜕𝜇𝜖𝜈 + 𝜕𝜈𝜖𝜇 = 2
𝑑 (𝜕 ⋅ 𝜖)𝜂𝜇𝜈
◻ 𝜖𝜈 + (1 − 2
𝑑) 𝜕𝜈(𝜕 ⋅ 𝜖) = 0
𝜕𝜇 ◻ 𝜖𝜈 + 𝜕𝜈 ◻ 𝜖𝜇 = 2
𝑑 𝜂𝜇𝜈 ◻ (𝜕 ⋅ 𝜖)
[𝜂𝜇𝜈 ◻ +(𝑑 − 2)𝜕𝜇𝜕𝜈]𝜕 ⋅ 𝜖 = 0
𝜖𝜇 = 𝑎𝜇translationes
𝜖𝜇 = 𝜔𝜈
𝜇𝑥𝜈 rotationes (𝜔𝜇𝜈 = −𝜔𝜈𝜇),
𝜖𝜇 = 𝜆𝑥𝜇 transformationes escalares
𝜖𝜇 = 𝑏𝜇𝑥2 − 2𝑥𝜇(𝑏 ⋅ 𝑥)
𝑑 + 1
2 𝑑(𝑑 − 1) + 1 + 𝑑 = 1
2 (𝑑 + 2)(𝑑 + 1)
𝜕1𝜖1 = 𝜕2𝜖2, 𝜕1𝜖2 = −𝜕2𝜖1
𝜕𝜖‾ = 0, 𝜕‾𝜖 = 0
𝑧 → 𝑓(𝑧) and 𝑧‾ → 𝑓‾(𝑧‾)
𝜖(𝑧) = − ∑ 𝑎𝑛𝑧𝑛+1
ℓ𝑛 = −𝑧𝑛+1𝜕𝑧
[ℓ𝑚, ℓ𝑛] = (𝑚 − 𝑛)ℓ𝑚+𝑛, [ℓ‾𝑚, ℓ‾𝑛] = (𝑚 − 𝑛)ℓ‾𝑚+𝑛
𝑧 → 𝑎𝑧 + 𝑏
𝑐𝑧 + 𝑑
pág. 16
Invariancia y covariancia de gauge.
𝑧 → 𝑓(𝑧) = 𝑎𝑧 + 𝑏
𝑐𝑧 + 𝑑 , 𝑧‾ → 𝑓‾(𝑧‾) = 𝑎‾𝑧‾ + 𝑏‾
𝑐‾𝑧‾ + 𝑑‾
Φ(𝑧, 𝑧‾) → (𝜕𝑓
𝜕𝑧)
ℎ
(𝜕𝑓‾
𝜕𝑧‾)
ℎ‾
Φ(𝑓(𝑧), 𝑓‾(𝑧‾))
Φ(𝑧, 𝑧‾)𝑑𝑧ℎ𝑑𝑧‾ℎ‾
⟨∏
𝑁
𝑖=1
Φ𝑖(𝑧𝑖, 𝑧‾𝑖)⟩ = ∏
𝑁
𝑖=1
(𝜕𝑓
𝜕𝑧)𝑧→𝑧𝑖
ℎ𝑖
(𝜕𝑓‾
𝜕𝑧‾)
𝑧‾→𝑧‾𝑖
ℎ‾ 𝑖
⟨∏
𝑁
𝑗=1
Φ𝑗 (𝑓(𝑧𝑗), 𝑓‾(𝑧‾𝑗))⟩ .
𝛿𝜖,𝜖‾ Φ(𝑧, 𝑧‾) = [(ℎ𝜕𝜖 + 𝜖𝜕) + (ℎ‾𝜕‾𝜖‾ + 𝜖‾𝜕‾)]Φ(𝑧, 𝑧‾)
𝛿𝜖,𝜖‾ 𝐺(2)(𝑧𝑖, 𝑧‾𝑖) = ⟨𝛿𝜖,𝜖‾ Φ1, Φ2⟩ + ⟨Φ1, 𝛿𝜖,𝜖‾ Φ2⟩ = 0
[(𝜖(𝑧1)𝜕𝑧1 + ℎ1𝜕𝜖(𝑧1) + 𝜖(𝑧2)𝜕𝑧2 + ℎ2𝜕𝜖(𝑧2)) + ( 𝔅barred terms )] 𝐺(2)(𝑧𝑖, 𝑧‾𝑖) = 0
𝐺(2)(𝑧𝑖, 𝑧‾𝑖) = 𝐶12
𝑧12
2ℎ𝑧‾12
2ℎ‾
𝐺(3)(𝑧𝑖, 𝑧‾𝑖) = 𝐶123
𝑧12
Δ12 𝑧23
Δ23 𝑧31
Δ31 𝑧‾12
Δ‾ 12 𝑧‾12
Δ‾ 12 𝑧‾12
Δ‾ 12
𝐺(4)(𝑧𝑖, 𝑧‾𝑖) = 𝑓(𝑥, 𝑥‾) ∏
𝑖<𝑗
𝑧𝑖𝑗
−(ℎ𝑖+ℎ𝑗)+ℎ/3 ∏
𝑖<𝑗
𝑧‾𝑖𝑗
−(ℎ‾𝑖+ℎ‾𝑗)+ℎ‾ /3
𝐺𝑁(𝑧1, 𝑧‾1, … 𝑧𝑁, 𝑧‾𝑁) = ⟨∏
𝑁
𝑖=1
Φ𝑖(𝑧𝑖, 𝑧‾𝑖)⟩
∑
𝑁
𝑖=1
𝜕𝑖𝐺𝑁 = 0
∑
𝑁
𝑖=1
(𝑧𝑖𝜕𝑖 + ℎ𝑖)𝐺𝑁 = 0
∑
𝑁
𝑖=1
(𝑧𝑖
2𝜕𝑖 + 2𝑧𝑖ℎ𝑖)𝐺𝑁 = 0
Invariancia y covariancia de gauge.
𝑧 → 𝑓(𝑧) = 𝑎𝑧 + 𝑏
𝑐𝑧 + 𝑑 , 𝑧‾ → 𝑓‾(𝑧‾) = 𝑎‾𝑧‾ + 𝑏‾
𝑐‾𝑧‾ + 𝑑‾
Φ(𝑧, 𝑧‾) → (𝜕𝑓
𝜕𝑧)
ℎ
(𝜕𝑓‾
𝜕𝑧‾)
ℎ‾
Φ(𝑓(𝑧), 𝑓‾(𝑧‾))
Φ(𝑧, 𝑧‾)𝑑𝑧ℎ𝑑𝑧‾ℎ‾
⟨∏
𝑁
𝑖=1
Φ𝑖(𝑧𝑖, 𝑧‾𝑖)⟩ = ∏
𝑁
𝑖=1
(𝜕𝑓
𝜕𝑧)𝑧→𝑧𝑖
ℎ𝑖
(𝜕𝑓‾
𝜕𝑧‾)
𝑧‾→𝑧‾𝑖
ℎ‾ 𝑖
⟨∏
𝑁
𝑗=1
Φ𝑗 (𝑓(𝑧𝑗), 𝑓‾(𝑧‾𝑗))⟩ .
𝛿𝜖,𝜖‾ Φ(𝑧, 𝑧‾) = [(ℎ𝜕𝜖 + 𝜖𝜕) + (ℎ‾𝜕‾𝜖‾ + 𝜖‾𝜕‾)]Φ(𝑧, 𝑧‾)
𝛿𝜖,𝜖‾ 𝐺(2)(𝑧𝑖, 𝑧‾𝑖) = ⟨𝛿𝜖,𝜖‾ Φ1, Φ2⟩ + ⟨Φ1, 𝛿𝜖,𝜖‾ Φ2⟩ = 0
[(𝜖(𝑧1)𝜕𝑧1 + ℎ1𝜕𝜖(𝑧1) + 𝜖(𝑧2)𝜕𝑧2 + ℎ2𝜕𝜖(𝑧2)) + ( 𝔅barred terms )] 𝐺(2)(𝑧𝑖, 𝑧‾𝑖) = 0
𝐺(2)(𝑧𝑖, 𝑧‾𝑖) = 𝐶12
𝑧12
2ℎ𝑧‾12
2ℎ‾
𝐺(3)(𝑧𝑖, 𝑧‾𝑖) = 𝐶123
𝑧12
Δ12 𝑧23
Δ23 𝑧31
Δ31 𝑧‾12
Δ‾ 12 𝑧‾12
Δ‾ 12 𝑧‾12
Δ‾ 12
𝐺(4)(𝑧𝑖, 𝑧‾𝑖) = 𝑓(𝑥, 𝑥‾) ∏
𝑖<𝑗
𝑧𝑖𝑗
−(ℎ𝑖+ℎ𝑗)+ℎ/3 ∏
𝑖<𝑗
𝑧‾𝑖𝑗
−(ℎ‾𝑖+ℎ‾𝑗)+ℎ‾ /3
𝐺𝑁(𝑧1, 𝑧‾1, … 𝑧𝑁, 𝑧‾𝑁) = ⟨∏
𝑁
𝑖=1
Φ𝑖(𝑧𝑖, 𝑧‾𝑖)⟩
∑
𝑁
𝑖=1
𝜕𝑖𝐺𝑁 = 0
∑
𝑁
𝑖=1
(𝑧𝑖𝜕𝑖 + ℎ𝑖)𝐺𝑁 = 0
∑
𝑁
𝑖=1
(𝑧𝑖
2𝜕𝑖 + 2𝑧𝑖ℎ𝑖)𝐺𝑁 = 0
pág. 17
Cuantización radial.
𝑧 = 𝑒𝜏+𝑖𝜎, 𝑧‾ = 𝑒𝜏−𝑖𝜎
𝐻 = ℓ0 + ℓ‾0
𝑇𝜇 𝜇 = 0
𝑇𝑧𝑧‾ = 𝑇𝑧‾𝑧 = 1
4 (𝑇00 + 𝑇11) = 1
4 𝑇𝜇 𝜇
𝜕𝑧𝑇𝑧‾𝑧‾ = 0 and 𝜕𝑧‾𝑇𝑧𝑧 = 0
𝑇(𝑧) ≡ 𝑇𝑧𝑧 and 𝑇‾(𝑧‾) ≡ 𝑇𝑧‾𝑧‾
𝑄𝜖 = 1
2𝜋𝑖 ∮ 𝑑𝑧𝜖(𝑧)𝑇(𝑧) , 𝑄𝜖‾ = 1
2𝜋𝑖 ∮ 𝑑𝑧‾𝜖‾(𝑧‾)𝑇‾(𝑧‾).
𝑧 → 𝑧 + 𝜖(𝑧), 𝑧‾ → 𝑧‾ + 𝜖‾(𝑧‾)
𝛿𝜖,𝜖‾ Φ(𝑧, 𝑧‾) = [𝑄𝜖 + 𝑄𝜖‾ , Φ(𝑧, 𝑧‾)]
𝑅(𝐴(𝑧)𝐵(𝑤)) = { 𝐴(𝑧)𝐵(𝑤) |𝑧| > |𝑤|
(−1)𝐹𝐵(𝑤)𝐴(𝑧) |𝑧| < |𝑤|.
[∫ 𝑑𝜎𝐵, 𝐴] = ∮ 𝑑𝑧𝑅(𝐵(𝑧)𝐴(𝑤))
𝛿𝜖,𝜖‾ Φ(𝑧, 𝑧‾) = 1
2𝜋𝑖 ∮ (𝑑𝑧𝜖(𝑧)𝑅(𝑇(𝑧)Φ(𝑤, 𝑤‾ )) + 𝑑𝑧‾𝜖‾(𝑧‾)𝑅(𝑇‾(𝑧‾)Φ(𝑤, 𝑤‾ )))
= [(ℎ𝜕𝜖(𝑤) + 𝜖(𝑤)𝜕) + (ℎ‾𝜕‾𝜖‾(𝑤‾ ) + 𝜖‾(𝑤‾ )𝜕‾)]Φ(𝑤, 𝑤‾ )
𝑅(𝑇(𝑧)Φ(𝑤, 𝑤‾ )) = ℎ
(𝑧 − 𝑤)2 Φ(𝑤, 𝑤‾ ) + 1
𝑧 − 𝑤 𝜕𝑤Φ(𝑤, 𝑤‾ ) + ⋯
𝑅(𝑇‾(𝑧‾)Φ(𝑤, 𝑤‾ )) = ℎ‾
(𝑧‾ − 𝑤‾ )2 Φ(𝑤, 𝑤‾ ) + 1
𝑧‾ − 𝑤‾ 𝜕𝑤‾ Φ(𝑤, 𝑤‾ ) + ⋯
𝐹𝑁(𝑧, 𝑧𝑖, 𝑧‾𝑖) = ⟨𝑇(𝑧) ∏
𝑁
𝑖=1
Φ𝑖(𝑧𝑖, 𝑧‾𝑖)⟩ ,
𝐹𝑁(𝑧, 𝑧𝑖, 𝑧‾𝑖) = ∑
𝑁
𝑖=1
( ℎ𝑖
(𝑧 − 𝑧𝑖)2 + 𝜕𝑧𝑖
𝑧 − 𝑧𝑖
) ⟨∏
𝑁
𝑖=1
Φ𝑖(𝑧𝑖, 𝑧‾𝑖)⟩ .
Φ𝑖(𝑧, 𝑧‾)Φ𝑗(𝑤, 𝑤‾ ) = ∑
𝑘
𝐶𝑖𝑗𝑘(𝑧 − 𝑤)ℎ𝑘−ℎ𝑖−ℎ𝑗 (𝑧‾ − 𝑤‾ )ℎ‾𝑘−ℎ‾𝑖−ℎ‾𝑗 Φ𝑘(𝑤, 𝑤‾ )
Cuantización radial.
𝑧 = 𝑒𝜏+𝑖𝜎, 𝑧‾ = 𝑒𝜏−𝑖𝜎
𝐻 = ℓ0 + ℓ‾0
𝑇𝜇 𝜇 = 0
𝑇𝑧𝑧‾ = 𝑇𝑧‾𝑧 = 1
4 (𝑇00 + 𝑇11) = 1
4 𝑇𝜇 𝜇
𝜕𝑧𝑇𝑧‾𝑧‾ = 0 and 𝜕𝑧‾𝑇𝑧𝑧 = 0
𝑇(𝑧) ≡ 𝑇𝑧𝑧 and 𝑇‾(𝑧‾) ≡ 𝑇𝑧‾𝑧‾
𝑄𝜖 = 1
2𝜋𝑖 ∮ 𝑑𝑧𝜖(𝑧)𝑇(𝑧) , 𝑄𝜖‾ = 1
2𝜋𝑖 ∮ 𝑑𝑧‾𝜖‾(𝑧‾)𝑇‾(𝑧‾).
𝑧 → 𝑧 + 𝜖(𝑧), 𝑧‾ → 𝑧‾ + 𝜖‾(𝑧‾)
𝛿𝜖,𝜖‾ Φ(𝑧, 𝑧‾) = [𝑄𝜖 + 𝑄𝜖‾ , Φ(𝑧, 𝑧‾)]
𝑅(𝐴(𝑧)𝐵(𝑤)) = { 𝐴(𝑧)𝐵(𝑤) |𝑧| > |𝑤|
(−1)𝐹𝐵(𝑤)𝐴(𝑧) |𝑧| < |𝑤|.
[∫ 𝑑𝜎𝐵, 𝐴] = ∮ 𝑑𝑧𝑅(𝐵(𝑧)𝐴(𝑤))
𝛿𝜖,𝜖‾ Φ(𝑧, 𝑧‾) = 1
2𝜋𝑖 ∮ (𝑑𝑧𝜖(𝑧)𝑅(𝑇(𝑧)Φ(𝑤, 𝑤‾ )) + 𝑑𝑧‾𝜖‾(𝑧‾)𝑅(𝑇‾(𝑧‾)Φ(𝑤, 𝑤‾ )))
= [(ℎ𝜕𝜖(𝑤) + 𝜖(𝑤)𝜕) + (ℎ‾𝜕‾𝜖‾(𝑤‾ ) + 𝜖‾(𝑤‾ )𝜕‾)]Φ(𝑤, 𝑤‾ )
𝑅(𝑇(𝑧)Φ(𝑤, 𝑤‾ )) = ℎ
(𝑧 − 𝑤)2 Φ(𝑤, 𝑤‾ ) + 1
𝑧 − 𝑤 𝜕𝑤Φ(𝑤, 𝑤‾ ) + ⋯
𝑅(𝑇‾(𝑧‾)Φ(𝑤, 𝑤‾ )) = ℎ‾
(𝑧‾ − 𝑤‾ )2 Φ(𝑤, 𝑤‾ ) + 1
𝑧‾ − 𝑤‾ 𝜕𝑤‾ Φ(𝑤, 𝑤‾ ) + ⋯
𝐹𝑁(𝑧, 𝑧𝑖, 𝑧‾𝑖) = ⟨𝑇(𝑧) ∏
𝑁
𝑖=1
Φ𝑖(𝑧𝑖, 𝑧‾𝑖)⟩ ,
𝐹𝑁(𝑧, 𝑧𝑖, 𝑧‾𝑖) = ∑
𝑁
𝑖=1
( ℎ𝑖
(𝑧 − 𝑧𝑖)2 + 𝜕𝑧𝑖
𝑧 − 𝑧𝑖
) ⟨∏
𝑁
𝑖=1
Φ𝑖(𝑧𝑖, 𝑧‾𝑖)⟩ .
Φ𝑖(𝑧, 𝑧‾)Φ𝑗(𝑤, 𝑤‾ ) = ∑
𝑘
𝐶𝑖𝑗𝑘(𝑧 − 𝑤)ℎ𝑘−ℎ𝑖−ℎ𝑗 (𝑧‾ − 𝑤‾ )ℎ‾𝑘−ℎ‾𝑖−ℎ‾𝑗 Φ𝑘(𝑤, 𝑤‾ )
pág. 18
Bosón libre.
𝑆 = 1
4𝜋 ∫ 𝑑2𝑧𝜕𝑋𝜕‾𝑋
⟨𝑋(𝑧, 𝑧‾)𝑋(𝑤, 𝑤‾ )⟩ = −log (|𝑧 − 𝑤|2𝜇2)
𝜕𝑧𝑋(𝑧)𝜕𝑤𝑋(𝑤) = 𝜕𝑧𝜕𝑤⟨𝑋𝑋⟩+: 𝜕𝑧𝑋𝜕𝑤𝑋:= − 1
(𝑧 − 𝑤)2 +: 𝜕𝑧𝑋𝜕𝑤𝑋:
𝑇(𝑧) = − 1
2 : 𝜕𝑋𝜕𝑋: = − 1
2 lim
𝑧→𝑤 [𝜕𝑧𝑋𝜕𝑤𝑋 + 1
(𝑧 − 𝑤)2]
𝑇‾(𝑧‾) = − 1
2 : 𝜕‾𝑋𝜕‾𝑋: = − 1
2 lim
𝑧‾→𝑤‾ [𝜕𝑧‾𝑋𝜕𝑤‾ 𝑋 + 1
(𝑧‾ − 𝑤‾ )2]
• Wick:
𝑇(𝑧)𝜕𝑋(𝑤) = − 1
2 : 𝜕𝑋(𝑧)𝜕𝑋(𝑧): 𝜕𝑋(𝑤) = −𝜕𝑋(𝑧)⟨𝜕𝑋(𝑧)𝜕𝑋(𝑤)⟩ + ⋯ = 𝜕𝑋(𝑧) 1
(𝑧 − 𝑤)2 + ⋯
= 𝜕𝑋(𝑤)
(𝑧 − 𝑤)2 + 1
𝑧 − 𝑤 𝜕2𝑋(𝑤) + ⋯
𝑇(𝑧)𝑉𝑎(𝑤, 𝑤‾ ) = − 1
2 : 𝜕𝑋(𝑧)𝜕𝑋(𝑧): ∑
∞
𝑛=0
𝑖𝑛𝑎𝑛
𝑛! : 𝑋𝑛(𝑤, 𝑤‾ ):
𝑇(𝑧)𝑉𝑎(𝑤) = − 1
2 [𝑖𝑎𝜕⟨𝑋𝑋⟩]2𝑒𝑖𝑎𝑋(𝑤) − 1
2 2𝑖𝑎: 𝜕𝑋(𝑧)𝜕⟨𝑋𝑋⟩𝑒𝑖𝑎𝑋(𝑤): + ⋯
= 𝑎2/2
(𝑧 − 𝑤)2 𝑒𝑖𝑎𝑋(𝑤) + 𝑖𝑎𝜕𝑋(𝑧)
𝑧 − 𝑤 𝑒𝑖𝑎𝑋(𝑤) + ⋯ = 𝑎2/2
(𝑧 − 𝑤)2 𝑉𝑎(𝑤) + 1
𝑧 − 𝑤 𝜕𝑉𝑎(𝑤) + ⋯
𝐺𝑁 = ⟨∏
𝑁
𝑖=1
𝑉𝑎𝑖 (𝑧𝑖, 𝑧‾𝑖)⟩ = exp [1
2 ∑
𝑁
𝑖,𝑗=1;𝑖≠𝑗
𝑎𝑖𝑎𝑗⟨𝑋(𝑧𝑖, 𝑧‾𝑖)𝑋(𝑧𝑗, 𝑧‾𝑗)⟩]
∑
𝑖
𝑎𝑖 = 0
⟨𝑉𝑎(𝑧)𝑉−𝑎(𝑤)⟩ = ⟨: 𝑒𝑖𝑎𝑋(𝑧): : 𝑒−𝑖𝑎𝑋(𝑤): ⟩= 𝑒−𝑎2log |𝑧−𝑤|2
= 1
|𝑧 − 𝑤|2𝑎2
𝑖𝜕𝑧𝑋𝑉𝑎(𝑤, 𝑤‾ ) = 𝑎 𝑉𝑎(𝑤, 𝑤‾ )
(𝑧 − 𝑤) + finito
Bosón libre.
𝑆 = 1
4𝜋 ∫ 𝑑2𝑧𝜕𝑋𝜕‾𝑋
⟨𝑋(𝑧, 𝑧‾)𝑋(𝑤, 𝑤‾ )⟩ = −log (|𝑧 − 𝑤|2𝜇2)
𝜕𝑧𝑋(𝑧)𝜕𝑤𝑋(𝑤) = 𝜕𝑧𝜕𝑤⟨𝑋𝑋⟩+: 𝜕𝑧𝑋𝜕𝑤𝑋:= − 1
(𝑧 − 𝑤)2 +: 𝜕𝑧𝑋𝜕𝑤𝑋:
𝑇(𝑧) = − 1
2 : 𝜕𝑋𝜕𝑋: = − 1
2 lim
𝑧→𝑤 [𝜕𝑧𝑋𝜕𝑤𝑋 + 1
(𝑧 − 𝑤)2]
𝑇‾(𝑧‾) = − 1
2 : 𝜕‾𝑋𝜕‾𝑋: = − 1
2 lim
𝑧‾→𝑤‾ [𝜕𝑧‾𝑋𝜕𝑤‾ 𝑋 + 1
(𝑧‾ − 𝑤‾ )2]
• Wick:
𝑇(𝑧)𝜕𝑋(𝑤) = − 1
2 : 𝜕𝑋(𝑧)𝜕𝑋(𝑧): 𝜕𝑋(𝑤) = −𝜕𝑋(𝑧)⟨𝜕𝑋(𝑧)𝜕𝑋(𝑤)⟩ + ⋯ = 𝜕𝑋(𝑧) 1
(𝑧 − 𝑤)2 + ⋯
= 𝜕𝑋(𝑤)
(𝑧 − 𝑤)2 + 1
𝑧 − 𝑤 𝜕2𝑋(𝑤) + ⋯
𝑇(𝑧)𝑉𝑎(𝑤, 𝑤‾ ) = − 1
2 : 𝜕𝑋(𝑧)𝜕𝑋(𝑧): ∑
∞
𝑛=0
𝑖𝑛𝑎𝑛
𝑛! : 𝑋𝑛(𝑤, 𝑤‾ ):
𝑇(𝑧)𝑉𝑎(𝑤) = − 1
2 [𝑖𝑎𝜕⟨𝑋𝑋⟩]2𝑒𝑖𝑎𝑋(𝑤) − 1
2 2𝑖𝑎: 𝜕𝑋(𝑧)𝜕⟨𝑋𝑋⟩𝑒𝑖𝑎𝑋(𝑤): + ⋯
= 𝑎2/2
(𝑧 − 𝑤)2 𝑒𝑖𝑎𝑋(𝑤) + 𝑖𝑎𝜕𝑋(𝑧)
𝑧 − 𝑤 𝑒𝑖𝑎𝑋(𝑤) + ⋯ = 𝑎2/2
(𝑧 − 𝑤)2 𝑉𝑎(𝑤) + 1
𝑧 − 𝑤 𝜕𝑉𝑎(𝑤) + ⋯
𝐺𝑁 = ⟨∏
𝑁
𝑖=1
𝑉𝑎𝑖 (𝑧𝑖, 𝑧‾𝑖)⟩ = exp [1
2 ∑
𝑁
𝑖,𝑗=1;𝑖≠𝑗
𝑎𝑖𝑎𝑗⟨𝑋(𝑧𝑖, 𝑧‾𝑖)𝑋(𝑧𝑗, 𝑧‾𝑗)⟩]
∑
𝑖
𝑎𝑖 = 0
⟨𝑉𝑎(𝑧)𝑉−𝑎(𝑤)⟩ = ⟨: 𝑒𝑖𝑎𝑋(𝑧): : 𝑒−𝑖𝑎𝑋(𝑤): ⟩= 𝑒−𝑎2log |𝑧−𝑤|2
= 1
|𝑧 − 𝑤|2𝑎2
𝑖𝜕𝑧𝑋𝑉𝑎(𝑤, 𝑤‾ ) = 𝑎 𝑉𝑎(𝑤, 𝑤‾ )
(𝑧 − 𝑤) + finito
pág. 19
Carga central – OPE.
𝑇(𝑧)𝑇(𝑤) = 𝑐/2
(𝑧 − 𝑤)4 + 2 𝑇(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝑇(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧)𝑇‾(𝑤‾ ) = regular
𝑇(𝑧)𝑇(𝑤) = 1
4 {2(𝜕𝜕⟨𝑋𝑋⟩)2 + 4: 𝜕𝑋(𝑧)𝜕𝑋(𝑤): 𝜕𝜕⟨𝑋𝑋⟩ + ⋯ }
= 1/2
(𝑧 − 𝑤)4 + 2
(𝑧 − 𝑤)2 𝑇(𝑤) + 1
𝑧 − 𝑤 𝜕𝑇(𝑤) + ⋯
𝑇˜ = − 1
2 : 𝜕𝑋𝜕𝑋: +𝑖𝑄𝜕2𝑋
Fermión libre.
𝑐 = 1 − 12𝑄2
∂̸ = 𝜎1𝜕1 + 𝜎2𝜕2 = ( 0 𝜕1 − 𝑖𝜕2
𝜕1 + 𝑖𝜕2 0 ) ∼ (0 𝜕
𝜕‾ 0) .
• Majorana espinor (𝜓
𝜓‾ ):
𝑆 = − 1
8𝜋 ∫ 𝑑2𝑧(𝜓𝜕‾𝜓 + 𝜓‾𝜕𝜓‾)
𝜕‾𝜓 = 𝜕𝜓‾ = 0
𝜓(𝑧)𝜓(𝑤) = 1
𝑧 − 𝑤 , 𝜓‾(𝑧‾)𝜓‾(𝑤‾ ) = 1
𝑧‾ − 𝑤‾
𝑇(𝑧) = − 1
2 : 𝜓(𝑧)𝜕𝜓(𝑧): , 𝑇‾ (𝑧‾) = − 1
2 : 𝜓‾(𝑧‾)𝜕‾𝜓‾(𝑧‾): .
𝑇(𝑧)𝑇(𝑤) = 1/4
(𝑧 − 𝑤)4 + 2
(𝑧 − 𝑤)2 𝑇(𝑤) + 1
𝑧 − 𝑤 𝜕𝑇(𝑤)
𝑇(𝑧) = ∑
𝑛∈ℤ
𝑧−𝑛−2𝐿𝑛 , 𝑇‾(𝑧‾) = ∑
𝑛∈ℤ
𝑧‾−𝑛−2𝐿‾𝑛
𝑤 = 𝜏 + 𝑖𝜎 → 𝑧 = 𝑒𝑤
Φcyl(𝑤) = ∑
𝑛∈ℤ
𝜙𝑛𝑒−𝑛𝑤 = ∑
𝑛∈ℤ
𝜙𝑛𝑒𝑖𝑛(𝑖𝜏−𝜎) = ∑
𝑛∈ℤ
𝜙𝑛𝑧−𝑛
Φ(𝑧) = ∑
𝑛∈ℤ
𝜙𝑛𝑧−𝑛−ℎ
𝑇(𝑧) → (𝑓′)2𝑇(𝑓(𝑧)) + 𝑐
12 [𝑓′′′
𝑓′ − 3
2 (𝑓′′
𝑓′ )
2
]
Carga central – OPE.
𝑇(𝑧)𝑇(𝑤) = 𝑐/2
(𝑧 − 𝑤)4 + 2 𝑇(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝑇(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧)𝑇‾(𝑤‾ ) = regular
𝑇(𝑧)𝑇(𝑤) = 1
4 {2(𝜕𝜕⟨𝑋𝑋⟩)2 + 4: 𝜕𝑋(𝑧)𝜕𝑋(𝑤): 𝜕𝜕⟨𝑋𝑋⟩ + ⋯ }
= 1/2
(𝑧 − 𝑤)4 + 2
(𝑧 − 𝑤)2 𝑇(𝑤) + 1
𝑧 − 𝑤 𝜕𝑇(𝑤) + ⋯
𝑇˜ = − 1
2 : 𝜕𝑋𝜕𝑋: +𝑖𝑄𝜕2𝑋
Fermión libre.
𝑐 = 1 − 12𝑄2
∂̸ = 𝜎1𝜕1 + 𝜎2𝜕2 = ( 0 𝜕1 − 𝑖𝜕2
𝜕1 + 𝑖𝜕2 0 ) ∼ (0 𝜕
𝜕‾ 0) .
• Majorana espinor (𝜓
𝜓‾ ):
𝑆 = − 1
8𝜋 ∫ 𝑑2𝑧(𝜓𝜕‾𝜓 + 𝜓‾𝜕𝜓‾)
𝜕‾𝜓 = 𝜕𝜓‾ = 0
𝜓(𝑧)𝜓(𝑤) = 1
𝑧 − 𝑤 , 𝜓‾(𝑧‾)𝜓‾(𝑤‾ ) = 1
𝑧‾ − 𝑤‾
𝑇(𝑧) = − 1
2 : 𝜓(𝑧)𝜕𝜓(𝑧): , 𝑇‾ (𝑧‾) = − 1
2 : 𝜓‾(𝑧‾)𝜕‾𝜓‾(𝑧‾): .
𝑇(𝑧)𝑇(𝑤) = 1/4
(𝑧 − 𝑤)4 + 2
(𝑧 − 𝑤)2 𝑇(𝑤) + 1
𝑧 − 𝑤 𝜕𝑇(𝑤)
𝑇(𝑧) = ∑
𝑛∈ℤ
𝑧−𝑛−2𝐿𝑛 , 𝑇‾(𝑧‾) = ∑
𝑛∈ℤ
𝑧‾−𝑛−2𝐿‾𝑛
𝑤 = 𝜏 + 𝑖𝜎 → 𝑧 = 𝑒𝑤
Φcyl(𝑤) = ∑
𝑛∈ℤ
𝜙𝑛𝑒−𝑛𝑤 = ∑
𝑛∈ℤ
𝜙𝑛𝑒𝑖𝑛(𝑖𝜏−𝜎) = ∑
𝑛∈ℤ
𝜙𝑛𝑧−𝑛
Φ(𝑧) = ∑
𝑛∈ℤ
𝜙𝑛𝑧−𝑛−ℎ
𝑇(𝑧) → (𝑓′)2𝑇(𝑓(𝑧)) + 𝑐
12 [𝑓′′′
𝑓′ − 3
2 (𝑓′′
𝑓′ )
2
]
pág. 20
𝐿𝑛 = ∮ 𝑑𝑧
2𝜋𝑖 𝑧𝑛+1𝑇(𝑧) , 𝐿‾𝑛 = ∮ 𝑑𝑧‾
2𝜋𝑖 𝑧‾𝑛+1𝑇‾(𝑧‾)
[𝐿𝑛, 𝐿𝑚] = (∮ 𝑑𝑧
2𝜋𝑖 ∮ 𝑑𝑤
2𝜋𝑖 − ∮ 𝑑𝑤
2𝜋𝑖 ∮ 𝑑𝑧
2𝜋𝑖) 𝑧𝑛+1𝑇(𝑧)𝑤𝑚+1𝑇(𝑤)
= ∮ 𝑑𝑤
2𝜋𝑖 ∮ 𝐶𝑤
𝑑𝑧
2𝜋𝑖 𝑧𝑛+1𝑤𝑚+1 ( 𝑐/2
(𝑧 − 𝑤)4 + 2𝑇(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝑇(𝑤)
𝑧 − 𝑤 + ⋯ )
= ∮ 𝑑𝑤
2𝜋𝑖 ( 𝑐
12 (𝑛 + 1)𝑛(𝑛 − 1)𝑤𝑛−2𝑤𝑚+1
+ 2(𝑛 + 1)𝑤𝑛𝑤𝑚+1𝑇(𝑤) + 𝑤𝑛+1𝑤𝑚+1𝜕𝑇(𝑤))
[𝐿𝑛, 𝐿𝑚] = (𝑛 − 𝑚)𝐿𝑛+𝑚 + 𝑐
12 (𝑛3 − 𝑛)𝛿𝑛+𝑚,0
[𝐿‾𝑛, 𝐿‾𝑚] = (𝑛 − 𝑚)𝐿‾𝑛+𝑚 + 𝑐‾
12 (𝑛3 − 𝑛)𝛿𝑛+𝑚,0
[𝐿𝑛, 𝐿‾𝑚] = 0
𝑇𝛼
𝛼 = 𝑐
96𝜋3 √𝑔𝑅(2)
∫ [𝐷𝑋]𝑔ˆ 𝑒−𝑆[𝑔ˆ𝛼𝛽,𝑋] = 𝑒−𝑐𝑆𝐿[𝑔𝛼𝛽,𝜙] ∫ [𝐷𝑋]𝑔𝑒−𝑆[𝑔𝛼𝛽,𝑋]
𝑆𝐿[𝑔𝛼𝛽, 𝜙] = 1
96𝜋 ∫ √det𝑔𝑔𝛼𝛽𝜕𝛼𝜙𝜕𝛽𝜙 + 1
48𝜋 ∫ √det𝑔𝑅(2)𝜙
Espacio de Hilbert – Einstein.
|𝐴in ⟩ = lim
𝜏→−∞ 𝐴(𝜏, 𝜎)|0⟩ = lim
𝑧→0 𝐴(𝑧, 𝑧‾)|0⟩.
𝐴˜(𝑤, 𝑤‾ ) = 𝐴(𝑓(𝑤), 𝑓‾(𝑤‾ ))(𝜕𝑓(𝑤))ℎ(𝜕‾𝑓‾(𝑤‾ ))ℎ‾
𝐴˜(𝑤, 𝑤˜ ) = 𝐴 ( 1
𝑤 , 1
𝑤‾ ) (−𝑤−2)ℎ(−𝑤‾ −2)ℎ‾
⟨𝐴out| = lim
𝑤,𝑤‾ →0 ⟨0|𝐴˜(𝑤, 𝑤‾ )
[𝐴(𝑧, 𝑧‾)]† = 𝐴 (1
𝑧‾ , 1
𝑧) 𝑧‾−2ℎ𝑧−2ℎ‾
⟨𝐴out | = lim
𝑤→0 ⟨0|𝐴˜(𝑤, 𝑤‾ ) = lim
𝑧→0 ⟨0|𝐴 (1
𝑧 , 1
𝑧‾) 𝑧‾−2ℎ𝑧−2ℎ‾ = lim
𝑧→0 ⟨0|[𝐴(𝑧, 𝑧‾)]† = |𝐴in ⟩†
𝑇†(𝑧) = ∑
𝑚
𝐿𝑚
†
𝑧‾𝑚+2 ≡ ∑
𝑚
𝐿𝑚
𝑧‾−𝑚−2
1
𝑧‾4
𝐿𝑚
† = 𝐿−𝑚
𝐿𝑛 = ∮ 𝑑𝑧
2𝜋𝑖 𝑧𝑛+1𝑇(𝑧) , 𝐿‾𝑛 = ∮ 𝑑𝑧‾
2𝜋𝑖 𝑧‾𝑛+1𝑇‾(𝑧‾)
[𝐿𝑛, 𝐿𝑚] = (∮ 𝑑𝑧
2𝜋𝑖 ∮ 𝑑𝑤
2𝜋𝑖 − ∮ 𝑑𝑤
2𝜋𝑖 ∮ 𝑑𝑧
2𝜋𝑖) 𝑧𝑛+1𝑇(𝑧)𝑤𝑚+1𝑇(𝑤)
= ∮ 𝑑𝑤
2𝜋𝑖 ∮ 𝐶𝑤
𝑑𝑧
2𝜋𝑖 𝑧𝑛+1𝑤𝑚+1 ( 𝑐/2
(𝑧 − 𝑤)4 + 2𝑇(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝑇(𝑤)
𝑧 − 𝑤 + ⋯ )
= ∮ 𝑑𝑤
2𝜋𝑖 ( 𝑐
12 (𝑛 + 1)𝑛(𝑛 − 1)𝑤𝑛−2𝑤𝑚+1
+ 2(𝑛 + 1)𝑤𝑛𝑤𝑚+1𝑇(𝑤) + 𝑤𝑛+1𝑤𝑚+1𝜕𝑇(𝑤))
[𝐿𝑛, 𝐿𝑚] = (𝑛 − 𝑚)𝐿𝑛+𝑚 + 𝑐
12 (𝑛3 − 𝑛)𝛿𝑛+𝑚,0
[𝐿‾𝑛, 𝐿‾𝑚] = (𝑛 − 𝑚)𝐿‾𝑛+𝑚 + 𝑐‾
12 (𝑛3 − 𝑛)𝛿𝑛+𝑚,0
[𝐿𝑛, 𝐿‾𝑚] = 0
𝑇𝛼
𝛼 = 𝑐
96𝜋3 √𝑔𝑅(2)
∫ [𝐷𝑋]𝑔ˆ 𝑒−𝑆[𝑔ˆ𝛼𝛽,𝑋] = 𝑒−𝑐𝑆𝐿[𝑔𝛼𝛽,𝜙] ∫ [𝐷𝑋]𝑔𝑒−𝑆[𝑔𝛼𝛽,𝑋]
𝑆𝐿[𝑔𝛼𝛽, 𝜙] = 1
96𝜋 ∫ √det𝑔𝑔𝛼𝛽𝜕𝛼𝜙𝜕𝛽𝜙 + 1
48𝜋 ∫ √det𝑔𝑅(2)𝜙
Espacio de Hilbert – Einstein.
|𝐴in ⟩ = lim
𝜏→−∞ 𝐴(𝜏, 𝜎)|0⟩ = lim
𝑧→0 𝐴(𝑧, 𝑧‾)|0⟩.
𝐴˜(𝑤, 𝑤‾ ) = 𝐴(𝑓(𝑤), 𝑓‾(𝑤‾ ))(𝜕𝑓(𝑤))ℎ(𝜕‾𝑓‾(𝑤‾ ))ℎ‾
𝐴˜(𝑤, 𝑤˜ ) = 𝐴 ( 1
𝑤 , 1
𝑤‾ ) (−𝑤−2)ℎ(−𝑤‾ −2)ℎ‾
⟨𝐴out| = lim
𝑤,𝑤‾ →0 ⟨0|𝐴˜(𝑤, 𝑤‾ )
[𝐴(𝑧, 𝑧‾)]† = 𝐴 (1
𝑧‾ , 1
𝑧) 𝑧‾−2ℎ𝑧−2ℎ‾
⟨𝐴out | = lim
𝑤→0 ⟨0|𝐴˜(𝑤, 𝑤‾ ) = lim
𝑧→0 ⟨0|𝐴 (1
𝑧 , 1
𝑧‾) 𝑧‾−2ℎ𝑧−2ℎ‾ = lim
𝑧→0 ⟨0|[𝐴(𝑧, 𝑧‾)]† = |𝐴in ⟩†
𝑇†(𝑧) = ∑
𝑚
𝐿𝑚
†
𝑧‾𝑚+2 ≡ ∑
𝑚
𝐿𝑚
𝑧‾−𝑚−2
1
𝑧‾4
𝐿𝑚
† = 𝐿−𝑚
pág. 21
𝑇(𝑧)|0⟩ = ∑
𝑚∈ℤ
𝐿𝑚𝑧−𝑚−2|0⟩
𝐿𝑚|0⟩ = 0, 𝑚 ≥ −1
⟨0|𝐿𝑚 = 0, 𝑚 ≤ 1
Φ𝑛>−ℎ|0⟩ = 0
[𝐿𝑛, Φ(𝑤)] = ∮ 𝑑𝑧
2𝜋𝑖 𝑧𝑛+1𝑇(𝑧)Φ(𝑤) = ℎ(𝑛 + 1)𝑤𝑛Φ(𝑤) + 𝑤𝑛+1𝜕Φ(𝑤)
|ℎ⟩ ≡ Φ(0)|0⟩.
𝐿𝑚>0|ℎ⟩ = 𝐿𝑚>0Φ(0)|0⟩ = [𝐿𝑚, Φ(0)]|0⟩ + Φ(0)𝐿𝑚>0|0⟩ = 0
|𝜒⟩ = 𝐿−𝑛1 𝐿−𝑛2 … 𝐿−𝑛𝑘 |ℎ⟩,
(𝐿−1Φ)(𝑧) ≡ ∮ 𝐶𝑧
𝑑𝑤
2𝜋𝑖 𝑇(𝑤)Φℎ(𝑧)
Φ𝜒(𝑧) = ∏
𝑘
𝑖=1
∮ 𝑑𝑤𝑖
2𝜋𝑖 (𝑤𝑖 − 𝑧)−𝑛𝑖+1𝑇(𝑤𝑖)Φℎ(𝑧)
[𝐿−1, 𝑂(𝑧, 𝑧‾)] = 𝜕𝑧𝑂(𝑧, 𝑧‾)
𝜒ℎ(𝑞) ≡ Tr [𝑞𝐿0− 𝑐
24]
𝜒ℎ(𝑞) = 𝑞ℎ−𝑐/24
∏∞
𝑛=1 (1 − 𝑞𝑛)
𝜒0(𝑞) = 𝑞−𝑐/24
∏∞
𝑛=2 (1 − 𝑞𝑛)
‖𝐿−𝑛|0⟩‖2 = ⟨0|𝐿−𝑛
† 𝐿−𝑛|0⟩ = ⟨0| [ 𝑐
12 (𝑛3 − 𝑛) + 2𝑛𝐿0] |0⟩ = 𝑐
12 (𝑛3 − 𝑛)
𝑐 = 1 − 6
𝑚(𝑚 + 1)
|𝜒⟩ = (𝐿−2 − 3
4 𝐿−1
2 ) |1/2⟩.
𝐽𝑎(𝑧)𝐽𝑏(𝑤) = 𝐺𝑎𝑏
(𝑧 − 𝑤)2 + 𝑖𝑓𝑎𝑏 𝑐𝐽𝑐(𝑤)
𝑧 − 𝑤 + finito
[𝐽𝑚
𝑎 , 𝐽𝑛
𝑏] = 𝑚𝐺𝑎𝑏𝛿𝑚+𝑛,0 + 𝑖𝑓𝑎𝑏 𝑐𝐽𝑚+𝑛
𝑐 .
𝑇(𝑧)𝐽𝑎(𝑤) = 𝐽𝑎(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐽𝑎(𝑤)
𝑧 − 𝑤
𝑇(𝑧)|0⟩ = ∑
𝑚∈ℤ
𝐿𝑚𝑧−𝑚−2|0⟩
𝐿𝑚|0⟩ = 0, 𝑚 ≥ −1
⟨0|𝐿𝑚 = 0, 𝑚 ≤ 1
Φ𝑛>−ℎ|0⟩ = 0
[𝐿𝑛, Φ(𝑤)] = ∮ 𝑑𝑧
2𝜋𝑖 𝑧𝑛+1𝑇(𝑧)Φ(𝑤) = ℎ(𝑛 + 1)𝑤𝑛Φ(𝑤) + 𝑤𝑛+1𝜕Φ(𝑤)
|ℎ⟩ ≡ Φ(0)|0⟩.
𝐿𝑚>0|ℎ⟩ = 𝐿𝑚>0Φ(0)|0⟩ = [𝐿𝑚, Φ(0)]|0⟩ + Φ(0)𝐿𝑚>0|0⟩ = 0
|𝜒⟩ = 𝐿−𝑛1 𝐿−𝑛2 … 𝐿−𝑛𝑘 |ℎ⟩,
(𝐿−1Φ)(𝑧) ≡ ∮ 𝐶𝑧
𝑑𝑤
2𝜋𝑖 𝑇(𝑤)Φℎ(𝑧)
Φ𝜒(𝑧) = ∏
𝑘
𝑖=1
∮ 𝑑𝑤𝑖
2𝜋𝑖 (𝑤𝑖 − 𝑧)−𝑛𝑖+1𝑇(𝑤𝑖)Φℎ(𝑧)
[𝐿−1, 𝑂(𝑧, 𝑧‾)] = 𝜕𝑧𝑂(𝑧, 𝑧‾)
𝜒ℎ(𝑞) ≡ Tr [𝑞𝐿0− 𝑐
24]
𝜒ℎ(𝑞) = 𝑞ℎ−𝑐/24
∏∞
𝑛=1 (1 − 𝑞𝑛)
𝜒0(𝑞) = 𝑞−𝑐/24
∏∞
𝑛=2 (1 − 𝑞𝑛)
‖𝐿−𝑛|0⟩‖2 = ⟨0|𝐿−𝑛
† 𝐿−𝑛|0⟩ = ⟨0| [ 𝑐
12 (𝑛3 − 𝑛) + 2𝑛𝐿0] |0⟩ = 𝑐
12 (𝑛3 − 𝑛)
𝑐 = 1 − 6
𝑚(𝑚 + 1)
|𝜒⟩ = (𝐿−2 − 3
4 𝐿−1
2 ) |1/2⟩.
𝐽𝑎(𝑧)𝐽𝑏(𝑤) = 𝐺𝑎𝑏
(𝑧 − 𝑤)2 + 𝑖𝑓𝑎𝑏 𝑐𝐽𝑐(𝑤)
𝑧 − 𝑤 + finito
[𝐽𝑚
𝑎 , 𝐽𝑛
𝑏] = 𝑚𝐺𝑎𝑏𝛿𝑚+𝑛,0 + 𝑖𝑓𝑎𝑏 𝑐𝐽𝑚+𝑛
𝑐 .
𝑇(𝑧)𝐽𝑎(𝑤) = 𝐽𝑎(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐽𝑎(𝑤)
𝑧 − 𝑤
pág. 22
𝑆 = 1
4𝜆2 ∫
𝑀2
𝑑2𝜉Tr(𝜕𝜇𝑔𝜕𝜇𝑔−1) + 𝑖𝑘
8𝜋 ∫
𝐵;𝜕𝐵=𝑀2
𝑑3𝜉Tr(𝜖𝛼𝛽𝛾𝑈𝛼𝑈𝛽𝑈𝛾)
𝑇𝐺 (𝑧) = 1
2(𝑘 + ℎ˜ ) : 𝐽𝑎(𝑧)𝐽𝑎(𝑧):
𝑐𝐺 = 𝑘𝐷𝐺
𝑘 + ℎ˜
𝐽𝑚>0
𝑎 |𝑅𝑖⟩ = 0 , 𝐽0
𝑎|𝑅𝑖⟩ = 𝑖(𝑇𝑅
𝑎)𝑖𝑗|𝑅𝑗⟩
𝐽𝑎(𝑧)𝑅𝑖(𝑤, 𝑤‾ ) = 𝑖 (𝑇𝑅
𝑎)𝑖𝑗
(𝑧 − 𝑤) 𝑅𝑗(𝑤, 𝑤‾ ) + ⋯
ℎ𝑅 = 𝐶𝑅
𝑘 + ℎ˜ ,
𝑇G/H(𝑧)𝐽H(𝑤) = regular , 𝑇G/H(𝑧)𝑇H(𝑤) = regular
Supersimetrías. Métrica de Ramond – Clifford.
𝑆 = − 1
8𝜋 ∫ 𝑑2𝑧𝜓𝑖𝜕‾𝜓𝑖
𝐽𝑖𝑗(𝑧) = 𝑖: 𝜓𝑖(𝑧)𝜓𝑗(𝑧): , 𝑖 < 𝑗
𝜓𝑖(𝑧)𝜓𝑗(𝑤) = 𝛿𝑖𝑗
𝑧 − 𝑤
𝐽𝑖𝑗(𝑧)𝐽𝑘𝑙(𝑤) = 𝐺𝑖𝑗,𝑘𝑙
(𝑧 − 𝑤)2 + 𝑖𝑓𝑚𝑛
𝑖𝑗,𝑘𝑙 𝐽𝑚𝑛(𝑤)
(𝑧 − 𝑤) + ⋯
2𝑓𝑖𝑗,𝑘𝑙 𝑚𝑛 = (𝛿𝑖𝑘𝛿𝑙𝑛 − 𝛿𝑖𝑙𝛿𝑘𝑛)𝛿𝑗𝑚 + (𝛿𝑗𝑙𝛿𝑘𝑛 − 𝛿𝑗𝑘𝛿𝑙𝑛)𝛿𝑖𝑚 − (𝑚 ↔ 𝑛)
𝑇(𝑧) = 1
2(𝑁 − 1) ∑
𝑁
𝑖<𝑗
: 𝐽𝑖𝑗(𝑧)𝐽𝑖𝑗(𝑧): .
𝑇(𝑧)𝑇(𝑤) = 𝑐𝐺 /2
(𝑧 − 𝑤)4 + 2𝑇(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝑇(𝑤)
𝑧 − 𝑤
𝑐𝐺 = 𝑘𝐷
𝑘 + ℎ˜
𝑐𝐺 = 𝑁(𝑁 − 1)/2
1 + 𝑁 − 2 = 𝑁
2 ,
𝑇(𝑧) = − 1
2 ∑
𝑁
𝑖=1
: 𝜓𝑖𝜕𝜓𝑖:
𝑆 = 1
4𝜆2 ∫
𝑀2
𝑑2𝜉Tr(𝜕𝜇𝑔𝜕𝜇𝑔−1) + 𝑖𝑘
8𝜋 ∫
𝐵;𝜕𝐵=𝑀2
𝑑3𝜉Tr(𝜖𝛼𝛽𝛾𝑈𝛼𝑈𝛽𝑈𝛾)
𝑇𝐺 (𝑧) = 1
2(𝑘 + ℎ˜ ) : 𝐽𝑎(𝑧)𝐽𝑎(𝑧):
𝑐𝐺 = 𝑘𝐷𝐺
𝑘 + ℎ˜
𝐽𝑚>0
𝑎 |𝑅𝑖⟩ = 0 , 𝐽0
𝑎|𝑅𝑖⟩ = 𝑖(𝑇𝑅
𝑎)𝑖𝑗|𝑅𝑗⟩
𝐽𝑎(𝑧)𝑅𝑖(𝑤, 𝑤‾ ) = 𝑖 (𝑇𝑅
𝑎)𝑖𝑗
(𝑧 − 𝑤) 𝑅𝑗(𝑤, 𝑤‾ ) + ⋯
ℎ𝑅 = 𝐶𝑅
𝑘 + ℎ˜ ,
𝑇G/H(𝑧)𝐽H(𝑤) = regular , 𝑇G/H(𝑧)𝑇H(𝑤) = regular
Supersimetrías. Métrica de Ramond – Clifford.
𝑆 = − 1
8𝜋 ∫ 𝑑2𝑧𝜓𝑖𝜕‾𝜓𝑖
𝐽𝑖𝑗(𝑧) = 𝑖: 𝜓𝑖(𝑧)𝜓𝑗(𝑧): , 𝑖 < 𝑗
𝜓𝑖(𝑧)𝜓𝑗(𝑤) = 𝛿𝑖𝑗
𝑧 − 𝑤
𝐽𝑖𝑗(𝑧)𝐽𝑘𝑙(𝑤) = 𝐺𝑖𝑗,𝑘𝑙
(𝑧 − 𝑤)2 + 𝑖𝑓𝑚𝑛
𝑖𝑗,𝑘𝑙 𝐽𝑚𝑛(𝑤)
(𝑧 − 𝑤) + ⋯
2𝑓𝑖𝑗,𝑘𝑙 𝑚𝑛 = (𝛿𝑖𝑘𝛿𝑙𝑛 − 𝛿𝑖𝑙𝛿𝑘𝑛)𝛿𝑗𝑚 + (𝛿𝑗𝑙𝛿𝑘𝑛 − 𝛿𝑗𝑘𝛿𝑙𝑛)𝛿𝑖𝑚 − (𝑚 ↔ 𝑛)
𝑇(𝑧) = 1
2(𝑁 − 1) ∑
𝑁
𝑖<𝑗
: 𝐽𝑖𝑗(𝑧)𝐽𝑖𝑗(𝑧): .
𝑇(𝑧)𝑇(𝑤) = 𝑐𝐺 /2
(𝑧 − 𝑤)4 + 2𝑇(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝑇(𝑤)
𝑧 − 𝑤
𝑐𝐺 = 𝑘𝐷
𝑘 + ℎ˜
𝑐𝐺 = 𝑁(𝑁 − 1)/2
1 + 𝑁 − 2 = 𝑁
2 ,
𝑇(𝑧) = − 1
2 ∑
𝑁
𝑖=1
: 𝜓𝑖𝜕𝜓𝑖:
pág. 23
ℎ𝑉 = (𝑁 − 1)/2
1 + 𝑁 − 2 = 1
2
𝐽𝑖𝑗(𝑧)𝜓𝑘(𝑤) = 𝑖 𝑇𝑘𝑙
𝑖𝑗
𝑧 − 𝑤 𝜓𝑙(𝑤) + ⋯ ,
𝜓𝑖 → −𝜓𝑖.
𝜓(𝜏 + 𝑖𝜎) = ∑
𝑛
𝜓𝑛𝑒−𝑛(𝜏+𝑖𝜎)
𝜓𝑖(𝑧) = ∑
𝑛
𝜓𝑛
𝑖 𝑧−𝑛−ℎ = ∑
𝑛
𝜓𝑛
𝑖 𝑧−𝑛−1/2
{𝜓𝑚
𝑖 , 𝜓𝑛
𝑗 } = 𝛿𝑖𝑗𝛿𝑚+𝑛,0
𝜓𝑛>0
𝑖 |0⟩ = 0
|𝑖⟩ = 𝜓−1
2
𝑖 |0⟩
{(−1)𝐹, 𝜓𝑛
𝑖 } = 0
𝐽−1
𝑖𝑗 |0⟩ = 𝑖𝜓−1
2
𝑖 𝜓−1
2
𝑗 |0⟩
𝐽−1
𝑖𝑗 |𝑘⟩ = 𝑖 [𝛿𝑗𝑘𝜓−3
2
𝑖 − 𝛿𝑖𝑘𝜓−3
2
𝑗 + 𝜓−1
2
𝑖 𝜓−1
2
𝑗 𝜓−1
2
𝑘 ] |0⟩.
Tr𝑁𝑆[𝑞𝐿0−𝑐/24] = 𝑞− 𝑁
48 ∏
∞
𝑛=1
(1 + 𝑞𝑛−1
2)
𝑁
Tr𝑁𝑆[𝑞𝐿0−𝑐/24] = [𝜗3
𝜂 ]
𝑁/2
Tr𝑁𝑆[(−1)𝐹𝑞𝐿0−𝑐/24] = 𝑞− 𝑁
48 ∏
∞
𝑛=1
(1 − 𝑞𝑛−1
2)
𝑁
= [𝜗4
𝜂 ]
𝑁/2
𝜒0 = Tr𝑁𝑆 [(1 + (−1)𝐹)
2 𝑞𝐿0−𝑐/24] = 1
2 ([𝜗3
𝜂 ]
𝑁/2
+ [𝜗4
𝜂 ]
𝑁/2
)
𝜒𝑉 = Tr𝑁𝑆 [(1 − (−1)𝐹)
2 𝑞𝐿0−𝑐/24] = 1
2 ([𝜗3
𝜂 ]
𝑁/2
− [𝜗4
𝜂 ]
𝑁/2
) .
𝜒𝑅(𝑣𝑖) = Tr𝑅 [𝑞𝐿0−𝑐/24𝑒2𝜋𝑖 ∑𝑖 𝑣𝑖𝐽0
𝑖
]
ℎ𝑉 = (𝑁 − 1)/2
1 + 𝑁 − 2 = 1
2
𝐽𝑖𝑗(𝑧)𝜓𝑘(𝑤) = 𝑖 𝑇𝑘𝑙
𝑖𝑗
𝑧 − 𝑤 𝜓𝑙(𝑤) + ⋯ ,
𝜓𝑖 → −𝜓𝑖.
𝜓(𝜏 + 𝑖𝜎) = ∑
𝑛
𝜓𝑛𝑒−𝑛(𝜏+𝑖𝜎)
𝜓𝑖(𝑧) = ∑
𝑛
𝜓𝑛
𝑖 𝑧−𝑛−ℎ = ∑
𝑛
𝜓𝑛
𝑖 𝑧−𝑛−1/2
{𝜓𝑚
𝑖 , 𝜓𝑛
𝑗 } = 𝛿𝑖𝑗𝛿𝑚+𝑛,0
𝜓𝑛>0
𝑖 |0⟩ = 0
|𝑖⟩ = 𝜓−1
2
𝑖 |0⟩
{(−1)𝐹, 𝜓𝑛
𝑖 } = 0
𝐽−1
𝑖𝑗 |0⟩ = 𝑖𝜓−1
2
𝑖 𝜓−1
2
𝑗 |0⟩
𝐽−1
𝑖𝑗 |𝑘⟩ = 𝑖 [𝛿𝑗𝑘𝜓−3
2
𝑖 − 𝛿𝑖𝑘𝜓−3
2
𝑗 + 𝜓−1
2
𝑖 𝜓−1
2
𝑗 𝜓−1
2
𝑘 ] |0⟩.
Tr𝑁𝑆[𝑞𝐿0−𝑐/24] = 𝑞− 𝑁
48 ∏
∞
𝑛=1
(1 + 𝑞𝑛−1
2)
𝑁
Tr𝑁𝑆[𝑞𝐿0−𝑐/24] = [𝜗3
𝜂 ]
𝑁/2
Tr𝑁𝑆[(−1)𝐹𝑞𝐿0−𝑐/24] = 𝑞− 𝑁
48 ∏
∞
𝑛=1
(1 − 𝑞𝑛−1
2)
𝑁
= [𝜗4
𝜂 ]
𝑁/2
𝜒0 = Tr𝑁𝑆 [(1 + (−1)𝐹)
2 𝑞𝐿0−𝑐/24] = 1
2 ([𝜗3
𝜂 ]
𝑁/2
+ [𝜗4
𝜂 ]
𝑁/2
)
𝜒𝑉 = Tr𝑁𝑆 [(1 − (−1)𝐹)
2 𝑞𝐿0−𝑐/24] = 1
2 ([𝜗3
𝜂 ]
𝑁/2
− [𝜗4
𝜂 ]
𝑁/2
) .
𝜒𝑅(𝑣𝑖) = Tr𝑅 [𝑞𝐿0−𝑐/24𝑒2𝜋𝑖 ∑𝑖 𝑣𝑖𝐽0
𝑖
]
pág. 24
𝜒0(𝑣𝑖) = 1
2 [∏
𝑁/2
𝑖=1
𝜗3(𝑣𝑖)
𝜂 + ∏
𝑁/2
𝑖=1
𝜗4(𝑣𝑖)
𝜂 ]
𝜒𝑉(𝑣𝑖) = 1
2 [∏
𝑁/2
𝑖=1
𝜗3(𝑣𝑖)
𝜂 − ∏
𝑁/2
𝑖=1
𝜗4(𝑣𝑖)
𝜂 ]
{𝜓0
𝑖 , 𝜓0
𝑗} = 𝛿𝑖𝑗
𝜓𝑚>0
𝑖 |𝑆ˆ𝛼⟩ = 0 , 𝜓0
𝑖 |𝑆ˆ𝛼⟩ = 𝛾𝛼𝛽
𝑖 |𝑆𝛽⟩
𝛾𝑁+1 = ∏
𝑁
𝑖=1
(𝜓0
𝑖 /√2) , {𝛾𝑁+1, 𝜓0
𝑖 } = 0 , [𝛾𝑁+1]2 = 1
(−1)𝐹 = 𝛾𝑁+1(−1)∑∞
𝑛=1 𝜓−𝑛
𝑖 𝜓𝑛
𝑖
(−1)𝐹|𝑆⟩ = |𝑆⟩ , (−1)𝐹|𝐶⟩ = −|𝐶⟩.
𝐺𝑅
𝑖𝑗(𝑧, 𝑤) = ⟨𝑆ˆ|𝜓𝑖(𝑧)𝜓𝑗(𝑤)|𝑆ˆ⟩
𝐺𝑅
𝑖𝑗(𝑧, 𝑤) = 𝛿𝑖𝑗 𝑧 + 𝑤
2√𝑧𝑤
1
𝑧 − 𝑤
⟨𝑋|𝑇(𝑧)|𝑋⟩ = ℎ
𝑧2
𝑇(𝑤) = lim
𝑧→𝑤 [− 1
2 ∑
𝑁
𝑖=1
𝜓𝑖(𝑧)𝜕𝑤𝜓𝑖(𝑤) + 𝑁
2(𝑧 − 𝑤)2]
⟨𝑆ˆ|𝑇(𝑧)|𝑆ˆ⟩ = 𝑁
16𝑧2
Tr𝑅[𝑞𝐿0−𝑐/24] = 2𝑁/2𝑞 𝑁
16− 𝑁
48 ∏
∞
𝑛=1
(1 + 𝑞𝑛)𝑁 = [𝜗2
𝜂 ]
𝑁/2
𝜒𝑆 = 𝜒𝐶 = 1
2 [𝜗2
𝜂 ]
𝑁/2
.
𝜒𝑆(𝑣𝑖) = 1
2 [∏
𝑁/2
𝑖=1
𝜗2(𝑣𝑖)
𝜂 + ∏
𝑁/2
𝑖=1
𝜗1(𝑣𝑖)
𝜂 ] ,
𝜒𝐶 (𝑣𝑖) = 1
2 [∏
𝑁/2
𝑖=1
𝜗2(𝑣𝑖)
𝜂 − ∏
𝑁/2
𝑖=1
𝜗1(𝑣𝑖)
𝜂 ]
|𝑆ˆ𝛼⟩ = lim
𝑧→0 𝑆ˆ𝛼(𝑧)|0⟩
𝜒0(𝑣𝑖) = 1
2 [∏
𝑁/2
𝑖=1
𝜗3(𝑣𝑖)
𝜂 + ∏
𝑁/2
𝑖=1
𝜗4(𝑣𝑖)
𝜂 ]
𝜒𝑉(𝑣𝑖) = 1
2 [∏
𝑁/2
𝑖=1
𝜗3(𝑣𝑖)
𝜂 − ∏
𝑁/2
𝑖=1
𝜗4(𝑣𝑖)
𝜂 ]
{𝜓0
𝑖 , 𝜓0
𝑗} = 𝛿𝑖𝑗
𝜓𝑚>0
𝑖 |𝑆ˆ𝛼⟩ = 0 , 𝜓0
𝑖 |𝑆ˆ𝛼⟩ = 𝛾𝛼𝛽
𝑖 |𝑆𝛽⟩
𝛾𝑁+1 = ∏
𝑁
𝑖=1
(𝜓0
𝑖 /√2) , {𝛾𝑁+1, 𝜓0
𝑖 } = 0 , [𝛾𝑁+1]2 = 1
(−1)𝐹 = 𝛾𝑁+1(−1)∑∞
𝑛=1 𝜓−𝑛
𝑖 𝜓𝑛
𝑖
(−1)𝐹|𝑆⟩ = |𝑆⟩ , (−1)𝐹|𝐶⟩ = −|𝐶⟩.
𝐺𝑅
𝑖𝑗(𝑧, 𝑤) = ⟨𝑆ˆ|𝜓𝑖(𝑧)𝜓𝑗(𝑤)|𝑆ˆ⟩
𝐺𝑅
𝑖𝑗(𝑧, 𝑤) = 𝛿𝑖𝑗 𝑧 + 𝑤
2√𝑧𝑤
1
𝑧 − 𝑤
⟨𝑋|𝑇(𝑧)|𝑋⟩ = ℎ
𝑧2
𝑇(𝑤) = lim
𝑧→𝑤 [− 1
2 ∑
𝑁
𝑖=1
𝜓𝑖(𝑧)𝜕𝑤𝜓𝑖(𝑤) + 𝑁
2(𝑧 − 𝑤)2]
⟨𝑆ˆ|𝑇(𝑧)|𝑆ˆ⟩ = 𝑁
16𝑧2
Tr𝑅[𝑞𝐿0−𝑐/24] = 2𝑁/2𝑞 𝑁
16− 𝑁
48 ∏
∞
𝑛=1
(1 + 𝑞𝑛)𝑁 = [𝜗2
𝜂 ]
𝑁/2
𝜒𝑆 = 𝜒𝐶 = 1
2 [𝜗2
𝜂 ]
𝑁/2
.
𝜒𝑆(𝑣𝑖) = 1
2 [∏
𝑁/2
𝑖=1
𝜗2(𝑣𝑖)
𝜂 + ∏
𝑁/2
𝑖=1
𝜗1(𝑣𝑖)
𝜂 ] ,
𝜒𝐶 (𝑣𝑖) = 1
2 [∏
𝑁/2
𝑖=1
𝜗2(𝑣𝑖)
𝜂 − ∏
𝑁/2
𝑖=1
𝜗1(𝑣𝑖)
𝜂 ]
|𝑆ˆ𝛼⟩ = lim
𝑧→0 𝑆ˆ𝛼(𝑧)|0⟩
pág. 25
𝜓𝑖(𝑧)𝑆ˆ𝛼(𝑤) = 𝛾𝛼𝛽
𝑖 𝑆ˆ𝛽(𝑤)
√𝑧 − 𝑤 + ⋯
𝐽𝑖𝑗(𝑧)𝑆ˆ𝛼(𝑤) = 𝑖
2 [𝛾𝑖, 𝛾𝑗]𝛼𝛽
𝑆ˆ𝛽(𝑤)
(𝑧 − 𝑤) + ⋯
𝑆ˆ𝛼(𝑧)𝑆ˆ𝛽(𝑤) = 𝛿𝛼𝛽
(𝑧 − 𝑤)𝑁/8 + 𝛾𝛼𝛽
𝑖 𝜓𝑖(𝑤)
(𝑧 − 𝑤)𝑁/8−1/2 + 𝑖
2 [𝛾𝑖, 𝛾𝑗]𝛼𝛽
𝐽𝑖𝑗(𝑤)
(𝑧 − 𝑤)𝑁/8−1 + ⋯
Supersimetría isotrópica. Métrica de Majorana.
𝑆 = 1
2𝜋 ∫ 𝑑2𝑧𝜕𝑋𝜕‾𝑋 + 1
2𝜋 ∫ 𝑑2𝑧(𝜓𝜕‾𝜓 + 𝜓‾𝜕𝜓‾)
𝛿𝑋 = 𝜖(𝑧)𝜓 , 𝛿𝜓 = −𝜖(𝑧)𝜕𝑋 , 𝛿𝜓‾ = 0
𝛿𝑋 = 𝜖‾(𝑧‾)𝜓‾ , 𝛿𝜓‾ = −𝜖‾(𝑧‾)𝜕‾𝑋 , 𝛿𝜓 = 0
𝐺(𝑧) = 𝑖𝜓𝜕𝑋 , 𝐺‾(𝑧‾) = 𝑖𝜓‾𝜕‾𝑋
𝐺(𝑧)𝐺(𝑤) = 1
(𝑧 − 𝑤)3 + 2 𝑇(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧)𝐺(𝑤) = 3
2
𝐺(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐺(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧) = − 1
2 : 𝜕𝑋𝜕𝑋: − 1
2 : 𝜓𝜕𝜓: .
𝐺(𝑧)𝐺(𝑤) = 𝑐ˆ
(𝑧 − 𝑤)3 + 2 𝑇(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧)𝐺(𝑤) = 3
2
𝐺(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐺(𝑤)
𝑧 − 𝑤 + ⋯
{𝐺𝑟, 𝐺𝑠} = 𝑐ˆ
2 (𝑟2 − 1
4) 𝛿𝑟+𝑠,0 + 2𝐿𝑟+𝑠
[𝐿𝑚, 𝐺𝑟] = (𝑚
2 − 𝑟) 𝐺𝑚+𝑟
𝜒𝑁=1
𝑁𝑆 = Tr[𝑞𝐿0−𝑐/24] = 𝑞ℎ−𝑐/24 ∏
∞
𝑛=1
1 + 𝑞𝑛−1
2
1 − 𝑞𝑛 .
{𝐺0, 𝐺0} = 2𝐿0 − 𝑐ˆ
8
𝜒𝑁=1
𝑅 = Tr[𝑞𝐿0−𝑐/24] = 𝑞ℎ−𝑐/24 ∏
∞
𝑛=1
1 + 𝑞𝑛
1 − 𝑞𝑛 .
𝐷𝜃 = 𝜕
𝜕𝜃 + 𝜃𝜕𝑧 , 𝐷‾ 𝜃‾ = 𝜕
𝜕𝜃‾ + 𝜃‾𝜕𝑧‾
𝑋ˆ (𝑧, 𝑧‾, 𝜃, 𝜃‾) = 𝑋 + 𝜃𝜓 + 𝜃‾𝜓‾ + 𝜃𝜃‾𝐹
𝜓𝑖(𝑧)𝑆ˆ𝛼(𝑤) = 𝛾𝛼𝛽
𝑖 𝑆ˆ𝛽(𝑤)
√𝑧 − 𝑤 + ⋯
𝐽𝑖𝑗(𝑧)𝑆ˆ𝛼(𝑤) = 𝑖
2 [𝛾𝑖, 𝛾𝑗]𝛼𝛽
𝑆ˆ𝛽(𝑤)
(𝑧 − 𝑤) + ⋯
𝑆ˆ𝛼(𝑧)𝑆ˆ𝛽(𝑤) = 𝛿𝛼𝛽
(𝑧 − 𝑤)𝑁/8 + 𝛾𝛼𝛽
𝑖 𝜓𝑖(𝑤)
(𝑧 − 𝑤)𝑁/8−1/2 + 𝑖
2 [𝛾𝑖, 𝛾𝑗]𝛼𝛽
𝐽𝑖𝑗(𝑤)
(𝑧 − 𝑤)𝑁/8−1 + ⋯
Supersimetría isotrópica. Métrica de Majorana.
𝑆 = 1
2𝜋 ∫ 𝑑2𝑧𝜕𝑋𝜕‾𝑋 + 1
2𝜋 ∫ 𝑑2𝑧(𝜓𝜕‾𝜓 + 𝜓‾𝜕𝜓‾)
𝛿𝑋 = 𝜖(𝑧)𝜓 , 𝛿𝜓 = −𝜖(𝑧)𝜕𝑋 , 𝛿𝜓‾ = 0
𝛿𝑋 = 𝜖‾(𝑧‾)𝜓‾ , 𝛿𝜓‾ = −𝜖‾(𝑧‾)𝜕‾𝑋 , 𝛿𝜓 = 0
𝐺(𝑧) = 𝑖𝜓𝜕𝑋 , 𝐺‾(𝑧‾) = 𝑖𝜓‾𝜕‾𝑋
𝐺(𝑧)𝐺(𝑤) = 1
(𝑧 − 𝑤)3 + 2 𝑇(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧)𝐺(𝑤) = 3
2
𝐺(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐺(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧) = − 1
2 : 𝜕𝑋𝜕𝑋: − 1
2 : 𝜓𝜕𝜓: .
𝐺(𝑧)𝐺(𝑤) = 𝑐ˆ
(𝑧 − 𝑤)3 + 2 𝑇(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧)𝐺(𝑤) = 3
2
𝐺(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐺(𝑤)
𝑧 − 𝑤 + ⋯
{𝐺𝑟, 𝐺𝑠} = 𝑐ˆ
2 (𝑟2 − 1
4) 𝛿𝑟+𝑠,0 + 2𝐿𝑟+𝑠
[𝐿𝑚, 𝐺𝑟] = (𝑚
2 − 𝑟) 𝐺𝑚+𝑟
𝜒𝑁=1
𝑁𝑆 = Tr[𝑞𝐿0−𝑐/24] = 𝑞ℎ−𝑐/24 ∏
∞
𝑛=1
1 + 𝑞𝑛−1
2
1 − 𝑞𝑛 .
{𝐺0, 𝐺0} = 2𝐿0 − 𝑐ˆ
8
𝜒𝑁=1
𝑅 = Tr[𝑞𝐿0−𝑐/24] = 𝑞ℎ−𝑐/24 ∏
∞
𝑛=1
1 + 𝑞𝑛
1 − 𝑞𝑛 .
𝐷𝜃 = 𝜕
𝜕𝜃 + 𝜃𝜕𝑧 , 𝐷‾ 𝜃‾ = 𝜕
𝜕𝜃‾ + 𝜃‾𝜕𝑧‾
𝑋ˆ (𝑧, 𝑧‾, 𝜃, 𝜃‾) = 𝑋 + 𝜃𝜓 + 𝜃‾𝜓‾ + 𝜃𝜃‾𝐹
pág. 26
𝑆 = 1
2𝜋 ∫ 𝑑2𝑧 ∫ 𝑑𝜃𝑑𝜃‾𝐷𝜃𝑋ˆ 𝐷‾ 𝜃‾ 𝑋ˆ
𝐺+(𝑧)𝐺−(𝑤) = 2𝑐
3
1
(𝑧 − 𝑤)3 + ( 2𝐽(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐽(𝑤)
𝑧 − 𝑤) + 2
𝑧 − 𝑤 𝑇(𝑤) + ⋯ ,
𝐺+(𝑧)𝐺+(𝑤) = regular , 𝐺−(𝑧)𝐺−(𝑤) = regular ,
𝑇(𝑧)𝐺±(𝑤) = 3
2
𝐺±(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐺±(𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽(𝑧)𝐺±(𝑤) = ± 𝐺±(𝑤)
𝑧 − 𝑤 + ⋯ ,
𝑇(𝑧)𝐽(𝑤) = 𝐽(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐽(𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽(𝑧)𝐽(𝑤) = 𝑐/3
(𝑧 − 𝑤)2 + ⋯
𝐺±(𝑒2𝜋𝑖𝑧) = 𝑒∓2𝜋𝑖𝛼𝐺±(𝑧)
𝐽𝑛
𝛼 = 𝐽𝑛 − 𝛼 𝑐
3 𝛿𝑛,0 , 𝐿𝑛
𝛼 = 𝐿𝑛 − 𝛼𝐽𝑛 + 𝛼2 𝑐
6 𝛿𝑛,0
𝐺𝑟+𝛼
𝛼,+ = 𝐺𝑟
+ , 𝐺𝑟−𝛼
𝛼,− = 𝐺𝑟
−
𝐺0
±𝐺0
± = 0 , {𝐺0
+, 𝐺0
−} = 2 (𝐿0 − 𝑐
24)
𝐽0
𝑅±
= 𝐽0
𝑁𝑆 ∓ 𝑐
6 , 𝐿0
𝑅±
− 𝑐
24 = 𝐿0
𝑁𝑆 − 1
2 𝐽0
𝑁𝑆
𝑂𝑞1 (𝑧)𝑂𝑞2 (𝑧) = 𝑂𝑞1+𝑞2 (𝑧)
𝐽𝑎(𝑧)𝐽𝑏(𝑤) = 𝑘
2
𝛿𝑎𝑏
(𝑧 − 𝑤)2 + 𝑖𝜖𝑎𝑏𝑐 𝐽𝑐(𝑤)
(𝑧 − 𝑤) + ⋯
𝐽𝑎(𝑧)𝐺𝛼(𝑤) = 1
2 𝜎𝛽𝛼
𝑎 𝐺𝛽(𝑤)
(𝑧 − 𝑤) + ⋯ , 𝐽𝑎(𝑧)𝐺‾𝛼(𝑤) = − 1
2 𝜎𝛼𝛽
𝑎 𝐺‾𝛽(𝑤)
(𝑧 − 𝑤) + ⋯
𝐺𝛼(𝑧)𝐺‾𝛽(𝑤) = 4𝑘𝛿𝛼𝛽
(𝑧 − 𝑤)3 + 2𝜎𝛽𝛼
𝑎 [ 2𝐽𝑎(𝑤)
(𝑧 − 𝑤)2 + 𝛿𝐽𝑎(𝑤)
(𝑧 − 𝑤)] + 2𝛿𝛼𝛽 𝑇(𝑤)
(𝑧 − 𝑤) + ⋯
𝐺𝛼(𝑧)𝐺𝛽(𝑤) = regular, 𝐺‾𝛼(𝑧)𝐺‾𝛽(𝑤) = regular
Campos fantasmas.
𝑆𝜆 = 1
𝜋 ∫ 𝑑2𝑧𝑏𝜕‾𝑐
𝑐(𝑧)𝑏(𝑤) = 1
𝑧 − 𝑤 , 𝑏(𝑧)𝑐(𝑤) = 𝜖
𝑧 − 𝑤
𝑐(𝑧) = ∑
𝑛∈𝑍
𝑧−𝑛−(1−𝜆)𝑐𝑛 , 𝑐𝑛
† = 𝑐−𝑛
𝑏(𝑧) = ∑
𝑛∈𝑍
𝑧−𝑛−𝜆𝑏𝑛 , 𝑏𝑛
† = 𝜖𝑏−𝑛
𝑆 = 1
2𝜋 ∫ 𝑑2𝑧 ∫ 𝑑𝜃𝑑𝜃‾𝐷𝜃𝑋ˆ 𝐷‾ 𝜃‾ 𝑋ˆ
𝐺+(𝑧)𝐺−(𝑤) = 2𝑐
3
1
(𝑧 − 𝑤)3 + ( 2𝐽(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐽(𝑤)
𝑧 − 𝑤) + 2
𝑧 − 𝑤 𝑇(𝑤) + ⋯ ,
𝐺+(𝑧)𝐺+(𝑤) = regular , 𝐺−(𝑧)𝐺−(𝑤) = regular ,
𝑇(𝑧)𝐺±(𝑤) = 3
2
𝐺±(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐺±(𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽(𝑧)𝐺±(𝑤) = ± 𝐺±(𝑤)
𝑧 − 𝑤 + ⋯ ,
𝑇(𝑧)𝐽(𝑤) = 𝐽(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝐽(𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽(𝑧)𝐽(𝑤) = 𝑐/3
(𝑧 − 𝑤)2 + ⋯
𝐺±(𝑒2𝜋𝑖𝑧) = 𝑒∓2𝜋𝑖𝛼𝐺±(𝑧)
𝐽𝑛
𝛼 = 𝐽𝑛 − 𝛼 𝑐
3 𝛿𝑛,0 , 𝐿𝑛
𝛼 = 𝐿𝑛 − 𝛼𝐽𝑛 + 𝛼2 𝑐
6 𝛿𝑛,0
𝐺𝑟+𝛼
𝛼,+ = 𝐺𝑟
+ , 𝐺𝑟−𝛼
𝛼,− = 𝐺𝑟
−
𝐺0
±𝐺0
± = 0 , {𝐺0
+, 𝐺0
−} = 2 (𝐿0 − 𝑐
24)
𝐽0
𝑅±
= 𝐽0
𝑁𝑆 ∓ 𝑐
6 , 𝐿0
𝑅±
− 𝑐
24 = 𝐿0
𝑁𝑆 − 1
2 𝐽0
𝑁𝑆
𝑂𝑞1 (𝑧)𝑂𝑞2 (𝑧) = 𝑂𝑞1+𝑞2 (𝑧)
𝐽𝑎(𝑧)𝐽𝑏(𝑤) = 𝑘
2
𝛿𝑎𝑏
(𝑧 − 𝑤)2 + 𝑖𝜖𝑎𝑏𝑐 𝐽𝑐(𝑤)
(𝑧 − 𝑤) + ⋯
𝐽𝑎(𝑧)𝐺𝛼(𝑤) = 1
2 𝜎𝛽𝛼
𝑎 𝐺𝛽(𝑤)
(𝑧 − 𝑤) + ⋯ , 𝐽𝑎(𝑧)𝐺‾𝛼(𝑤) = − 1
2 𝜎𝛼𝛽
𝑎 𝐺‾𝛽(𝑤)
(𝑧 − 𝑤) + ⋯
𝐺𝛼(𝑧)𝐺‾𝛽(𝑤) = 4𝑘𝛿𝛼𝛽
(𝑧 − 𝑤)3 + 2𝜎𝛽𝛼
𝑎 [ 2𝐽𝑎(𝑤)
(𝑧 − 𝑤)2 + 𝛿𝐽𝑎(𝑤)
(𝑧 − 𝑤)] + 2𝛿𝛼𝛽 𝑇(𝑤)
(𝑧 − 𝑤) + ⋯
𝐺𝛼(𝑧)𝐺𝛽(𝑤) = regular, 𝐺‾𝛼(𝑧)𝐺‾𝛽(𝑤) = regular
Campos fantasmas.
𝑆𝜆 = 1
𝜋 ∫ 𝑑2𝑧𝑏𝜕‾𝑐
𝑐(𝑧)𝑏(𝑤) = 1
𝑧 − 𝑤 , 𝑏(𝑧)𝑐(𝑤) = 𝜖
𝑧 − 𝑤
𝑐(𝑧) = ∑
𝑛∈𝑍
𝑧−𝑛−(1−𝜆)𝑐𝑛 , 𝑐𝑛
† = 𝑐−𝑛
𝑏(𝑧) = ∑
𝑛∈𝑍
𝑧−𝑛−𝜆𝑏𝑛 , 𝑏𝑛
† = 𝜖𝑏−𝑛
pág. 27
𝑐𝑚𝑏𝑛 + 𝜖𝑏𝑛𝑐𝑚 = 𝛿𝑚+𝑛,0 , 𝑐𝑚𝑐𝑛 + 𝜖𝑐𝑛𝑐𝑚 = 𝑏𝑚𝑏𝑛 + 𝜖𝑏𝑛𝑏𝑚 = 0
NS: 𝑏𝑛, 𝑛 ∈ ℤ − 𝜆, 𝑐𝑛, 𝑛 ∈ ℤ + 𝜆
R: 𝑏𝑛, 𝑛 ∈ 1
2 + ℤ − 𝜆, 𝑐𝑛, 𝑛 ∈ 1
2 + ℤ + 𝜆
𝑇 = −𝜆𝑏𝜕𝑐 + (1 − 𝜆)(𝜕𝑏)𝑐
𝑐 = −2𝜖(6𝜆2 − 6𝜆 + 1) = 𝜖(1 − 3𝑄2) , 𝑄 = 𝜖(1 − 2𝜆)
𝐽(𝑧) = −: 𝑏(𝑧)𝑐(𝑧): = ∑
𝑛∈ℤ
𝑧−𝑛−1𝐽𝑛
𝐽(𝑧)𝐽(𝑤) = 𝜖
(𝑧 − 𝑤)2 + ⋯
𝐽(𝑧)𝑏(𝑤) = − 𝑏(𝑤)
𝑧 − 𝑤 + ⋯ , 𝐽(𝑧)𝑐(𝑤) = 𝑐(𝑤)
𝑧 − 𝑤 …
𝑇(𝑧)𝐽(𝑤) = 𝑄
(𝑧 − 𝑤)3 + 𝐽(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝑤𝐽(𝑤)
𝑧 − 𝑤 + ⋯
[𝐿𝑚, 𝐽𝑛] = −𝑛𝐽𝑚+𝑛 + 𝑄
2 𝑚(𝑚 + 1)𝛿𝑚+𝑛,0
[𝐿1, 𝐽−1] = 𝐽0 + 𝑄 , 𝑄 + 𝐽0
† = [𝐿1, 𝐽−1]† = −𝐽0
ℤzero modes of 𝑐 − ℤ# zero modes of 𝑏 = − 𝜖
2 𝑄𝜒
𝑏𝑛>−𝜆|0⟩ = 𝑐𝑛>𝜆−1|0⟩ = 0
𝐽(𝑧) = −𝜕𝜙 , ⟨𝜙(𝑧)𝜙(𝑤)⟩ = −log (𝑧 − 𝑤)
𝑇ˆ = 1
2 : 𝐽2: + 1
2 𝑄𝜕𝐽 = 1
2 (𝜕𝜙)2 − 𝑄
2 𝜕2𝜙.
𝑆𝑄 = 1
2𝜋 ∫ 𝑑2𝑧 [𝜕𝜙𝜕‾𝜙 − 𝑄
4 √𝑔𝑅(2)𝜙]
𝑇 = 𝑇ˆ + 𝑇𝜂𝜉
𝑇(𝑧): 𝑒𝑞𝜙(𝑤): = [− 𝑞(𝑞 + 𝑄)
(𝑧 − 𝑤)2 + 1
𝑧 − 𝑤 𝜕𝑤] : 𝑒𝑞𝜙(𝑤): + ⋯
𝐽(𝑧): 𝑒𝑞𝜙(𝑤): = 𝑞
𝑧 − 𝑤 : 𝑒𝑞𝜙(𝑤): … → [𝐽0, : 𝑒𝑞𝜙(𝑤): ] = 𝑞: 𝑒𝑞𝜙(𝑤): .
𝑐(𝑧) = 𝑒𝜙(𝑧)𝜂(𝑧) , 𝑏(𝑧) = 𝑒−𝜙(𝑧)𝜕𝜉(𝑧)
𝑔𝑖𝑗 = 1
𝜏2
( 1 𝜏1
𝜏1 |𝜏|2)
𝑐𝑚𝑏𝑛 + 𝜖𝑏𝑛𝑐𝑚 = 𝛿𝑚+𝑛,0 , 𝑐𝑚𝑐𝑛 + 𝜖𝑐𝑛𝑐𝑚 = 𝑏𝑚𝑏𝑛 + 𝜖𝑏𝑛𝑏𝑚 = 0
NS: 𝑏𝑛, 𝑛 ∈ ℤ − 𝜆, 𝑐𝑛, 𝑛 ∈ ℤ + 𝜆
R: 𝑏𝑛, 𝑛 ∈ 1
2 + ℤ − 𝜆, 𝑐𝑛, 𝑛 ∈ 1
2 + ℤ + 𝜆
𝑇 = −𝜆𝑏𝜕𝑐 + (1 − 𝜆)(𝜕𝑏)𝑐
𝑐 = −2𝜖(6𝜆2 − 6𝜆 + 1) = 𝜖(1 − 3𝑄2) , 𝑄 = 𝜖(1 − 2𝜆)
𝐽(𝑧) = −: 𝑏(𝑧)𝑐(𝑧): = ∑
𝑛∈ℤ
𝑧−𝑛−1𝐽𝑛
𝐽(𝑧)𝐽(𝑤) = 𝜖
(𝑧 − 𝑤)2 + ⋯
𝐽(𝑧)𝑏(𝑤) = − 𝑏(𝑤)
𝑧 − 𝑤 + ⋯ , 𝐽(𝑧)𝑐(𝑤) = 𝑐(𝑤)
𝑧 − 𝑤 …
𝑇(𝑧)𝐽(𝑤) = 𝑄
(𝑧 − 𝑤)3 + 𝐽(𝑤)
(𝑧 − 𝑤)2 + 𝜕𝑤𝐽(𝑤)
𝑧 − 𝑤 + ⋯
[𝐿𝑚, 𝐽𝑛] = −𝑛𝐽𝑚+𝑛 + 𝑄
2 𝑚(𝑚 + 1)𝛿𝑚+𝑛,0
[𝐿1, 𝐽−1] = 𝐽0 + 𝑄 , 𝑄 + 𝐽0
† = [𝐿1, 𝐽−1]† = −𝐽0
ℤzero modes of 𝑐 − ℤ# zero modes of 𝑏 = − 𝜖
2 𝑄𝜒
𝑏𝑛>−𝜆|0⟩ = 𝑐𝑛>𝜆−1|0⟩ = 0
𝐽(𝑧) = −𝜕𝜙 , ⟨𝜙(𝑧)𝜙(𝑤)⟩ = −log (𝑧 − 𝑤)
𝑇ˆ = 1
2 : 𝐽2: + 1
2 𝑄𝜕𝐽 = 1
2 (𝜕𝜙)2 − 𝑄
2 𝜕2𝜙.
𝑆𝑄 = 1
2𝜋 ∫ 𝑑2𝑧 [𝜕𝜙𝜕‾𝜙 − 𝑄
4 √𝑔𝑅(2)𝜙]
𝑇 = 𝑇ˆ + 𝑇𝜂𝜉
𝑇(𝑧): 𝑒𝑞𝜙(𝑤): = [− 𝑞(𝑞 + 𝑄)
(𝑧 − 𝑤)2 + 1
𝑧 − 𝑤 𝜕𝑤] : 𝑒𝑞𝜙(𝑤): + ⋯
𝐽(𝑧): 𝑒𝑞𝜙(𝑤): = 𝑞
𝑧 − 𝑤 : 𝑒𝑞𝜙(𝑤): … → [𝐽0, : 𝑒𝑞𝜙(𝑤): ] = 𝑞: 𝑒𝑞𝜙(𝑤): .
𝑐(𝑧) = 𝑒𝜙(𝑧)𝜂(𝑧) , 𝑏(𝑧) = 𝑒−𝜙(𝑧)𝜕𝜉(𝑧)
𝑔𝑖𝑗 = 1
𝜏2
( 1 𝜏1
𝜏1 |𝜏|2)
pág. 28
𝑑𝑠2 = 𝑔𝑖𝑗𝑑𝜎𝑖𝑑𝜎𝑗 = 1
𝜏2
|𝑑𝜎1 + 𝜏𝑑𝜎2|2 = 𝑑𝑤𝑑𝑤‾
𝜏2
𝑤 = 𝜎1 + 𝜏𝜎2 , 𝑤‾ = 𝜎1 + 𝜏‾𝜎2
𝑤 → 𝑤 + 1 , 𝑤 → 𝑤 + 𝜏
𝑇: 𝜏 → 𝜏 + 1
𝑇𝑆𝑇: 𝜏 → 𝜏
𝜏 + 1 .
𝑆: 𝜏 → − 1
𝜏 , 𝑆2 = 1 , (𝑆𝑇)3 = 1
𝜏′ = 𝑎𝜏 + 𝑏
𝑐𝜏 + 𝑑 ↔ 𝐴 = (𝑎 𝑏
𝑐 𝑑)
𝐿0
𝑐𝑦𝑙 = 𝐿0 − 𝑐
24 , 𝐿‾0
𝑐𝑦𝑙 = 𝐿‾0 − 𝑐‾
24 ,
∫ 𝑒−𝑆 = Tr[𝑒−2𝜋𝜏2𝐻𝑒2𝜋𝑖𝜏1𝑃] = Tr [𝑒2𝜋𝑖𝜏𝐿0
𝑐𝑦𝑙
𝑒−2𝜋𝑖𝜏‾𝐿‾0
𝑐𝑦𝑙
]
= Tr[𝑞𝐿0−𝑐/24𝑞‾𝐿‾0−𝑐‾/24]
Escalares compactos.
𝑆 = 1
4𝜋 ∫ 𝑑2𝜎√𝑔𝑔𝑖𝑗𝜕𝑖𝑋𝜕𝑗𝑋 = 1
4𝜋 ∫
1
0
𝑑𝜎1 ∫
1
0
𝑑𝜎2
1
𝜏2
|𝜏𝜕1𝑋 − 𝜕2𝑋|2 = − 1
4𝜋 ∫ 𝑑2𝜎𝑋 ◻ 𝑋
◻= 1
𝜏2
|𝜏𝜕1 − 𝜕2|2.
𝑍(𝑅) = ∫ 𝐷𝑋𝑒−𝑆
𝑋class = 2𝜋𝑅(𝑛𝜎1 + 𝑚𝜎2) , 𝑚, 𝑛 ∈ ℤ
𝑋𝑐𝑙𝑎𝑠𝑠(𝜎1 + 1, 𝜎2) = 𝑋(𝜎1, 𝜎2) + 2𝜋𝑛𝑅, 𝑋𝑐𝑙𝑎𝑠𝑠(𝜎1, 𝜎2 + 1) = 𝑋(𝜎1, 𝜎2) + 2𝜋𝑚𝑅
𝑆𝑚,𝑛 = 𝜋𝑅2
𝜏2
|𝑚 − 𝑛𝜏|2
𝑍(𝑅) = ∑
𝑚,𝑛∈ℤ
∫ 𝐷𝜒𝑒−𝑆𝑚,𝑛−𝑆(𝜒)
= ∑
𝑚,𝑛∈ℤ
𝑒−𝑆𝑚,𝑛 ∫ 𝐷𝜒𝑒−𝑆(𝜒)
◻ 𝜓𝑖 = −𝜆𝑖𝜓𝑖
𝑑𝑠2 = 𝑔𝑖𝑗𝑑𝜎𝑖𝑑𝜎𝑗 = 1
𝜏2
|𝑑𝜎1 + 𝜏𝑑𝜎2|2 = 𝑑𝑤𝑑𝑤‾
𝜏2
𝑤 = 𝜎1 + 𝜏𝜎2 , 𝑤‾ = 𝜎1 + 𝜏‾𝜎2
𝑤 → 𝑤 + 1 , 𝑤 → 𝑤 + 𝜏
𝑇: 𝜏 → 𝜏 + 1
𝑇𝑆𝑇: 𝜏 → 𝜏
𝜏 + 1 .
𝑆: 𝜏 → − 1
𝜏 , 𝑆2 = 1 , (𝑆𝑇)3 = 1
𝜏′ = 𝑎𝜏 + 𝑏
𝑐𝜏 + 𝑑 ↔ 𝐴 = (𝑎 𝑏
𝑐 𝑑)
𝐿0
𝑐𝑦𝑙 = 𝐿0 − 𝑐
24 , 𝐿‾0
𝑐𝑦𝑙 = 𝐿‾0 − 𝑐‾
24 ,
∫ 𝑒−𝑆 = Tr[𝑒−2𝜋𝜏2𝐻𝑒2𝜋𝑖𝜏1𝑃] = Tr [𝑒2𝜋𝑖𝜏𝐿0
𝑐𝑦𝑙
𝑒−2𝜋𝑖𝜏‾𝐿‾0
𝑐𝑦𝑙
]
= Tr[𝑞𝐿0−𝑐/24𝑞‾𝐿‾0−𝑐‾/24]
Escalares compactos.
𝑆 = 1
4𝜋 ∫ 𝑑2𝜎√𝑔𝑔𝑖𝑗𝜕𝑖𝑋𝜕𝑗𝑋 = 1
4𝜋 ∫
1
0
𝑑𝜎1 ∫
1
0
𝑑𝜎2
1
𝜏2
|𝜏𝜕1𝑋 − 𝜕2𝑋|2 = − 1
4𝜋 ∫ 𝑑2𝜎𝑋 ◻ 𝑋
◻= 1
𝜏2
|𝜏𝜕1 − 𝜕2|2.
𝑍(𝑅) = ∫ 𝐷𝑋𝑒−𝑆
𝑋class = 2𝜋𝑅(𝑛𝜎1 + 𝑚𝜎2) , 𝑚, 𝑛 ∈ ℤ
𝑋𝑐𝑙𝑎𝑠𝑠(𝜎1 + 1, 𝜎2) = 𝑋(𝜎1, 𝜎2) + 2𝜋𝑛𝑅, 𝑋𝑐𝑙𝑎𝑠𝑠(𝜎1, 𝜎2 + 1) = 𝑋(𝜎1, 𝜎2) + 2𝜋𝑚𝑅
𝑆𝑚,𝑛 = 𝜋𝑅2
𝜏2
|𝑚 − 𝑛𝜏|2
𝑍(𝑅) = ∑
𝑚,𝑛∈ℤ
∫ 𝐷𝜒𝑒−𝑆𝑚,𝑛−𝑆(𝜒)
= ∑
𝑚,𝑛∈ℤ
𝑒−𝑆𝑚,𝑛 ∫ 𝐷𝜒𝑒−𝑆(𝜒)
◻ 𝜓𝑖 = −𝜆𝑖𝜓𝑖
pág. 29
𝜓𝑚1,𝑚2 = 𝑒2𝜋𝑖(𝑚1𝜎1+𝑚2𝜎2) , 𝜆𝑚1,𝑚2 = 4𝜋2
𝜏2
|𝑚1𝜏 − 𝑚2|2
∫ 𝑑2𝜎𝜓𝑚1,𝑚2 𝜓𝑛1,𝑛2 = 𝛿𝑚1+𝑛1,0𝛿𝑚2+𝑛2,0
𝛿𝜒 = ∑
𝑚1,𝑚2∈ℤ
′𝐴𝑚1,𝑚2 𝜓𝑚1,𝑚2
𝑆(𝜒) = 1
4𝜋 ∑
′
𝑚1,𝑚2
𝜆𝑚1,𝑚2 |𝐴𝑚1,𝑚2 |2
‖𝛿𝑋‖ = ∫ 𝑑2𝜎√det𝐺(𝑑𝜒)2 = ∑
𝑚1,𝑚2
′|𝑑𝐴𝑚1,𝑚2 |2
∫ 𝐷𝜒 = ∫
2𝜋𝑅
0
𝑑𝜒0 ∏
𝑚1,𝑚2
𝑑𝐴𝑚1,𝑚2
2𝜋
∫ 𝐷𝜒𝑒−𝑆(𝜒) = 2𝜋𝑅
∏′
𝑚1,𝑚2 𝜆𝑚1,𝑚2
1/2 = 2𝜋𝑅
√det′ ◻
det′ ◻= 4𝜋2𝜏2𝜂2(𝜏)𝜂‾2(𝜏‾)
𝑍(𝑅) = 𝑅
√𝜏2|𝜂|2 ∑
𝑚,𝑛∈𝑍
𝑒−𝜋𝑅2
𝜏2 |𝑚−𝑛𝜏|2
𝑍(𝑅) = ∑
𝑚,𝑛∈ℤ
𝑞𝑃𝐿
2
2 𝑞‾𝑃𝑅
2
2
𝜂𝜂‾
𝑃𝐿 = 1
√2 (𝑚
𝑅 + 𝑛𝑅) , 𝑃𝑅 = 1
√2 (𝑚
𝑅 − 𝑛𝑅)
𝐽(𝑧) = 𝑖𝜕𝑋 , 𝐽‾(𝑧‾) = 𝑖𝜕‾𝑋
𝐽(𝑧)𝐽(𝑤) = 1
(𝑧 − 𝑤)2 + finite , 𝐽‾(𝑧‾)𝐽‾(𝑤‾ ) = 1
(𝑧‾ − 𝑤‾ )2 + finito
• Sugawara:
𝑇(𝑧) = − 1
2 (𝜕𝑋)2 = 1
2 : 𝐽2: , 𝑇‾ (𝑧‾) = − 1
2 (𝜕‾𝑋)2 = 1
2 : 𝐽‾2: .
Δ = 1
2 𝑄𝐿
2 , Δ‾ = 1
2 𝑄𝑅
2
𝜒𝑄𝐿,𝑄𝑅 (𝑞, 𝑞‾) = Tr[𝑞𝐿0−1/24𝑞‾𝐿‾0−1/24] = 𝑞𝑄𝐿
2/2𝑞‾𝑄𝑅
2/2
𝜂𝜂‾
𝜓𝑚1,𝑚2 = 𝑒2𝜋𝑖(𝑚1𝜎1+𝑚2𝜎2) , 𝜆𝑚1,𝑚2 = 4𝜋2
𝜏2
|𝑚1𝜏 − 𝑚2|2
∫ 𝑑2𝜎𝜓𝑚1,𝑚2 𝜓𝑛1,𝑛2 = 𝛿𝑚1+𝑛1,0𝛿𝑚2+𝑛2,0
𝛿𝜒 = ∑
𝑚1,𝑚2∈ℤ
′𝐴𝑚1,𝑚2 𝜓𝑚1,𝑚2
𝑆(𝜒) = 1
4𝜋 ∑
′
𝑚1,𝑚2
𝜆𝑚1,𝑚2 |𝐴𝑚1,𝑚2 |2
‖𝛿𝑋‖ = ∫ 𝑑2𝜎√det𝐺(𝑑𝜒)2 = ∑
𝑚1,𝑚2
′|𝑑𝐴𝑚1,𝑚2 |2
∫ 𝐷𝜒 = ∫
2𝜋𝑅
0
𝑑𝜒0 ∏
𝑚1,𝑚2
𝑑𝐴𝑚1,𝑚2
2𝜋
∫ 𝐷𝜒𝑒−𝑆(𝜒) = 2𝜋𝑅
∏′
𝑚1,𝑚2 𝜆𝑚1,𝑚2
1/2 = 2𝜋𝑅
√det′ ◻
det′ ◻= 4𝜋2𝜏2𝜂2(𝜏)𝜂‾2(𝜏‾)
𝑍(𝑅) = 𝑅
√𝜏2|𝜂|2 ∑
𝑚,𝑛∈𝑍
𝑒−𝜋𝑅2
𝜏2 |𝑚−𝑛𝜏|2
𝑍(𝑅) = ∑
𝑚,𝑛∈ℤ
𝑞𝑃𝐿
2
2 𝑞‾𝑃𝑅
2
2
𝜂𝜂‾
𝑃𝐿 = 1
√2 (𝑚
𝑅 + 𝑛𝑅) , 𝑃𝑅 = 1
√2 (𝑚
𝑅 − 𝑛𝑅)
𝐽(𝑧) = 𝑖𝜕𝑋 , 𝐽‾(𝑧‾) = 𝑖𝜕‾𝑋
𝐽(𝑧)𝐽(𝑤) = 1
(𝑧 − 𝑤)2 + finite , 𝐽‾(𝑧‾)𝐽‾(𝑤‾ ) = 1
(𝑧‾ − 𝑤‾ )2 + finito
• Sugawara:
𝑇(𝑧) = − 1
2 (𝜕𝑋)2 = 1
2 : 𝐽2: , 𝑇‾ (𝑧‾) = − 1
2 (𝜕‾𝑋)2 = 1
2 : 𝐽‾2: .
Δ = 1
2 𝑄𝐿
2 , Δ‾ = 1
2 𝑄𝑅
2
𝜒𝑄𝐿,𝑄𝑅 (𝑞, 𝑞‾) = Tr[𝑞𝐿0−1/24𝑞‾𝐿‾0−1/24] = 𝑞𝑄𝐿
2/2𝑞‾𝑄𝑅
2/2
𝜂𝜂‾
pág. 30
𝐽0|𝑚, 𝑛⟩ = 𝑃𝐿|𝑚, 𝑛⟩ , 𝐽‾0|𝑚, 𝑛⟩ = 𝑃𝑅|𝑚, 𝑛⟩
𝑉𝑚,𝑛 =: exp [𝑖𝑝𝐿𝑋 + 𝑖𝑝𝑅𝑋‾]: ,
𝐽(𝑧)𝑉𝑚,𝑛(𝑤, 𝑤‾ ) = 𝑝𝐿
𝑉𝑚,𝑛(𝑤, 𝑤‾ )
𝑧 − 𝑤 + ⋯
𝐽‾(𝑧‾)𝑉𝑚,𝑛(𝑤, 𝑤‾ ) = 𝑝𝑅
𝑉𝑚,𝑛(𝑤, 𝑤‾ )
𝑧‾ − 𝑤‾ + ⋯
⟨∏
𝑁
𝑖=1
𝑉𝑚𝑖,𝑛𝑖 (𝑧𝑖, 𝑧‾𝑖)⟩ = ∏
𝑁
𝑖<𝑗
𝑧𝑖𝑗
𝑝𝐿
𝑖 𝑝𝐿
𝑗
𝑧‾𝑖𝑗
𝑝𝑅
𝑖 𝑝𝑅
𝑗
,
: 𝑒𝑖𝑎𝜙(𝑧): : 𝑒𝑖𝑏𝜙(𝑤): = (𝑧 − 𝑤)𝑎𝑏: 𝑒𝑖𝑎𝜙(𝑧)+𝑖𝑏𝜙(𝑤): = (𝑧 − 𝑤)𝑎𝑏[: 𝑒𝑖(𝑎+𝑏)𝜙(𝑤): +𝒪(𝑧 − 𝑤)]
[𝑉𝑚1,𝑛1 ] ⋅ [𝑉𝑚2,𝑛2 ] ∼ [𝑉𝑚1+𝑚2,𝑛1+𝑛2 ]
Δ − Δ‾ − 𝑐 − 𝑐‾
24 = ℑinteger
lim
𝑅→∞
𝑍(𝑅)
𝑅 = 1
√𝜏2𝜂𝜂‾
Δ(𝜎1, 𝜎2) ≡ ⟨𝛿𝜒(𝜎1, 𝜎2)𝛿𝜒(0,0)⟩ = − ∑
′
𝑚,𝑛
1
|𝑚𝜏 − 𝑛|2 𝑒2𝜋𝑖(𝑚𝜎1+𝑛𝜎2)
◻ Δ(𝜎1, 𝜎2) = 4𝜋2
𝜏2
[𝛿(𝜎1)𝛿(𝜎2) − 1]
Δ(𝜎1, 𝜎2) = −log 𝐺(𝑧, 𝑧‾) , 𝐺 = 𝑒−2𝜋𝐼𝑚𝑧2
𝜏2 |𝜗1(𝑧)
𝜗1
′ (0)|
2
𝑆 = 1
4𝜋 ∫ 𝑑2𝜎√det𝑔𝑔𝑎𝑏𝐺𝑖𝑗𝜕𝑎𝑋𝑖𝜕𝑏𝑋𝑗 + 1
4𝜋 ∫ 𝑑2𝜎𝜖𝑎𝑏𝐵𝑖𝑗𝜕𝑎𝑋𝑖𝜕𝑏𝑋𝑗,
ℤd, d(𝐺, 𝐵) = √det𝐺
(√𝜏2𝜂𝜂‾)𝑁 ∑
𝑚⃗⃗⃗ ,𝑛⃗
𝑒−𝜋(𝐺𝑖𝑗+𝐵𝑖𝑗)
𝜏2 (𝑚𝑖+𝑛𝑖𝜏)(𝑚𝑗+𝑛𝑗𝜏‾)
ℤd, d(𝐺, 𝐵) = Γ𝑑,𝑑(𝐺, 𝐵)
𝜂𝑑𝜂‾𝑑 = ∑
𝑚⃗⃗⃗ ,𝑛⃗ ∈ℤ𝑁
𝑞1
2𝑃𝐿
2
𝑞‾1
2𝑃𝑅
2
𝜂𝑁𝜂‾𝑁
𝑃𝐿,𝑅
2 ≡ 𝑃𝐿,𝑅
𝑖 𝐺𝑖𝑗𝑃𝐿,𝑅
𝑗
𝑃𝐿
𝑖 = 𝐺𝑖𝑗
√2 (𝑚𝑗 + (𝐵𝑗𝑘 + 𝐺𝑗𝑘)𝑛𝑘) , 𝑃𝑅
𝑖 = 𝐺𝑖𝑗
√2 (𝑚𝑗 + (𝐵𝑗𝑘 − 𝐺𝑗𝑘)𝑛𝑘)
𝐽0|𝑚, 𝑛⟩ = 𝑃𝐿|𝑚, 𝑛⟩ , 𝐽‾0|𝑚, 𝑛⟩ = 𝑃𝑅|𝑚, 𝑛⟩
𝑉𝑚,𝑛 =: exp [𝑖𝑝𝐿𝑋 + 𝑖𝑝𝑅𝑋‾]: ,
𝐽(𝑧)𝑉𝑚,𝑛(𝑤, 𝑤‾ ) = 𝑝𝐿
𝑉𝑚,𝑛(𝑤, 𝑤‾ )
𝑧 − 𝑤 + ⋯
𝐽‾(𝑧‾)𝑉𝑚,𝑛(𝑤, 𝑤‾ ) = 𝑝𝑅
𝑉𝑚,𝑛(𝑤, 𝑤‾ )
𝑧‾ − 𝑤‾ + ⋯
⟨∏
𝑁
𝑖=1
𝑉𝑚𝑖,𝑛𝑖 (𝑧𝑖, 𝑧‾𝑖)⟩ = ∏
𝑁
𝑖<𝑗
𝑧𝑖𝑗
𝑝𝐿
𝑖 𝑝𝐿
𝑗
𝑧‾𝑖𝑗
𝑝𝑅
𝑖 𝑝𝑅
𝑗
,
: 𝑒𝑖𝑎𝜙(𝑧): : 𝑒𝑖𝑏𝜙(𝑤): = (𝑧 − 𝑤)𝑎𝑏: 𝑒𝑖𝑎𝜙(𝑧)+𝑖𝑏𝜙(𝑤): = (𝑧 − 𝑤)𝑎𝑏[: 𝑒𝑖(𝑎+𝑏)𝜙(𝑤): +𝒪(𝑧 − 𝑤)]
[𝑉𝑚1,𝑛1 ] ⋅ [𝑉𝑚2,𝑛2 ] ∼ [𝑉𝑚1+𝑚2,𝑛1+𝑛2 ]
Δ − Δ‾ − 𝑐 − 𝑐‾
24 = ℑinteger
lim
𝑅→∞
𝑍(𝑅)
𝑅 = 1
√𝜏2𝜂𝜂‾
Δ(𝜎1, 𝜎2) ≡ ⟨𝛿𝜒(𝜎1, 𝜎2)𝛿𝜒(0,0)⟩ = − ∑
′
𝑚,𝑛
1
|𝑚𝜏 − 𝑛|2 𝑒2𝜋𝑖(𝑚𝜎1+𝑛𝜎2)
◻ Δ(𝜎1, 𝜎2) = 4𝜋2
𝜏2
[𝛿(𝜎1)𝛿(𝜎2) − 1]
Δ(𝜎1, 𝜎2) = −log 𝐺(𝑧, 𝑧‾) , 𝐺 = 𝑒−2𝜋𝐼𝑚𝑧2
𝜏2 |𝜗1(𝑧)
𝜗1
′ (0)|
2
𝑆 = 1
4𝜋 ∫ 𝑑2𝜎√det𝑔𝑔𝑎𝑏𝐺𝑖𝑗𝜕𝑎𝑋𝑖𝜕𝑏𝑋𝑗 + 1
4𝜋 ∫ 𝑑2𝜎𝜖𝑎𝑏𝐵𝑖𝑗𝜕𝑎𝑋𝑖𝜕𝑏𝑋𝑗,
ℤd, d(𝐺, 𝐵) = √det𝐺
(√𝜏2𝜂𝜂‾)𝑁 ∑
𝑚⃗⃗⃗ ,𝑛⃗
𝑒−𝜋(𝐺𝑖𝑗+𝐵𝑖𝑗)
𝜏2 (𝑚𝑖+𝑛𝑖𝜏)(𝑚𝑗+𝑛𝑗𝜏‾)
ℤd, d(𝐺, 𝐵) = Γ𝑑,𝑑(𝐺, 𝐵)
𝜂𝑑𝜂‾𝑑 = ∑
𝑚⃗⃗⃗ ,𝑛⃗ ∈ℤ𝑁
𝑞1
2𝑃𝐿
2
𝑞‾1
2𝑃𝑅
2
𝜂𝑁𝜂‾𝑁
𝑃𝐿,𝑅
2 ≡ 𝑃𝐿,𝑅
𝑖 𝐺𝑖𝑗𝑃𝐿,𝑅
𝑗
𝑃𝐿
𝑖 = 𝐺𝑖𝑗
√2 (𝑚𝑗 + (𝐵𝑗𝑘 + 𝐺𝑗𝑘)𝑛𝑘) , 𝑃𝑅
𝑖 = 𝐺𝑖𝑗
√2 (𝑚𝑗 + (𝐵𝑗𝑘 − 𝐺𝑗𝑘)𝑛𝑘)
pág. 31
𝐽𝑖(𝑧) = 𝑖𝜕𝑋𝑖 , 𝐽‾𝑖 = 𝑖𝜕‾𝑋𝑖
𝐽𝑖(𝑧)𝐽𝑗(𝑤) = 𝐺𝑖𝑗
(𝑧 − 𝑤)2 + ⋯
𝑇(𝑧) = − 1
2 𝐺𝑖𝑗𝜕𝑋𝑖𝜕𝑋𝑗 = 1
2 𝐺𝑖𝑗: 𝐽𝑖𝐽𝑗: .
Δ = 1
2 𝐺𝑖𝑗𝑄𝐿
𝑖 𝑄𝐿
𝑗 , Δ‾ = 1
2 𝐺𝑖𝑗𝑄𝑅
𝑖 𝑄𝑅
𝑗
Supersimetría de Higgs.
Δ = 1
4 (𝑚 + 𝑛)2 , Δ‾ = 1
4 (𝑚 − 𝑛)2
𝐽±(𝑧) = 1
√2 : 𝑒±𝑖√2𝑋(𝑧):
𝐽3(𝑧) = 1
√2 𝐽(𝑧) = 𝑖
√2 𝜕𝑋(𝑧)
𝐽3(𝑧)𝐽±(𝑤) = ± 𝐽±(𝑤)
𝑧 − 𝑤 + ⋯ , 𝐽+(𝑧)𝐽+(𝑤) = ⋯
𝐽−(𝑧)𝐽−(𝑤) = ⋯ ,
𝐽+(𝑧)𝐽−(𝑤) = 1/2
(𝑧 − 𝑤)2 + 𝐽3(𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽3(𝑧)𝐽3(𝑤) = 1/2
(𝑧 − 𝑤)2 + ⋯
𝑎−1
𝜇 𝑎‾−1
𝜈 |0⟩ , 𝑎−1
𝜇 𝑎‾−1
25 |0⟩ , 𝑎−1
25 𝑎‾−1
𝜇 |0⟩ , 𝑎−1
25 𝑎‾−1
25 |0⟩,
|𝐴𝜇
±⟩ = 𝑎‾𝜇|𝑚 = ±1, 𝑛 = ±1⟩
|𝐴‾𝜇
±⟩ = 𝑎𝜇|𝑚 = ±1, 𝑛 = ∓1⟩
Dualidad T.
𝐻 = 𝐿0 + 𝐿‾0 = 1
2 (𝑚2
𝑅2 + 𝑛2𝑅2) , 𝑃 = 𝐿0 − 𝐿‾0 = 𝑚𝑛
𝑅 → 1
𝑅 , 𝑚 ↔ 𝑛
𝑃𝐿 → 𝑃𝐿 , 𝑃𝑅 → −𝑃𝑅
𝐽(𝑧) → 𝐽(𝑧) , 𝐽‾(𝑧‾) → −𝐽‾(𝑧‾)
Ω = (𝐴 𝐵
𝐶 𝐷)
𝐿 = ( 0 𝟏𝑁
𝟏𝑁 0 )
𝐽𝑖(𝑧) = 𝑖𝜕𝑋𝑖 , 𝐽‾𝑖 = 𝑖𝜕‾𝑋𝑖
𝐽𝑖(𝑧)𝐽𝑗(𝑤) = 𝐺𝑖𝑗
(𝑧 − 𝑤)2 + ⋯
𝑇(𝑧) = − 1
2 𝐺𝑖𝑗𝜕𝑋𝑖𝜕𝑋𝑗 = 1
2 𝐺𝑖𝑗: 𝐽𝑖𝐽𝑗: .
Δ = 1
2 𝐺𝑖𝑗𝑄𝐿
𝑖 𝑄𝐿
𝑗 , Δ‾ = 1
2 𝐺𝑖𝑗𝑄𝑅
𝑖 𝑄𝑅
𝑗
Supersimetría de Higgs.
Δ = 1
4 (𝑚 + 𝑛)2 , Δ‾ = 1
4 (𝑚 − 𝑛)2
𝐽±(𝑧) = 1
√2 : 𝑒±𝑖√2𝑋(𝑧):
𝐽3(𝑧) = 1
√2 𝐽(𝑧) = 𝑖
√2 𝜕𝑋(𝑧)
𝐽3(𝑧)𝐽±(𝑤) = ± 𝐽±(𝑤)
𝑧 − 𝑤 + ⋯ , 𝐽+(𝑧)𝐽+(𝑤) = ⋯
𝐽−(𝑧)𝐽−(𝑤) = ⋯ ,
𝐽+(𝑧)𝐽−(𝑤) = 1/2
(𝑧 − 𝑤)2 + 𝐽3(𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽3(𝑧)𝐽3(𝑤) = 1/2
(𝑧 − 𝑤)2 + ⋯
𝑎−1
𝜇 𝑎‾−1
𝜈 |0⟩ , 𝑎−1
𝜇 𝑎‾−1
25 |0⟩ , 𝑎−1
25 𝑎‾−1
𝜇 |0⟩ , 𝑎−1
25 𝑎‾−1
25 |0⟩,
|𝐴𝜇
±⟩ = 𝑎‾𝜇|𝑚 = ±1, 𝑛 = ±1⟩
|𝐴‾𝜇
±⟩ = 𝑎𝜇|𝑚 = ±1, 𝑛 = ∓1⟩
Dualidad T.
𝐻 = 𝐿0 + 𝐿‾0 = 1
2 (𝑚2
𝑅2 + 𝑛2𝑅2) , 𝑃 = 𝐿0 − 𝐿‾0 = 𝑚𝑛
𝑅 → 1
𝑅 , 𝑚 ↔ 𝑛
𝑃𝐿 → 𝑃𝐿 , 𝑃𝑅 → −𝑃𝑅
𝐽(𝑧) → 𝐽(𝑧) , 𝐽‾(𝑧‾) → −𝐽‾(𝑧‾)
Ω = (𝐴 𝐵
𝐶 𝐷)
𝐿 = ( 0 𝟏𝑁
𝟏𝑁 0 )
pág. 32
Ω𝑇𝐿Ω = 𝐿
𝐸 → (𝐴𝐸 + 𝐵)(𝐶𝐸 + 𝐷)−1 , (𝑚⃗⃗
𝑛⃗ ) → Ω (𝑚⃗⃗
𝑛⃗ )
𝐺𝑖𝑗 = 𝑇2
𝑈2
( 1 𝑈1
𝑈1 𝑈1
2 + 𝑈2
2) , 𝐵𝑖𝑗 = ( 0 𝑇1
−𝑇1 0 ) ,
Γ2,2(𝑇, 𝑈) = ∑
𝑚⃗⃗⃗ ,𝑛⃗
exp [− 𝜋𝜏2
𝑇2𝑈2
| − 𝑚1𝑈 + 𝑚2 + 𝑇(𝑛1 + 𝑈𝑛2)|2 + 2𝜋𝑖𝜏(𝑚1𝑛1 + 𝑚2𝑛2)]
∫ 𝑒−𝑆 = (det𝜕)𝑁/2
𝜆𝐴𝐴 ∼ ((𝑚1 + 1
2) 𝜏 + (𝑚2 + 1
2)) , 𝑚1,2 ∈ ℤ
(det𝜕)𝐴𝐴 = 𝜗3(𝜏)
𝜂(𝜏)
𝜆𝐴𝑃 ∼ ((𝑚1 + 1
2) 𝜏 + 𝑚2) , 𝑚1,2 ∈ ℤ
(det𝜕)𝐴𝑃 = 𝜗4(𝜏)
𝜂(𝜏)
𝜆𝑃𝐴 ∼ (𝑚1𝜏 + (𝑚2 + 1
2)) , 𝑚1,2 ∈ ℤ
(det𝜕)𝑃𝐴 = 𝜗2(𝜏)
𝜂(𝜏)
(det𝜕) [𝑎
𝑏] = 𝜗 [𝑎
𝑏] (𝜏)
𝜂(𝜏)
𝑍𝑁
fermionic = 1
2 ∑
1
𝑎,𝑏=0
|
𝜗 [𝑎
𝑏]
𝜂 |
𝑁
𝑍𝑁
fermionic = |𝜒0|2 + |𝜒𝑉|2 + |𝜒𝑆|2 + |𝜒𝐶 |2
• Szegö:
⟨𝜓𝑖(𝑧)𝜓𝑗(0)⟩ = 𝛿𝑖𝑗𝑆 [𝑎
𝑏] (𝑧) , 𝑆 [𝑎
𝑏] (𝑧) = 𝜗 [𝑎
𝑏] (𝑧)𝜗1
′ (0)
𝜗1(𝑧)𝜗 [𝑎
𝑏] (0)
𝑞−𝑁/24 ∏
∞
𝑛=1
(1 − 𝑞𝑛)𝑁 = 𝜂𝑁 = [ 1
2𝜋
𝜕𝑣𝜗1(𝑣)|𝑣=0
𝜂 ]
𝑁/2
Ω𝑇𝐿Ω = 𝐿
𝐸 → (𝐴𝐸 + 𝐵)(𝐶𝐸 + 𝐷)−1 , (𝑚⃗⃗
𝑛⃗ ) → Ω (𝑚⃗⃗
𝑛⃗ )
𝐺𝑖𝑗 = 𝑇2
𝑈2
( 1 𝑈1
𝑈1 𝑈1
2 + 𝑈2
2) , 𝐵𝑖𝑗 = ( 0 𝑇1
−𝑇1 0 ) ,
Γ2,2(𝑇, 𝑈) = ∑
𝑚⃗⃗⃗ ,𝑛⃗
exp [− 𝜋𝜏2
𝑇2𝑈2
| − 𝑚1𝑈 + 𝑚2 + 𝑇(𝑛1 + 𝑈𝑛2)|2 + 2𝜋𝑖𝜏(𝑚1𝑛1 + 𝑚2𝑛2)]
∫ 𝑒−𝑆 = (det𝜕)𝑁/2
𝜆𝐴𝐴 ∼ ((𝑚1 + 1
2) 𝜏 + (𝑚2 + 1
2)) , 𝑚1,2 ∈ ℤ
(det𝜕)𝐴𝐴 = 𝜗3(𝜏)
𝜂(𝜏)
𝜆𝐴𝑃 ∼ ((𝑚1 + 1
2) 𝜏 + 𝑚2) , 𝑚1,2 ∈ ℤ
(det𝜕)𝐴𝑃 = 𝜗4(𝜏)
𝜂(𝜏)
𝜆𝑃𝐴 ∼ (𝑚1𝜏 + (𝑚2 + 1
2)) , 𝑚1,2 ∈ ℤ
(det𝜕)𝑃𝐴 = 𝜗2(𝜏)
𝜂(𝜏)
(det𝜕) [𝑎
𝑏] = 𝜗 [𝑎
𝑏] (𝜏)
𝜂(𝜏)
𝑍𝑁
fermionic = 1
2 ∑
1
𝑎,𝑏=0
|
𝜗 [𝑎
𝑏]
𝜂 |
𝑁
𝑍𝑁
fermionic = |𝜒0|2 + |𝜒𝑉|2 + |𝜒𝑆|2 + |𝜒𝐶 |2
• Szegö:
⟨𝜓𝑖(𝑧)𝜓𝑗(0)⟩ = 𝛿𝑖𝑗𝑆 [𝑎
𝑏] (𝑧) , 𝑆 [𝑎
𝑏] (𝑧) = 𝜗 [𝑎
𝑏] (𝑧)𝜗1
′ (0)
𝜗1(𝑧)𝜗 [𝑎
𝑏] (0)
𝑞−𝑁/24 ∏
∞
𝑛=1
(1 − 𝑞𝑛)𝑁 = 𝜂𝑁 = [ 1
2𝜋
𝜕𝑣𝜗1(𝑣)|𝑣=0
𝜂 ]
𝑁/2
pág. 33
⟨∏
𝑁
𝑘=1
𝜓𝑖𝑘 (𝑧𝑘)⟩
odd
= 𝜖𝑖1,…,𝑖𝑁 𝜂𝑁
Bosonización Majorana-Weyl 𝝍𝒊(𝒛):
𝜓𝑖(𝑧)𝜓𝑗(𝑤) = 𝛿𝑖𝑗
𝑧 − 𝑤 + ⋯
𝜓 = 1
√2 (𝜓1 + 𝑖𝜓2) , 𝜓‾ = 1
√2 (𝜓1 − 𝑖𝜓2)
𝐽(𝑧) =: 𝜓𝜓‾: , 𝐽(𝑧)𝐽(𝑤) = 1
(𝑧 − 𝑤)2 + ⋯
𝐽(𝑧)𝜓(𝑤) = 𝜓(𝑤)
𝑧 − 𝑤 + ⋯ , 𝐽(𝑧)𝜓‾(𝑤) = − 𝜓‾(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧) = − 1
2 : 𝜓𝑖𝜕𝜓𝑖: = 1
2 : 𝐽2: .
𝐽(𝑧) = 𝑖𝜕𝑋 , 𝜓 =: 𝑒𝑖𝑋: , 𝜓‾ =: 𝑒−𝑖𝑋:
• Poisson:
|𝜗 [𝑎
𝑏]|2
= 1
√2𝜏2
∑
𝑚,𝑛∈𝑍
exp [− 𝜋
2𝜏2
|𝑛 − 𝑏 + 𝜏(𝑚 − 𝑎)|2 + 𝑖𝜋𝑚𝑛]
= 1
√2𝜏2
∑
𝑚,𝑛∈𝑍
exp [− 𝜋
2𝜏2
|𝑛 + 𝜏𝑚|2 + 𝑖𝜋(𝑚 + 𝑎)(𝑛 + 𝑏)]
𝑍 = 1
2 ∑
1
𝑎,𝑏=0
|𝜗 [𝑎
𝑏]
𝜂 |
2
= 1
2√2𝜏2
∑
1
𝑎,𝑏=0
∑
𝑚,𝑛∈ℤ
exp [− 𝜋
2𝜏2
|𝑛 + 𝜏𝑚|2 + 𝑖𝜋(𝑚 + 𝑎)(𝑛 + 𝑏)]
𝑍Dirac = 1
√2𝜏2
∑
𝑚,𝑛∈ℤ
exp [− 𝜋
2𝜏2
|𝑛 + 𝜏𝑚|2]
Orbifolds.
𝑔 [∏
𝑁
𝑖=1
𝑎−𝑛𝑖 ∏
𝑁‾
𝑗=1
𝑎‾𝑛‾𝑗 |𝑚, 𝑛⟩] = (−1)𝑁+𝑁‾ ∏
𝑁
𝑖=1
𝑎−𝑛𝑖 ∏
𝑁‾
𝑗=1
𝑎‾𝑛‾𝑗 | − 𝑚, −𝑛⟩.
𝑍(𝑅)invariant = 1
2 𝑍(𝑅) + 1
2 Tr[𝑔𝑞𝐿0−1/24𝑞‾𝐿‾0−1/24].
1
2 Tr[𝑔𝑞𝐿0−1/24𝑞‾𝐿‾0−1/24] = 1
2 (𝑞𝑞‾)−1/24 ∏
∞
𝑛=1
1
(1 + 𝑞𝑛)(1 + 𝑞‾𝑛) = | 𝜂
𝜗2
|
⟨∏
𝑁
𝑘=1
𝜓𝑖𝑘 (𝑧𝑘)⟩
odd
= 𝜖𝑖1,…,𝑖𝑁 𝜂𝑁
Bosonización Majorana-Weyl 𝝍𝒊(𝒛):
𝜓𝑖(𝑧)𝜓𝑗(𝑤) = 𝛿𝑖𝑗
𝑧 − 𝑤 + ⋯
𝜓 = 1
√2 (𝜓1 + 𝑖𝜓2) , 𝜓‾ = 1
√2 (𝜓1 − 𝑖𝜓2)
𝐽(𝑧) =: 𝜓𝜓‾: , 𝐽(𝑧)𝐽(𝑤) = 1
(𝑧 − 𝑤)2 + ⋯
𝐽(𝑧)𝜓(𝑤) = 𝜓(𝑤)
𝑧 − 𝑤 + ⋯ , 𝐽(𝑧)𝜓‾(𝑤) = − 𝜓‾(𝑤)
𝑧 − 𝑤 + ⋯
𝑇(𝑧) = − 1
2 : 𝜓𝑖𝜕𝜓𝑖: = 1
2 : 𝐽2: .
𝐽(𝑧) = 𝑖𝜕𝑋 , 𝜓 =: 𝑒𝑖𝑋: , 𝜓‾ =: 𝑒−𝑖𝑋:
• Poisson:
|𝜗 [𝑎
𝑏]|2
= 1
√2𝜏2
∑
𝑚,𝑛∈𝑍
exp [− 𝜋
2𝜏2
|𝑛 − 𝑏 + 𝜏(𝑚 − 𝑎)|2 + 𝑖𝜋𝑚𝑛]
= 1
√2𝜏2
∑
𝑚,𝑛∈𝑍
exp [− 𝜋
2𝜏2
|𝑛 + 𝜏𝑚|2 + 𝑖𝜋(𝑚 + 𝑎)(𝑛 + 𝑏)]
𝑍 = 1
2 ∑
1
𝑎,𝑏=0
|𝜗 [𝑎
𝑏]
𝜂 |
2
= 1
2√2𝜏2
∑
1
𝑎,𝑏=0
∑
𝑚,𝑛∈ℤ
exp [− 𝜋
2𝜏2
|𝑛 + 𝜏𝑚|2 + 𝑖𝜋(𝑚 + 𝑎)(𝑛 + 𝑏)]
𝑍Dirac = 1
√2𝜏2
∑
𝑚,𝑛∈ℤ
exp [− 𝜋
2𝜏2
|𝑛 + 𝜏𝑚|2]
Orbifolds.
𝑔 [∏
𝑁
𝑖=1
𝑎−𝑛𝑖 ∏
𝑁‾
𝑗=1
𝑎‾𝑛‾𝑗 |𝑚, 𝑛⟩] = (−1)𝑁+𝑁‾ ∏
𝑁
𝑖=1
𝑎−𝑛𝑖 ∏
𝑁‾
𝑗=1
𝑎‾𝑛‾𝑗 | − 𝑚, −𝑛⟩.
𝑍(𝑅)invariant = 1
2 𝑍(𝑅) + 1
2 Tr[𝑔𝑞𝐿0−1/24𝑞‾𝐿‾0−1/24].
1
2 Tr[𝑔𝑞𝐿0−1/24𝑞‾𝐿‾0−1/24] = 1
2 (𝑞𝑞‾)−1/24 ∏
∞
𝑛=1
1
(1 + 𝑞𝑛)(1 + 𝑞‾𝑛) = | 𝜂
𝜗2
|
pág. 34
𝑍(𝑅)invariant = 1
2 𝑍(𝑅) + | 𝜂
𝜗2
| .
𝑋(𝜎, 𝜏) = 𝑥0 + 𝑖
√4𝜋𝑇 ∑
𝑛∈𝑍
( 𝑎𝑛+1/2
𝑛 + 1/2 𝑒𝑖(𝑛+1/2)(𝜎+𝜏) + 𝑎‾𝑛+1/2
𝑛 + 1/2 𝑒−𝑖(𝑛+1/2)(𝜎−𝜏)) .
𝑎𝑛+1/2|𝐻0,𝜋⟩ = 𝑎‾𝑛+1/2|𝐻0,𝜋⟩ = 0 𝑛 ≥ 0
𝑍twisted = 1
2 Tr[(1 + 𝑔)𝑞𝐿0−1/24𝑞‾𝐿‾0−1/24]
= 1
2
1
(𝑞𝑞‾)48 [∏
∞
𝑛=1
1
(1 − 𝑞𝑛−1
2) (1 − 𝑞‾𝑛−1
2)
+ ∏
∞
𝑛=1
1
(1 + 𝑞𝑛−1
2) (1 + 𝑞‾𝑛−1
2)
]
= | 𝜂
𝜗4
| + | 𝜂
𝜗3
|
𝑍orb (𝑅) = 𝑍untwisted + 𝑍twisted = 1
2 𝑍(𝑅) + | 𝜂
𝜗2
| + | 𝜂
𝜗4
| + | 𝜂
𝜗3
|
𝑍orb = 1
2 ∑
1
ℎ,𝑔=0
𝑍 [ℎ
𝑔]
𝑍 [ℎ
𝑔] = 2 | 𝜂
𝜗 [1 − ℎ
1 − 𝑔]
| , (ℎ, 𝑔) ≠ (0,0)
𝜏 → 𝜏 + 1: 𝑍 [ℎ
𝑔] → 𝑍 [ ℎ
ℎ + 𝑔]
𝜏 → − 1
𝜏 : 𝑍 [ℎ
𝑔] → 𝑍 [𝑔
ℎ]
[𝐻0] ⋅ [𝐻0] ∼ ∑
𝑛,𝑚
𝐶2𝑚,2𝑛[𝑉2𝑚,2𝑛
+ ] + 𝐶2𝑚,2𝑛+1[𝑉2𝑚,2𝑛+1
+ ]
[𝐻𝜋] ⋅ [𝐻𝜋] ∼ ∑
𝑛,𝑚
𝐶2𝑚,2𝑛[𝑉2𝑚,2𝑛
+ ] − 𝐶2𝑚,2𝑛+1[𝑉2𝑚,2𝑛+1
+ ]
[𝐻0] ⋅ [𝐻𝜋] ∼ ∑
𝑛,𝑚
𝐶2𝑚+1,2𝑛[𝑉2𝑚+1,2𝑛
+ ]
𝐶𝑚,𝑛 = √22−2(ℎ𝑚,𝑛+ℎ‾𝑚,𝑛) , 𝐶0,0 = 1
ℎ𝑚,𝑛 = (𝑚/𝑅 + 𝑛𝑅)2/4 , ℎ‾𝑚,𝑛 = (𝑚/𝑅 − 𝑛𝑅)2/4
(𝐻0, 𝐻𝜋, 𝑉𝑚,𝑛
+ ) → (−𝐻0, 𝐻𝜋, (−1)𝑚𝑉𝑚,𝑛
+ )
(𝐻0, 𝐻𝜋, 𝑉𝑚,𝑛
+ ) → (𝐻𝜋, 𝐻0, (−1)𝑛𝑉𝑚,𝑛
+ )
𝑍(𝑅)invariant = 1
2 𝑍(𝑅) + | 𝜂
𝜗2
| .
𝑋(𝜎, 𝜏) = 𝑥0 + 𝑖
√4𝜋𝑇 ∑
𝑛∈𝑍
( 𝑎𝑛+1/2
𝑛 + 1/2 𝑒𝑖(𝑛+1/2)(𝜎+𝜏) + 𝑎‾𝑛+1/2
𝑛 + 1/2 𝑒−𝑖(𝑛+1/2)(𝜎−𝜏)) .
𝑎𝑛+1/2|𝐻0,𝜋⟩ = 𝑎‾𝑛+1/2|𝐻0,𝜋⟩ = 0 𝑛 ≥ 0
𝑍twisted = 1
2 Tr[(1 + 𝑔)𝑞𝐿0−1/24𝑞‾𝐿‾0−1/24]
= 1
2
1
(𝑞𝑞‾)48 [∏
∞
𝑛=1
1
(1 − 𝑞𝑛−1
2) (1 − 𝑞‾𝑛−1
2)
+ ∏
∞
𝑛=1
1
(1 + 𝑞𝑛−1
2) (1 + 𝑞‾𝑛−1
2)
]
= | 𝜂
𝜗4
| + | 𝜂
𝜗3
|
𝑍orb (𝑅) = 𝑍untwisted + 𝑍twisted = 1
2 𝑍(𝑅) + | 𝜂
𝜗2
| + | 𝜂
𝜗4
| + | 𝜂
𝜗3
|
𝑍orb = 1
2 ∑
1
ℎ,𝑔=0
𝑍 [ℎ
𝑔]
𝑍 [ℎ
𝑔] = 2 | 𝜂
𝜗 [1 − ℎ
1 − 𝑔]
| , (ℎ, 𝑔) ≠ (0,0)
𝜏 → 𝜏 + 1: 𝑍 [ℎ
𝑔] → 𝑍 [ ℎ
ℎ + 𝑔]
𝜏 → − 1
𝜏 : 𝑍 [ℎ
𝑔] → 𝑍 [𝑔
ℎ]
[𝐻0] ⋅ [𝐻0] ∼ ∑
𝑛,𝑚
𝐶2𝑚,2𝑛[𝑉2𝑚,2𝑛
+ ] + 𝐶2𝑚,2𝑛+1[𝑉2𝑚,2𝑛+1
+ ]
[𝐻𝜋] ⋅ [𝐻𝜋] ∼ ∑
𝑛,𝑚
𝐶2𝑚,2𝑛[𝑉2𝑚,2𝑛
+ ] − 𝐶2𝑚,2𝑛+1[𝑉2𝑚,2𝑛+1
+ ]
[𝐻0] ⋅ [𝐻𝜋] ∼ ∑
𝑛,𝑚
𝐶2𝑚+1,2𝑛[𝑉2𝑚+1,2𝑛
+ ]
𝐶𝑚,𝑛 = √22−2(ℎ𝑚,𝑛+ℎ‾𝑚,𝑛) , 𝐶0,0 = 1
ℎ𝑚,𝑛 = (𝑚/𝑅 + 𝑛𝑅)2/4 , ℎ‾𝑚,𝑛 = (𝑚/𝑅 − 𝑛𝑅)2/4
(𝐻0, 𝐻𝜋, 𝑉𝑚,𝑛
+ ) → (−𝐻0, 𝐻𝜋, (−1)𝑚𝑉𝑚,𝑛
+ )
(𝐻0, 𝐻𝜋, 𝑉𝑚,𝑛
+ ) → (𝐻𝜋, 𝐻0, (−1)𝑛𝑉𝑚,𝑛
+ )
pág. 35
(𝐻0, 𝐻𝜋, 𝑉𝑚,𝑛
+ ) → (−𝐻0, −𝐻𝜋, 𝑉𝑚,𝑛
+ )
𝑍orb(𝑅) = 𝑍orb(1/𝑅)
(𝐻0
𝐻𝜋) → 1
√2 (1 1
1 −1) (𝐻0
𝐻𝜋)
𝑍Ising = 1
2 [|𝜗2
𝜂 | + |𝜗3
𝜂 | + |𝜗4
𝜂 |] ,
𝑍 [0
1] = ∑
𝑚,𝑛∈ℤ
(−1)𝑚
exp [𝑖𝜋𝜏
2 (𝑚
𝑅 + 𝑛𝑅)2
− 𝑖𝜋𝜏‾
2 (𝑚
𝑅 − 𝑛𝑅)2
]
𝜂𝜂‾
𝑍 [1
0] = ∑
𝑚,𝑛∈ℤ
exp [𝑖𝜋𝜏
2 (𝑚
𝑅 + (𝑛 + 1
2) 𝑅)
2
− 𝑖𝜋𝜏‾
2 (𝑚
𝑅 − (𝑛 + 1
2) 𝑅)
2
]
𝜂𝜂‾ .
𝑍 [1
1] = ∑
𝑚,𝑛∈ℤ
(−1)𝑚
exp [𝑖𝜋𝜏
2 (𝑚
𝑅 + (𝑛 + 1
2) 𝑅)
2
− 𝑖𝜋𝜏‾
2 (𝑚
𝑅 − (𝑛 + 1
2) 𝑅)
2
]
𝜂𝜂‾ .
𝑍 [ℎ
𝑔] = ∑
𝑚,𝑛∈ℤ
(−1)𝑔𝑚
exp [𝑖𝜋𝜏
2 (𝑚
𝑅 + (𝑛 + ℎ
2) 𝑅)
2
− 𝑖𝜋𝜏‾
2 (𝑚
𝑅 − (𝑛 + ℎ
2) 𝑅)
2
]
𝜂𝜂‾
𝑍 [ℎ
𝑔] = 𝑅
√𝜏2𝜂𝜂‾ ∑
𝑚,𝑛,∈ℤ
exp [− 𝜋𝑅2
𝜏2
|𝑚 + 𝑔
2 + (𝑛 + ℎ
2) 𝜏|
2
]
𝑔translation = exp [2𝜋𝑖 ∑
𝑑
𝑖=1
(𝑚𝑖𝜃𝑖 + 𝑛𝑖𝜙𝑖)]
CFT en superficies de Riemann. Amplitudes escalares y operadores de vértice.
⟨1⟩𝑔=2 = ∑
𝑖
𝑞ℎ𝑖−𝑐/24𝑞‾ℎ‾𝑖−𝑐‾/24⟨𝜙𝑖⟩𝑔=1⟨𝜙𝑖⟩𝑔=1
𝑆𝑃 = 1
4𝜋𝛼′ ∫ 𝑑2𝜎𝜕𝑋𝜇𝜕‾𝑋𝜈𝜂𝜇𝜈
⟨𝑋𝜇(𝑧, 𝑧‾)𝑋𝜈(𝑤, 𝑤‾ )⟩ = − 𝛼′
4 𝜂𝜇𝜈log |𝑧 − 𝑤|2
𝑇 = − 2
𝛼′ 𝜂𝜇𝜈𝜕𝑋𝜇𝜕𝑋𝜈
𝑇(𝑧)𝑂(𝑤, 𝑤‾ ) = −𝑖𝑝𝜇𝜖𝜇𝜈
𝛼′
4
𝜕‾𝑥𝜈𝑉𝑝
(𝑧 − 𝑤)3 + (1 + 𝛼′𝑝2
4 ) 𝑂(𝑤, 𝑤‾ )
(𝑧 − 𝑤)2 + 𝜕𝑤𝑂(𝑤, 𝑤‾ )
𝑧 − 𝑤 + ⋯
(𝐻0, 𝐻𝜋, 𝑉𝑚,𝑛
+ ) → (−𝐻0, −𝐻𝜋, 𝑉𝑚,𝑛
+ )
𝑍orb(𝑅) = 𝑍orb(1/𝑅)
(𝐻0
𝐻𝜋) → 1
√2 (1 1
1 −1) (𝐻0
𝐻𝜋)
𝑍Ising = 1
2 [|𝜗2
𝜂 | + |𝜗3
𝜂 | + |𝜗4
𝜂 |] ,
𝑍 [0
1] = ∑
𝑚,𝑛∈ℤ
(−1)𝑚
exp [𝑖𝜋𝜏
2 (𝑚
𝑅 + 𝑛𝑅)2
− 𝑖𝜋𝜏‾
2 (𝑚
𝑅 − 𝑛𝑅)2
]
𝜂𝜂‾
𝑍 [1
0] = ∑
𝑚,𝑛∈ℤ
exp [𝑖𝜋𝜏
2 (𝑚
𝑅 + (𝑛 + 1
2) 𝑅)
2
− 𝑖𝜋𝜏‾
2 (𝑚
𝑅 − (𝑛 + 1
2) 𝑅)
2
]
𝜂𝜂‾ .
𝑍 [1
1] = ∑
𝑚,𝑛∈ℤ
(−1)𝑚
exp [𝑖𝜋𝜏
2 (𝑚
𝑅 + (𝑛 + 1
2) 𝑅)
2
− 𝑖𝜋𝜏‾
2 (𝑚
𝑅 − (𝑛 + 1
2) 𝑅)
2
]
𝜂𝜂‾ .
𝑍 [ℎ
𝑔] = ∑
𝑚,𝑛∈ℤ
(−1)𝑔𝑚
exp [𝑖𝜋𝜏
2 (𝑚
𝑅 + (𝑛 + ℎ
2) 𝑅)
2
− 𝑖𝜋𝜏‾
2 (𝑚
𝑅 − (𝑛 + ℎ
2) 𝑅)
2
]
𝜂𝜂‾
𝑍 [ℎ
𝑔] = 𝑅
√𝜏2𝜂𝜂‾ ∑
𝑚,𝑛,∈ℤ
exp [− 𝜋𝑅2
𝜏2
|𝑚 + 𝑔
2 + (𝑛 + ℎ
2) 𝜏|
2
]
𝑔translation = exp [2𝜋𝑖 ∑
𝑑
𝑖=1
(𝑚𝑖𝜃𝑖 + 𝑛𝑖𝜙𝑖)]
CFT en superficies de Riemann. Amplitudes escalares y operadores de vértice.
⟨1⟩𝑔=2 = ∑
𝑖
𝑞ℎ𝑖−𝑐/24𝑞‾ℎ‾𝑖−𝑐‾/24⟨𝜙𝑖⟩𝑔=1⟨𝜙𝑖⟩𝑔=1
𝑆𝑃 = 1
4𝜋𝛼′ ∫ 𝑑2𝜎𝜕𝑋𝜇𝜕‾𝑋𝜈𝜂𝜇𝜈
⟨𝑋𝜇(𝑧, 𝑧‾)𝑋𝜈(𝑤, 𝑤‾ )⟩ = − 𝛼′
4 𝜂𝜇𝜈log |𝑧 − 𝑤|2
𝑇 = − 2
𝛼′ 𝜂𝜇𝜈𝜕𝑋𝜇𝜕𝑋𝜈
𝑇(𝑧)𝑂(𝑤, 𝑤‾ ) = −𝑖𝑝𝜇𝜖𝜇𝜈
𝛼′
4
𝜕‾𝑥𝜈𝑉𝑝
(𝑧 − 𝑤)3 + (1 + 𝛼′𝑝2
4 ) 𝑂(𝑤, 𝑤‾ )
(𝑧 − 𝑤)2 + 𝜕𝑤𝑂(𝑤, 𝑤‾ )
𝑧 − 𝑤 + ⋯
pág. 36
𝑝𝜇𝜖𝜇𝜈 = 𝑝𝜈𝜖𝜇𝜈 = 0
Λ4 = ∫
ℱ
𝑑2𝜏
𝜏2
2 𝑍bosonic (𝜏, 𝜏‾) = ∫
ℱ
𝑑2𝜏
𝜏2
2
1
(√𝜏2𝜂𝜂‾)24
𝜒 = 2(1 − 𝑔) − 𝐵 − 𝐶.
Acción stress – tensor – energía - momentum. Centro de masa. Partícula Supermasiva u Oscura.
𝑆𝑃 = 1
4𝜋𝛼′ ∫ 𝑑2𝜉 [√𝑔𝑔𝑎𝑏𝐺𝜇𝜈(𝑋) + 𝜖𝑎𝑏𝐵𝜇𝜈(𝑋)]𝜕𝑎𝑋𝜇𝜕𝑏𝑋𝜈 + + 1
8𝜋 ∫ 𝑑2𝜉√𝑔𝑅(2)Φ(𝑋)
𝜒 = 1
4𝜋 ∫ √𝑔𝑅(2)
𝑔spacetime dimension = 𝑒Φ0/2
𝑇𝑎
𝑎
√𝑔 = 𝛽Φ
96𝜋3 𝑅(2) + 1
2𝜋 (𝛽𝜇𝜈
𝐺 𝑔𝑎𝑏 + 𝛽𝜇𝜈
𝐵 𝜖𝑎𝑏)𝜕𝑎𝑋𝜇𝜕𝑏𝑋𝜈
𝛽𝜇𝜈
𝐺
𝛼′ = 𝑅𝜇𝜈 − 1
4 𝐻𝜇𝜌𝜎𝐻𝜈
𝜌𝜎 + ∇𝜇∇𝜈Φ + 𝒪(𝛼′)
𝛽𝜇𝜈
𝐵
𝛼′ = ∇𝜇[𝑒−Φ𝐻𝜇𝜈𝜌] + 𝒪(𝛼′)
𝛽Φ = 𝐷 − 26 + 3𝛼′ [(∇Φ)2 − 2 ◻ Φ − 𝑅 + 1
12 𝐻2] + 𝒪(𝛼′2)
𝐻𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 + 𝜕𝜈𝐵𝜌𝜇 + 𝜕𝜌𝐵𝜇𝜈
𝛽Φ = 𝛽𝜇𝜈
𝐺 = 𝛽𝜇𝜈
𝐵 = 0
𝛼′𝐷−2𝑆tree ∼ ∫ 𝑑𝐷𝑥√−det𝐺𝑒−Φ [𝑅 + (∇Φ)2 − 1
12 𝐻2 + 𝐷 − 26
3 ] + 𝒪(𝛼′)
𝐺𝜇𝜈
𝐸 = 𝑒− 2Φ
𝐷−2𝐺𝜇𝜈
𝑆𝐸
tree ∼ 1
𝜅2 ∫ 𝑑𝐷𝑥√𝐺𝐸 [𝑅 − 1
𝐷 − 2 (∇Φ)2 − 𝑒−4Φ/(𝐷−2)
12 𝐻2
+𝑒2Φ/(𝐷−2) 𝐷 − 26
3 ] + 𝒪(𝛼′)
𝜅 = 𝑔spacetime dimension 𝛼′(𝐷−2)/2
Supersimetría de Einstein – Polyakov – Virasoro – Majorana - Weyl. Superconformidad de gauge.
𝑆𝑃
𝐼𝐼 = 1
4𝜋𝛼′ ∫ √𝑔 [𝑔𝑎𝑏𝜕𝑎𝑋𝜇𝜕𝑏𝑋𝜇 + 𝑖
2 𝜓𝜇 ∂̸𝜓𝜇 + 𝑖
2 (𝜒𝑎𝛾𝑏𝛾𝑎𝜓𝜇) (𝜕𝑏𝑋𝜇 − 𝑖
4 𝜒𝑏𝜓𝜇)]
𝑝𝜇𝜖𝜇𝜈 = 𝑝𝜈𝜖𝜇𝜈 = 0
Λ4 = ∫
ℱ
𝑑2𝜏
𝜏2
2 𝑍bosonic (𝜏, 𝜏‾) = ∫
ℱ
𝑑2𝜏
𝜏2
2
1
(√𝜏2𝜂𝜂‾)24
𝜒 = 2(1 − 𝑔) − 𝐵 − 𝐶.
Acción stress – tensor – energía - momentum. Centro de masa. Partícula Supermasiva u Oscura.
𝑆𝑃 = 1
4𝜋𝛼′ ∫ 𝑑2𝜉 [√𝑔𝑔𝑎𝑏𝐺𝜇𝜈(𝑋) + 𝜖𝑎𝑏𝐵𝜇𝜈(𝑋)]𝜕𝑎𝑋𝜇𝜕𝑏𝑋𝜈 + + 1
8𝜋 ∫ 𝑑2𝜉√𝑔𝑅(2)Φ(𝑋)
𝜒 = 1
4𝜋 ∫ √𝑔𝑅(2)
𝑔spacetime dimension = 𝑒Φ0/2
𝑇𝑎
𝑎
√𝑔 = 𝛽Φ
96𝜋3 𝑅(2) + 1
2𝜋 (𝛽𝜇𝜈
𝐺 𝑔𝑎𝑏 + 𝛽𝜇𝜈
𝐵 𝜖𝑎𝑏)𝜕𝑎𝑋𝜇𝜕𝑏𝑋𝜈
𝛽𝜇𝜈
𝐺
𝛼′ = 𝑅𝜇𝜈 − 1
4 𝐻𝜇𝜌𝜎𝐻𝜈
𝜌𝜎 + ∇𝜇∇𝜈Φ + 𝒪(𝛼′)
𝛽𝜇𝜈
𝐵
𝛼′ = ∇𝜇[𝑒−Φ𝐻𝜇𝜈𝜌] + 𝒪(𝛼′)
𝛽Φ = 𝐷 − 26 + 3𝛼′ [(∇Φ)2 − 2 ◻ Φ − 𝑅 + 1
12 𝐻2] + 𝒪(𝛼′2)
𝐻𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 + 𝜕𝜈𝐵𝜌𝜇 + 𝜕𝜌𝐵𝜇𝜈
𝛽Φ = 𝛽𝜇𝜈
𝐺 = 𝛽𝜇𝜈
𝐵 = 0
𝛼′𝐷−2𝑆tree ∼ ∫ 𝑑𝐷𝑥√−det𝐺𝑒−Φ [𝑅 + (∇Φ)2 − 1
12 𝐻2 + 𝐷 − 26
3 ] + 𝒪(𝛼′)
𝐺𝜇𝜈
𝐸 = 𝑒− 2Φ
𝐷−2𝐺𝜇𝜈
𝑆𝐸
tree ∼ 1
𝜅2 ∫ 𝑑𝐷𝑥√𝐺𝐸 [𝑅 − 1
𝐷 − 2 (∇Φ)2 − 𝑒−4Φ/(𝐷−2)
12 𝐻2
+𝑒2Φ/(𝐷−2) 𝐷 − 26
3 ] + 𝒪(𝛼′)
𝜅 = 𝑔spacetime dimension 𝛼′(𝐷−2)/2
Supersimetría de Einstein – Polyakov – Virasoro – Majorana - Weyl. Superconformidad de gauge.
𝑆𝑃
𝐼𝐼 = 1
4𝜋𝛼′ ∫ √𝑔 [𝑔𝑎𝑏𝜕𝑎𝑋𝜇𝜕𝑏𝑋𝜇 + 𝑖
2 𝜓𝜇 ∂̸𝜓𝜇 + 𝑖
2 (𝜒𝑎𝛾𝑏𝛾𝑎𝜓𝜇) (𝜕𝑏𝑋𝜇 − 𝑖
4 𝜒𝑏𝜓𝜇)]
pág. 37
𝛿𝑔𝑎𝑏 = 𝑖𝜖(𝛾𝑎𝜒𝑏 + 𝛾𝑏𝜒𝑎) , 𝛿𝜒𝑎 = 2∇𝑎𝜖,
𝛿𝑋𝜇 = 𝑖𝜖𝜓𝜇 , 𝛿𝜓𝜇 = 𝛾𝑎 (𝜕𝑎𝑋𝜇 − 𝑖
2 𝜒𝑎𝜓𝜇) 𝜖 , 𝛿𝜓‾𝜇 = 0,
𝑔𝑎𝑏 = 𝑒𝜙𝛿𝑎𝑏 , 𝜒𝑎 = 𝛾𝑎𝜁
𝐺matter = 𝑖𝜓𝜇𝜕𝑋𝜇 , 𝐺‾matter = 𝑖𝜓‾𝜇𝜕‾𝑋𝜇
𝑗𝐵𝑅𝑆𝑇 = 𝛾𝐺matter + 𝑐𝑇matter + 1
2 (𝑐𝑇ghost + 𝛾𝐺ghost)
𝐺matter = 𝑖𝜓𝜇𝜕𝑋𝜇 , 𝑇matter = − 1
2 𝜕𝑋𝜇𝜕𝑋𝜇 − 1
2 𝜓𝜇𝜕𝜓𝜇,
𝐺ghost = −𝑖 (𝑐𝜕𝛽 − 1
2 𝛾𝑏 + 3
2 𝜕𝑐𝛽) , 𝑇ghost = 𝑇𝑏𝑐 − 1
2 𝛾𝜕𝛽 − 3
2 𝜕𝛾𝛽.
𝑄 = 1
2𝜋𝑖 [∮ 𝑑𝑧𝑗𝐵𝑅𝑆𝑇 + ∮ 𝑑𝑧‾𝑗‾𝐵𝑅𝑆𝑇]
𝑋+ = 𝑥+ + 𝑝+𝜏 , 𝜓+ = 𝜓‾+ = 0
0 = {𝐺0, 𝐺0} = 2 (𝐿0 − 𝐷 − 2
16 )
𝐿0 = 𝐿‾0 , 𝐿0 − 1
2 = 0
(−1)𝐹 = exp [𝑖𝜋 ∑
𝑟∈𝑍+1/2
𝜓𝑟
𝑖 𝜓−𝑟
𝑖 ] .
(−1)𝐹 = ∏
9
𝜇=0
𝜓0
𝜇exp [𝑖𝜋 ∑
∞
𝑛=1
𝜓𝑛
𝑖 𝜓−𝑛
𝑖 ] = Γ11exp [𝑖𝜋 ∑
∞
𝑛=1
𝜓𝑛
𝑖 𝜓−𝑛
𝑖 ]
{(−1)𝐹𝐿 , 𝐺0} = 0 , {(−1)𝐹𝑅 , 𝐺‾0} = 0
{Γ11, ∂̸} = 0
𝑍𝐼𝐼𝐵 = (𝜒𝑉 − 𝜒𝑆)(𝜒‾𝑉 − 𝜒‾𝑆)
(√𝜏2𝜂𝜂‾)8
𝑍𝐼𝐼𝐵 = 1
(√𝜏2𝜂𝜂‾)8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 1
2 ∑
1
𝑎‾,𝑏‾=0
(−1)𝑎‾+𝑏‾+𝑎‾𝑏‾ 𝜗4 [𝑎
𝑏] 𝜗‾4 [𝑎‾
𝑏‾]
𝜂4𝜂‾4 ,
𝑍𝐼𝐼𝐴 = 1
(√𝜏2𝜂𝜂‾)8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏 1
2 ∑
1
𝑎‾ ,𝑏‾=0
(−1)𝑎‾+𝑏‾+𝑎‾𝑏‾ 𝜗4 [𝑎
𝑏] 𝜗‾4 [𝑎‾
𝑏‾]
𝜂4𝜂‾4 .
𝛿𝑔𝑎𝑏 = 𝑖𝜖(𝛾𝑎𝜒𝑏 + 𝛾𝑏𝜒𝑎) , 𝛿𝜒𝑎 = 2∇𝑎𝜖,
𝛿𝑋𝜇 = 𝑖𝜖𝜓𝜇 , 𝛿𝜓𝜇 = 𝛾𝑎 (𝜕𝑎𝑋𝜇 − 𝑖
2 𝜒𝑎𝜓𝜇) 𝜖 , 𝛿𝜓‾𝜇 = 0,
𝑔𝑎𝑏 = 𝑒𝜙𝛿𝑎𝑏 , 𝜒𝑎 = 𝛾𝑎𝜁
𝐺matter = 𝑖𝜓𝜇𝜕𝑋𝜇 , 𝐺‾matter = 𝑖𝜓‾𝜇𝜕‾𝑋𝜇
𝑗𝐵𝑅𝑆𝑇 = 𝛾𝐺matter + 𝑐𝑇matter + 1
2 (𝑐𝑇ghost + 𝛾𝐺ghost)
𝐺matter = 𝑖𝜓𝜇𝜕𝑋𝜇 , 𝑇matter = − 1
2 𝜕𝑋𝜇𝜕𝑋𝜇 − 1
2 𝜓𝜇𝜕𝜓𝜇,
𝐺ghost = −𝑖 (𝑐𝜕𝛽 − 1
2 𝛾𝑏 + 3
2 𝜕𝑐𝛽) , 𝑇ghost = 𝑇𝑏𝑐 − 1
2 𝛾𝜕𝛽 − 3
2 𝜕𝛾𝛽.
𝑄 = 1
2𝜋𝑖 [∮ 𝑑𝑧𝑗𝐵𝑅𝑆𝑇 + ∮ 𝑑𝑧‾𝑗‾𝐵𝑅𝑆𝑇]
𝑋+ = 𝑥+ + 𝑝+𝜏 , 𝜓+ = 𝜓‾+ = 0
0 = {𝐺0, 𝐺0} = 2 (𝐿0 − 𝐷 − 2
16 )
𝐿0 = 𝐿‾0 , 𝐿0 − 1
2 = 0
(−1)𝐹 = exp [𝑖𝜋 ∑
𝑟∈𝑍+1/2
𝜓𝑟
𝑖 𝜓−𝑟
𝑖 ] .
(−1)𝐹 = ∏
9
𝜇=0
𝜓0
𝜇exp [𝑖𝜋 ∑
∞
𝑛=1
𝜓𝑛
𝑖 𝜓−𝑛
𝑖 ] = Γ11exp [𝑖𝜋 ∑
∞
𝑛=1
𝜓𝑛
𝑖 𝜓−𝑛
𝑖 ]
{(−1)𝐹𝐿 , 𝐺0} = 0 , {(−1)𝐹𝑅 , 𝐺‾0} = 0
{Γ11, ∂̸} = 0
𝑍𝐼𝐼𝐵 = (𝜒𝑉 − 𝜒𝑆)(𝜒‾𝑉 − 𝜒‾𝑆)
(√𝜏2𝜂𝜂‾)8
𝑍𝐼𝐼𝐵 = 1
(√𝜏2𝜂𝜂‾)8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 1
2 ∑
1
𝑎‾,𝑏‾=0
(−1)𝑎‾+𝑏‾+𝑎‾𝑏‾ 𝜗4 [𝑎
𝑏] 𝜗‾4 [𝑎‾
𝑏‾]
𝜂4𝜂‾4 ,
𝑍𝐼𝐼𝐴 = 1
(√𝜏2𝜂𝜂‾)8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏 1
2 ∑
1
𝑎‾ ,𝑏‾=0
(−1)𝑎‾+𝑏‾+𝑎‾𝑏‾ 𝜗4 [𝑎
𝑏] 𝜗‾4 [𝑎‾
𝑏‾]
𝜂4𝜂‾4 .
pág. 38
Estados supermasivos y pluridimensiones por deformación de superficie. Métrica de Majorana.
{Γ𝜇, Γ𝜈} = −2𝜂𝜇𝜈 , 𝜂𝜇𝜈 = (− + + ⋯ +)
Γ𝜇 = 𝜂𝜇𝜈Γ𝜈 Γ𝜇 = 𝜂𝜇𝜈Γ𝜈
Γ0Γ𝜇
†Γ0 = Γ𝜇 , Γ0Γ𝜇Γ0 = −Γ𝜇
𝑇
Γ11 = Γ0 … Γ9 , (Γ11)2 = 1 , {Γ11, Γ𝜇} = 0;
Γ𝜇1…𝜇𝑘 = 1
𝑘! Γ[𝜇1 … Γ𝜇𝑘] = 1
𝑘! (Γ𝜇1 … Γ𝜇𝑘 ± permutations ).
Γ11Γ𝜇1…𝜇𝑘 = (−1)[𝑘
2]
(10 − 𝑘)! 𝜖𝜇1…𝜇10 Γ𝜇𝑘+1…𝜇10
Γ𝜇1…𝜇𝑘 Γ11 = (−1)[𝑘+1
2 ]
(10 − 𝑘)! 𝜖𝜇1…𝜇10 Γ𝜇𝑘+1…𝜇10
Γ𝜇Γ𝜈1…𝜈𝑘 = Γ𝜇𝜈1…𝜈𝑘 − 1
(𝑘 − 1)! 𝜂𝜇[𝜈1 Γ𝜈2…𝜈𝑘]
Γ𝜈1…𝜈𝑘 Γ𝜇 = Γ𝜈1…𝜈𝑘𝜇 − 1
(𝑘 − 1)! 𝜂𝜇[𝜈𝑘 Γ𝜈1…𝜈𝑘−1]
𝐹𝛼𝛽 = 𝑆𝛼(𝑖Γ0)𝛽𝛾𝑆˜𝛾
Γ11𝐹 = 𝐹 , 𝐹Γ11 = −𝜉𝐹
𝐹𝛼𝛽 = ∑
10
𝑘=0
𝑖𝑘
𝑘! 𝐹𝜇1…𝜇𝑘 (Γ𝜇1…𝜇𝑘 )𝛼𝛽
𝐹𝜇1…𝜇𝑘 = (−1)[𝑘+1
2 ]
(10 − 𝑘)! 𝜖𝜇1…𝜇10 𝐹𝜇𝑘+1…𝜇10
𝐹𝜇1…𝜇𝑘 = 𝜉 (−1)[𝑘
2]+1
(10 − 𝑘)! 𝜖𝜇1…𝜇10 𝐹𝜇𝑘+1…𝜇10
(𝑝𝜇Γ𝜇)𝐹 = 𝐹(𝑝𝜇Γ𝜇) = 0
𝑝[𝜇𝐹𝜈1…𝜈𝑘] = 𝑝𝜇𝐹𝜇𝜈2…𝜈𝑘 = 0
𝑑𝐹 = 𝑑∗𝐹 = 0
𝐹𝜇1…𝜇𝑘 = 1
(𝑘 − 1)! 𝜕[𝜇1 𝐶𝜇2…𝜇𝑘]
𝐹(𝑘) = 𝑑𝐶(𝑘−1)
𝑋(𝜎 + 2𝜋) = 𝑋(2𝜋 − 𝜎)
Estados supermasivos y pluridimensiones por deformación de superficie. Métrica de Majorana.
{Γ𝜇, Γ𝜈} = −2𝜂𝜇𝜈 , 𝜂𝜇𝜈 = (− + + ⋯ +)
Γ𝜇 = 𝜂𝜇𝜈Γ𝜈 Γ𝜇 = 𝜂𝜇𝜈Γ𝜈
Γ0Γ𝜇
†Γ0 = Γ𝜇 , Γ0Γ𝜇Γ0 = −Γ𝜇
𝑇
Γ11 = Γ0 … Γ9 , (Γ11)2 = 1 , {Γ11, Γ𝜇} = 0;
Γ𝜇1…𝜇𝑘 = 1
𝑘! Γ[𝜇1 … Γ𝜇𝑘] = 1
𝑘! (Γ𝜇1 … Γ𝜇𝑘 ± permutations ).
Γ11Γ𝜇1…𝜇𝑘 = (−1)[𝑘
2]
(10 − 𝑘)! 𝜖𝜇1…𝜇10 Γ𝜇𝑘+1…𝜇10
Γ𝜇1…𝜇𝑘 Γ11 = (−1)[𝑘+1
2 ]
(10 − 𝑘)! 𝜖𝜇1…𝜇10 Γ𝜇𝑘+1…𝜇10
Γ𝜇Γ𝜈1…𝜈𝑘 = Γ𝜇𝜈1…𝜈𝑘 − 1
(𝑘 − 1)! 𝜂𝜇[𝜈1 Γ𝜈2…𝜈𝑘]
Γ𝜈1…𝜈𝑘 Γ𝜇 = Γ𝜈1…𝜈𝑘𝜇 − 1
(𝑘 − 1)! 𝜂𝜇[𝜈𝑘 Γ𝜈1…𝜈𝑘−1]
𝐹𝛼𝛽 = 𝑆𝛼(𝑖Γ0)𝛽𝛾𝑆˜𝛾
Γ11𝐹 = 𝐹 , 𝐹Γ11 = −𝜉𝐹
𝐹𝛼𝛽 = ∑
10
𝑘=0
𝑖𝑘
𝑘! 𝐹𝜇1…𝜇𝑘 (Γ𝜇1…𝜇𝑘 )𝛼𝛽
𝐹𝜇1…𝜇𝑘 = (−1)[𝑘+1
2 ]
(10 − 𝑘)! 𝜖𝜇1…𝜇10 𝐹𝜇𝑘+1…𝜇10
𝐹𝜇1…𝜇𝑘 = 𝜉 (−1)[𝑘
2]+1
(10 − 𝑘)! 𝜖𝜇1…𝜇10 𝐹𝜇𝑘+1…𝜇10
(𝑝𝜇Γ𝜇)𝐹 = 𝐹(𝑝𝜇Γ𝜇) = 0
𝑝[𝜇𝐹𝜈1…𝜈𝑘] = 𝑝𝜇𝐹𝜇𝜈2…𝜈𝑘 = 0
𝑑𝐹 = 𝑑∗𝐹 = 0
𝐹𝜇1…𝜇𝑘 = 1
(𝑘 − 1)! 𝜕[𝜇1 𝐶𝜇2…𝜇𝑘]
𝐹(𝑘) = 𝑑𝐶(𝑘−1)
𝑋(𝜎 + 2𝜋) = 𝑋(2𝜋 − 𝜎)
pág. 39
𝑍compact (𝑞‾) = ∑
𝐿16
𝑞‾𝑝‾𝑅
2
2
𝜂‾16 = Γ‾16(𝑞‾)
𝜂‾16
𝑍heterotic = 1
(√𝜏2𝜂𝜂‾)8
Γ‾16
𝜂‾16
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗 [𝑎
𝑏]4
𝜂4
𝜏 → − 1
𝜏 : Γ‾16 → 𝜏‾8Γ‾16
Γ‾E8×E8 = (Γ‾8)2 = [1
2 ∑
𝑎,𝑏=0,1
𝜗‾ [𝑎
𝑏]8
= 1 + 2 ⋅ 240𝑞‾ + 𝒪(𝑞‾2).
Γ‾O(32)/Z2 = 1
2 ∑
𝑎,𝑏=0,1
𝜗‾ [𝑎
𝑏]16
= 1 + 480𝑞‾ + 𝒪(𝑞‾2)
𝐽‾𝑖𝑗 = 𝑖𝜓‾𝑖𝜓‾ 𝑗
𝐿0 = 1
2 , 𝐿‾0 = 1
𝐺0 = 0 , 𝐿‾0 = 1
𝑍fermions [ℎ
𝑔] = 1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏+𝑎𝑔+𝑏ℎ+𝑔ℎ 𝜗4 [𝑎
𝑏]
𝜂4
𝑍‾𝐸8 [ℎ
𝑔] = 1
2 ∑
1
𝛾,𝛿=0
(−1)𝛾𝑔+𝛿ℎ 𝜗‾8 [𝛾
𝛿]
𝜂‾8
𝑍O(16)×O(16)
heterotic = 1
2 ∑
1
ℎ,𝑔=0
𝑍‾𝐸8 [ℎ
𝑔]
2
(√𝜏2𝜂𝜂‾)8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏+𝑎𝑔+𝑏ℎ+𝑔ℎ 𝜗4 [𝑎
𝑏]
𝜂4 .
𝑋ˆ𝜇(𝑧, 𝜃) = 𝑋𝜇(𝑧) + 𝜃𝜓𝜇(𝑧)
∫ 𝑑𝑧 ∫ 𝑑𝜃𝑉(𝑧, 𝜃) = ∫ 𝑑𝑧 ∫ 𝑑𝜃(𝑉0(𝑧) + 𝜃𝑉−1(𝑧)) = ∫ 𝑑𝑧𝑉−1
𝑉boson (𝜖, 𝑝, 𝑧, 𝜃) = 𝜖𝜇: 𝐷𝑋ˆ𝜇𝑒𝑖𝑝⋅𝑋ˆ
𝑉0
boson = 𝜖𝜇𝜓𝜇𝑒𝑖𝑝⋅𝑋 , 𝑉−1
boson (𝜖, 𝑝, 𝑧) = 𝜖𝜇: (𝜕𝑋𝜇 + 𝑖𝑝 ⋅ 𝜓𝜓𝜇)𝑒𝑖𝑝⋅𝑋: ,
𝑉−1
boson (𝜖, 𝑝, 𝑧) = [𝑄BRST, 𝜉(𝑧)𝑒−𝜙(𝑧)𝜖 ⋅ 𝜓𝑒𝑖𝑝⋅𝑋]
𝑉−1/2
fermion(𝑢, 𝑝, 𝑧) = 𝑢𝛼(𝑝): 𝑒−𝜙(𝑧)/2𝑆𝛼(𝑧)𝑒𝑖𝑝⋅𝑋: ,
𝑍compact (𝑞‾) = ∑
𝐿16
𝑞‾𝑝‾𝑅
2
2
𝜂‾16 = Γ‾16(𝑞‾)
𝜂‾16
𝑍heterotic = 1
(√𝜏2𝜂𝜂‾)8
Γ‾16
𝜂‾16
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗 [𝑎
𝑏]4
𝜂4
𝜏 → − 1
𝜏 : Γ‾16 → 𝜏‾8Γ‾16
Γ‾E8×E8 = (Γ‾8)2 = [1
2 ∑
𝑎,𝑏=0,1
𝜗‾ [𝑎
𝑏]8
= 1 + 2 ⋅ 240𝑞‾ + 𝒪(𝑞‾2).
Γ‾O(32)/Z2 = 1
2 ∑
𝑎,𝑏=0,1
𝜗‾ [𝑎
𝑏]16
= 1 + 480𝑞‾ + 𝒪(𝑞‾2)
𝐽‾𝑖𝑗 = 𝑖𝜓‾𝑖𝜓‾ 𝑗
𝐿0 = 1
2 , 𝐿‾0 = 1
𝐺0 = 0 , 𝐿‾0 = 1
𝑍fermions [ℎ
𝑔] = 1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏+𝑎𝑔+𝑏ℎ+𝑔ℎ 𝜗4 [𝑎
𝑏]
𝜂4
𝑍‾𝐸8 [ℎ
𝑔] = 1
2 ∑
1
𝛾,𝛿=0
(−1)𝛾𝑔+𝛿ℎ 𝜗‾8 [𝛾
𝛿]
𝜂‾8
𝑍O(16)×O(16)
heterotic = 1
2 ∑
1
ℎ,𝑔=0
𝑍‾𝐸8 [ℎ
𝑔]
2
(√𝜏2𝜂𝜂‾)8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏+𝑎𝑔+𝑏ℎ+𝑔ℎ 𝜗4 [𝑎
𝑏]
𝜂4 .
𝑋ˆ𝜇(𝑧, 𝜃) = 𝑋𝜇(𝑧) + 𝜃𝜓𝜇(𝑧)
∫ 𝑑𝑧 ∫ 𝑑𝜃𝑉(𝑧, 𝜃) = ∫ 𝑑𝑧 ∫ 𝑑𝜃(𝑉0(𝑧) + 𝜃𝑉−1(𝑧)) = ∫ 𝑑𝑧𝑉−1
𝑉boson (𝜖, 𝑝, 𝑧, 𝜃) = 𝜖𝜇: 𝐷𝑋ˆ𝜇𝑒𝑖𝑝⋅𝑋ˆ
𝑉0
boson = 𝜖𝜇𝜓𝜇𝑒𝑖𝑝⋅𝑋 , 𝑉−1
boson (𝜖, 𝑝, 𝑧) = 𝜖𝜇: (𝜕𝑋𝜇 + 𝑖𝑝 ⋅ 𝜓𝜓𝜇)𝑒𝑖𝑝⋅𝑋: ,
𝑉−1
boson (𝜖, 𝑝, 𝑧) = [𝑄BRST, 𝜉(𝑧)𝑒−𝜙(𝑧)𝜖 ⋅ 𝜓𝑒𝑖𝑝⋅𝑋]
𝑉−1/2
fermion(𝑢, 𝑝, 𝑧) = 𝑢𝛼(𝑝): 𝑒−𝜙(𝑧)/2𝑆𝛼(𝑧)𝑒𝑖𝑝⋅𝑋: ,
pág. 40
𝑉1/2
fermion (𝑢, 𝑝) = [𝑄BRST, 𝜉(𝑧)𝑉−1/2
fermion (𝑢, 𝑝, 𝑧)] = 𝑢𝛼(𝑝)𝑒𝜙/2𝑆𝛼𝑒𝑖𝑝⋅𝑋 + ⋯
𝑄𝛼 = 1
2𝜋𝑖 ∮ 𝑑𝑧: 𝑒−𝜙(𝑧)/2𝑆𝛼(𝑧)
[𝑄𝛼, 𝑉−1/2
fermion (𝑢, 𝑝, 𝑧)] = 𝑉−1
boson (𝜖𝜇 = 𝑢𝛽𝛾𝛽𝛼
𝜇 , 𝑝, 𝑧) ,
[𝑄𝛼, 𝑉0
boson (𝜖, 𝑝, 𝑧)] = 𝑉−1/2
fermion (𝑢𝛽 = 𝑖𝑝𝜇𝜖𝜈(𝛾𝜇𝜈)𝛼
𝛽, 𝑝, 𝑧)
Acciones supersimétricas y de supergravedad. Métrica de Yang – Mills - Chern-Simons.
Superinvariancias y supercovariancias.
𝛿𝜖𝜙 ∼ 𝜙𝑚𝜓𝜖 , 𝛿𝜖𝜓 ∼ 𝜕𝜙𝑚𝜖 + 𝜙𝑚𝜓2𝜖
𝐿YM = − 1
4 𝐹𝜇𝜈
𝑎 𝐹𝑎,𝜇𝜈 − 𝜒‾𝑎Γ𝜇𝐷𝜇𝜒𝑎
𝐹𝜇𝜈
𝑎 = 𝜕𝜇𝐴𝜈
𝑎 − 𝜕𝜈𝐴𝜇
𝑎 + 𝑔𝑓𝑏𝑐
𝑎 𝐴𝜇
𝑏𝐴𝜈
𝑐
𝐷𝜇𝜒𝑎 = 𝜕𝜇𝜒𝑎 + 𝑔𝑓𝑏𝑐
𝑎 𝐴𝜇
𝑏𝜒𝑐
𝐿SUGRA
𝑁=1 = − 1
2𝜅2 𝑅 − 3
4 𝜙−3
2𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌 − 9
16𝜅2
𝜕𝜇𝜙𝜕𝜇𝜙
𝜙2 − 1
2 𝜓‾𝜇Γ𝜇𝜈𝜌∇𝜈𝜓𝜌 − 1
2 𝜆‾Γ𝜇∇𝜇𝜆
− 3√2
8
𝜕𝜈
𝜙 𝜓‾𝜇Γ𝜈Γ𝜇𝜆
+ √2𝜅
16 𝜙−3/4𝐻𝜈𝜌𝜎[𝜓‾𝜇Γ𝜇𝜈𝜌𝜎𝜏𝜓𝜏 + +6𝜓‾𝜈Γ𝜌𝜓𝜎 − √2𝜓‾𝜇Γ𝜈𝜌𝜎Γ𝜇𝜆] + ( 𝔽𝑓𝑒𝑟𝑚𝑖𝑜𝑛 )4
𝐿SUGRA+YM
𝑁=1 = 𝐿SUGRA
𝑁=1 ′ + 𝜙−3/4𝐿YM
′
𝐻ˆ𝜇𝜈𝜌 = 𝐻𝜇𝜈𝜌 − 𝜅
√2 𝜔𝜇𝜈𝜌
𝐶𝑆
𝜔𝜇𝜈𝜌
𝐶𝑆 = 𝐴𝜇
𝑎𝐹𝜈𝜌
𝑎 − 𝑔
3 𝑓𝑎𝑏𝑐𝐴𝜇
𝑎𝐴𝜈
𝑏𝐴𝜌
𝑐 + cyclic
𝛿𝐵 = 𝜅
√2 Tr[Λ𝑑𝐴]
𝐿𝐷=11 = 1
2𝜅2 [𝑅 − 1
2 ⋅ 4! 𝐺4
2] − 𝑖𝜓‾𝜇Γ𝜇𝜈𝜌∇˜ 𝜈𝜓𝜌 + 1
2𝜅2(144)2 𝐺4 ∧ 𝐺4 ∧ 𝐶ˆ
+ 1
192 [𝜓‾𝜇Γ𝜇𝜈𝜌𝜎𝜏𝑣𝜓𝑣 + 12𝜓‾𝜈Γ𝜌𝜎𝜓𝜏](𝐺 + 𝐺ˆ )𝜈𝜌𝜎𝜏
𝜔˜ 𝜇,𝑎𝑏 = 𝜔𝜇,𝑎𝑏 + 𝑖𝜅2
4 [−𝜓‾𝜈Γ𝜈𝜇𝑎𝑏𝜌𝜓𝜌 + 2(𝜓‾𝜇Γ𝑏𝜓𝑎 − 𝜓‾𝜇Γ𝑎𝜓𝑏 + 𝜓‾𝑏Γ𝜇𝜓𝑎)]
𝐺𝜇𝜈𝜌𝜎 = 𝜕𝜇𝐶ˆ𝜈𝜌𝜎 − 𝜕𝜈𝐶ˆ𝜌𝜎𝜇 + 𝜕𝜌𝐶ˆ𝜎𝜇𝜈 − 𝜕𝜎𝐶ˆ𝜇𝜈𝜌
𝑉1/2
fermion (𝑢, 𝑝) = [𝑄BRST, 𝜉(𝑧)𝑉−1/2
fermion (𝑢, 𝑝, 𝑧)] = 𝑢𝛼(𝑝)𝑒𝜙/2𝑆𝛼𝑒𝑖𝑝⋅𝑋 + ⋯
𝑄𝛼 = 1
2𝜋𝑖 ∮ 𝑑𝑧: 𝑒−𝜙(𝑧)/2𝑆𝛼(𝑧)
[𝑄𝛼, 𝑉−1/2
fermion (𝑢, 𝑝, 𝑧)] = 𝑉−1
boson (𝜖𝜇 = 𝑢𝛽𝛾𝛽𝛼
𝜇 , 𝑝, 𝑧) ,
[𝑄𝛼, 𝑉0
boson (𝜖, 𝑝, 𝑧)] = 𝑉−1/2
fermion (𝑢𝛽 = 𝑖𝑝𝜇𝜖𝜈(𝛾𝜇𝜈)𝛼
𝛽, 𝑝, 𝑧)
Acciones supersimétricas y de supergravedad. Métrica de Yang – Mills - Chern-Simons.
Superinvariancias y supercovariancias.
𝛿𝜖𝜙 ∼ 𝜙𝑚𝜓𝜖 , 𝛿𝜖𝜓 ∼ 𝜕𝜙𝑚𝜖 + 𝜙𝑚𝜓2𝜖
𝐿YM = − 1
4 𝐹𝜇𝜈
𝑎 𝐹𝑎,𝜇𝜈 − 𝜒‾𝑎Γ𝜇𝐷𝜇𝜒𝑎
𝐹𝜇𝜈
𝑎 = 𝜕𝜇𝐴𝜈
𝑎 − 𝜕𝜈𝐴𝜇
𝑎 + 𝑔𝑓𝑏𝑐
𝑎 𝐴𝜇
𝑏𝐴𝜈
𝑐
𝐷𝜇𝜒𝑎 = 𝜕𝜇𝜒𝑎 + 𝑔𝑓𝑏𝑐
𝑎 𝐴𝜇
𝑏𝜒𝑐
𝐿SUGRA
𝑁=1 = − 1
2𝜅2 𝑅 − 3
4 𝜙−3
2𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌 − 9
16𝜅2
𝜕𝜇𝜙𝜕𝜇𝜙
𝜙2 − 1
2 𝜓‾𝜇Γ𝜇𝜈𝜌∇𝜈𝜓𝜌 − 1
2 𝜆‾Γ𝜇∇𝜇𝜆
− 3√2
8
𝜕𝜈
𝜙 𝜓‾𝜇Γ𝜈Γ𝜇𝜆
+ √2𝜅
16 𝜙−3/4𝐻𝜈𝜌𝜎[𝜓‾𝜇Γ𝜇𝜈𝜌𝜎𝜏𝜓𝜏 + +6𝜓‾𝜈Γ𝜌𝜓𝜎 − √2𝜓‾𝜇Γ𝜈𝜌𝜎Γ𝜇𝜆] + ( 𝔽𝑓𝑒𝑟𝑚𝑖𝑜𝑛 )4
𝐿SUGRA+YM
𝑁=1 = 𝐿SUGRA
𝑁=1 ′ + 𝜙−3/4𝐿YM
′
𝐻ˆ𝜇𝜈𝜌 = 𝐻𝜇𝜈𝜌 − 𝜅
√2 𝜔𝜇𝜈𝜌
𝐶𝑆
𝜔𝜇𝜈𝜌
𝐶𝑆 = 𝐴𝜇
𝑎𝐹𝜈𝜌
𝑎 − 𝑔
3 𝑓𝑎𝑏𝑐𝐴𝜇
𝑎𝐴𝜈
𝑏𝐴𝜌
𝑐 + cyclic
𝛿𝐵 = 𝜅
√2 Tr[Λ𝑑𝐴]
𝐿𝐷=11 = 1
2𝜅2 [𝑅 − 1
2 ⋅ 4! 𝐺4
2] − 𝑖𝜓‾𝜇Γ𝜇𝜈𝜌∇˜ 𝜈𝜓𝜌 + 1
2𝜅2(144)2 𝐺4 ∧ 𝐺4 ∧ 𝐶ˆ
+ 1
192 [𝜓‾𝜇Γ𝜇𝜈𝜌𝜎𝜏𝑣𝜓𝑣 + 12𝜓‾𝜈Γ𝜌𝜎𝜓𝜏](𝐺 + 𝐺ˆ )𝜈𝜌𝜎𝜏
𝜔˜ 𝜇,𝑎𝑏 = 𝜔𝜇,𝑎𝑏 + 𝑖𝜅2
4 [−𝜓‾𝜈Γ𝜈𝜇𝑎𝑏𝜌𝜓𝜌 + 2(𝜓‾𝜇Γ𝑏𝜓𝑎 − 𝜓‾𝜇Γ𝑎𝜓𝑏 + 𝜓‾𝑏Γ𝜇𝜓𝑎)]
𝐺𝜇𝜈𝜌𝜎 = 𝜕𝜇𝐶ˆ𝜈𝜌𝜎 − 𝜕𝜈𝐶ˆ𝜌𝜎𝜇 + 𝜕𝜌𝐶ˆ𝜎𝜇𝜈 − 𝜕𝜎𝐶ˆ𝜇𝜈𝜌
pág. 41
𝐺˜𝜇𝜈𝜌𝜎 = 𝐺𝜇𝜈𝜌𝜎 − 6𝜅2𝜓‾[𝜇Γ𝜈𝜌𝜓𝜎]
𝐺𝜇𝜈 = (𝑔𝜇𝜈 + 𝑒2𝜎𝐴𝜇𝐴𝜈 𝑒2𝜎𝐴𝜇
𝑒2𝜎𝐴𝜇 𝑒2𝜎 )
𝐶𝜇𝜈𝜌 = 𝐶ˆ𝜇𝜈𝜌 − (𝐶ˆ𝜈𝜌,11𝐴𝜇 + cyclic ) , 𝐵𝜇𝜈 = 𝐶ˆ𝜇𝜈,11
𝑆𝐼𝐼𝐴 = 1
2𝜅2 ∫ 𝑑10𝑥√𝑔𝑒𝜎 [𝑅 − 1
2 ⋅ 4! 𝐺ˆ 2 − 1
2 ⋅ 3! 𝑒−2𝜎𝐻2 − 1
4 𝑒2𝜎𝐹2] + 1
2𝜅2(48)2 ∫ 𝐵 ∧ 𝐺 ∧ 𝐺
𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 , 𝐻𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 + cíclico
𝐺ˆ𝜇𝜈𝜌𝜎 = 𝐺𝜇𝜈𝜌𝜎 + (𝐹𝜇𝜈𝐵𝜌𝜎 + 5 permutaciones ).
𝑆˜10 = 1
2𝜅2 ∫ 𝑑10𝑥√𝑔𝑒−Φ [(𝑅 + (∇Φ)2 − 1
12 𝐻2) − 1
2 ⋅ 4! 𝐺ˆ 2 − 1
4 𝐹2]+ 1
2𝜅2(48)2 ∫ 𝐵 ∧ 𝐺 ∧ 𝐺
𝑆 = 𝜒 + 𝑖𝑒−𝜙/2
𝑆 → 𝑎𝑆 + 𝑏
𝑐𝑆 + 𝑑 , (𝐵𝜇𝜈
𝑁
𝐵𝜇𝜈
𝑅 ) → ( 𝑑 −𝑐
−𝑏 𝑎 ) (𝐵𝜇𝜈
𝑁
𝐵𝜇𝜈
𝑅 )
𝑆𝐼𝐼𝐵 = 1
2𝜅2 ∫ 𝑑10𝑥√−det𝑔 [𝑅 − 1
2
𝜕𝑆𝜕𝑆‾
𝑆2
2 − 1
12
|𝐻𝑅 + 𝑆𝐻𝑁|2
𝑆2
]
𝐽𝜇 = 𝛿Γeff
𝛿𝐴𝜇
, 𝑇𝜇𝜈 = 1
√−𝑔
𝛿Γeff
𝛿𝑔𝜇𝜈
𝛿ΛΓeff = Tr ∫ 𝐷𝜇Λ 𝛿Γeff
𝛿𝐴𝜇
= Tr ∫ Λ𝐷𝜇
𝛿Γeff
𝛿𝐴𝜇
= ∫ Tr[Λ𝐷𝜇𝐽𝜇]
𝛿𝑑𝑖𝑓𝑓Γeff = ∫ (∇𝜇𝜖𝜈 + ∇𝜈𝜖𝜇) 𝛿Γeff
𝛿𝑔𝜇𝜈
= ∫ 𝜖𝜇∇𝜈𝑇𝜇𝜈
𝛿Γ|gauge ∼ ∫ 𝑑10𝑥[𝑐1Tr[Λ𝐹0
5] + 𝑐2Tr[Λ𝐹0]Tr[𝐹0
4] + 𝑐3Tr[Λ𝐹0](Tr[𝐹0
2])2]
𝛿Γ|grav ∼ ∫ 𝑑10𝑥[𝑑1Tr[Θ𝑅0
5] + 𝑑2Tr[Θ𝑅0]Tr[𝑅0
4] + 𝑑3Tr[Θ𝑅0](Tr[𝑅0
2])2]
𝛿Γ|mixed ∼ ∫ 𝑑10𝑥[𝑒1Tr[Λ𝐹0]Tr[𝑅0
4] + 𝑒2Tr[Θ𝑅0]Tr[𝐹0
4]
+ +𝑒3Tr[Θ𝑅0](Tr[𝐹0
2])2 + 𝑒4Tr[Λ𝐹0](Tr[𝑅0])2]
• Wess-Zumino:
𝛿Λ1 𝐺(Λ2) − 𝛿Λ2 𝐺(Λ1) = 𝐺([Λ1, Λ2])
𝐺˜𝜇𝜈𝜌𝜎 = 𝐺𝜇𝜈𝜌𝜎 − 6𝜅2𝜓‾[𝜇Γ𝜈𝜌𝜓𝜎]
𝐺𝜇𝜈 = (𝑔𝜇𝜈 + 𝑒2𝜎𝐴𝜇𝐴𝜈 𝑒2𝜎𝐴𝜇
𝑒2𝜎𝐴𝜇 𝑒2𝜎 )
𝐶𝜇𝜈𝜌 = 𝐶ˆ𝜇𝜈𝜌 − (𝐶ˆ𝜈𝜌,11𝐴𝜇 + cyclic ) , 𝐵𝜇𝜈 = 𝐶ˆ𝜇𝜈,11
𝑆𝐼𝐼𝐴 = 1
2𝜅2 ∫ 𝑑10𝑥√𝑔𝑒𝜎 [𝑅 − 1
2 ⋅ 4! 𝐺ˆ 2 − 1
2 ⋅ 3! 𝑒−2𝜎𝐻2 − 1
4 𝑒2𝜎𝐹2] + 1
2𝜅2(48)2 ∫ 𝐵 ∧ 𝐺 ∧ 𝐺
𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 , 𝐻𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 + cíclico
𝐺ˆ𝜇𝜈𝜌𝜎 = 𝐺𝜇𝜈𝜌𝜎 + (𝐹𝜇𝜈𝐵𝜌𝜎 + 5 permutaciones ).
𝑆˜10 = 1
2𝜅2 ∫ 𝑑10𝑥√𝑔𝑒−Φ [(𝑅 + (∇Φ)2 − 1
12 𝐻2) − 1
2 ⋅ 4! 𝐺ˆ 2 − 1
4 𝐹2]+ 1
2𝜅2(48)2 ∫ 𝐵 ∧ 𝐺 ∧ 𝐺
𝑆 = 𝜒 + 𝑖𝑒−𝜙/2
𝑆 → 𝑎𝑆 + 𝑏
𝑐𝑆 + 𝑑 , (𝐵𝜇𝜈
𝑁
𝐵𝜇𝜈
𝑅 ) → ( 𝑑 −𝑐
−𝑏 𝑎 ) (𝐵𝜇𝜈
𝑁
𝐵𝜇𝜈
𝑅 )
𝑆𝐼𝐼𝐵 = 1
2𝜅2 ∫ 𝑑10𝑥√−det𝑔 [𝑅 − 1
2
𝜕𝑆𝜕𝑆‾
𝑆2
2 − 1
12
|𝐻𝑅 + 𝑆𝐻𝑁|2
𝑆2
]
𝐽𝜇 = 𝛿Γeff
𝛿𝐴𝜇
, 𝑇𝜇𝜈 = 1
√−𝑔
𝛿Γeff
𝛿𝑔𝜇𝜈
𝛿ΛΓeff = Tr ∫ 𝐷𝜇Λ 𝛿Γeff
𝛿𝐴𝜇
= Tr ∫ Λ𝐷𝜇
𝛿Γeff
𝛿𝐴𝜇
= ∫ Tr[Λ𝐷𝜇𝐽𝜇]
𝛿𝑑𝑖𝑓𝑓Γeff = ∫ (∇𝜇𝜖𝜈 + ∇𝜈𝜖𝜇) 𝛿Γeff
𝛿𝑔𝜇𝜈
= ∫ 𝜖𝜇∇𝜈𝑇𝜇𝜈
𝛿Γ|gauge ∼ ∫ 𝑑10𝑥[𝑐1Tr[Λ𝐹0
5] + 𝑐2Tr[Λ𝐹0]Tr[𝐹0
4] + 𝑐3Tr[Λ𝐹0](Tr[𝐹0
2])2]
𝛿Γ|grav ∼ ∫ 𝑑10𝑥[𝑑1Tr[Θ𝑅0
5] + 𝑑2Tr[Θ𝑅0]Tr[𝑅0
4] + 𝑑3Tr[Θ𝑅0](Tr[𝑅0
2])2]
𝛿Γ|mixed ∼ ∫ 𝑑10𝑥[𝑒1Tr[Λ𝐹0]Tr[𝑅0
4] + 𝑒2Tr[Θ𝑅0]Tr[𝐹0
4]
+ +𝑒3Tr[Θ𝑅0](Tr[𝐹0
2])2 + 𝑒4Tr[Λ𝐹0](Tr[𝑅0])2]
• Wess-Zumino:
𝛿Λ1 𝐺(Λ2) − 𝛿Λ2 𝐺(Λ1) = 𝐺([Λ1, Λ2])
pág. 42
𝛿𝐹 = [𝐹, Λ] , 𝛿𝑅 = [𝑅, Θ]
𝑑Tr[𝑅𝑚] = 𝑑Tr[𝐹𝑚] = 0
𝐼𝐷+2(𝑅, 𝐹) = 𝑑Ω𝐷+1(𝜔, 𝐴)
𝛿ΛΩ𝐷+1(𝜔, 𝐴) = 𝑑Ω𝐷(𝜔, 𝐴, Λ)
• Mecanismo de Green-Schwarz:
𝛿Γ|reduc ∼ ∫ 𝑑10𝑥(Tr[Λ𝐹0] + Tr[Θ𝑅0])(𝑎1Tr[𝐹0
4]
+ 𝑎2Tr[𝑅0
4]+𝑎3(Tr[𝐹0
2])2 + 𝑎4(Tr[𝑅0
2])2 + 𝑎5Tr[𝐹0
2]Tr[𝑅0
2])
𝐻ˆ = 𝑑𝐵 + Ω𝐶𝑆(𝐴) + Ω𝐶𝑆(𝜔)
𝛿ΛΩ𝐶𝑆(𝐴) = 𝑑Tr[Λ𝑑𝐴] , 𝛿ΘΩ𝐶𝑆(𝜔) = 𝑑Tr[Θ𝑑𝜔]
𝛿𝐵 = −Tr[Λ𝐹0 + Θ𝑅0]
Γcounter ∼ ∫ 𝑑10𝑥𝐵(𝑎1Tr[𝐹0
4] + 𝑎2Tr[𝑅0
4] + 𝑎3(Tr[𝐹0
2])2 +
+𝑎4(Tr[𝑅0
2])2 + 𝑎5Tr[𝐹0
2]Tr[𝑅0
2])
𝐹𝜇1…𝜇𝐷/2 = ± 𝑖
(𝐷/2)! 𝜖𝜇1…𝜇𝐷 𝐹𝜇𝐷/2+1…𝜇𝐷
𝑅0 =
(
0 𝑥1 0 0 ⋯
−𝑥1 0 0 0 ⋯
0 0 0 𝑥2 ⋯
0 0 −𝑥2 0 ⋯
⋯ ⋯ ⋯ ⋯ ⋯
⋯ 0 𝑥𝐷/2
⋯ −𝑥𝐷/2 0 )
𝐼ˆ1/2(𝑅) = ∏
𝐷/2
𝑖=1
( 𝑥𝑖/2
sinh (𝑥𝑖/2))
𝐼3/2(𝑅) = 𝐼ˆ1/2(𝑅) (−1 + 2 ∑
𝐷/2
𝑖=1
cosh (𝑥𝑖))
𝐼𝐴(𝑅) = − 1
8 ∏
𝐷/2
𝑖=1
( 𝑥𝑖
tanh (𝑥𝑖))
𝐼1/2(𝑅, 𝐹) = Tr[𝑒𝑖𝐹]𝐼ˆ1/2(𝑅)
𝛿𝐹 = [𝐹, Λ] , 𝛿𝑅 = [𝑅, Θ]
𝑑Tr[𝑅𝑚] = 𝑑Tr[𝐹𝑚] = 0
𝐼𝐷+2(𝑅, 𝐹) = 𝑑Ω𝐷+1(𝜔, 𝐴)
𝛿ΛΩ𝐷+1(𝜔, 𝐴) = 𝑑Ω𝐷(𝜔, 𝐴, Λ)
• Mecanismo de Green-Schwarz:
𝛿Γ|reduc ∼ ∫ 𝑑10𝑥(Tr[Λ𝐹0] + Tr[Θ𝑅0])(𝑎1Tr[𝐹0
4]
+ 𝑎2Tr[𝑅0
4]+𝑎3(Tr[𝐹0
2])2 + 𝑎4(Tr[𝑅0
2])2 + 𝑎5Tr[𝐹0
2]Tr[𝑅0
2])
𝐻ˆ = 𝑑𝐵 + Ω𝐶𝑆(𝐴) + Ω𝐶𝑆(𝜔)
𝛿ΛΩ𝐶𝑆(𝐴) = 𝑑Tr[Λ𝑑𝐴] , 𝛿ΘΩ𝐶𝑆(𝜔) = 𝑑Tr[Θ𝑑𝜔]
𝛿𝐵 = −Tr[Λ𝐹0 + Θ𝑅0]
Γcounter ∼ ∫ 𝑑10𝑥𝐵(𝑎1Tr[𝐹0
4] + 𝑎2Tr[𝑅0
4] + 𝑎3(Tr[𝐹0
2])2 +
+𝑎4(Tr[𝑅0
2])2 + 𝑎5Tr[𝐹0
2]Tr[𝑅0
2])
𝐹𝜇1…𝜇𝐷/2 = ± 𝑖
(𝐷/2)! 𝜖𝜇1…𝜇𝐷 𝐹𝜇𝐷/2+1…𝜇𝐷
𝑅0 =
(
0 𝑥1 0 0 ⋯
−𝑥1 0 0 0 ⋯
0 0 0 𝑥2 ⋯
0 0 −𝑥2 0 ⋯
⋯ ⋯ ⋯ ⋯ ⋯
⋯ 0 𝑥𝐷/2
⋯ −𝑥𝐷/2 0 )
𝐼ˆ1/2(𝑅) = ∏
𝐷/2
𝑖=1
( 𝑥𝑖/2
sinh (𝑥𝑖/2))
𝐼3/2(𝑅) = 𝐼ˆ1/2(𝑅) (−1 + 2 ∑
𝐷/2
𝑖=1
cosh (𝑥𝑖))
𝐼𝐴(𝑅) = − 1
8 ∏
𝐷/2
𝑖=1
( 𝑥𝑖
tanh (𝑥𝑖))
𝐼1/2(𝑅, 𝐹) = Tr[𝑒𝑖𝐹]𝐼ˆ1/2(𝑅)
pág. 43
𝐼1/2(𝑅, 𝐹)|12− form = − Tr[𝐹6]
720 + Tr[𝐹4]Tr[𝑅2]
24 ⋅ 48 +
− Tr[𝐹2]
256 (Tr[𝑅4]
45 + (Tr[𝑅2])2
36 ) +
+ 𝑛
64 (Tr[𝑅6]
5670 + Tr[𝑅2]Tr[𝑅4]
4320 + (Tr[𝑅2])3
10368 )
𝐼3/2(𝑅)|12− form = − 495
64 (Tr[𝑅6]
5670 + Tr[𝑅2]Tr[𝑅4]
4320 + (Tr[𝑅2])3
10368 ) +
+ Tr[𝑅2]
384 (Tr[𝑅4] + (Tr[𝑅2])2
4 )
𝐼𝐴(𝑅)|12− form = 𝐼ˆ1/2(𝑅)|12− form − 𝐼3/2(𝑅)|12− form
2𝐼𝑁=1 = 𝐼3/2(𝑅) − 𝐼1/2(𝑅) + 𝐼1/2(𝑅, 𝐹)
96𝐼total = − Tr[𝐹6]
15 + Tr[𝑅2]Tr[𝐹4]
24 + Tr[𝑅2]Tr[𝑅4]
8 + (Tr[𝑅2])3
32 −
− Tr[𝐹2]
960 (4Tr[𝑅4] + 5(Tr[𝑅2])2)
Tr[𝐹6] = 1
48 Tr[𝐹2]Tr[𝐹4] − 1
14400 (Tr[𝐹2])3.
96𝐼total = (Tr[𝑅2] − 1
30 Tr[𝐹2]) 𝑋8
𝑋8 = Tr[𝐹4]
24 − (Tr[𝐹2])2
720 − Tr[𝐹2]Tr[𝑅2]
240 + Tr[𝑅4]
8 + (Tr[𝑅2])2
32 ,
Tr[𝐹6] = (𝑁 − 32)tr[𝐹6] + 15tr[𝐹2]tr[𝐹4]
Tr[𝐹4] = (𝑁 − 8)tr[𝐹4] + 3(tr[𝐹2])2 , Tr[𝐹2] = (𝑁 − 2)tr[𝐹2]
Tr[𝐹6] = 1
7200 (Tr[𝐹2])3 , Tr[𝐹4] = 1
100 (Tr[𝐹2])2
𝑑𝐻ˆ = tr[𝑅2] − 1
30 Tr[𝐹2]
∫ tr[𝑅2] = 1
30 ∫ Tr[𝐹2]
tr𝑆[𝐹6] = 16tr[𝐹6] − 15tr[𝐹2]tr[𝐹4] + 15
4 (tr[𝐹2])3
tr𝑆[𝐹4] = −8tr[𝐹4] + 6(tr[𝐹2])2 , tr𝑆[𝐹2] = 16tr[𝐹2]
Compactificación y rompimiento de supersimetría. Métrica de Kaluza-Klein.
𝑍𝐷
heterotic = Γ10−𝐷,10−𝐷(𝐺, 𝐵)Γ‾𝐻
𝜏2
𝐷−2
2 𝜂8𝜂‾8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗4 [𝑎
𝑏]
𝜂4 ,
𝐼1/2(𝑅, 𝐹)|12− form = − Tr[𝐹6]
720 + Tr[𝐹4]Tr[𝑅2]
24 ⋅ 48 +
− Tr[𝐹2]
256 (Tr[𝑅4]
45 + (Tr[𝑅2])2
36 ) +
+ 𝑛
64 (Tr[𝑅6]
5670 + Tr[𝑅2]Tr[𝑅4]
4320 + (Tr[𝑅2])3
10368 )
𝐼3/2(𝑅)|12− form = − 495
64 (Tr[𝑅6]
5670 + Tr[𝑅2]Tr[𝑅4]
4320 + (Tr[𝑅2])3
10368 ) +
+ Tr[𝑅2]
384 (Tr[𝑅4] + (Tr[𝑅2])2
4 )
𝐼𝐴(𝑅)|12− form = 𝐼ˆ1/2(𝑅)|12− form − 𝐼3/2(𝑅)|12− form
2𝐼𝑁=1 = 𝐼3/2(𝑅) − 𝐼1/2(𝑅) + 𝐼1/2(𝑅, 𝐹)
96𝐼total = − Tr[𝐹6]
15 + Tr[𝑅2]Tr[𝐹4]
24 + Tr[𝑅2]Tr[𝑅4]
8 + (Tr[𝑅2])3
32 −
− Tr[𝐹2]
960 (4Tr[𝑅4] + 5(Tr[𝑅2])2)
Tr[𝐹6] = 1
48 Tr[𝐹2]Tr[𝐹4] − 1
14400 (Tr[𝐹2])3.
96𝐼total = (Tr[𝑅2] − 1
30 Tr[𝐹2]) 𝑋8
𝑋8 = Tr[𝐹4]
24 − (Tr[𝐹2])2
720 − Tr[𝐹2]Tr[𝑅2]
240 + Tr[𝑅4]
8 + (Tr[𝑅2])2
32 ,
Tr[𝐹6] = (𝑁 − 32)tr[𝐹6] + 15tr[𝐹2]tr[𝐹4]
Tr[𝐹4] = (𝑁 − 8)tr[𝐹4] + 3(tr[𝐹2])2 , Tr[𝐹2] = (𝑁 − 2)tr[𝐹2]
Tr[𝐹6] = 1
7200 (Tr[𝐹2])3 , Tr[𝐹4] = 1
100 (Tr[𝐹2])2
𝑑𝐻ˆ = tr[𝑅2] − 1
30 Tr[𝐹2]
∫ tr[𝑅2] = 1
30 ∫ Tr[𝐹2]
tr𝑆[𝐹6] = 16tr[𝐹6] − 15tr[𝐹2]tr[𝐹4] + 15
4 (tr[𝐹2])3
tr𝑆[𝐹4] = −8tr[𝐹4] + 6(tr[𝐹2])2 , tr𝑆[𝐹2] = 16tr[𝐹2]
Compactificación y rompimiento de supersimetría. Métrica de Kaluza-Klein.
𝑍𝐷
heterotic = Γ10−𝐷,10−𝐷(𝐺, 𝐵)Γ‾𝐻
𝜏2
𝐷−2
2 𝜂8𝜂‾8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗4 [𝑎
𝑏]
𝜂4 ,
pág. 44
𝑉Higgs ∼ 𝑓𝑎 𝑏𝑐𝑓𝑎 𝑏′𝑐′ 𝐺𝛼𝛾𝐺𝛽𝛿𝑌𝛼
𝑏𝑌𝛽
𝑐𝑌𝛾
𝑏′
𝑌𝛿
𝑐′
[𝑚2]𝑎𝑏 ∼ 𝐺𝛼𝛽𝑓𝑑
𝑐𝑎𝑓𝑑
𝑐𝑏′
𝑌𝛼
𝑑𝑌𝛽
𝑑′
𝑍𝐷
heterotic = Γ10−𝐷,26−𝐷(𝐺, 𝐵, 𝑌)
𝜏2
𝐷−2
2 𝜂8𝜂‾8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗4 [𝑎
𝑏]
𝜂4 ,
𝛼′8𝑆10−𝑑
heterotic = ∫ 𝑑10𝑥√−det𝐺10𝑒−Φ [𝑅 + (∇Φ)2 − 1
12 𝐻ˆ 2 − 1
4 Tr[𝐹2]] + 𝒪(𝛼′)
𝐹𝜇𝜈
𝐼 = 𝜕𝜇𝐴𝜈
𝐼 − 𝜕𝜈𝐴𝜇
𝐼
𝐻ˆ𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 − 1
2 ∑
𝐼
𝐴𝜇
𝐼 𝐹𝜈𝜌
𝐼 + cyclic
𝑆𝐷
heterotic = ∫ 𝑑𝐷 𝑥√−det𝐺𝑒−Φ [𝑅 + 𝜕𝜇Φ𝜕𝜇Φ − 1
12 𝐻ˆ 𝜇𝜈𝜌𝐻ˆ𝜇𝜈𝜌 −
− 1
4 (𝑀ˆ −1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1)]
𝐻ˆ𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 − 1
2 𝐿𝑖𝑗𝐴𝜇
𝑖 𝐹𝜈𝜌
𝑗 + cyclic
𝑀ˆ → Ω𝑀ˆ Ω𝑇 , 𝐴𝜇 → Ω ⋅ 𝐴𝜇
𝑆𝐷
heterotic = ∫ 𝑑𝐷𝑥√−det𝐺𝐸 [𝑅 − 1
𝐷 − 2 𝜕𝜇Φ𝜕𝜇Φ
− 𝑒− 4Φ
𝐷−2
12 𝐻ˆ 𝜇𝜈𝜌𝐻ˆ𝜇𝜈𝜌 − 𝑒− 2Φ
𝐷−2
4 (𝑀ˆ −1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1)]
𝑒−2𝜙𝐻ˆ𝜇𝜈𝜌 = 𝜖𝜇𝜈𝜌
𝜎
√−det𝑔𝐸
∇𝜎𝑎
𝜖𝜇𝜈𝜌𝜎
√−det𝑔𝐸
𝜕𝜇𝐻ˆ𝜈𝜌𝜎 = −𝐿𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹˜ 𝑗,𝜇𝜈,
𝐹˜𝜇𝜈 = 1
2
𝜖𝜇𝜈𝜌𝜎
√−det𝑔𝐸
𝐹𝜌𝜎,
∇𝜇𝑒2𝜙∇𝜇𝑎 = − 1
4 𝐹𝜇𝜈
𝑖 𝐹˜ 𝑗,𝜇𝜈
𝑉Higgs ∼ 𝑓𝑎 𝑏𝑐𝑓𝑎 𝑏′𝑐′ 𝐺𝛼𝛾𝐺𝛽𝛿𝑌𝛼
𝑏𝑌𝛽
𝑐𝑌𝛾
𝑏′
𝑌𝛿
𝑐′
[𝑚2]𝑎𝑏 ∼ 𝐺𝛼𝛽𝑓𝑑
𝑐𝑎𝑓𝑑
𝑐𝑏′
𝑌𝛼
𝑑𝑌𝛽
𝑑′
𝑍𝐷
heterotic = Γ10−𝐷,26−𝐷(𝐺, 𝐵, 𝑌)
𝜏2
𝐷−2
2 𝜂8𝜂‾8
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗4 [𝑎
𝑏]
𝜂4 ,
𝛼′8𝑆10−𝑑
heterotic = ∫ 𝑑10𝑥√−det𝐺10𝑒−Φ [𝑅 + (∇Φ)2 − 1
12 𝐻ˆ 2 − 1
4 Tr[𝐹2]] + 𝒪(𝛼′)
𝐹𝜇𝜈
𝐼 = 𝜕𝜇𝐴𝜈
𝐼 − 𝜕𝜈𝐴𝜇
𝐼
𝐻ˆ𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 − 1
2 ∑
𝐼
𝐴𝜇
𝐼 𝐹𝜈𝜌
𝐼 + cyclic
𝑆𝐷
heterotic = ∫ 𝑑𝐷 𝑥√−det𝐺𝑒−Φ [𝑅 + 𝜕𝜇Φ𝜕𝜇Φ − 1
12 𝐻ˆ 𝜇𝜈𝜌𝐻ˆ𝜇𝜈𝜌 −
− 1
4 (𝑀ˆ −1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1)]
𝐻ˆ𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 − 1
2 𝐿𝑖𝑗𝐴𝜇
𝑖 𝐹𝜈𝜌
𝑗 + cyclic
𝑀ˆ → Ω𝑀ˆ Ω𝑇 , 𝐴𝜇 → Ω ⋅ 𝐴𝜇
𝑆𝐷
heterotic = ∫ 𝑑𝐷𝑥√−det𝐺𝐸 [𝑅 − 1
𝐷 − 2 𝜕𝜇Φ𝜕𝜇Φ
− 𝑒− 4Φ
𝐷−2
12 𝐻ˆ 𝜇𝜈𝜌𝐻ˆ𝜇𝜈𝜌 − 𝑒− 2Φ
𝐷−2
4 (𝑀ˆ −1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1)]
𝑒−2𝜙𝐻ˆ𝜇𝜈𝜌 = 𝜖𝜇𝜈𝜌
𝜎
√−det𝑔𝐸
∇𝜎𝑎
𝜖𝜇𝜈𝜌𝜎
√−det𝑔𝐸
𝜕𝜇𝐻ˆ𝜈𝜌𝜎 = −𝐿𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹˜ 𝑗,𝜇𝜈,
𝐹˜𝜇𝜈 = 1
2
𝜖𝜇𝜈𝜌𝜎
√−det𝑔𝐸
𝐹𝜌𝜎,
∇𝜇𝑒2𝜙∇𝜇𝑎 = − 1
4 𝐹𝜇𝜈
𝑖 𝐹˜ 𝑗,𝜇𝜈
pág. 45
𝑆˜𝐷=4
heterotic = ∫ 𝑑4𝑥√−det𝑔𝐸 [𝑅 − 1
2 𝜕𝜇𝜙𝜕𝜇𝜙 − 1
2 𝑒2𝜙𝜕𝜇𝑎𝜕𝜇𝑎 ± 1
4 𝑒−𝜙(𝑀−1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗,𝜇𝜈
+ 1
4 𝑎𝐿𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹˜ 𝑗,𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝑆 = 𝑎 + 𝑖𝑒−𝜙
𝑆˜𝐷=4
heterotic = ∫ 𝑑4𝑥√−det𝑔𝐸 [𝑅 − 1
2
𝜕𝜇𝑆𝜕𝜇𝑆‾
Im𝑆2 − 1
4 Im𝑆(𝑀−1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗,𝜇𝜈 + 1
4 Re𝑆𝐿𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹˜ 𝑗,𝜇𝜈
+ 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝛿𝜓𝜇 = ∇𝜇𝜖 + √2
32 𝑒2Φ(𝛾𝜇𝛾5 ⊗ 𝐻)𝜖,
𝛿𝜓𝑚 = ∇𝑚𝜖 + √2
32 𝑒2Φ(𝛾𝑚𝐻 − 12𝐻𝑚)𝜖,
𝛿𝜆 = √2(𝛾𝑚∇𝑚Φ)𝜖 + 1
8 𝑒2Φ𝐻𝜖,
𝛿𝜒𝑎 = − 1
4 𝑒Φ𝐹𝑚,𝑛
𝑎 𝛾𝑚𝑛𝜖,
𝐻 = 𝐻𝑚𝑛𝑟𝛾𝑚𝑛𝑟 , 𝐻𝑚 = 𝐻𝑚𝑛𝑟𝛾𝑛𝑟
Γ𝜇 = 𝛾𝜇 ⊗ 𝟏6 , Γ𝑚 = 𝛾5 ⊗ 𝛾𝑚
𝛾5 = 𝑖
4! 𝜖𝜇𝜈𝜌𝜎𝛾𝜇𝜈𝜌𝜎 , 𝛾 = 𝑖
6! √detg𝜖𝑚𝑛𝑟𝑝𝑞𝑠𝛾𝑚𝑛𝑟𝑝𝑞𝑠
∇𝑚𝜉 = 0
𝐹𝑚𝑛
𝑎 𝛾𝑚𝑛𝜉 = 0
𝑅[𝑚𝑛
𝑟𝑠 𝑅𝑝𝑞]𝑟𝑠 = 1
30 𝐹[𝑚𝑛
𝑎 𝐹𝑝𝑞]
𝑎
𝐺𝑀𝑁 ∼ ℎ𝜇𝜈(𝑥) ⊗ 𝜙(𝑦) + 𝐴𝜇(𝑥) ⊗ 𝑓𝑚(𝑦) + Φ(𝑥) ⊗ ℎ𝑚𝑛(𝑦)
◻𝑦 𝜙(𝑦) = 0
• Lichnerowicz:
− ◻ ℎ𝑚𝑛 + 2𝑅𝑚𝑛𝑟𝑠ℎ𝑟𝑠 = 0 , ∇𝑚ℎ𝑚𝑛 = 𝑔𝑚𝑛ℎ𝑚𝑛 = 0.
◻ 𝑓𝑚𝑛 − 𝑅𝑚𝑛𝑟𝑠𝑓𝑟𝑠 =◻ 𝑓𝑚𝑛 + 2𝑅𝑚𝑟𝑠𝑛𝑓𝑟𝑠 = 0,
∇𝑚𝐴𝑛𝑝 + ∇𝑝𝐴𝑚𝑛 + ∇𝑛𝐴𝑝𝑚 = 0, ∇𝑚𝐴𝑚𝑛 = 0
ℎ𝑚𝑛 = 𝐴𝑚
𝑝 𝑆𝑝𝑚 + 𝐴𝑛
𝑝 𝑆𝑝𝑚
𝐵𝑀𝑁 ∼ 𝐵𝜇𝜈(𝑥) ⊗ 𝜙(𝑦) + 𝐵𝜇(𝑥) ⊗ 𝑓𝑚(𝑦) + Φ(𝑥) ⊗ 𝐵𝑚𝑛(𝑦)
𝑆˜𝐷=4
heterotic = ∫ 𝑑4𝑥√−det𝑔𝐸 [𝑅 − 1
2 𝜕𝜇𝜙𝜕𝜇𝜙 − 1
2 𝑒2𝜙𝜕𝜇𝑎𝜕𝜇𝑎 ± 1
4 𝑒−𝜙(𝑀−1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗,𝜇𝜈
+ 1
4 𝑎𝐿𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹˜ 𝑗,𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝑆 = 𝑎 + 𝑖𝑒−𝜙
𝑆˜𝐷=4
heterotic = ∫ 𝑑4𝑥√−det𝑔𝐸 [𝑅 − 1
2
𝜕𝜇𝑆𝜕𝜇𝑆‾
Im𝑆2 − 1
4 Im𝑆(𝑀−1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗,𝜇𝜈 + 1
4 Re𝑆𝐿𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹˜ 𝑗,𝜇𝜈
+ 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝛿𝜓𝜇 = ∇𝜇𝜖 + √2
32 𝑒2Φ(𝛾𝜇𝛾5 ⊗ 𝐻)𝜖,
𝛿𝜓𝑚 = ∇𝑚𝜖 + √2
32 𝑒2Φ(𝛾𝑚𝐻 − 12𝐻𝑚)𝜖,
𝛿𝜆 = √2(𝛾𝑚∇𝑚Φ)𝜖 + 1
8 𝑒2Φ𝐻𝜖,
𝛿𝜒𝑎 = − 1
4 𝑒Φ𝐹𝑚,𝑛
𝑎 𝛾𝑚𝑛𝜖,
𝐻 = 𝐻𝑚𝑛𝑟𝛾𝑚𝑛𝑟 , 𝐻𝑚 = 𝐻𝑚𝑛𝑟𝛾𝑛𝑟
Γ𝜇 = 𝛾𝜇 ⊗ 𝟏6 , Γ𝑚 = 𝛾5 ⊗ 𝛾𝑚
𝛾5 = 𝑖
4! 𝜖𝜇𝜈𝜌𝜎𝛾𝜇𝜈𝜌𝜎 , 𝛾 = 𝑖
6! √detg𝜖𝑚𝑛𝑟𝑝𝑞𝑠𝛾𝑚𝑛𝑟𝑝𝑞𝑠
∇𝑚𝜉 = 0
𝐹𝑚𝑛
𝑎 𝛾𝑚𝑛𝜉 = 0
𝑅[𝑚𝑛
𝑟𝑠 𝑅𝑝𝑞]𝑟𝑠 = 1
30 𝐹[𝑚𝑛
𝑎 𝐹𝑝𝑞]
𝑎
𝐺𝑀𝑁 ∼ ℎ𝜇𝜈(𝑥) ⊗ 𝜙(𝑦) + 𝐴𝜇(𝑥) ⊗ 𝑓𝑚(𝑦) + Φ(𝑥) ⊗ ℎ𝑚𝑛(𝑦)
◻𝑦 𝜙(𝑦) = 0
• Lichnerowicz:
− ◻ ℎ𝑚𝑛 + 2𝑅𝑚𝑛𝑟𝑠ℎ𝑟𝑠 = 0 , ∇𝑚ℎ𝑚𝑛 = 𝑔𝑚𝑛ℎ𝑚𝑛 = 0.
◻ 𝑓𝑚𝑛 − 𝑅𝑚𝑛𝑟𝑠𝑓𝑟𝑠 =◻ 𝑓𝑚𝑛 + 2𝑅𝑚𝑟𝑠𝑛𝑓𝑟𝑠 = 0,
∇𝑚𝐴𝑛𝑝 + ∇𝑝𝐴𝑚𝑛 + ∇𝑛𝐴𝑝𝑚 = 0, ∇𝑚𝐴𝑚𝑛 = 0
ℎ𝑚𝑛 = 𝐴𝑚
𝑝 𝑆𝑝𝑚 + 𝐴𝑛
𝑝 𝑆𝑝𝑚
𝐵𝑀𝑁 ∼ 𝐵𝜇𝜈(𝑥) ⊗ 𝜙(𝑦) + 𝐵𝜇(𝑥) ⊗ 𝑓𝑚(𝑦) + Φ(𝑥) ⊗ 𝐵𝑚𝑛(𝑦)
pág. 46
𝑆K3
𝐼𝐼𝐴 = ∫ 𝑑6𝑥√−det𝐺6𝑒−Φ [𝑅 + ∇𝜇Φ∇𝜇Φ − 1
12 𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌
+ 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1) + 1
4 ∫ 𝑑6𝑥√−det𝐺(𝑀ˆ −1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽𝜇𝜈]
+ 1
16 ∫ 𝑑6𝑥𝜖𝜇𝜈𝜌𝜎𝜏𝑣𝐵𝜇𝜈𝐹𝜌𝜎
𝐼 𝐿ˆ 𝐼𝐽𝐹𝜏𝑣
𝐽
Espacio – tiempo cuántico supersimétrico.
{𝑄𝛼
𝐼 , 𝑄𝛽
𝐽 } = 𝜖𝛼𝛽𝑍𝐼𝐽
{𝑄‾𝛼˙
𝐼 , 𝑄𝛽˙
𝐽} = 𝜖𝛼˙ 𝛽˙ 𝑍‾𝐼𝐽
{𝑄𝛼
𝐼 , 𝑄‾𝛼˙
𝐽 } = 𝛿𝐼𝐽𝜎𝛼𝛼˙
𝜇 𝑃𝜇
𝑄𝛼
𝐼 = 1
2𝜋𝑖 ∮ 𝑑𝑧𝑒−𝜙/2𝑆𝛼Σ𝐼 , 𝑄‾𝛼˙
𝐼 = 1
2𝜋𝑖 ∮ 𝑑𝑧𝑒−𝜙/2𝐶𝛼˙ Σ‾𝐼
: 𝑒𝑞1𝜙(𝑧): : 𝑒𝑞2𝜙(𝑤): = (𝑧 − 𝑤)−𝑞1𝑞2 : 𝑒(𝑞1+𝑞2)𝜙(𝑤): + ⋯ ,
𝑆𝛼(𝑧)𝐶𝛼˙ (𝑤) = 𝜎𝛼𝛼˙
𝜇 𝜓𝜇(𝑤) + 𝒪(𝑧 − 𝑤),
𝑆𝛼(𝑧)𝑆𝛽(𝑤) = 𝜖𝛼𝛽
√𝑧 − 𝑤 + 𝒪(√𝑧 − 𝑤),
𝐶𝛼˙ (𝑧)𝐶𝛽˙ (𝑤) = 𝜖𝛼˙ 𝛽˙
√𝑧 − 𝑤 + 𝒪(√𝑧 − 𝑤),
Σ𝐼(𝑧)Σ‾ 𝐽(𝑤) = 𝛿𝐼𝐽
(𝑧 − 𝑤)3/4 + (𝑧 − 𝑤)1/4𝐽𝐼𝐽(𝑤) + ⋯
Σ𝐼(𝑧)Σ𝐽(𝑤) = (𝑧 − 𝑤)−1/4Ψ𝐼𝐽(𝑤) + ⋯
Σ‾𝐼(𝑧)Σ‾ 𝐽(𝑤) = (𝑧 − 𝑤)−1/4Ψ‾ 𝐼𝐽(𝑤) + ⋯
𝐺int (𝑧)Σ𝐼(𝑤) ∼ (𝑧 − 𝑤)−1/2 , 𝐺int (𝑧)Σ‾𝐼(𝑤) ∼ (𝑧 − 𝑤)−1/2
𝜆𝐼(𝑧) = Σ𝐼(𝑧)𝑒𝑖𝑋/2 , 𝜆‾(𝑧) = Σ‾ 𝐼(𝑧)𝑒−𝑖𝑋/2
𝜆𝐼(𝑧)𝜆‾𝐽(𝑤) = 𝛿𝐼𝐽
𝑧 − 𝑤 + 𝐽ˆ𝐼𝐽 + 𝒪(𝑧 − 𝑤),
𝜆𝐼(𝑧)𝜆𝐽(𝑤) = 𝑒𝑖𝑋Ψ𝐼𝐽 + 𝒪(𝑧 − 𝑤),
𝜆‾𝐼(𝑧)𝜆‾𝐽(𝑤) = 𝑒−𝑖𝑋Ψ‾ 𝐼𝐽 + 𝒪(𝑧 − 𝑤),
𝐽ˆ𝐼𝐼(𝑧)𝐽ˆ𝐽𝐽(𝑤) = 𝛿𝐼𝐽
(𝑧 − 𝑤)2 + regular
𝐽𝐼𝐼(𝑧)𝐽𝐽𝐽(𝑤) = 𝛿𝐼𝐽 − 3/4
(𝑧 − 𝑤)2 + regular
𝐽 = 2𝐽11 , 𝐽(𝑧)𝐽(𝑤) = 3
(𝑧 − 𝑤)2 + regular
𝑆K3
𝐼𝐼𝐴 = ∫ 𝑑6𝑥√−det𝐺6𝑒−Φ [𝑅 + ∇𝜇Φ∇𝜇Φ − 1
12 𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌
+ 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1) + 1
4 ∫ 𝑑6𝑥√−det𝐺(𝑀ˆ −1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽𝜇𝜈]
+ 1
16 ∫ 𝑑6𝑥𝜖𝜇𝜈𝜌𝜎𝜏𝑣𝐵𝜇𝜈𝐹𝜌𝜎
𝐼 𝐿ˆ 𝐼𝐽𝐹𝜏𝑣
𝐽
Espacio – tiempo cuántico supersimétrico.
{𝑄𝛼
𝐼 , 𝑄𝛽
𝐽 } = 𝜖𝛼𝛽𝑍𝐼𝐽
{𝑄‾𝛼˙
𝐼 , 𝑄𝛽˙
𝐽} = 𝜖𝛼˙ 𝛽˙ 𝑍‾𝐼𝐽
{𝑄𝛼
𝐼 , 𝑄‾𝛼˙
𝐽 } = 𝛿𝐼𝐽𝜎𝛼𝛼˙
𝜇 𝑃𝜇
𝑄𝛼
𝐼 = 1
2𝜋𝑖 ∮ 𝑑𝑧𝑒−𝜙/2𝑆𝛼Σ𝐼 , 𝑄‾𝛼˙
𝐼 = 1
2𝜋𝑖 ∮ 𝑑𝑧𝑒−𝜙/2𝐶𝛼˙ Σ‾𝐼
: 𝑒𝑞1𝜙(𝑧): : 𝑒𝑞2𝜙(𝑤): = (𝑧 − 𝑤)−𝑞1𝑞2 : 𝑒(𝑞1+𝑞2)𝜙(𝑤): + ⋯ ,
𝑆𝛼(𝑧)𝐶𝛼˙ (𝑤) = 𝜎𝛼𝛼˙
𝜇 𝜓𝜇(𝑤) + 𝒪(𝑧 − 𝑤),
𝑆𝛼(𝑧)𝑆𝛽(𝑤) = 𝜖𝛼𝛽
√𝑧 − 𝑤 + 𝒪(√𝑧 − 𝑤),
𝐶𝛼˙ (𝑧)𝐶𝛽˙ (𝑤) = 𝜖𝛼˙ 𝛽˙
√𝑧 − 𝑤 + 𝒪(√𝑧 − 𝑤),
Σ𝐼(𝑧)Σ‾ 𝐽(𝑤) = 𝛿𝐼𝐽
(𝑧 − 𝑤)3/4 + (𝑧 − 𝑤)1/4𝐽𝐼𝐽(𝑤) + ⋯
Σ𝐼(𝑧)Σ𝐽(𝑤) = (𝑧 − 𝑤)−1/4Ψ𝐼𝐽(𝑤) + ⋯
Σ‾𝐼(𝑧)Σ‾ 𝐽(𝑤) = (𝑧 − 𝑤)−1/4Ψ‾ 𝐼𝐽(𝑤) + ⋯
𝐺int (𝑧)Σ𝐼(𝑤) ∼ (𝑧 − 𝑤)−1/2 , 𝐺int (𝑧)Σ‾𝐼(𝑤) ∼ (𝑧 − 𝑤)−1/2
𝜆𝐼(𝑧) = Σ𝐼(𝑧)𝑒𝑖𝑋/2 , 𝜆‾(𝑧) = Σ‾ 𝐼(𝑧)𝑒−𝑖𝑋/2
𝜆𝐼(𝑧)𝜆‾𝐽(𝑤) = 𝛿𝐼𝐽
𝑧 − 𝑤 + 𝐽ˆ𝐼𝐽 + 𝒪(𝑧 − 𝑤),
𝜆𝐼(𝑧)𝜆𝐽(𝑤) = 𝑒𝑖𝑋Ψ𝐼𝐽 + 𝒪(𝑧 − 𝑤),
𝜆‾𝐼(𝑧)𝜆‾𝐽(𝑤) = 𝑒−𝑖𝑋Ψ‾ 𝐼𝐽 + 𝒪(𝑧 − 𝑤),
𝐽ˆ𝐼𝐼(𝑧)𝐽ˆ𝐽𝐽(𝑤) = 𝛿𝐼𝐽
(𝑧 − 𝑤)2 + regular
𝐽𝐼𝐼(𝑧)𝐽𝐽𝐽(𝑤) = 𝛿𝐼𝐽 − 3/4
(𝑧 − 𝑤)2 + regular
𝐽 = 2𝐽11 , 𝐽(𝑧)𝐽(𝑤) = 3
(𝑧 − 𝑤)2 + regular
pág. 47
⟨𝐽(𝑧1)Σ(𝑧2)Σ‾(𝑧3)⟩ = 3
2
𝑧23
1/4
𝑧12𝑧13
𝐽 = 𝑖√3𝜕Φ , Σ = 𝑒𝑖√3Φ/2𝑊+ , Σ‾ = 𝑒−𝑖√3Φ/2𝑊−
𝐺𝑖𝑛𝑡 = ∑
𝑞≥0
𝑒𝑖𝑞Φ𝑇(𝑞) + 𝑒−𝑞Φ𝑇(−𝑞)
𝐽(𝑧)𝐺±(𝑤) = ± 𝐺±(𝑤)
(𝑧 − 𝑤) + ⋯
|ℎ, 𝑞; ℎ‾⟩ ∶ |0,0; 0⟩, |0,0; 1⟩𝐼, |1/2, ±1; 1⟩𝑖
𝐽𝑠(𝑧)𝐽𝑠(𝑤) = 1
(𝑧 − 𝑤)2 + ⋯
𝐽3(𝑧)𝐽3(𝑤) = 1/2
(𝑧 − 𝑤)2 + ⋯
𝐽𝑠(𝑧)𝐽3(𝑤) = ⋯
𝐽𝑠 = 𝑖𝜕𝜙 , 𝐽3 = 𝑖
√2 𝜕𝜒
Σ1 = exp [𝑖
2 𝜙 + 𝑖
√2 𝜒] , Σ2 = exp [𝑖
2 𝜙 − 𝑖
√2 𝜒]
Σ‾1 = exp [− 𝑖
2 𝜙 − 𝑖
√2 𝜒] , Σ‾2 = exp [− 𝑖
2 𝜙 + 𝑖
√2 𝜒] .
𝐺int = 𝐺(2) + 𝐺(4), 𝐺(2) = 𝐺(2)
+ + 𝐺(2)
−
𝐺(4) = 𝐺(4)
+ + 𝐺(4)
− ,
𝐽𝑠(𝑧)𝐺(2)
± (𝑤) = ± 𝐺(2)
± (𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽3(𝑧)𝐺(4)
± (𝑤) = ± 1
2
𝐺(4)
± (𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽𝑠(𝑧)𝐺(4)
± (𝑤) = finite , 𝐽3(𝑧)𝐺(2)
± (𝑤) = finite
𝐺(2)
± = 𝑒±𝑖𝜙𝑍±.
𝜓0 = 1
√2 (𝜓3 + 𝑖𝜓4), 𝜓1 = 1
√2 (𝜓5 + 𝑖𝜓6)
𝜓2 = 1
√2 (𝜓7 + 𝑖𝜓8), 𝜓3 = 1
√2 (𝜓9 + 𝑖𝜓10)
⟨𝜓𝐼(𝑧)𝜓‾ 𝐽(𝑤)⟩ = 𝛿𝐼𝐽
𝑧 − 𝑤 , ⟨𝜓𝐼(𝑧)𝜓𝐽(𝑤)⟩ = ⟨𝜓‾𝐼(𝑧)𝜓‾ 𝐽(𝑤)⟩ = 0
𝐽𝐼(𝑧) = 𝑖𝜕𝑧𝜙𝐼(𝑧) , ⟨𝜙𝐼(𝑧)𝜙𝐽(𝑤)⟩ = −𝛿𝐼𝐽log (𝑧 − 𝑤)
𝜓𝐼 =: 𝑒𝑖𝜙𝐼
: , 𝜓‾𝐼 =: 𝑒−𝑖𝜙𝐼
:
⟨𝐽(𝑧1)Σ(𝑧2)Σ‾(𝑧3)⟩ = 3
2
𝑧23
1/4
𝑧12𝑧13
𝐽 = 𝑖√3𝜕Φ , Σ = 𝑒𝑖√3Φ/2𝑊+ , Σ‾ = 𝑒−𝑖√3Φ/2𝑊−
𝐺𝑖𝑛𝑡 = ∑
𝑞≥0
𝑒𝑖𝑞Φ𝑇(𝑞) + 𝑒−𝑞Φ𝑇(−𝑞)
𝐽(𝑧)𝐺±(𝑤) = ± 𝐺±(𝑤)
(𝑧 − 𝑤) + ⋯
|ℎ, 𝑞; ℎ‾⟩ ∶ |0,0; 0⟩, |0,0; 1⟩𝐼, |1/2, ±1; 1⟩𝑖
𝐽𝑠(𝑧)𝐽𝑠(𝑤) = 1
(𝑧 − 𝑤)2 + ⋯
𝐽3(𝑧)𝐽3(𝑤) = 1/2
(𝑧 − 𝑤)2 + ⋯
𝐽𝑠(𝑧)𝐽3(𝑤) = ⋯
𝐽𝑠 = 𝑖𝜕𝜙 , 𝐽3 = 𝑖
√2 𝜕𝜒
Σ1 = exp [𝑖
2 𝜙 + 𝑖
√2 𝜒] , Σ2 = exp [𝑖
2 𝜙 − 𝑖
√2 𝜒]
Σ‾1 = exp [− 𝑖
2 𝜙 − 𝑖
√2 𝜒] , Σ‾2 = exp [− 𝑖
2 𝜙 + 𝑖
√2 𝜒] .
𝐺int = 𝐺(2) + 𝐺(4), 𝐺(2) = 𝐺(2)
+ + 𝐺(2)
−
𝐺(4) = 𝐺(4)
+ + 𝐺(4)
− ,
𝐽𝑠(𝑧)𝐺(2)
± (𝑤) = ± 𝐺(2)
± (𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽3(𝑧)𝐺(4)
± (𝑤) = ± 1
2
𝐺(4)
± (𝑤)
𝑧 − 𝑤 + ⋯ ,
𝐽𝑠(𝑧)𝐺(4)
± (𝑤) = finite , 𝐽3(𝑧)𝐺(2)
± (𝑤) = finite
𝐺(2)
± = 𝑒±𝑖𝜙𝑍±.
𝜓0 = 1
√2 (𝜓3 + 𝑖𝜓4), 𝜓1 = 1
√2 (𝜓5 + 𝑖𝜓6)
𝜓2 = 1
√2 (𝜓7 + 𝑖𝜓8), 𝜓3 = 1
√2 (𝜓9 + 𝑖𝜓10)
⟨𝜓𝐼(𝑧)𝜓‾ 𝐽(𝑤)⟩ = 𝛿𝐼𝐽
𝑧 − 𝑤 , ⟨𝜓𝐼(𝑧)𝜓𝐽(𝑤)⟩ = ⟨𝜓‾𝐼(𝑧)𝜓‾ 𝐽(𝑤)⟩ = 0
𝐽𝐼(𝑧) = 𝑖𝜕𝑧𝜙𝐼(𝑧) , ⟨𝜙𝐼(𝑧)𝜙𝐽(𝑤)⟩ = −𝛿𝐼𝐽log (𝑧 − 𝑤)
𝜓𝐼 =: 𝑒𝑖𝜙𝐼
: , 𝜓‾𝐼 =: 𝑒−𝑖𝜙𝐼
:
pág. 48
𝑉(𝜖𝐼) =: exp [𝑖
2 ∑
3
𝐼=0
𝜖𝐼𝜙𝐼] :
𝜓𝐼 → 𝑒2𝜋𝑖𝜃𝐼
𝜓𝐼 , 𝜓‾𝐼 → 𝑒−2𝜋𝑖𝜃𝐼
𝜓‾𝐼
𝜙𝐼 → 𝜙𝐼 + 2𝜋𝜃𝐼
𝑉±,𝜖 = 𝜕‾𝑋±𝑉𝑆(𝜖)𝑒𝑖𝑝⋅𝑋 , 𝑋± = 1
√2 (𝑋3 ± 𝑖𝑋4)
𝐺int = ∑
10
𝑖=5
𝜓𝑖𝜕𝑋𝑖
𝜙2 → 𝜙2 + 𝜋 , 𝜙3 → 𝜙3 − 𝜋
[𝟏𝟐𝟎] → [𝟑, 𝟏, 𝟏] ⊕ [𝟏, 𝟑, 𝟏] ⊕ [𝟏, 𝟏, 𝟔𝟔] ⊕ [𝟐, 𝟏, 𝟏𝟐] ⊕ [𝟏, 𝟐, 𝟏𝟐]
∈ SU(2) × SU(2) × O(12)[𝟏𝟐𝟖] → [𝟐, 𝟏, 𝟑𝟐] ⊕ [𝟏, 𝟐, 𝟑𝟐]
∈ SU(2) × SU(2) × O(12)
E8 ∋ [𝟐𝟒𝟖] → [𝟏, 𝟏𝟑𝟑] ⊕ [𝟑, 𝟏] ⊕ [𝟐, 𝟓𝟔] ∈ SU(2) × E7
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4
𝑍(4,4) [0
0] = Γ4,4
𝜂4𝜂‾4 , 𝑍(4,4) [ℎ
𝑔] = 24 𝜂2𝜂‾2
𝜗2 [1 − ℎ
1 − 𝑔] 𝜗‾2 [1 − ℎ
1 − 𝑔]
, (ℎ, 𝑔) ≠ (0,0)
1
2 ∑
1
𝛾,𝛿=0
𝜗‾ [𝛾 + ℎ
𝛿 + 𝑔] 𝜗‾ [𝛾 − ℎ
𝛿 − 𝑔] 𝜗‾6 [𝛾
𝛿]
𝜂‾8
𝑍𝑁=2
heterotic = 1
2 ∑
1
ℎ,𝑔=0
Γ2,2Γ‾𝐸8 𝑍(4,4) [ℎ
𝑔]
𝜏2𝜂4𝜂‾12
1
2 ∑
1
𝛾,𝛿=0
𝜗‾ [𝛾 + ℎ
𝛿 + 𝑔] 𝜗‾ [𝛾 − ℎ
𝛿 − 𝑔] 𝜗‾6 [𝛾
𝛿]
𝜂‾8
× 1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4
𝐽𝑎 = − 𝑖
2 [𝜓0𝜓𝑎 + 1
2 𝜖𝑎𝑏𝑐𝜓𝑏𝜓𝑐] , 𝐽˜𝑎 = − 𝑖
2 [𝜓0𝜓𝑎 − 1
2 𝜖𝑎𝑏𝑐𝜓𝑏𝜓𝑐] .
𝑉𝛼𝛽
± = ±𝑖(𝛿𝛼𝛽𝜓0 ± 𝑖𝜎𝛼𝛽
𝑎 𝜓𝑎)
𝑉(𝜖𝐼) =: exp [𝑖
2 ∑
3
𝐼=0
𝜖𝐼𝜙𝐼] :
𝜓𝐼 → 𝑒2𝜋𝑖𝜃𝐼
𝜓𝐼 , 𝜓‾𝐼 → 𝑒−2𝜋𝑖𝜃𝐼
𝜓‾𝐼
𝜙𝐼 → 𝜙𝐼 + 2𝜋𝜃𝐼
𝑉±,𝜖 = 𝜕‾𝑋±𝑉𝑆(𝜖)𝑒𝑖𝑝⋅𝑋 , 𝑋± = 1
√2 (𝑋3 ± 𝑖𝑋4)
𝐺int = ∑
10
𝑖=5
𝜓𝑖𝜕𝑋𝑖
𝜙2 → 𝜙2 + 𝜋 , 𝜙3 → 𝜙3 − 𝜋
[𝟏𝟐𝟎] → [𝟑, 𝟏, 𝟏] ⊕ [𝟏, 𝟑, 𝟏] ⊕ [𝟏, 𝟏, 𝟔𝟔] ⊕ [𝟐, 𝟏, 𝟏𝟐] ⊕ [𝟏, 𝟐, 𝟏𝟐]
∈ SU(2) × SU(2) × O(12)[𝟏𝟐𝟖] → [𝟐, 𝟏, 𝟑𝟐] ⊕ [𝟏, 𝟐, 𝟑𝟐]
∈ SU(2) × SU(2) × O(12)
E8 ∋ [𝟐𝟒𝟖] → [𝟏, 𝟏𝟑𝟑] ⊕ [𝟑, 𝟏] ⊕ [𝟐, 𝟓𝟔] ∈ SU(2) × E7
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4
𝑍(4,4) [0
0] = Γ4,4
𝜂4𝜂‾4 , 𝑍(4,4) [ℎ
𝑔] = 24 𝜂2𝜂‾2
𝜗2 [1 − ℎ
1 − 𝑔] 𝜗‾2 [1 − ℎ
1 − 𝑔]
, (ℎ, 𝑔) ≠ (0,0)
1
2 ∑
1
𝛾,𝛿=0
𝜗‾ [𝛾 + ℎ
𝛿 + 𝑔] 𝜗‾ [𝛾 − ℎ
𝛿 − 𝑔] 𝜗‾6 [𝛾
𝛿]
𝜂‾8
𝑍𝑁=2
heterotic = 1
2 ∑
1
ℎ,𝑔=0
Γ2,2Γ‾𝐸8 𝑍(4,4) [ℎ
𝑔]
𝜏2𝜂4𝜂‾12
1
2 ∑
1
𝛾,𝛿=0
𝜗‾ [𝛾 + ℎ
𝛿 + 𝑔] 𝜗‾ [𝛾 − ℎ
𝛿 − 𝑔] 𝜗‾6 [𝛾
𝛿]
𝜂‾8
× 1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4
𝐽𝑎 = − 𝑖
2 [𝜓0𝜓𝑎 + 1
2 𝜖𝑎𝑏𝑐𝜓𝑏𝜓𝑐] , 𝐽˜𝑎 = − 𝑖
2 [𝜓0𝜓𝑎 − 1
2 𝜖𝑎𝑏𝑐𝜓𝑏𝜓𝑐] .
𝑉𝛼𝛽
± = ±𝑖(𝛿𝛼𝛽𝜓0 ± 𝑖𝜎𝛼𝛽
𝑎 𝜓𝑎)
pág. 49
𝑉𝛼𝛾
+ (𝑧)𝑉𝛾𝛽
+ (𝑤) = 𝑉𝛼𝛾
− (𝑧)𝑉𝛾𝛽
− (𝑤) = 𝛿𝛼𝛽
𝑧 − 𝑤 − 2𝜎𝛼𝛽
𝑎 (𝐽𝑎(𝑤) − 𝐽˜𝑎(𝑤)) + 𝒪(𝑧 − 𝑤)
𝑉𝛼𝛾
+ (𝑧)𝑉𝛾𝛽
− (𝑤) = 3𝛿𝛼𝛽
𝑧 − 𝑤 + 4𝜎𝛼𝛽
𝑎 𝐽˜𝑎(𝑤) + 𝒪(𝑧 − 𝑤)
𝑉𝛼𝛾
− (𝑧)𝑉𝛾𝛽
+ (𝑤) = 3𝛿𝛼𝛽
𝑧 − 𝑤 − 4𝜎𝛼𝛽
𝑎 𝐽𝑎(𝑤) + 𝒪(𝑧 − 𝑤)
𝑍𝑁=2
heterotic = 1
2 ∑
1
ℎ,𝑔=0
Γ2,18(𝜖) [ℎ
𝑔] 𝑍(4,4) [ℎ
𝑔]
𝜏2𝜂4𝜂‾20
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4
𝐺 =
𝑇2 − 𝑊2
𝐼𝑊2
𝐼
2𝑈2
𝑈2
( 1 𝑈1
𝑈1 |𝑈|2)
𝐵 = (𝑇1 − 𝑊1
𝐼𝑊2
𝐼
2𝑈2
) ( 0 1
−1 0)
• Kähler:
𝑓 = 𝑆 (𝑇𝑈 − 1
2 𝑊𝐼𝑊𝐼) , 𝐾 = −log (𝑆2) − log [𝑈2𝑇2 − 1
2 𝑊2
𝐼𝑊2
𝐼]
𝜏2𝐵2 = 𝜏2⟨𝜆2⟩ = Γ2,18 [0
1] 𝜗‾3
2𝜗‾4
2
𝜂‾24 − Γ2,18 [1
0] 𝜗‾2
2𝜗‾3
2
𝜂‾24 − Γ2,18 [1
1] 𝜗‾2
2𝜗‾4
2
𝜂‾24
=
Γ2,18 [0
0] + Γ2,18 [0
1]
2 𝐹‾1 −
Γ2,18 [0
0] − Γ2,18 [0
1]
2 𝐹‾1 +
Γ2,18 [1
0] + Γ2,18 [1
0]
2 𝐹‾+
−
Γ2,18 [1
0] − Γ2,18 [1
0]
2 𝐹‾−
𝐹‾1 = 𝜗‾3
2𝜗‾4
2
𝜂‾24 , 𝐹‾± = 𝜗‾2
2(𝜗‾3
2 ± 𝜗‾4
2)
𝜂‾24
𝜏 → 𝜏 + 1 ∶ 𝐵2 → 𝐵2 , 𝜏 → − 1
𝜏 ∶ 𝐵2 → 𝜏2𝐵2
𝐹1 = 1
𝑞 + ∑
∞
𝑛=0
𝑑1(𝑛)𝑞𝑛 = 1
𝑞 + 16 + 156𝑞 + 𝒪(𝑞2)
𝐹+ = 8
𝑞3/4 + 𝑞1/4 ∑
∞
𝑛=0
𝑑+(𝑛)𝑞𝑛 = 8
𝑞3/4 + 8𝑞1/4(30 + 481𝑞 + 𝒪(𝑞2))
𝐹− = 32
𝑞1/4 + 𝑞3/4 ∑
∞
𝑛=0
𝑑−(𝑛)𝑞𝑛 = 32
𝑞1/4 + 32𝑞3/4(26 + 375𝑞 + 𝒪(𝑞2))
𝑀2 = |−𝑚1𝑈 + 𝑚2 + 𝑇𝑛1 + (𝑇𝑈 − 1
2 𝑊⃗⃗⃗ 2) 𝑛2 + 𝑊⃗⃗⃗ ⋅ 𝑄⃗ |
2
4𝑆2 (𝑇2𝑈2 − 1
2 Im𝑊⃗⃗⃗ 2)
𝑉𝛼𝛾
+ (𝑧)𝑉𝛾𝛽
+ (𝑤) = 𝑉𝛼𝛾
− (𝑧)𝑉𝛾𝛽
− (𝑤) = 𝛿𝛼𝛽
𝑧 − 𝑤 − 2𝜎𝛼𝛽
𝑎 (𝐽𝑎(𝑤) − 𝐽˜𝑎(𝑤)) + 𝒪(𝑧 − 𝑤)
𝑉𝛼𝛾
+ (𝑧)𝑉𝛾𝛽
− (𝑤) = 3𝛿𝛼𝛽
𝑧 − 𝑤 + 4𝜎𝛼𝛽
𝑎 𝐽˜𝑎(𝑤) + 𝒪(𝑧 − 𝑤)
𝑉𝛼𝛾
− (𝑧)𝑉𝛾𝛽
+ (𝑤) = 3𝛿𝛼𝛽
𝑧 − 𝑤 − 4𝜎𝛼𝛽
𝑎 𝐽𝑎(𝑤) + 𝒪(𝑧 − 𝑤)
𝑍𝑁=2
heterotic = 1
2 ∑
1
ℎ,𝑔=0
Γ2,18(𝜖) [ℎ
𝑔] 𝑍(4,4) [ℎ
𝑔]
𝜏2𝜂4𝜂‾20
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4
𝐺 =
𝑇2 − 𝑊2
𝐼𝑊2
𝐼
2𝑈2
𝑈2
( 1 𝑈1
𝑈1 |𝑈|2)
𝐵 = (𝑇1 − 𝑊1
𝐼𝑊2
𝐼
2𝑈2
) ( 0 1
−1 0)
• Kähler:
𝑓 = 𝑆 (𝑇𝑈 − 1
2 𝑊𝐼𝑊𝐼) , 𝐾 = −log (𝑆2) − log [𝑈2𝑇2 − 1
2 𝑊2
𝐼𝑊2
𝐼]
𝜏2𝐵2 = 𝜏2⟨𝜆2⟩ = Γ2,18 [0
1] 𝜗‾3
2𝜗‾4
2
𝜂‾24 − Γ2,18 [1
0] 𝜗‾2
2𝜗‾3
2
𝜂‾24 − Γ2,18 [1
1] 𝜗‾2
2𝜗‾4
2
𝜂‾24
=
Γ2,18 [0
0] + Γ2,18 [0
1]
2 𝐹‾1 −
Γ2,18 [0
0] − Γ2,18 [0
1]
2 𝐹‾1 +
Γ2,18 [1
0] + Γ2,18 [1
0]
2 𝐹‾+
−
Γ2,18 [1
0] − Γ2,18 [1
0]
2 𝐹‾−
𝐹‾1 = 𝜗‾3
2𝜗‾4
2
𝜂‾24 , 𝐹‾± = 𝜗‾2
2(𝜗‾3
2 ± 𝜗‾4
2)
𝜂‾24
𝜏 → 𝜏 + 1 ∶ 𝐵2 → 𝐵2 , 𝜏 → − 1
𝜏 ∶ 𝐵2 → 𝜏2𝐵2
𝐹1 = 1
𝑞 + ∑
∞
𝑛=0
𝑑1(𝑛)𝑞𝑛 = 1
𝑞 + 16 + 156𝑞 + 𝒪(𝑞2)
𝐹+ = 8
𝑞3/4 + 𝑞1/4 ∑
∞
𝑛=0
𝑑+(𝑛)𝑞𝑛 = 8
𝑞3/4 + 8𝑞1/4(30 + 481𝑞 + 𝒪(𝑞2))
𝐹− = 32
𝑞1/4 + 𝑞3/4 ∑
∞
𝑛=0
𝑑−(𝑛)𝑞𝑛 = 32
𝑞1/4 + 32𝑞3/4(26 + 375𝑞 + 𝒪(𝑞2))
𝑀2 = |−𝑚1𝑈 + 𝑚2 + 𝑇𝑛1 + (𝑇𝑈 − 1
2 𝑊⃗⃗⃗ 2) 𝑛2 + 𝑊⃗⃗⃗ ⋅ 𝑄⃗ |
2
4𝑆2 (𝑇2𝑈2 − 1
2 Im𝑊⃗⃗⃗ 2)
pág. 50
𝜌 = 𝑚⃗⃗ ⋅ 𝜖𝑅 + 𝑛⃗ ⋅ 𝜖𝐿 − 𝑄⃗ ⋅ 𝜁
𝑠 = 𝑚⃗⃗ ⋅ 𝑛⃗ − 1
2 𝑄⃗ ⋅ 𝑄⃗
𝑀2 = | (𝑚1 + 1
2 𝜖𝐿
1) 𝑈 − (𝑚2 + 1
2 𝜖𝐿
2) − 𝑇 (𝑛1 + 1
2 𝜖𝑅
1 ) − (𝑇𝑈 − 1
2 𝑊⃗⃗⃗ 2) (𝑛2 + 1
2 𝜖𝑅
2)
+
− 𝑊⃗⃗⃗ ⋅ (𝑄⃗ + 1
2 𝜁 )|
2
/4𝑆2 (𝑇2𝑈2 − 1
2 Im𝑊⃗⃗⃗ 2)(12.4.36)
𝑠′ = (𝑚⃗⃗ + 𝜖𝐿
2 ) ⋅ (𝑛⃗ + 𝜖𝑅
2 ) − 1
2 (𝑄⃗ + 𝜁
2) ⋅ (𝑄⃗ + 𝜁
2) ,
𝑚𝐿
2 = 1
4 (1
𝑅 + 𝑛𝑅)
2
, 𝑚𝑅
2 = 1
4 (1
𝑅 − 𝑛𝑅)
2
𝑍𝑍2×𝑍2
𝑁=1 = 1
𝜏2𝜂2𝜂‾2
1
4 ∑
1
ℎ1,𝑔1=0,ℎ2,𝑔2=0
1
2 ∑
1
𝛼,𝛽=0
(−)𝛼+𝛽+𝛼𝛽
×
𝜗 [𝛼
𝛽]
𝜂
𝜗 [𝛼 + ℎ1
𝛽 + 𝑔1
]
𝜂
𝜗 [𝛼 + ℎ2
𝛽 + 𝑔2
]
𝜂
𝜗 [𝛼 − ℎ1 − ℎ2
𝛽 − 𝑔1 − 𝑔2
]
𝜂
Γ‾8
𝜂‾8 𝑍2,2
1 [ℎ1
𝑔1
] × 𝑍2,2
2 [ℎ2
𝑔2
] 𝑍2,2
3 [ℎ1 + ℎ2
𝑔1 + 𝑔2
]
× 1
2 ∑
1
𝛼‾ ,𝛽‾ =0
𝜗‾ [𝛼‾
𝛽‾]
5
𝜂‾5
𝜗‾ [𝛼‾ + ℎ1
𝛽‾ + 𝑔1
]
𝜂‾
𝜗‾ [𝛼‾ + ℎ2
𝛽‾ + 𝑔2
]
𝜂‾
𝜗‾ [𝛼‾ − ℎ1 − ℎ2
𝛽‾ − 𝑔1 − 𝑔2
]
𝜂‾
𝑍6−𝑑
𝐼𝐼−𝜆 = 1
2 ∑
1
ℎ,𝑔=0
𝑍(4,4) [ℎ
𝑔]
𝜏2
2𝜂4𝜂‾4 × 1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4 ×
1
2 ∑
1
𝑎‾,𝑏‾ =0
(−1)𝑎‾+𝑏‾+𝜆𝑎‾𝑏‾ 𝜗‾2 [𝑎‾
𝑏‾] 𝜗‾ [𝑎‾ + ℎ
𝑏‾ + 𝑔] 𝜗‾ [𝑎‾ − ℎ
𝑏‾ − 𝑔]
𝜂‾4
𝜒 = 1
𝑔 [𝜒(𝑀) − 𝜒(𝐹)] + 𝜒(𝑁).
1
𝑔𝑖
2|
tree
= 𝑘𝑖
𝑔spacetime dimensions
2 = 𝑘𝑖𝑆2
𝑉𝐺
𝜇,𝑎 ∼ (𝜕𝑋𝜇 + 𝑖(𝑝 ⋅ 𝜓)𝜓𝜇)𝐽‾𝑎𝑒𝑖𝑝⋅𝑋
𝜌 = 𝑚⃗⃗ ⋅ 𝜖𝑅 + 𝑛⃗ ⋅ 𝜖𝐿 − 𝑄⃗ ⋅ 𝜁
𝑠 = 𝑚⃗⃗ ⋅ 𝑛⃗ − 1
2 𝑄⃗ ⋅ 𝑄⃗
𝑀2 = | (𝑚1 + 1
2 𝜖𝐿
1) 𝑈 − (𝑚2 + 1
2 𝜖𝐿
2) − 𝑇 (𝑛1 + 1
2 𝜖𝑅
1 ) − (𝑇𝑈 − 1
2 𝑊⃗⃗⃗ 2) (𝑛2 + 1
2 𝜖𝑅
2)
+
− 𝑊⃗⃗⃗ ⋅ (𝑄⃗ + 1
2 𝜁 )|
2
/4𝑆2 (𝑇2𝑈2 − 1
2 Im𝑊⃗⃗⃗ 2)(12.4.36)
𝑠′ = (𝑚⃗⃗ + 𝜖𝐿
2 ) ⋅ (𝑛⃗ + 𝜖𝑅
2 ) − 1
2 (𝑄⃗ + 𝜁
2) ⋅ (𝑄⃗ + 𝜁
2) ,
𝑚𝐿
2 = 1
4 (1
𝑅 + 𝑛𝑅)
2
, 𝑚𝑅
2 = 1
4 (1
𝑅 − 𝑛𝑅)
2
𝑍𝑍2×𝑍2
𝑁=1 = 1
𝜏2𝜂2𝜂‾2
1
4 ∑
1
ℎ1,𝑔1=0,ℎ2,𝑔2=0
1
2 ∑
1
𝛼,𝛽=0
(−)𝛼+𝛽+𝛼𝛽
×
𝜗 [𝛼
𝛽]
𝜂
𝜗 [𝛼 + ℎ1
𝛽 + 𝑔1
]
𝜂
𝜗 [𝛼 + ℎ2
𝛽 + 𝑔2
]
𝜂
𝜗 [𝛼 − ℎ1 − ℎ2
𝛽 − 𝑔1 − 𝑔2
]
𝜂
Γ‾8
𝜂‾8 𝑍2,2
1 [ℎ1
𝑔1
] × 𝑍2,2
2 [ℎ2
𝑔2
] 𝑍2,2
3 [ℎ1 + ℎ2
𝑔1 + 𝑔2
]
× 1
2 ∑
1
𝛼‾ ,𝛽‾ =0
𝜗‾ [𝛼‾
𝛽‾]
5
𝜂‾5
𝜗‾ [𝛼‾ + ℎ1
𝛽‾ + 𝑔1
]
𝜂‾
𝜗‾ [𝛼‾ + ℎ2
𝛽‾ + 𝑔2
]
𝜂‾
𝜗‾ [𝛼‾ − ℎ1 − ℎ2
𝛽‾ − 𝑔1 − 𝑔2
]
𝜂‾
𝑍6−𝑑
𝐼𝐼−𝜆 = 1
2 ∑
1
ℎ,𝑔=0
𝑍(4,4) [ℎ
𝑔]
𝜏2
2𝜂4𝜂‾4 × 1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4 ×
1
2 ∑
1
𝑎‾,𝑏‾ =0
(−1)𝑎‾+𝑏‾+𝜆𝑎‾𝑏‾ 𝜗‾2 [𝑎‾
𝑏‾] 𝜗‾ [𝑎‾ + ℎ
𝑏‾ + 𝑔] 𝜗‾ [𝑎‾ − ℎ
𝑏‾ − 𝑔]
𝜂‾4
𝜒 = 1
𝑔 [𝜒(𝑀) − 𝜒(𝐹)] + 𝜒(𝑁).
1
𝑔𝑖
2|
tree
= 𝑘𝑖
𝑔spacetime dimensions
2 = 𝑘𝑖𝑆2
𝑉𝐺
𝜇,𝑎 ∼ (𝜕𝑋𝜇 + 𝑖(𝑝 ⋅ 𝜓)𝜓𝜇)𝐽‾𝑎𝑒𝑖𝑝⋅𝑋
pág. 51
∫ 𝑑4𝑥 1
𝑔2(𝑇𝑖) 𝐹𝜇𝜈
𝑎 𝐹𝑎,𝜇𝜈
𝑉modulus
𝐼𝐽 = (𝜕𝑋𝐼 + 𝑖(𝑝 ⋅ 𝜓)𝜓𝐼)𝜕‾𝑋𝐽𝑒𝑖𝑝⋅𝑋
𝐼1− loop = ∫ ⟨𝑉𝑎,𝜇(𝑝1, 𝑧)𝑉𝑏,𝜈(𝑝2, 𝑤)𝑉modulus
𝐼𝐽 (𝑝3, 0)⟩ ∼ 𝛿𝑎𝑏(𝑝1 ⋅ 𝑝2𝜂𝜇𝜈 − 𝑝1
𝜇𝑝2
𝜈)𝐹𝐼𝐽(𝑇, 𝑈) + 𝒪(𝑝4)
𝐹𝐼𝐽 = ∫
ℱ
𝑑2𝜏
𝜏2
2 ∫ 𝑑2𝑧
𝜏2
∫ 𝑑2𝑤⟨𝜓(𝑧)𝜓(𝑤)⟩2⟨𝐽‾𝑎(𝑧‾)𝐽‾𝑏(𝑤‾ )⟩⟨𝜕𝑋𝐼(0)𝜕‾𝑋𝐽(0)⟩
𝑆 [𝑎
𝑏] (𝑧) = ⟨𝜓(𝑧)𝜓(0)⟩|𝑏
𝑎 = 𝜗 [𝑎
𝑏] (𝑧)𝜗1
′ (0)
𝜗1(𝑧)𝜗 𝑏
𝑎](0) = 1
𝑧 + ⋯
𝑆2 [𝑎
𝑏] (𝑧) = 𝒫(𝑧) + 4𝜋𝑖𝜕𝜏log 𝜃 [𝑎
𝑏] (𝜏)
𝜂(𝜏)
⟨⟨𝜓(𝑧)𝜓(0)⟩⟩ = 1
2 ∑
(𝑎,𝑏)≠(1,1)
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4 𝑆2 [𝑎
𝑏] (𝑧)
= 4𝜋2𝜂2𝜗 [1 + ℎ
1 + 𝑔] 𝜗 [1 − ℎ
1 − 𝑔]
⟨𝐽‾𝑎(𝑧‾)𝐽‾𝑏(0)⟩ = 𝑘𝛿𝑎𝑏
4𝜋2 𝜕‾𝑧‾
2log 𝜗‾1(𝑧‾) + Tr[𝐽0
𝑎𝐽0
𝑏]= 𝛿𝑎𝑏 ( 𝑘
4𝜋2 𝜕‾𝑧‾
2log 𝜗‾1(𝑧‾) + Tr[𝑄2])
⟨𝜕𝑋𝐼 (0)𝜕‾𝑋𝐽(0)⟩ = √det𝐺
(√𝜏2𝜂𝜂‾)2 ∑
𝑚⃗⃗⃗ ,𝑛⃗
(𝑚𝐼 + 𝑛𝐼𝜏)(𝑚𝐽 + 𝑛𝐽𝜏‾) ×
× exp [− 𝜋(𝐺𝐾𝐿 + 𝐵𝐾𝐿)
𝜏2
(𝑚𝐾 + 𝑛𝐾𝜏)(𝑚𝐿 + 𝑛𝐿𝜏‾)]
𝑉𝑇𝑖 = 𝑣𝐼𝐽(𝑇𝑖)𝜕𝑋𝐼𝜕‾𝑋𝐽
𝑣(𝑇) = − 𝑖
2𝑈2
(1 𝑈
𝑈‾ |𝑈|2) , 𝑣(𝑈) = 𝑖𝑇2
𝑈2
2 (1 𝑈‾
𝑈‾ 𝑈‾ 2)
𝑣(𝑇‾) = 𝑣(𝑇), 𝑣(𝑈‾ ) = 𝑣(𝑈)
⟨𝑉𝑇𝑖 ⟩ = − 𝜏2
2𝜋 𝜕𝑇𝑖
Γ2,2
𝜂2𝜂‾2
𝜕
𝜕𝑇𝑖
16𝜋2
𝑔𝑖
2 |
1−loop
∼ 𝜕
𝜕𝑇𝑖
∫
ℱ
𝑑2𝜏
𝜏2
2
𝜏2Γ2,2
𝜂‾4 Tr𝑅
𝑖𝑛𝑡 [(−1)𝐹 (𝑄𝑖
2 − 𝑘𝑖
4𝜋𝜏2
)] + constant
⟨𝑋(𝑧, 𝑧‾)𝑋(0)⟩ = −log |𝜗1(𝑧)|2 + 2𝜋 Im𝑧2
𝜏2
∫ 𝑑4𝑥 1
𝑔2(𝑇𝑖) 𝐹𝜇𝜈
𝑎 𝐹𝑎,𝜇𝜈
𝑉modulus
𝐼𝐽 = (𝜕𝑋𝐼 + 𝑖(𝑝 ⋅ 𝜓)𝜓𝐼)𝜕‾𝑋𝐽𝑒𝑖𝑝⋅𝑋
𝐼1− loop = ∫ ⟨𝑉𝑎,𝜇(𝑝1, 𝑧)𝑉𝑏,𝜈(𝑝2, 𝑤)𝑉modulus
𝐼𝐽 (𝑝3, 0)⟩ ∼ 𝛿𝑎𝑏(𝑝1 ⋅ 𝑝2𝜂𝜇𝜈 − 𝑝1
𝜇𝑝2
𝜈)𝐹𝐼𝐽(𝑇, 𝑈) + 𝒪(𝑝4)
𝐹𝐼𝐽 = ∫
ℱ
𝑑2𝜏
𝜏2
2 ∫ 𝑑2𝑧
𝜏2
∫ 𝑑2𝑤⟨𝜓(𝑧)𝜓(𝑤)⟩2⟨𝐽‾𝑎(𝑧‾)𝐽‾𝑏(𝑤‾ )⟩⟨𝜕𝑋𝐼(0)𝜕‾𝑋𝐽(0)⟩
𝑆 [𝑎
𝑏] (𝑧) = ⟨𝜓(𝑧)𝜓(0)⟩|𝑏
𝑎 = 𝜗 [𝑎
𝑏] (𝑧)𝜗1
′ (0)
𝜗1(𝑧)𝜗 𝑏
𝑎](0) = 1
𝑧 + ⋯
𝑆2 [𝑎
𝑏] (𝑧) = 𝒫(𝑧) + 4𝜋𝑖𝜕𝜏log 𝜃 [𝑎
𝑏] (𝜏)
𝜂(𝜏)
⟨⟨𝜓(𝑧)𝜓(0)⟩⟩ = 1
2 ∑
(𝑎,𝑏)≠(1,1)
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4 𝑆2 [𝑎
𝑏] (𝑧)
= 4𝜋2𝜂2𝜗 [1 + ℎ
1 + 𝑔] 𝜗 [1 − ℎ
1 − 𝑔]
⟨𝐽‾𝑎(𝑧‾)𝐽‾𝑏(0)⟩ = 𝑘𝛿𝑎𝑏
4𝜋2 𝜕‾𝑧‾
2log 𝜗‾1(𝑧‾) + Tr[𝐽0
𝑎𝐽0
𝑏]= 𝛿𝑎𝑏 ( 𝑘
4𝜋2 𝜕‾𝑧‾
2log 𝜗‾1(𝑧‾) + Tr[𝑄2])
⟨𝜕𝑋𝐼 (0)𝜕‾𝑋𝐽(0)⟩ = √det𝐺
(√𝜏2𝜂𝜂‾)2 ∑
𝑚⃗⃗⃗ ,𝑛⃗
(𝑚𝐼 + 𝑛𝐼𝜏)(𝑚𝐽 + 𝑛𝐽𝜏‾) ×
× exp [− 𝜋(𝐺𝐾𝐿 + 𝐵𝐾𝐿)
𝜏2
(𝑚𝐾 + 𝑛𝐾𝜏)(𝑚𝐿 + 𝑛𝐿𝜏‾)]
𝑉𝑇𝑖 = 𝑣𝐼𝐽(𝑇𝑖)𝜕𝑋𝐼𝜕‾𝑋𝐽
𝑣(𝑇) = − 𝑖
2𝑈2
(1 𝑈
𝑈‾ |𝑈|2) , 𝑣(𝑈) = 𝑖𝑇2
𝑈2
2 (1 𝑈‾
𝑈‾ 𝑈‾ 2)
𝑣(𝑇‾) = 𝑣(𝑇), 𝑣(𝑈‾ ) = 𝑣(𝑈)
⟨𝑉𝑇𝑖 ⟩ = − 𝜏2
2𝜋 𝜕𝑇𝑖
Γ2,2
𝜂2𝜂‾2
𝜕
𝜕𝑇𝑖
16𝜋2
𝑔𝑖
2 |
1−loop
∼ 𝜕
𝜕𝑇𝑖
∫
ℱ
𝑑2𝜏
𝜏2
2
𝜏2Γ2,2
𝜂‾4 Tr𝑅
𝑖𝑛𝑡 [(−1)𝐹 (𝑄𝑖
2 − 𝑘𝑖
4𝜋𝜏2
)] + constant
⟨𝑋(𝑧, 𝑧‾)𝑋(0)⟩ = −log |𝜗1(𝑧)|2 + 2𝜋 Im𝑧2
𝜏2
pág. 52
∫ 𝑑2𝑧
𝜏2
(𝑆2[ 𝑏
𝑎](𝑧) − ⟨𝑋𝜕𝑋⟩2) ( 𝑘
4𝜋2 𝜕‾2log 𝜗‾1(𝑧‾) + Tr[𝑄2])
= 4𝜋𝑖𝜕𝜏log 𝜗 [𝑎
𝑏]
𝜂 (Tr[𝑄2] − 𝑘
4𝜋𝜏2
)
𝑍2
𝐼 = 16𝜋2
𝑔𝐼
2 |
1− loop
= 1
4𝜋2 ∫
ℱ
𝑑2𝜏
𝜏2
1
𝜂2𝜂‾2 ∑
even
4𝜋𝑖𝜕𝜏 (𝜗 [𝑎
𝑏]
𝜂 ) Trint [𝑄𝐼
2 − 𝑘𝐼
4𝜋𝜏2
] [𝑎
𝑏]
16𝜋2
𝑔𝐼
2 |
1− loop
IR
= ∫
ℱ
𝑑2𝜏
𝜏2
Str𝑄𝐼
2 ( 1
12 − 𝑠2)
16𝜋2
𝑔𝐼
2 |
1− loop
IR
= 𝑏𝐼log (𝜇2𝛼′) + finite
𝑏𝐼 = Str𝑄𝐼
2 ( 1
12 − 𝑠2)|massless
𝑉grav = 𝜖𝜇𝜈(𝜕𝑋𝜇 + 𝑖𝑝 ⋅ 𝜓𝜓𝜇)𝜕‾𝑋𝜈
∫ 𝑑2𝑧
𝜏2
⟨𝑋𝜕‾𝑧‾
2𝑋⟩ = ∫ 𝑑2𝑧
𝜏2
(𝜕‾𝑧‾
2log 𝜗‾1(𝑧‾) + 𝜋
𝜏2
) = 0.
𝑑𝑠2 = 𝐺𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈 = (𝑑𝑋0)2 + 𝑁
4 (𝑑𝛼2 + 𝑑𝛽2 + 𝑑𝛾2 + 2sin (𝛽/√𝛼′)𝑑𝛼𝑑𝛾)
𝐵𝜇𝜈𝑑𝑋𝜇 ∧ 𝑑𝑋𝜈 = 𝑁
2 cos (𝛽/√𝛼′)𝑑𝛼 ∧ 𝑑𝛾 , Φ = 𝑋0√𝛼′
√𝑁 + 2
𝐿0 = − 1
2𝛼′ + 𝐸2 + 1
4𝛼′(𝑁 + 2) + 𝑗(𝑗 + 1)
𝛼′(𝑁 + 2) + ⋯
𝜇2 = 𝑀spacetime dimensions
2
2(𝑁 + 2) , 𝑀spacetime dimensions = 1
√𝛼′ .
𝑍(𝜇) = Γ(𝜇/𝑀spacetime dimensions )𝑍(0),
Γ(𝜇/𝑀spacetime dimensions) = 4√𝑥 𝜕
𝜕𝑥 [𝜌(𝑥) − 𝜌(𝑥/4)]|𝑥=𝑁+2
𝜌(𝑥) = √𝑥 ∑
𝑚,𝑛∈𝑍
exp [− 𝜋𝑥
𝜏2
|𝑚 + 𝑛𝜏|2]
𝛿𝐼 = ∫ 𝑑2𝑧(𝐴𝜇
𝑎(𝑋)𝜕𝑋𝜇 + 𝐹𝜇𝜈
𝑎 𝜓𝜇𝜓𝜈)𝐽‾𝑎
𝛿𝐼 = ∫ 𝑑2𝑧𝐵𝑎(𝐽3 + 𝑖𝜓1𝜓2)𝐽‾𝑎
∫ 𝑑2𝑧
𝜏2
(𝑆2[ 𝑏
𝑎](𝑧) − ⟨𝑋𝜕𝑋⟩2) ( 𝑘
4𝜋2 𝜕‾2log 𝜗‾1(𝑧‾) + Tr[𝑄2])
= 4𝜋𝑖𝜕𝜏log 𝜗 [𝑎
𝑏]
𝜂 (Tr[𝑄2] − 𝑘
4𝜋𝜏2
)
𝑍2
𝐼 = 16𝜋2
𝑔𝐼
2 |
1− loop
= 1
4𝜋2 ∫
ℱ
𝑑2𝜏
𝜏2
1
𝜂2𝜂‾2 ∑
even
4𝜋𝑖𝜕𝜏 (𝜗 [𝑎
𝑏]
𝜂 ) Trint [𝑄𝐼
2 − 𝑘𝐼
4𝜋𝜏2
] [𝑎
𝑏]
16𝜋2
𝑔𝐼
2 |
1− loop
IR
= ∫
ℱ
𝑑2𝜏
𝜏2
Str𝑄𝐼
2 ( 1
12 − 𝑠2)
16𝜋2
𝑔𝐼
2 |
1− loop
IR
= 𝑏𝐼log (𝜇2𝛼′) + finite
𝑏𝐼 = Str𝑄𝐼
2 ( 1
12 − 𝑠2)|massless
𝑉grav = 𝜖𝜇𝜈(𝜕𝑋𝜇 + 𝑖𝑝 ⋅ 𝜓𝜓𝜇)𝜕‾𝑋𝜈
∫ 𝑑2𝑧
𝜏2
⟨𝑋𝜕‾𝑧‾
2𝑋⟩ = ∫ 𝑑2𝑧
𝜏2
(𝜕‾𝑧‾
2log 𝜗‾1(𝑧‾) + 𝜋
𝜏2
) = 0.
𝑑𝑠2 = 𝐺𝜇𝜈𝑑𝑥𝜇𝑑𝑥𝜈 = (𝑑𝑋0)2 + 𝑁
4 (𝑑𝛼2 + 𝑑𝛽2 + 𝑑𝛾2 + 2sin (𝛽/√𝛼′)𝑑𝛼𝑑𝛾)
𝐵𝜇𝜈𝑑𝑋𝜇 ∧ 𝑑𝑋𝜈 = 𝑁
2 cos (𝛽/√𝛼′)𝑑𝛼 ∧ 𝑑𝛾 , Φ = 𝑋0√𝛼′
√𝑁 + 2
𝐿0 = − 1
2𝛼′ + 𝐸2 + 1
4𝛼′(𝑁 + 2) + 𝑗(𝑗 + 1)
𝛼′(𝑁 + 2) + ⋯
𝜇2 = 𝑀spacetime dimensions
2
2(𝑁 + 2) , 𝑀spacetime dimensions = 1
√𝛼′ .
𝑍(𝜇) = Γ(𝜇/𝑀spacetime dimensions )𝑍(0),
Γ(𝜇/𝑀spacetime dimensions) = 4√𝑥 𝜕
𝜕𝑥 [𝜌(𝑥) − 𝜌(𝑥/4)]|𝑥=𝑁+2
𝜌(𝑥) = √𝑥 ∑
𝑚,𝑛∈𝑍
exp [− 𝜋𝑥
𝜏2
|𝑚 + 𝑛𝜏|2]
𝛿𝐼 = ∫ 𝑑2𝑧(𝐴𝜇
𝑎(𝑋)𝜕𝑋𝜇 + 𝐹𝜇𝜈
𝑎 𝜓𝜇𝜓𝜈)𝐽‾𝑎
𝛿𝐼 = ∫ 𝑑2𝑧𝐵𝑎(𝐽3 + 𝑖𝜓1𝜓2)𝐽‾𝑎
pág. 53
𝑘𝐼
16𝜋2
𝑔spacetime dimensions
2 + 𝑍2
𝐼 (𝜇/𝑀spacetime dimensions )
16𝜋2
𝑔𝐼 bare
2 + 𝑏𝐼(4𝜋)𝜖 ∫
∞
0
𝑑𝑡
𝑡1−𝜖 ΓEFT ( 𝜇
√𝜋𝑀spacetime dimensions
, 𝑡)
16𝜋2
𝑔𝐼 bare
2 = 16𝜋2
𝑔𝐼
2(𝜇) − 𝑏𝐼(4𝜋)𝜖 ∫
∞
0
𝑑𝑡
𝑡1−𝜖 𝑒−𝑡𝜇2/𝑀2
16𝜋2
𝑔𝐼
2(𝜇)|
𝐷𝑅
= 𝑘𝐼
16𝜋2
𝑔spacetime dimensions
2 + 𝑍2
𝐼 (𝜇/𝑀spacetime dimensions ) − 𝑏𝐼(2𝛾 + 2),
∫
ℱ
𝑑2𝜏
𝜏2
Γ(𝜇/𝑀spacetime dimensions ) = log 𝑀spacetime dimensions
2
𝜇2 + log 2𝑒𝛾+3
𝜋√27 + 𝒪 ( 𝜇
𝑀spacetime dimensions
)
16𝜋2
𝑔𝐼
2(𝜇)|
𝐷𝑅
= 𝑘𝐼
16𝜋2
𝑔spacetime dimensions
2 + 𝑏𝐼log 𝑀spacetime dimensions
2
𝜇2 + 𝑏𝐼log 2𝑒1−𝛾
𝜋√27 + Δ𝐼,
Δ𝐼 = ∫
ℱ
𝑑2𝜏
𝜏2
[ 1
|𝜂|4 ∑
even
𝑖
𝜋 𝜕𝜏 (𝜗 [𝑎
𝑏]
𝜂 ) Trint [𝑄𝐼
2 − 𝑘𝐼
4𝜋𝜏2
] [𝑎
𝑏] − 𝑏𝐼] .
Umbrales gravitacionales.
Δgrav = ∫
ℱ
𝑑2𝜏
𝜏2
[ 1
|𝜂|4 ∑
even
𝑖
𝜋 𝜕𝜏 (𝜗 [𝑎
𝑏]
𝜂 ) 𝐸‾ˆ2
12 𝐶int [𝑎
𝑏] − 𝑏grav ]
𝜁𝑈(1) = 𝜖𝜇𝜈
1 𝜖𝜌
2 ∫ 𝛿2𝑧
𝜏2
⟨(𝜕𝑥𝜇 + 𝑖𝑝1 ⋅ 𝜓𝜓𝜇)𝜕‾𝑋𝜈𝑒𝑖𝑝1⋅𝑋|𝑧 ×
× 𝜓𝜌𝐽‾𝑒𝑖𝑝2⋅𝑋|0∮ 𝑑𝑤(𝜓𝜎𝜕𝑋𝜎 + 𝐺int)|𝑤⟩
𝜁𝑈(1) = 𝜖𝜇𝜈
1 𝜖𝜌
2𝜖𝑎𝜇𝜌𝜎𝑝1
𝑎⟨𝜕𝑋𝜎𝜕‾𝑋𝜈(𝑧‾)⟩⟨𝐽‾⟩ + 𝒪(𝑝2)
𝜁𝑈(1) ∼ ∫
ℱ
𝑑2𝜏
𝜏2
2
1
𝜂‾2 Tr[(−1)𝐹𝑄]𝑅
𝜁𝑈(1) ∼ ∑
𝑖, massless
𝑞𝑖
𝑆 = ∫ √det𝐺 [− 1
12 𝑒−2𝜙𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌 + 𝜁𝐵 ∧ 𝐹]
𝑆˜ = ∫ √det𝐺𝑒2𝜙(𝜕𝜇𝑎 + 𝜁𝐴𝜇)2
𝑉𝐷 ∼ 𝜁𝑒𝜙 (𝑒−𝜙 + ∑
𝑖
𝑞𝑖ℎ𝑖|𝑐𝑖|2)
2
𝑘𝐼
16𝜋2
𝑔spacetime dimensions
2 + 𝑍2
𝐼 (𝜇/𝑀spacetime dimensions )
16𝜋2
𝑔𝐼 bare
2 + 𝑏𝐼(4𝜋)𝜖 ∫
∞
0
𝑑𝑡
𝑡1−𝜖 ΓEFT ( 𝜇
√𝜋𝑀spacetime dimensions
, 𝑡)
16𝜋2
𝑔𝐼 bare
2 = 16𝜋2
𝑔𝐼
2(𝜇) − 𝑏𝐼(4𝜋)𝜖 ∫
∞
0
𝑑𝑡
𝑡1−𝜖 𝑒−𝑡𝜇2/𝑀2
16𝜋2
𝑔𝐼
2(𝜇)|
𝐷𝑅
= 𝑘𝐼
16𝜋2
𝑔spacetime dimensions
2 + 𝑍2
𝐼 (𝜇/𝑀spacetime dimensions ) − 𝑏𝐼(2𝛾 + 2),
∫
ℱ
𝑑2𝜏
𝜏2
Γ(𝜇/𝑀spacetime dimensions ) = log 𝑀spacetime dimensions
2
𝜇2 + log 2𝑒𝛾+3
𝜋√27 + 𝒪 ( 𝜇
𝑀spacetime dimensions
)
16𝜋2
𝑔𝐼
2(𝜇)|
𝐷𝑅
= 𝑘𝐼
16𝜋2
𝑔spacetime dimensions
2 + 𝑏𝐼log 𝑀spacetime dimensions
2
𝜇2 + 𝑏𝐼log 2𝑒1−𝛾
𝜋√27 + Δ𝐼,
Δ𝐼 = ∫
ℱ
𝑑2𝜏
𝜏2
[ 1
|𝜂|4 ∑
even
𝑖
𝜋 𝜕𝜏 (𝜗 [𝑎
𝑏]
𝜂 ) Trint [𝑄𝐼
2 − 𝑘𝐼
4𝜋𝜏2
] [𝑎
𝑏] − 𝑏𝐼] .
Umbrales gravitacionales.
Δgrav = ∫
ℱ
𝑑2𝜏
𝜏2
[ 1
|𝜂|4 ∑
even
𝑖
𝜋 𝜕𝜏 (𝜗 [𝑎
𝑏]
𝜂 ) 𝐸‾ˆ2
12 𝐶int [𝑎
𝑏] − 𝑏grav ]
𝜁𝑈(1) = 𝜖𝜇𝜈
1 𝜖𝜌
2 ∫ 𝛿2𝑧
𝜏2
⟨(𝜕𝑥𝜇 + 𝑖𝑝1 ⋅ 𝜓𝜓𝜇)𝜕‾𝑋𝜈𝑒𝑖𝑝1⋅𝑋|𝑧 ×
× 𝜓𝜌𝐽‾𝑒𝑖𝑝2⋅𝑋|0∮ 𝑑𝑤(𝜓𝜎𝜕𝑋𝜎 + 𝐺int)|𝑤⟩
𝜁𝑈(1) = 𝜖𝜇𝜈
1 𝜖𝜌
2𝜖𝑎𝜇𝜌𝜎𝑝1
𝑎⟨𝜕𝑋𝜎𝜕‾𝑋𝜈(𝑧‾)⟩⟨𝐽‾⟩ + 𝒪(𝑝2)
𝜁𝑈(1) ∼ ∫
ℱ
𝑑2𝜏
𝜏2
2
1
𝜂‾2 Tr[(−1)𝐹𝑄]𝑅
𝜁𝑈(1) ∼ ∑
𝑖, massless
𝑞𝑖
𝑆 = ∫ √det𝐺 [− 1
12 𝑒−2𝜙𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌 + 𝜁𝐵 ∧ 𝐹]
𝑆˜ = ∫ √det𝐺𝑒2𝜙(𝜕𝜇𝑎 + 𝜁𝐴𝜇)2
𝑉𝐷 ∼ 𝜁𝑒𝜙 (𝑒−𝜙 + ∑
𝑖
𝑞𝑖ℎ𝑖|𝑐𝑖|2)
2
pág. 54
𝑖
2𝜋
1
2 ∑
even
(−1)𝑎+𝑏+𝑎𝑏𝜕𝜏 (𝜗 [𝑎
𝑏]
𝜂 )
𝜗 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂3
𝑍4,4 [ℎ
𝑔]
|𝜂|4 = 4 𝜂2
𝜗‾ [1 + ℎ
1 + 𝑔] 𝜗‾ [1 − ℎ
1 − 𝑔]
,
𝜒‾0
E8 (𝑣𝑖) = 1
2 ∑
1
𝑎,𝑏=0
∏8
𝑖=1 𝜗‾ [𝑎
𝑏] (𝑣𝑖)
𝜂‾8
[ 1
(2𝜋𝑖)2 𝜕𝑣1
2 − 1
4𝜋𝜏2
] 𝜒‾0
𝐸8 (𝑣𝑖)|
𝑣𝑖=0
= 1
12 (𝐸‾ˆ2𝐸‾4 − 𝐸‾6)
1
2 ∑
(ℎ,𝑔)≠(0,0)
∑
1
𝑎,𝑏=0
𝜗‾ [𝑎
𝑏] √6 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗‾ [𝑎 − ℎ
𝑏 − 𝑔]
𝜗‾ [1 + ℎ
1 + 𝑔] 𝜗‾ [1 − ℎ
1 − 𝑔]
= − 1
4
𝐸‾6
𝜂‾6
ΔE8 = ∫
ℱ
𝑑2𝜏
𝜏2
[− 1
12 Γ2,2
𝐸‾ˆ2𝐸‾4𝐸‾6 − 𝐸‾6
2
𝜂‾24 + 60]
[Tr𝑄E7
2 − 1
4𝜋𝜏2
] = [ 1
(2𝜋𝑖)2 𝜕𝑣
2 − 1
4𝜋𝜏2
] 1
2 ∑
𝑎,𝑏
𝜗‾ [𝑎
𝑏] (𝑣)𝜗‾5 [𝑎
𝑏] 𝜗‾ [𝑎 + ℎ
𝑏 + 𝑔] 𝜗‾ [𝑎 − ℎ
𝑏 − 𝑔]
𝜂‾8 |
𝑣=0
ΔE7 = ∫
ℱ
𝑑2𝜏
𝜏2
[− 1
12 Γ2,2
𝐸‾ˆ2𝐸‾4𝐸‾6 − 𝐸‾4
3
𝜂‾24 − 84]
ΔE8 − ΔE7 = −144Δ , Δ = ∫
ℱ
𝑑2𝜏
𝜏2
(Γ2,2 − 1)
Δ = −log [4𝜋2𝑇2𝑈2|𝜂(𝑇)𝜂(𝑢)|4 ∣]
lim
𝑇2→∞ Δ = 𝜋
3 𝑇2 + 𝒪(log 𝑇2)
ΔE8
𝑁=1 = ∫
ℱ
𝑑2𝜏
𝜏2
[− 1
12 ∑
3
𝑖=1
Γ2,2(𝑇𝑖, 𝑈𝑖) 𝐸‾ˆ2𝐸‾4𝐸‾6 − 𝐸‾6
2
𝜂‾24 + 3
2 60]
ΔE6
𝑁=1 = ∫
ℱ
𝑑2𝜏
𝜏2
[− 1
12 ∑
3
𝑖=1
Γ2,2(𝑇𝑖, 𝑈𝑖) 𝐸‾ˆ2𝐸‾4𝐸‾6 − 𝐸‾4
3
𝜂‾24 − 3
2 84] .
𝑍𝑁=4→𝑁=2 = 1
2 ∑
1
ℎ,𝑔=0
1
𝜏2|𝜂|4
Γ2,2 [ℎ
𝑔]
|𝜂|4
Γ‾E8
𝜂‾8 𝑍(4,4) [ℎ
𝑔] 1
2 ∑
1
𝛾,𝛿=0
𝜗‾ [𝛾 + ℎ
𝛿 + 𝑔] 𝜗‾ [𝛾 − ℎ
𝛿 − 𝑔] 𝜗‾6 [𝛾
𝛿]
𝜂‾8
× 1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4
𝑖
2𝜋
1
2 ∑
even
(−1)𝑎+𝑏+𝑎𝑏𝜕𝜏 (𝜗 [𝑎
𝑏]
𝜂 )
𝜗 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂3
𝑍4,4 [ℎ
𝑔]
|𝜂|4 = 4 𝜂2
𝜗‾ [1 + ℎ
1 + 𝑔] 𝜗‾ [1 − ℎ
1 − 𝑔]
,
𝜒‾0
E8 (𝑣𝑖) = 1
2 ∑
1
𝑎,𝑏=0
∏8
𝑖=1 𝜗‾ [𝑎
𝑏] (𝑣𝑖)
𝜂‾8
[ 1
(2𝜋𝑖)2 𝜕𝑣1
2 − 1
4𝜋𝜏2
] 𝜒‾0
𝐸8 (𝑣𝑖)|
𝑣𝑖=0
= 1
12 (𝐸‾ˆ2𝐸‾4 − 𝐸‾6)
1
2 ∑
(ℎ,𝑔)≠(0,0)
∑
1
𝑎,𝑏=0
𝜗‾ [𝑎
𝑏] √6 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗‾ [𝑎 − ℎ
𝑏 − 𝑔]
𝜗‾ [1 + ℎ
1 + 𝑔] 𝜗‾ [1 − ℎ
1 − 𝑔]
= − 1
4
𝐸‾6
𝜂‾6
ΔE8 = ∫
ℱ
𝑑2𝜏
𝜏2
[− 1
12 Γ2,2
𝐸‾ˆ2𝐸‾4𝐸‾6 − 𝐸‾6
2
𝜂‾24 + 60]
[Tr𝑄E7
2 − 1
4𝜋𝜏2
] = [ 1
(2𝜋𝑖)2 𝜕𝑣
2 − 1
4𝜋𝜏2
] 1
2 ∑
𝑎,𝑏
𝜗‾ [𝑎
𝑏] (𝑣)𝜗‾5 [𝑎
𝑏] 𝜗‾ [𝑎 + ℎ
𝑏 + 𝑔] 𝜗‾ [𝑎 − ℎ
𝑏 − 𝑔]
𝜂‾8 |
𝑣=0
ΔE7 = ∫
ℱ
𝑑2𝜏
𝜏2
[− 1
12 Γ2,2
𝐸‾ˆ2𝐸‾4𝐸‾6 − 𝐸‾4
3
𝜂‾24 − 84]
ΔE8 − ΔE7 = −144Δ , Δ = ∫
ℱ
𝑑2𝜏
𝜏2
(Γ2,2 − 1)
Δ = −log [4𝜋2𝑇2𝑈2|𝜂(𝑇)𝜂(𝑢)|4 ∣]
lim
𝑇2→∞ Δ = 𝜋
3 𝑇2 + 𝒪(log 𝑇2)
ΔE8
𝑁=1 = ∫
ℱ
𝑑2𝜏
𝜏2
[− 1
12 ∑
3
𝑖=1
Γ2,2(𝑇𝑖, 𝑈𝑖) 𝐸‾ˆ2𝐸‾4𝐸‾6 − 𝐸‾6
2
𝜂‾24 + 3
2 60]
ΔE6
𝑁=1 = ∫
ℱ
𝑑2𝜏
𝜏2
[− 1
12 ∑
3
𝑖=1
Γ2,2(𝑇𝑖, 𝑈𝑖) 𝐸‾ˆ2𝐸‾4𝐸‾6 − 𝐸‾4
3
𝜂‾24 − 3
2 84] .
𝑍𝑁=4→𝑁=2 = 1
2 ∑
1
ℎ,𝑔=0
1
𝜏2|𝜂|4
Γ2,2 [ℎ
𝑔]
|𝜂|4
Γ‾E8
𝜂‾8 𝑍(4,4) [ℎ
𝑔] 1
2 ∑
1
𝛾,𝛿=0
𝜗‾ [𝛾 + ℎ
𝛿 + 𝑔] 𝜗‾ [𝛾 − ℎ
𝛿 − 𝑔] 𝜗‾6 [𝛾
𝛿]
𝜂‾8
× 1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏
𝜗2 [𝑎
𝑏] 𝜗 [𝑎 + ℎ
𝑏 + 𝑔] 𝜗 [𝑎 − ℎ
𝑏 − 𝑔]
𝜂4
pág. 55
𝑚3/2
2 = |𝑈|2
𝑇2𝑈2
𝑏E8 = −60 , 𝑏E7 = −12 , 𝑏SU(2) = 52
Δ𝐼 = 𝑏𝐼Δ + (𝑏˜𝐼
3 − 𝑏𝐼) 𝛿 − 𝑘𝐼𝑌
Δ = ∫
ℱ
𝑑2𝜏
𝜏2
[∑
′
ℎ,𝑔
Γ2,2 [ℎ
𝑔] − 1] = −log [𝜋2
4 |𝜗4(𝑇)|4|𝜗2(𝑈)|4𝑇2𝑈2]
𝛿 = ∫
ℱ
𝑑2𝜏
𝜏2
∑
′
ℎ,𝑔
Γ2,2 [ℎ
𝑔] 𝜎‾ [ℎ
𝑔]
𝑌 = ∫
ℱ
𝑑2𝜏
𝜏2
∑
′
ℎ,𝑔
Γ2,2 [ 1
12
𝐸‾ˆ2
𝜂‾24 Ω‾ [ℎ
𝑔] + 𝜌‾ [ℎ
𝑔] + 40𝜎‾ [ℎ
𝑔]]
Ω [0
1] = 1
2 𝐸4𝜗3
4𝜗4
4(𝜗3
4 + 𝜗4
4),
𝜎 [ℎ
𝑔] = − 1
4
𝜗12 [ℎ
𝑔]
𝜂12 ,
𝜌 [0
1] = 𝑓(1 − 𝑥) , 𝜌 [1
0] = 𝑓(𝑥) , 𝜌 [1
1] = 𝑓(𝑥/(𝑥 − 1)),
𝑓(𝑥) = 4(8 − 49𝑥 + 66𝑥2 − 49𝑥3 + 8𝑥4)
3𝑥(1 − 𝑥)2
Δ𝐼 = ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2
𝜂‾24 (Tr[𝑄𝐼
2] − 𝑘𝐼
4𝜋𝜏2
) Ω‾ − 𝑏𝐼]
Δgrav = ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2
𝜂‾24
𝐸‾ˆ2
12 Ω‾ − 𝑏grav ]
Ω‾ = 𝜉𝐸‾4𝐸‾6
Δ𝐼 = ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2 (𝜉𝑘𝐼
12
𝐸‾ˆ2𝐸‾4𝐸‾6
𝜂‾24 + 𝐴𝐼𝑗‾ + 𝐵𝐼) − 𝑏𝐼]
Δgrav = 𝜉 ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2
𝐸‾ˆ2𝐸‾4𝐸‾6
12𝜂‾24 − 𝑏grav ]
𝐴𝐼 = − 𝜉𝑘𝐼
12
744𝐴𝐼 + 𝐵𝐼 − 𝑏𝐼 + 𝑘𝑖𝑏grav = 0
𝑚3/2
2 = |𝑈|2
𝑇2𝑈2
𝑏E8 = −60 , 𝑏E7 = −12 , 𝑏SU(2) = 52
Δ𝐼 = 𝑏𝐼Δ + (𝑏˜𝐼
3 − 𝑏𝐼) 𝛿 − 𝑘𝐼𝑌
Δ = ∫
ℱ
𝑑2𝜏
𝜏2
[∑
′
ℎ,𝑔
Γ2,2 [ℎ
𝑔] − 1] = −log [𝜋2
4 |𝜗4(𝑇)|4|𝜗2(𝑈)|4𝑇2𝑈2]
𝛿 = ∫
ℱ
𝑑2𝜏
𝜏2
∑
′
ℎ,𝑔
Γ2,2 [ℎ
𝑔] 𝜎‾ [ℎ
𝑔]
𝑌 = ∫
ℱ
𝑑2𝜏
𝜏2
∑
′
ℎ,𝑔
Γ2,2 [ 1
12
𝐸‾ˆ2
𝜂‾24 Ω‾ [ℎ
𝑔] + 𝜌‾ [ℎ
𝑔] + 40𝜎‾ [ℎ
𝑔]]
Ω [0
1] = 1
2 𝐸4𝜗3
4𝜗4
4(𝜗3
4 + 𝜗4
4),
𝜎 [ℎ
𝑔] = − 1
4
𝜗12 [ℎ
𝑔]
𝜂12 ,
𝜌 [0
1] = 𝑓(1 − 𝑥) , 𝜌 [1
0] = 𝑓(𝑥) , 𝜌 [1
1] = 𝑓(𝑥/(𝑥 − 1)),
𝑓(𝑥) = 4(8 − 49𝑥 + 66𝑥2 − 49𝑥3 + 8𝑥4)
3𝑥(1 − 𝑥)2
Δ𝐼 = ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2
𝜂‾24 (Tr[𝑄𝐼
2] − 𝑘𝐼
4𝜋𝜏2
) Ω‾ − 𝑏𝐼]
Δgrav = ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2
𝜂‾24
𝐸‾ˆ2
12 Ω‾ − 𝑏grav ]
Ω‾ = 𝜉𝐸‾4𝐸‾6
Δ𝐼 = ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2 (𝜉𝑘𝐼
12
𝐸‾ˆ2𝐸‾4𝐸‾6
𝜂‾24 + 𝐴𝐼𝑗‾ + 𝐵𝐼) − 𝑏𝐼]
Δgrav = 𝜉 ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2
𝐸‾ˆ2𝐸‾4𝐸‾6
12𝜂‾24 − 𝑏grav ]
𝐴𝐼 = − 𝜉𝑘𝐼
12
744𝐴𝐼 + 𝐵𝐼 − 𝑏𝐼 + 𝑘𝑖𝑏grav = 0
pág. 56
𝑏grav = 22 − 𝑁𝑉 + 𝑁𝐻
12
Δ𝐼 = 𝑏𝐼Δ − 𝑘𝐼𝑌
Δ = ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2(𝑇, 𝑈) − 1]
= −log (4𝜋2|𝜂(𝑇)|4|𝜂(𝑈)|4Im𝑇Im𝑈)
𝑌 = 1
12 ∫
ℱ
𝑑2𝜏
𝜏2
Γ2,2(𝑇, 𝑈) [𝐸‾ˆ2𝐸‾4𝐸‾6
𝜂‾24 − 𝑗‾ + 1008]
Δgrav = − ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2
𝐸‾ˆ2𝐸‾4𝐸‾6
12𝜂‾24 − 22]
16𝜋2
𝑔𝐼
2(𝜇) = 𝑘𝐼
16𝜋2
𝑔renorm
2 + 𝑏𝐼log 𝑀𝑠
2
𝜇2 + Δˆ 𝐼
𝑔renorm
2 = 𝑔string
2
1 − 𝑌
16𝜋2 𝑔string
2
𝑀string = 𝑀𝑃𝑔renorm
√1 + 𝑌
16𝜋2 𝑔renorm
2
1
𝑔𝐼
2 = 𝑘𝐼
𝑔𝑈
2
16𝜋2
𝑔𝐼
2(𝜇) = 𝑘𝐼
16𝜋2
𝑔𝑈
2 + 𝑏𝐼log 𝑀𝑈
2
𝜇2
𝑔𝑈 = 𝑔renorm = 𝑔spacetime dimensions
√1 − 𝑔spacetime dimensions
2 𝑌
16𝜋2
𝑀𝑈
2 = 2𝑒1−𝛾
𝜋√27 𝑒Δ𝑀𝑃
2𝑔spacetime dimensions
2 = 2𝑒1−𝛾
𝜋√27 𝑒Δ𝑀𝑃
2 𝑔𝑈
√1 + 𝑔𝑈
2 𝑌
16𝜋2
𝑍𝐷=9
O(32) = 1
(√𝜏2𝜂𝜂‾)7
Γ1,17(𝑅, 𝑌𝐼)
𝜂𝜂‾17
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗4 [𝑎
𝑏]
𝜂4 ,
𝑏grav = 22 − 𝑁𝑉 + 𝑁𝐻
12
Δ𝐼 = 𝑏𝐼Δ − 𝑘𝐼𝑌
Δ = ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2(𝑇, 𝑈) − 1]
= −log (4𝜋2|𝜂(𝑇)|4|𝜂(𝑈)|4Im𝑇Im𝑈)
𝑌 = 1
12 ∫
ℱ
𝑑2𝜏
𝜏2
Γ2,2(𝑇, 𝑈) [𝐸‾ˆ2𝐸‾4𝐸‾6
𝜂‾24 − 𝑗‾ + 1008]
Δgrav = − ∫
ℱ
𝑑2𝜏
𝜏2
[Γ2,2
𝐸‾ˆ2𝐸‾4𝐸‾6
12𝜂‾24 − 22]
16𝜋2
𝑔𝐼
2(𝜇) = 𝑘𝐼
16𝜋2
𝑔renorm
2 + 𝑏𝐼log 𝑀𝑠
2
𝜇2 + Δˆ 𝐼
𝑔renorm
2 = 𝑔string
2
1 − 𝑌
16𝜋2 𝑔string
2
𝑀string = 𝑀𝑃𝑔renorm
√1 + 𝑌
16𝜋2 𝑔renorm
2
1
𝑔𝐼
2 = 𝑘𝐼
𝑔𝑈
2
16𝜋2
𝑔𝐼
2(𝜇) = 𝑘𝐼
16𝜋2
𝑔𝑈
2 + 𝑏𝐼log 𝑀𝑈
2
𝜇2
𝑔𝑈 = 𝑔renorm = 𝑔spacetime dimensions
√1 − 𝑔spacetime dimensions
2 𝑌
16𝜋2
𝑀𝑈
2 = 2𝑒1−𝛾
𝜋√27 𝑒Δ𝑀𝑃
2𝑔spacetime dimensions
2 = 2𝑒1−𝛾
𝜋√27 𝑒Δ𝑀𝑃
2 𝑔𝑈
√1 + 𝑔𝑈
2 𝑌
16𝜋2
𝑍𝐷=9
O(32) = 1
(√𝜏2𝜂𝜂‾)7
Γ1,17(𝑅, 𝑌𝐼)
𝜂𝜂‾17
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗4 [𝑎
𝑏]
𝜂4 ,
pág. 57
Γ1,17(𝑅) = 𝑅 ∑
𝑚,𝑛∈𝑍
exp [− 𝜋𝑅2
𝜏2
|𝑚 + 𝜏𝑛|2] 1
2 ∑
𝑎,𝑏
𝜗‾8 [𝑎
𝑏] 𝜗‾8 [ 𝑎 + 𝑛
𝑏 + 𝑚]
= 1
2 ∑
1
ℎ,𝑔=0
Γ1,1(2𝑅) [ℎ
𝑔] 1
2 ∑
𝑎,𝑏
𝜗‾8 [𝑎
𝑏] 𝜗‾8 [𝑎 + ℎ
𝑏 + 𝑔]
Γ1,1(𝑅) [ℎ
𝑔] = 𝑅 ∑
𝑚,𝑛∈𝑍
exp [− 𝜋𝑅2
𝜏2
|(𝑚 + 𝑔
2) + 𝜏 (𝑛 + ℎ
2)|
2
]
= 1
𝑅 ∑
𝑚,𝑛∈𝑍
(−1)𝑚ℎ+𝑛𝑔exp [− 𝜋
𝜏2𝑅2 |𝑚 + 𝜏𝑛|2] .
𝜕𝑋9 → 𝜕𝑋9 , 𝜓9 → 𝜓9 , 𝜕‾𝑋9 → −𝜕‾𝑋9 , 𝜓‾9 → −𝜓‾9.
Tensores antisimétricos y supermembranas.
𝐴𝑝 ≡ 𝐴𝜇1𝜇2…𝜇𝑝 𝑑𝑥𝜇1 ∧ … ∧ 𝑑𝑥𝜇𝑝
𝐴𝑝 → 𝐴𝑝 + 𝑑Λ𝑝−1,
𝐹𝑝+1 = 𝑑𝐴𝑝
𝑑∗𝐹𝑝+1 = 0
exp [𝑖𝑄𝑝 ∫
world-volume
𝐴𝑝+1] = exp [𝑖𝑄𝑝 ∫ 𝐴𝜇0…𝜇𝑝 𝑑𝑥𝜇0 ∧ … ∧ 𝑑𝑥𝜇𝑝 ]
𝑑𝐴˜𝐷−𝑝−3 = 𝐹˜𝐷−𝑝−2 = ∗𝐹𝑝+2 = ∗𝑑𝐴𝑝+1
Φ = ∫
𝑆𝐷−𝑝−2
∗𝐹𝑝+2 = ∫
𝑆𝐷−𝑝−3
𝐴˜𝐷−𝑝−3
Φ𝑄˜𝐷−𝑝−4 = 𝑄𝑝𝑄˜𝐷−𝑝−4 = 2𝜋𝑁 , 𝑛 ∈ 𝑍
𝑀𝑚,𝑛
2 = |𝑚 + 𝑛𝜏|2
𝜏2
𝑀𝑚0,𝑛0 = ∑
𝑁
𝑖=1
√𝑀𝑖
2 + 𝑝𝑖
2 ≥ ∑
𝑁
𝑖=1
𝑀𝑖
𝑀𝑚0,𝑛0 ≥ ∑
𝑁
𝑖=1
𝑀𝑚𝑖,𝑛𝑖
‖𝑣0‖ ≤ ∑
𝑁
𝑖=1
‖𝑣𝑖‖
Γ1,17(𝑅) = 𝑅 ∑
𝑚,𝑛∈𝑍
exp [− 𝜋𝑅2
𝜏2
|𝑚 + 𝜏𝑛|2] 1
2 ∑
𝑎,𝑏
𝜗‾8 [𝑎
𝑏] 𝜗‾8 [ 𝑎 + 𝑛
𝑏 + 𝑚]
= 1
2 ∑
1
ℎ,𝑔=0
Γ1,1(2𝑅) [ℎ
𝑔] 1
2 ∑
𝑎,𝑏
𝜗‾8 [𝑎
𝑏] 𝜗‾8 [𝑎 + ℎ
𝑏 + 𝑔]
Γ1,1(𝑅) [ℎ
𝑔] = 𝑅 ∑
𝑚,𝑛∈𝑍
exp [− 𝜋𝑅2
𝜏2
|(𝑚 + 𝑔
2) + 𝜏 (𝑛 + ℎ
2)|
2
]
= 1
𝑅 ∑
𝑚,𝑛∈𝑍
(−1)𝑚ℎ+𝑛𝑔exp [− 𝜋
𝜏2𝑅2 |𝑚 + 𝜏𝑛|2] .
𝜕𝑋9 → 𝜕𝑋9 , 𝜓9 → 𝜓9 , 𝜕‾𝑋9 → −𝜕‾𝑋9 , 𝜓‾9 → −𝜓‾9.
Tensores antisimétricos y supermembranas.
𝐴𝑝 ≡ 𝐴𝜇1𝜇2…𝜇𝑝 𝑑𝑥𝜇1 ∧ … ∧ 𝑑𝑥𝜇𝑝
𝐴𝑝 → 𝐴𝑝 + 𝑑Λ𝑝−1,
𝐹𝑝+1 = 𝑑𝐴𝑝
𝑑∗𝐹𝑝+1 = 0
exp [𝑖𝑄𝑝 ∫
world-volume
𝐴𝑝+1] = exp [𝑖𝑄𝑝 ∫ 𝐴𝜇0…𝜇𝑝 𝑑𝑥𝜇0 ∧ … ∧ 𝑑𝑥𝜇𝑝 ]
𝑑𝐴˜𝐷−𝑝−3 = 𝐹˜𝐷−𝑝−2 = ∗𝐹𝑝+2 = ∗𝑑𝐴𝑝+1
Φ = ∫
𝑆𝐷−𝑝−2
∗𝐹𝑝+2 = ∫
𝑆𝐷−𝑝−3
𝐴˜𝐷−𝑝−3
Φ𝑄˜𝐷−𝑝−4 = 𝑄𝑝𝑄˜𝐷−𝑝−4 = 2𝜋𝑁 , 𝑛 ∈ 𝑍
𝑀𝑚,𝑛
2 = |𝑚 + 𝑛𝜏|2
𝜏2
𝑀𝑚0,𝑛0 = ∑
𝑁
𝑖=1
√𝑀𝑖
2 + 𝑝𝑖
2 ≥ ∑
𝑁
𝑖=1
𝑀𝑖
𝑀𝑚0,𝑛0 ≥ ∑
𝑁
𝑖=1
𝑀𝑚𝑖,𝑛𝑖
‖𝑣0‖ ≤ ∑
𝑁
𝑖=1
‖𝑣𝑖‖
pág. 58
N - Dimensiones y p - supermembranas.
𝑆het = ∫ 𝑑10𝑥√𝐺𝑒−Φ [𝑅 + (∇Φ)2 − 1
12 𝐻ˆ 2 − 1
4 𝐹2]
𝑆𝐼 = ∫ 𝑑10𝑥√𝐺 [𝑒−Φ(𝑅 + (∇Φ)2) − 1
4 𝑒−Φ/2𝐹2 − 1
12 𝐻ˆ 2]
𝑆𝐸
het = ∫ 𝑑10𝑥√𝑔 [𝑅 − 1
8 (∇Φ)2 − 1
4 𝑒−Φ/4𝐹2 − 1
12 𝑒−Φ/2𝐻ˆ 2]
𝑆𝐸
𝐼 = ∫ 𝑑10𝑥√𝑔 [𝑅 − 1
8 (∇Φ)2 − 1
4 𝑒Φ/4𝐹2 − 1
12 𝑒Φ/2𝐻ˆ 2]
NN NS sector 𝜓 + 𝜓‾|𝜎=0 = 𝜓 − 𝜓‾|𝜎=𝜋 = 0
NN R sector 𝜓 − 𝜓‾|𝜎=0 = 𝜓 − 𝜓‾|𝜎=𝜋 = 0.
DDNS sector 𝜓 − 𝜓‾|𝜎=0 = 𝜓 + 𝜓‾|𝜎=𝜋 = 0
DDR sector 𝜓 + 𝜓‾|𝜎=0 = 𝜓 + 𝜓‾|𝜎=𝜋 = 0
𝑋𝐼(𝜎, 𝜏) = 𝑥𝐼 + 𝑤𝐼𝜎 + 2 ∑
𝑛≠0
𝑎𝑛
𝐼
𝑛 𝑒𝑖𝑛𝜏sin (𝑛𝜎),
𝑋𝜇(𝜎, 𝜏) = 𝑥𝜇 + 𝑝𝜇𝜏 − 2𝑖 ∑
𝑛≠0
𝑎𝑛
𝜇
𝑛 𝑒𝑖𝑛𝜏cos (𝑛𝜎).
𝜓𝐼(𝜎, 𝜏) = ∑
𝑛∈𝑍
𝑏𝑛+1/2
𝐼 𝑒𝑖(𝑛+1/2)(𝜎+𝜏) , 𝜓𝜇(𝜎, 𝜏) = ∑
𝑛∈𝑍
𝑏𝑛+1/2
𝜇 𝑒𝑖(𝑛+1/2)(𝜎+𝜏),
𝜓𝐼(𝜎, 𝜏) = ∑
𝑛∈𝑍
𝑏𝑛
𝐼 𝑒𝑖𝑛(𝜎+𝜏) , 𝜓𝜇(𝜎, 𝜏) = ∑
𝑛∈𝑍
𝑏𝑛
𝜇𝑒𝑖𝑛(𝜎+𝜏).
𝑏‾𝑛+1/2
𝐼 = 𝑏𝑛+1/2
𝐼 , 𝑏‾𝑛
𝐼 = −𝑏𝑛
𝐼
𝑏‾𝑛+1/2
𝜇 = −𝑏𝑛+1/2
𝜇 , 𝑏‾𝑛
𝜇 = 𝑏𝑛
𝜇.
Ω𝑏−1/2
𝜇 |0⟩ = 𝑏‾−1/2
𝜇 |0⟩ = −𝑏−1/2
𝜇 |0⟩,
Ω𝑏−1/2
𝐼 |0⟩ = 𝑏‾−1/2
𝐼 |0⟩ = 𝑏−1/2
𝐼 |0⟩.
Γ11|𝑅⟩ = |𝑅⟩
Ω|𝑅⟩ = −Γ2 … Γ9|𝑅⟩ = |𝑅⟩
Γ0Γ1|𝑅⟩ = −|𝑅⟩
𝜕𝜏𝑋𝐼|𝜎=0 = 0, 𝜕𝜎𝑋𝐼|𝜎=𝜋 = 0,
DNNS sector 𝜓 + 𝜓‾|𝜎=0 = 𝜓 + 𝜓‾|𝜎=𝜋 = 0,
DNR sector 𝜓 − 𝜓‾|𝜎=0 = 𝜓 + 𝜓‾|𝜎=𝜋 = 0,
N - Dimensiones y p - supermembranas.
𝑆het = ∫ 𝑑10𝑥√𝐺𝑒−Φ [𝑅 + (∇Φ)2 − 1
12 𝐻ˆ 2 − 1
4 𝐹2]
𝑆𝐼 = ∫ 𝑑10𝑥√𝐺 [𝑒−Φ(𝑅 + (∇Φ)2) − 1
4 𝑒−Φ/2𝐹2 − 1
12 𝐻ˆ 2]
𝑆𝐸
het = ∫ 𝑑10𝑥√𝑔 [𝑅 − 1
8 (∇Φ)2 − 1
4 𝑒−Φ/4𝐹2 − 1
12 𝑒−Φ/2𝐻ˆ 2]
𝑆𝐸
𝐼 = ∫ 𝑑10𝑥√𝑔 [𝑅 − 1
8 (∇Φ)2 − 1
4 𝑒Φ/4𝐹2 − 1
12 𝑒Φ/2𝐻ˆ 2]
NN NS sector 𝜓 + 𝜓‾|𝜎=0 = 𝜓 − 𝜓‾|𝜎=𝜋 = 0
NN R sector 𝜓 − 𝜓‾|𝜎=0 = 𝜓 − 𝜓‾|𝜎=𝜋 = 0.
DDNS sector 𝜓 − 𝜓‾|𝜎=0 = 𝜓 + 𝜓‾|𝜎=𝜋 = 0
DDR sector 𝜓 + 𝜓‾|𝜎=0 = 𝜓 + 𝜓‾|𝜎=𝜋 = 0
𝑋𝐼(𝜎, 𝜏) = 𝑥𝐼 + 𝑤𝐼𝜎 + 2 ∑
𝑛≠0
𝑎𝑛
𝐼
𝑛 𝑒𝑖𝑛𝜏sin (𝑛𝜎),
𝑋𝜇(𝜎, 𝜏) = 𝑥𝜇 + 𝑝𝜇𝜏 − 2𝑖 ∑
𝑛≠0
𝑎𝑛
𝜇
𝑛 𝑒𝑖𝑛𝜏cos (𝑛𝜎).
𝜓𝐼(𝜎, 𝜏) = ∑
𝑛∈𝑍
𝑏𝑛+1/2
𝐼 𝑒𝑖(𝑛+1/2)(𝜎+𝜏) , 𝜓𝜇(𝜎, 𝜏) = ∑
𝑛∈𝑍
𝑏𝑛+1/2
𝜇 𝑒𝑖(𝑛+1/2)(𝜎+𝜏),
𝜓𝐼(𝜎, 𝜏) = ∑
𝑛∈𝑍
𝑏𝑛
𝐼 𝑒𝑖𝑛(𝜎+𝜏) , 𝜓𝜇(𝜎, 𝜏) = ∑
𝑛∈𝑍
𝑏𝑛
𝜇𝑒𝑖𝑛(𝜎+𝜏).
𝑏‾𝑛+1/2
𝐼 = 𝑏𝑛+1/2
𝐼 , 𝑏‾𝑛
𝐼 = −𝑏𝑛
𝐼
𝑏‾𝑛+1/2
𝜇 = −𝑏𝑛+1/2
𝜇 , 𝑏‾𝑛
𝜇 = 𝑏𝑛
𝜇.
Ω𝑏−1/2
𝜇 |0⟩ = 𝑏‾−1/2
𝜇 |0⟩ = −𝑏−1/2
𝜇 |0⟩,
Ω𝑏−1/2
𝐼 |0⟩ = 𝑏‾−1/2
𝐼 |0⟩ = 𝑏−1/2
𝐼 |0⟩.
Γ11|𝑅⟩ = |𝑅⟩
Ω|𝑅⟩ = −Γ2 … Γ9|𝑅⟩ = |𝑅⟩
Γ0Γ1|𝑅⟩ = −|𝑅⟩
𝜕𝜏𝑋𝐼|𝜎=0 = 0, 𝜕𝜎𝑋𝐼|𝜎=𝜋 = 0,
DNNS sector 𝜓 + 𝜓‾|𝜎=0 = 𝜓 + 𝜓‾|𝜎=𝜋 = 0,
DNR sector 𝜓 − 𝜓‾|𝜎=0 = 𝜓 + 𝜓‾|𝜎=𝜋 = 0,
pág. 59
𝑏−1/2
𝜇 |𝑝; 𝑖, 𝑗⟩ , 𝑏−1/2
𝐼 |𝑝; 𝑖, 𝑗⟩
𝑆˜𝐼𝐼𝐴 = 1
2𝜅10
2 [∫ 𝑑10𝑥√𝑔𝑒−Φ [(𝑅 + (∇Φ)2 − 1
12 𝐻2) − 1
2 ⋅ 4! 𝐺ˆ 2 − 1
4 𝐹2] + 1
(48)2 ∫ 𝐵 ∧ 𝐺 ∧ 𝐺]
{𝑄𝛼
1 , 𝑄𝛼˙
2} = 𝛿𝛼𝛼˙ 𝑊
𝑀 ≥ 𝑐0
𝜆 |𝑊|
𝑀 = 𝑐
𝜆 |𝑛| , 𝑛 ∈ 𝑍.
𝑅 = 𝜆2/3
𝑒𝜙/2 = 𝜆 + 𝑐
𝑟8 , 𝜒 = 𝜒0 + 𝑖 𝑐
𝜆(𝜆𝑟8 + 𝑐) ,
𝑑𝑠2 = 𝐴(𝑟)−3/4[−(𝑑𝑥0)2 + (𝑑𝑥1)2] + 𝐴(𝑟)1/4𝑑𝑦 ⋅ 𝑑𝑦
𝑆 = 𝜒0 + 𝑖 𝑒−𝜙0/2
√𝐴(𝑟)
𝐵1 = 0 , 𝐵01
2 = 1
√Δ𝐴(𝑟)
𝐴(𝑟) = 1 + 𝑄√Δ
3𝑟6 , 𝑄 = 3𝜅2𝑇
𝜋4 , Δ = 𝑒𝜙0/2[𝜒0
2 + 𝑒−𝜙0 ].
𝑇˜ = 𝑇√Δ.
𝑇𝑝,𝑞 = 𝑇 |𝑝 + 𝑞𝑆|
√𝑆2
𝑀𝐵
2 = 𝑚2
𝑅𝐵
2 + (2𝜋𝑅𝐵𝑛𝑇𝑝,𝑞)2 + 4𝜋𝑇𝑝,𝑞(𝑁𝐿 + 𝑁𝑅)
𝑀𝐵
2|BPS = ( 𝑚
𝑅𝐵
+ 2𝜋𝑅𝐵𝑛𝑇𝑝,𝑞)
2
𝑀𝐵
2|𝐵𝑃𝑆 = ( 𝑚
𝑅𝐵
+ 2𝜋𝑅𝐵𝑇 |𝑛1 + 𝑛2𝑆|
√𝑆2
)
2
𝑀11
2 = (𝑚(2𝜋𝑅11)2𝜏2𝑇11)2 + |𝑛1 + 𝑛2𝜏|2
𝑅11
2 𝜏2
2 + ⋯
𝑆 = 𝜏 , 1
𝑅𝐵
2 = 𝑇𝑇11𝐴11
3/2 , 𝛽 = 2𝜋𝑅11
√𝜏2𝑇11
𝑇
𝑏−1/2
𝜇 |𝑝; 𝑖, 𝑗⟩ , 𝑏−1/2
𝐼 |𝑝; 𝑖, 𝑗⟩
𝑆˜𝐼𝐼𝐴 = 1
2𝜅10
2 [∫ 𝑑10𝑥√𝑔𝑒−Φ [(𝑅 + (∇Φ)2 − 1
12 𝐻2) − 1
2 ⋅ 4! 𝐺ˆ 2 − 1
4 𝐹2] + 1
(48)2 ∫ 𝐵 ∧ 𝐺 ∧ 𝐺]
{𝑄𝛼
1 , 𝑄𝛼˙
2} = 𝛿𝛼𝛼˙ 𝑊
𝑀 ≥ 𝑐0
𝜆 |𝑊|
𝑀 = 𝑐
𝜆 |𝑛| , 𝑛 ∈ 𝑍.
𝑅 = 𝜆2/3
𝑒𝜙/2 = 𝜆 + 𝑐
𝑟8 , 𝜒 = 𝜒0 + 𝑖 𝑐
𝜆(𝜆𝑟8 + 𝑐) ,
𝑑𝑠2 = 𝐴(𝑟)−3/4[−(𝑑𝑥0)2 + (𝑑𝑥1)2] + 𝐴(𝑟)1/4𝑑𝑦 ⋅ 𝑑𝑦
𝑆 = 𝜒0 + 𝑖 𝑒−𝜙0/2
√𝐴(𝑟)
𝐵1 = 0 , 𝐵01
2 = 1
√Δ𝐴(𝑟)
𝐴(𝑟) = 1 + 𝑄√Δ
3𝑟6 , 𝑄 = 3𝜅2𝑇
𝜋4 , Δ = 𝑒𝜙0/2[𝜒0
2 + 𝑒−𝜙0 ].
𝑇˜ = 𝑇√Δ.
𝑇𝑝,𝑞 = 𝑇 |𝑝 + 𝑞𝑆|
√𝑆2
𝑀𝐵
2 = 𝑚2
𝑅𝐵
2 + (2𝜋𝑅𝐵𝑛𝑇𝑝,𝑞)2 + 4𝜋𝑇𝑝,𝑞(𝑁𝐿 + 𝑁𝑅)
𝑀𝐵
2|BPS = ( 𝑚
𝑅𝐵
+ 2𝜋𝑅𝐵𝑛𝑇𝑝,𝑞)
2
𝑀𝐵
2|𝐵𝑃𝑆 = ( 𝑚
𝑅𝐵
+ 2𝜋𝑅𝐵𝑇 |𝑛1 + 𝑛2𝑆|
√𝑆2
)
2
𝑀11
2 = (𝑚(2𝜋𝑅11)2𝜏2𝑇11)2 + |𝑛1 + 𝑛2𝜏|2
𝑅11
2 𝜏2
2 + ⋯
𝑆 = 𝜏 , 1
𝑅𝐵
2 = 𝑇𝑇11𝐴11
3/2 , 𝛽 = 2𝜋𝑅11
√𝜏2𝑇11
𝑇
pág. 60
𝒜 = 2𝑉𝑝+1 ∫ 𝑑𝑝+1𝑘
(2𝜋)𝑝+1 ∫
∞
0
𝑑𝑡
2𝑡 𝑒−2𝜋𝛼′𝑡𝑘2−𝑡 |𝑌|2
2𝜋𝛼′ 1
𝜂12(𝑖𝑡)
1
2 ∑
𝑎,𝑏
(−1)𝑎+𝑏+𝑎𝑏𝜗4 [𝑎
𝑏] (𝑖𝑡)
= 2𝑉𝑝+1 ∫
∞
0
𝑑𝑡
2𝑡 (8𝜋2𝛼′𝑡)−𝑝+1
2 𝑒−𝑡 |𝑌|2
2𝜋𝛼′ 1
𝜂12(𝑖𝑡)
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏𝜗4 [𝑎
𝑏] (𝑖𝑡)
𝒜|supermassive.
spacetime dimensions = 8(1 − 1)𝑉𝑝+1 ∫
∞
0
𝑑𝑡
𝑡 (8𝜋2𝛼′𝑡)−𝑝+1
2 𝑡4𝑒−𝑡|𝑌|2
2𝜋𝛼′
= 2𝜋(1 − 1)𝑉𝑝+1(4𝜋2𝛼′)3−𝑝𝐺9−𝑝(|𝑌|)
𝐺𝑑(|𝑌|) = 1
4𝜋𝑑/2 ∫
∞
0
𝑑𝑡
𝑡(4−𝑑)/2 𝑒−𝑡|𝑌|2
𝑆 = 𝛼𝑝
2 ∫ 𝐹𝑝+2
∗ 𝐹𝑝+2 + 𝑖𝑇𝑝 ∫
branes
𝐴𝑝+1
𝒜|field theory = (𝑖𝑇𝑝)2
𝛼𝑝
𝑉𝑝+1𝐺9−𝑝(|𝑌|)
𝑇𝑝
2
𝛼𝑝
= 2𝜋(4𝜋2𝛼′)3−𝑝
𝑇𝑝𝑇6−𝑝
𝛼𝑝
= 2𝜋𝑛
2𝜅10
2 = (2𝜋)7𝛼′4
𝑆𝑝 = −𝑇𝑝 ∫
𝑊𝑝+1
𝑑𝑝+1𝜉𝑒−Φ/2√det𝐺ˆ − 𝑖𝑇𝑝 ∫ 𝐴𝑝+1
𝐺ˆ𝛼𝛽 = 𝐺𝜇𝜈
𝜕𝑋𝜇
𝜕𝜉𝛼
𝜕𝑋𝜈
𝜕𝜉𝛽
∫ 𝐴𝑝+1 = 1
(𝑝 + 1)! ∫ 𝑑𝑝+1𝜉𝐴𝜇1⋯𝜇𝑝+1
𝜕𝑋𝜇1
𝜕𝜉𝛼1 ⋯ 𝜕𝑋𝜇𝑝+1
𝜕𝜉𝛼𝑝+1 𝜖𝛼1⋯𝛼𝑝+1
2𝜅10
2 𝑇𝑝𝑇6−𝑝 = 2𝜋𝑛
𝑇𝑝 = 1
(2𝜋)𝑝(𝛼′)(𝑝+1)/2
2𝜅11
2 = (2𝜋)8(𝛼′)9/2
𝑇2
𝑀 = 𝑇2 = 1
(2𝜋)2(𝛼′)3/2
2𝜅11
2 𝑇2
𝑀𝑇5
𝑀 = 2𝜋 → 𝑇5
𝑀 = 1
(2𝜋)5(𝛼′)3
2𝜋√𝛼′𝑇5
𝑀 = 𝑇4
𝒜 = 2𝑉𝑝+1 ∫ 𝑑𝑝+1𝑘
(2𝜋)𝑝+1 ∫
∞
0
𝑑𝑡
2𝑡 𝑒−2𝜋𝛼′𝑡𝑘2−𝑡 |𝑌|2
2𝜋𝛼′ 1
𝜂12(𝑖𝑡)
1
2 ∑
𝑎,𝑏
(−1)𝑎+𝑏+𝑎𝑏𝜗4 [𝑎
𝑏] (𝑖𝑡)
= 2𝑉𝑝+1 ∫
∞
0
𝑑𝑡
2𝑡 (8𝜋2𝛼′𝑡)−𝑝+1
2 𝑒−𝑡 |𝑌|2
2𝜋𝛼′ 1
𝜂12(𝑖𝑡)
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏𝜗4 [𝑎
𝑏] (𝑖𝑡)
𝒜|supermassive.
spacetime dimensions = 8(1 − 1)𝑉𝑝+1 ∫
∞
0
𝑑𝑡
𝑡 (8𝜋2𝛼′𝑡)−𝑝+1
2 𝑡4𝑒−𝑡|𝑌|2
2𝜋𝛼′
= 2𝜋(1 − 1)𝑉𝑝+1(4𝜋2𝛼′)3−𝑝𝐺9−𝑝(|𝑌|)
𝐺𝑑(|𝑌|) = 1
4𝜋𝑑/2 ∫
∞
0
𝑑𝑡
𝑡(4−𝑑)/2 𝑒−𝑡|𝑌|2
𝑆 = 𝛼𝑝
2 ∫ 𝐹𝑝+2
∗ 𝐹𝑝+2 + 𝑖𝑇𝑝 ∫
branes
𝐴𝑝+1
𝒜|field theory = (𝑖𝑇𝑝)2
𝛼𝑝
𝑉𝑝+1𝐺9−𝑝(|𝑌|)
𝑇𝑝
2
𝛼𝑝
= 2𝜋(4𝜋2𝛼′)3−𝑝
𝑇𝑝𝑇6−𝑝
𝛼𝑝
= 2𝜋𝑛
2𝜅10
2 = (2𝜋)7𝛼′4
𝑆𝑝 = −𝑇𝑝 ∫
𝑊𝑝+1
𝑑𝑝+1𝜉𝑒−Φ/2√det𝐺ˆ − 𝑖𝑇𝑝 ∫ 𝐴𝑝+1
𝐺ˆ𝛼𝛽 = 𝐺𝜇𝜈
𝜕𝑋𝜇
𝜕𝜉𝛼
𝜕𝑋𝜈
𝜕𝜉𝛽
∫ 𝐴𝑝+1 = 1
(𝑝 + 1)! ∫ 𝑑𝑝+1𝜉𝐴𝜇1⋯𝜇𝑝+1
𝜕𝑋𝜇1
𝜕𝜉𝛼1 ⋯ 𝜕𝑋𝜇𝑝+1
𝜕𝜉𝛼𝑝+1 𝜖𝛼1⋯𝛼𝑝+1
2𝜅10
2 𝑇𝑝𝑇6−𝑝 = 2𝜋𝑛
𝑇𝑝 = 1
(2𝜋)𝑝(𝛼′)(𝑝+1)/2
2𝜅11
2 = (2𝜋)8(𝛼′)9/2
𝑇2
𝑀 = 𝑇2 = 1
(2𝜋)2(𝛼′)3/2
2𝜅11
2 𝑇2
𝑀𝑇5
𝑀 = 2𝜋 → 𝑇5
𝑀 = 1
(2𝜋)5(𝛼′)3
2𝜋√𝛼′𝑇5
𝑀 = 𝑇4
pág. 61
𝑆𝐵𝐼 = ∫ 𝑑10𝑥𝑒−Φ/2√det(𝛿𝜇𝜈 + 2𝜋𝛼′𝐹𝜇𝜈)
𝑆𝐵 = 𝑖
2𝜋𝛼′ ∫
𝑀2
𝑑2𝜉𝜖𝛼𝛽𝐵𝜇𝜈𝜕𝑎𝑥𝜇𝜕𝛽𝑥𝜈 − 𝑖
2 ∫
𝐵1
𝑑𝑠𝐴𝜇𝜕𝑠𝑥𝜇
𝛿𝑆𝐵 = 𝑖
𝜋𝛼′ ∫
𝐵1
𝑑𝑠Λ𝜇𝜕𝑠𝑥𝜇
ℱ𝜇𝜈 = 2𝜋𝛼′𝐹𝜇𝜈 − 𝐵𝜇𝜈 = 2𝜋𝛼′(𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇) − 𝐵𝜇𝜈
𝑆𝑝 = −𝑇𝑝 ∫
𝑊𝑝+1
𝑑𝑝+1𝜉𝑒−Φ/2√det(𝐺ˆ + ℱ) − 𝑖𝑇𝑝 ∫ 𝐴𝑝+1
𝑆𝑝 = −𝑇𝑝 ∫
𝑊𝑝+1
𝑑𝑝+1𝜉𝑒−Φ
2 √det(𝐺ˆ + ℱ) ± 𝑖𝑇𝑝 ∫ 𝐴
𝑆1 = − 1
2𝜋𝛼′ [∫ 𝑑2𝜉 |𝑆|
√𝑆2
√det(𝐺ˆ + ℱ) + 𝑖 ∫ (𝐵𝑁 + 𝑖𝑆1
2𝜋 ℱ)]
𝑒−Φ/2 → √𝛼′
𝑅 𝑒−Φ/2
𝑇𝑝+1 = 𝑇𝑝
2𝜋√𝛼′
𝑆𝑝
𝑁 = −𝑇𝑝Str ∫
𝑊𝑝+1
𝑑𝑝+1𝜉𝑒−Φ/2(𝐹𝜇𝜈
2 + 2𝐹𝜇𝐼
2 + 𝐹𝐼𝐽
2 )
𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 + [𝐴𝜇, 𝐴𝜈]
𝐹𝜇𝐼 = 𝜕𝜇𝑋𝐼 + [𝐴𝜇, 𝑋𝐼] , 𝐹𝐼𝐽 = [𝑋𝐼, 𝑋𝐽]
𝑆𝐷=6
het = ∫ 𝑑6𝑥√−𝐺 [𝑅 − 1
4 𝜕𝜇Φ𝜕𝜇Φ − 𝑒−Φ
12 𝐻ˆ 𝜇𝜈𝜌𝐻ˆ𝜇𝜈𝜌 − 𝑒−Φ
2
4 (𝑀ˆ −1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1)]
𝑆𝐷=6
𝐼𝐼𝐴 = ∫ 𝑑6𝑥√−𝐺 [𝑅 − 1
4 𝜕𝜇Φ𝜕𝜇Φ
− 1
12 𝑒−Φ𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌 − 1
4 𝑒Φ/2(𝑀ˆ −1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1)]
+ 1
16 ∫ 𝑑6𝑥𝜖𝜇𝜈𝜌𝜎𝜏𝜀𝐵𝜇𝜈𝐹𝜌𝜎
𝑖 𝐿ˆ 𝑖𝑗𝐹𝜏𝜀
𝑗
Φ′ = −Φ , 𝐺𝜇𝜈
′ = 𝐺𝜇𝜈 , 𝑀ˆ ′ = 𝑀ˆ , 𝐴𝜇
′𝑖 = 𝐴𝜇
𝑖
𝑒−Φ𝐻ˆ𝜇𝜈𝜌 = 1
6
𝜖𝜇𝜈𝜌𝜎𝜏𝜀
√−𝐺 𝐻𝜎𝜏𝜀
′
𝑆𝐵𝐼 = ∫ 𝑑10𝑥𝑒−Φ/2√det(𝛿𝜇𝜈 + 2𝜋𝛼′𝐹𝜇𝜈)
𝑆𝐵 = 𝑖
2𝜋𝛼′ ∫
𝑀2
𝑑2𝜉𝜖𝛼𝛽𝐵𝜇𝜈𝜕𝑎𝑥𝜇𝜕𝛽𝑥𝜈 − 𝑖
2 ∫
𝐵1
𝑑𝑠𝐴𝜇𝜕𝑠𝑥𝜇
𝛿𝑆𝐵 = 𝑖
𝜋𝛼′ ∫
𝐵1
𝑑𝑠Λ𝜇𝜕𝑠𝑥𝜇
ℱ𝜇𝜈 = 2𝜋𝛼′𝐹𝜇𝜈 − 𝐵𝜇𝜈 = 2𝜋𝛼′(𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇) − 𝐵𝜇𝜈
𝑆𝑝 = −𝑇𝑝 ∫
𝑊𝑝+1
𝑑𝑝+1𝜉𝑒−Φ/2√det(𝐺ˆ + ℱ) − 𝑖𝑇𝑝 ∫ 𝐴𝑝+1
𝑆𝑝 = −𝑇𝑝 ∫
𝑊𝑝+1
𝑑𝑝+1𝜉𝑒−Φ
2 √det(𝐺ˆ + ℱ) ± 𝑖𝑇𝑝 ∫ 𝐴
𝑆1 = − 1
2𝜋𝛼′ [∫ 𝑑2𝜉 |𝑆|
√𝑆2
√det(𝐺ˆ + ℱ) + 𝑖 ∫ (𝐵𝑁 + 𝑖𝑆1
2𝜋 ℱ)]
𝑒−Φ/2 → √𝛼′
𝑅 𝑒−Φ/2
𝑇𝑝+1 = 𝑇𝑝
2𝜋√𝛼′
𝑆𝑝
𝑁 = −𝑇𝑝Str ∫
𝑊𝑝+1
𝑑𝑝+1𝜉𝑒−Φ/2(𝐹𝜇𝜈
2 + 2𝐹𝜇𝐼
2 + 𝐹𝐼𝐽
2 )
𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 + [𝐴𝜇, 𝐴𝜈]
𝐹𝜇𝐼 = 𝜕𝜇𝑋𝐼 + [𝐴𝜇, 𝑋𝐼] , 𝐹𝐼𝐽 = [𝑋𝐼, 𝑋𝐽]
𝑆𝐷=6
het = ∫ 𝑑6𝑥√−𝐺 [𝑅 − 1
4 𝜕𝜇Φ𝜕𝜇Φ − 𝑒−Φ
12 𝐻ˆ 𝜇𝜈𝜌𝐻ˆ𝜇𝜈𝜌 − 𝑒−Φ
2
4 (𝑀ˆ −1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1)]
𝑆𝐷=6
𝐼𝐼𝐴 = ∫ 𝑑6𝑥√−𝐺 [𝑅 − 1
4 𝜕𝜇Φ𝜕𝜇Φ
− 1
12 𝑒−Φ𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌 − 1
4 𝑒Φ/2(𝑀ˆ −1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1)]
+ 1
16 ∫ 𝑑6𝑥𝜖𝜇𝜈𝜌𝜎𝜏𝜀𝐵𝜇𝜈𝐹𝜌𝜎
𝑖 𝐿ˆ 𝑖𝑗𝐹𝜏𝜀
𝑗
Φ′ = −Φ , 𝐺𝜇𝜈
′ = 𝐺𝜇𝜈 , 𝑀ˆ ′ = 𝑀ˆ , 𝐴𝜇
′𝑖 = 𝐴𝜇
𝑖
𝑒−Φ𝐻ˆ𝜇𝜈𝜌 = 1
6
𝜖𝜇𝜈𝜌𝜎𝜏𝜀
√−𝐺 𝐻𝜎𝜏𝜀
′
pág. 62
𝜙 = Φ − 1
2 log [det𝐺𝛼𝛽]
𝑆𝐷=4
het = ∫ 𝑑4𝑥√−𝑔𝑒−𝜙[𝑅 + 𝐿𝐵 + 𝐿gauge + 𝐿scalar ]
ℒ𝑔+𝜙 = 𝑅 + 𝜕𝜇𝜙𝜕𝜇𝜙,
ℒ𝐵 = − 1
12 𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌,
𝐻𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 − 1
2 [𝐵𝜇𝛼𝐹𝜈𝜌
𝐴,𝛼 + 𝐴𝜇
𝛼𝐹𝑎,𝜈𝜌
𝐵 + 𝐿ˆ 𝑖𝑗𝐴𝜇
𝑖 𝐹𝜈𝜌
𝑗 ] + cyclic
≡ 𝜕𝜇𝐵𝜈𝜌 − 1
2 𝐿𝐼𝐽𝐴𝜇
𝐼 𝐹𝜈𝜌
𝐽 + cyclic
𝐿 =
(
0 0 1 0 0⃗
0 0 0 1 0⃗
1 0 0 0 0⃗
0 1 0 0 0⃗
0⃗ 0⃗ 0⃗ 0⃗ 𝐿ˆ )
𝐶𝛼𝛽 = 𝜖𝛼𝛽𝐵 − 1
2 𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝑌𝛽
𝑗
ℒgauge = − 1
4 {[(𝑀ˆ −1)𝑖𝑗 + 𝐿ˆ 𝑘𝑖𝐿ˆ 𝑙𝑗𝑌𝛼
𝑘𝐺𝛼𝛽𝑌𝛽
𝑙] 𝐹𝜇𝜈
𝑖 𝐹𝑗,𝜇𝜈 + 𝐺𝛼𝛽𝐹𝛼,𝜇𝜈
𝐵 𝐹𝐵,𝛽
𝜇𝜈
+ [𝐺𝛼𝛽 + 𝐶𝛾𝛼𝐺𝛾𝛿𝐶𝛿𝛽 + 𝑌𝛼
𝑖 (𝑀ˆ −1)𝑖𝑗𝑌𝛽
𝑗] 𝐹𝜇𝜈
𝐴,𝑎𝐹𝐴
𝛽,𝜇𝜈 − 2𝐺𝛼𝛾𝐶𝛾𝛽𝐹𝛼,𝜇𝜈
𝐵 𝐹𝐴,𝛽,𝜇𝜈
− 2𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝐺𝛼𝛽𝐹𝜇𝜈
𝑗 𝐹𝛽
𝐵,𝜇𝜈 +2 (𝑌𝛼
𝑖 (𝑀ˆ −1)𝑖𝑗 + 𝐶𝛾𝛼𝐺𝛾𝛽𝐿ˆ 𝑖𝑗𝑌𝛽
𝑖) 𝐹𝜇𝜈
𝑎,𝐴𝐹𝑗,𝜇𝜈}
≡ − 1
4 (𝑀−1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈
ℒscalar = 𝜕𝜇𝜙𝜕𝜇𝜙 + 1
8 Tr[𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1] + 1
2 𝐺𝛼𝛽(𝑀ˆ −1)𝑖𝑗𝜕𝜇𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗 + 1
4 𝜕𝜇𝐺𝛼𝛽𝜕𝜇𝐺𝛼𝛽
+ 1
2det𝐺 [𝜕𝜇𝐵 + 𝜖𝛼𝛽𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗] [𝜕𝜇𝐵 + 𝜖𝛼𝛽𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗]
= 𝜕𝜇𝜙𝜕𝜇𝜙 + 1
8 Tr[𝜕𝜇𝑀𝜕𝜇𝑀−1]
𝑔𝜇𝜈 → 𝑒−𝜙𝑔𝜇𝜈
𝑆𝐷=4
het,E = ∫ 𝑑4𝑥√−𝑔 [𝑅 − 1
2 𝜕𝜇𝜙𝜕𝜇𝜙
− 1
12 𝑒−2𝜙𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌 − 1
4 𝑒−𝜙(𝑀−1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝜙 = Φ − 1
2 log [det𝐺𝛼𝛽]
𝑆𝐷=4
het = ∫ 𝑑4𝑥√−𝑔𝑒−𝜙[𝑅 + 𝐿𝐵 + 𝐿gauge + 𝐿scalar ]
ℒ𝑔+𝜙 = 𝑅 + 𝜕𝜇𝜙𝜕𝜇𝜙,
ℒ𝐵 = − 1
12 𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌,
𝐻𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 − 1
2 [𝐵𝜇𝛼𝐹𝜈𝜌
𝐴,𝛼 + 𝐴𝜇
𝛼𝐹𝑎,𝜈𝜌
𝐵 + 𝐿ˆ 𝑖𝑗𝐴𝜇
𝑖 𝐹𝜈𝜌
𝑗 ] + cyclic
≡ 𝜕𝜇𝐵𝜈𝜌 − 1
2 𝐿𝐼𝐽𝐴𝜇
𝐼 𝐹𝜈𝜌
𝐽 + cyclic
𝐿 =
(
0 0 1 0 0⃗
0 0 0 1 0⃗
1 0 0 0 0⃗
0 1 0 0 0⃗
0⃗ 0⃗ 0⃗ 0⃗ 𝐿ˆ )
𝐶𝛼𝛽 = 𝜖𝛼𝛽𝐵 − 1
2 𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝑌𝛽
𝑗
ℒgauge = − 1
4 {[(𝑀ˆ −1)𝑖𝑗 + 𝐿ˆ 𝑘𝑖𝐿ˆ 𝑙𝑗𝑌𝛼
𝑘𝐺𝛼𝛽𝑌𝛽
𝑙] 𝐹𝜇𝜈
𝑖 𝐹𝑗,𝜇𝜈 + 𝐺𝛼𝛽𝐹𝛼,𝜇𝜈
𝐵 𝐹𝐵,𝛽
𝜇𝜈
+ [𝐺𝛼𝛽 + 𝐶𝛾𝛼𝐺𝛾𝛿𝐶𝛿𝛽 + 𝑌𝛼
𝑖 (𝑀ˆ −1)𝑖𝑗𝑌𝛽
𝑗] 𝐹𝜇𝜈
𝐴,𝑎𝐹𝐴
𝛽,𝜇𝜈 − 2𝐺𝛼𝛾𝐶𝛾𝛽𝐹𝛼,𝜇𝜈
𝐵 𝐹𝐴,𝛽,𝜇𝜈
− 2𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝐺𝛼𝛽𝐹𝜇𝜈
𝑗 𝐹𝛽
𝐵,𝜇𝜈 +2 (𝑌𝛼
𝑖 (𝑀ˆ −1)𝑖𝑗 + 𝐶𝛾𝛼𝐺𝛾𝛽𝐿ˆ 𝑖𝑗𝑌𝛽
𝑖) 𝐹𝜇𝜈
𝑎,𝐴𝐹𝑗,𝜇𝜈}
≡ − 1
4 (𝑀−1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈
ℒscalar = 𝜕𝜇𝜙𝜕𝜇𝜙 + 1
8 Tr[𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1] + 1
2 𝐺𝛼𝛽(𝑀ˆ −1)𝑖𝑗𝜕𝜇𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗 + 1
4 𝜕𝜇𝐺𝛼𝛽𝜕𝜇𝐺𝛼𝛽
+ 1
2det𝐺 [𝜕𝜇𝐵 + 𝜖𝛼𝛽𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗] [𝜕𝜇𝐵 + 𝜖𝛼𝛽𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗]
= 𝜕𝜇𝜙𝜕𝜇𝜙 + 1
8 Tr[𝜕𝜇𝑀𝜕𝜇𝑀−1]
𝑔𝜇𝜈 → 𝑒−𝜙𝑔𝜇𝜈
𝑆𝐷=4
het,E = ∫ 𝑑4𝑥√−𝑔 [𝑅 − 1
2 𝜕𝜇𝜙𝜕𝜇𝜙
− 1
12 𝑒−2𝜙𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌 − 1
4 𝑒−𝜙(𝑀−1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
pág. 63
𝑒−2𝜙𝐻𝜇𝜈𝜌 = 𝜖𝜇𝜈𝜌
𝜎
√−𝑔 𝜕𝜎𝑎
𝑆˜𝐷=4
het = ∫ 𝑑4𝑥√−𝑔 [𝑅 − 1
2 𝜕𝜇𝜙𝜕𝜇𝜙 − 1
2 𝑒2𝜙𝜕𝜇𝑎𝜕𝜇𝑎 − 1
4 𝑒−𝜙(𝑀−1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈
+ 1
4 𝑎𝐿𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹˜ 𝐽,𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝐹˜𝜇𝜈 = 1
2
𝜖𝜇𝜈𝜌𝜎
√−𝑔 𝐹𝜌𝜎
𝑆 = 𝑎 + 𝑖𝑒−𝜙
𝑆˜𝐷=4
het = ∫ 𝑑4𝑥√−𝑔 [𝑅 − 1
2
𝜕𝜇𝑆𝜕𝜇𝑆‾
Im𝑆2 − 1
4 Im𝑆(𝑀−1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈 + + 1
4 Re𝑆𝐿𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹˜ 𝐽,𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝑒−2𝜙𝐻𝜇𝜈𝜌 = 𝜖𝜇𝜈𝜌
𝜎
√−𝑔 [𝜕𝜎𝑎 + 1
2 𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝛿𝜎𝑌𝛽
𝑗𝜖𝛼𝛽]
𝑆˜𝐷=4
𝐼𝐼𝐴 = ∫ 𝑑4𝑥√−𝑔[𝑅 + ℒgauge
even + ℒgauge
odd + ℒscalar ]
ℒgauge
even = − 1
4 ∫ 𝑑4𝑥√−𝑔 [𝑒−𝜙𝐺𝛼𝛽(𝐹𝛼,𝜇𝜈
𝐵 − 𝐵𝛼𝛾𝐹𝜇𝜈
𝐴,𝛾) (𝐹𝛽
𝐵,𝜇𝜈 − 𝐵𝛼𝛿 𝐹𝐴
𝛿,𝜇𝜈)
+ 𝑒−𝜙𝐺𝛼𝛽𝐹𝜇𝜈
𝐴,𝛼𝐹𝐴
𝛽,𝜇𝜈 √det𝐺𝛼𝛽(𝑀ˆ −1)𝑖𝑗(𝐹𝜇𝜈
𝑖 + 𝑌𝛼
𝑖 𝐹𝜇𝜈
𝐴,𝛼) (𝐹𝑗,𝜇𝜈 + 𝑌𝛽
𝑗𝐹𝐴
𝛽,𝜇𝜈)]
ℒgauge
odd = 1
2 ∫ 𝑑4𝑥𝜖𝜇𝜈𝜌𝜎 [1
4 𝑎𝐹𝛼,𝜇𝜈
𝐵 𝐹𝜌𝜎
𝐴,𝛼
+ 1
2 𝜖𝛼𝛽𝐿ˆ 𝑖𝑗𝑌𝛽
𝑖𝐹𝛼,𝜇𝜈
𝐵 (𝐹𝜌𝜎
𝑗 + 1
2 𝑌𝛾
𝑗𝐹𝜌𝜎
𝐴,𝛾) − 1
8 𝜖𝛼𝛽𝐿ˆ 𝑖𝑗𝐵𝛼𝛽(𝐹𝜇𝜈
𝑖 + 𝑌𝛾
𝑖𝐹𝜇𝜈
𝐴,𝛾)(𝐹𝜌𝜎
𝑗 + 𝑌𝛿
𝑗𝐹𝜌𝜎
𝐴,𝛿 )]
ℒscalar = − 1
2 (𝜕𝜙)2 + 1
4 𝜕𝜇𝐺𝛼𝛽𝜕𝜇𝐺𝛼𝛽 − 1
2det𝐺 𝜕𝜇𝐵𝜕𝜇𝐵 + 1
8 Tr[𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1]
− 1
2 𝑒2𝜙 (𝜕𝜇𝑎 + 1
2 𝐿ˆ 𝑖𝑗𝜖𝛼𝛽𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗)
2
− 1
2 𝑒𝜙√det𝐺𝛼𝛽𝑀ˆ −1)
𝑖𝑗
𝐺𝛼𝛽𝜕𝜇𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗
𝑒−𝜙 = √det𝐺𝛼𝛽
′ , 𝑒−𝜙′
= √det𝐺𝛼𝛽
𝐺𝛼𝛽
√det𝐺𝛼𝛽
= 𝐺𝛼𝛽
′
√det𝐺𝛼𝛽
′
, 𝐴𝜇
′𝛼 = 𝐴𝜇
𝛼
𝑔𝜇𝜈 = 𝑔𝜇𝜈
′ Einstein frame
𝑀ˆ ′ = 𝑀ˆ , 𝐴𝜇
𝑖 = 𝐴𝜇
′𝑖 , 𝑌𝛼
𝑖 = 𝑌𝛼
′𝑖
𝑒−2𝜙𝐻𝜇𝜈𝜌 = 𝜖𝜇𝜈𝜌
𝜎
√−𝑔 𝜕𝜎𝑎
𝑆˜𝐷=4
het = ∫ 𝑑4𝑥√−𝑔 [𝑅 − 1
2 𝜕𝜇𝜙𝜕𝜇𝜙 − 1
2 𝑒2𝜙𝜕𝜇𝑎𝜕𝜇𝑎 − 1
4 𝑒−𝜙(𝑀−1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈
+ 1
4 𝑎𝐿𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹˜ 𝐽,𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝐹˜𝜇𝜈 = 1
2
𝜖𝜇𝜈𝜌𝜎
√−𝑔 𝐹𝜌𝜎
𝑆 = 𝑎 + 𝑖𝑒−𝜙
𝑆˜𝐷=4
het = ∫ 𝑑4𝑥√−𝑔 [𝑅 − 1
2
𝜕𝜇𝑆𝜕𝜇𝑆‾
Im𝑆2 − 1
4 Im𝑆(𝑀−1)𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈 + + 1
4 Re𝑆𝐿𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹˜ 𝐽,𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝑒−2𝜙𝐻𝜇𝜈𝜌 = 𝜖𝜇𝜈𝜌
𝜎
√−𝑔 [𝜕𝜎𝑎 + 1
2 𝐿ˆ 𝑖𝑗𝑌𝛼
𝑖 𝛿𝜎𝑌𝛽
𝑗𝜖𝛼𝛽]
𝑆˜𝐷=4
𝐼𝐼𝐴 = ∫ 𝑑4𝑥√−𝑔[𝑅 + ℒgauge
even + ℒgauge
odd + ℒscalar ]
ℒgauge
even = − 1
4 ∫ 𝑑4𝑥√−𝑔 [𝑒−𝜙𝐺𝛼𝛽(𝐹𝛼,𝜇𝜈
𝐵 − 𝐵𝛼𝛾𝐹𝜇𝜈
𝐴,𝛾) (𝐹𝛽
𝐵,𝜇𝜈 − 𝐵𝛼𝛿 𝐹𝐴
𝛿,𝜇𝜈)
+ 𝑒−𝜙𝐺𝛼𝛽𝐹𝜇𝜈
𝐴,𝛼𝐹𝐴
𝛽,𝜇𝜈 √det𝐺𝛼𝛽(𝑀ˆ −1)𝑖𝑗(𝐹𝜇𝜈
𝑖 + 𝑌𝛼
𝑖 𝐹𝜇𝜈
𝐴,𝛼) (𝐹𝑗,𝜇𝜈 + 𝑌𝛽
𝑗𝐹𝐴
𝛽,𝜇𝜈)]
ℒgauge
odd = 1
2 ∫ 𝑑4𝑥𝜖𝜇𝜈𝜌𝜎 [1
4 𝑎𝐹𝛼,𝜇𝜈
𝐵 𝐹𝜌𝜎
𝐴,𝛼
+ 1
2 𝜖𝛼𝛽𝐿ˆ 𝑖𝑗𝑌𝛽
𝑖𝐹𝛼,𝜇𝜈
𝐵 (𝐹𝜌𝜎
𝑗 + 1
2 𝑌𝛾
𝑗𝐹𝜌𝜎
𝐴,𝛾) − 1
8 𝜖𝛼𝛽𝐿ˆ 𝑖𝑗𝐵𝛼𝛽(𝐹𝜇𝜈
𝑖 + 𝑌𝛾
𝑖𝐹𝜇𝜈
𝐴,𝛾)(𝐹𝜌𝜎
𝑗 + 𝑌𝛿
𝑗𝐹𝜌𝜎
𝐴,𝛿 )]
ℒscalar = − 1
2 (𝜕𝜙)2 + 1
4 𝜕𝜇𝐺𝛼𝛽𝜕𝜇𝐺𝛼𝛽 − 1
2det𝐺 𝜕𝜇𝐵𝜕𝜇𝐵 + 1
8 Tr[𝜕𝜇𝑀ˆ 𝜕𝜇𝑀ˆ −1]
− 1
2 𝑒2𝜙 (𝜕𝜇𝑎 + 1
2 𝐿ˆ 𝑖𝑗𝜖𝛼𝛽𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗)
2
− 1
2 𝑒𝜙√det𝐺𝛼𝛽𝑀ˆ −1)
𝑖𝑗
𝐺𝛼𝛽𝜕𝜇𝑌𝛼
𝑖 𝜕𝜇𝑌𝛽
𝑗
𝑒−𝜙 = √det𝐺𝛼𝛽
′ , 𝑒−𝜙′
= √det𝐺𝛼𝛽
𝐺𝛼𝛽
√det𝐺𝛼𝛽
= 𝐺𝛼𝛽
′
√det𝐺𝛼𝛽
′
, 𝐴𝜇
′𝛼 = 𝐴𝜇
𝛼
𝑔𝜇𝜈 = 𝑔𝜇𝜈
′ Einstein frame
𝑀ˆ ′ = 𝑀ˆ , 𝐴𝜇
𝑖 = 𝐴𝜇
′𝑖 , 𝑌𝛼
𝑖 = 𝑌𝛼
′𝑖
pág. 64
𝐴 = 𝐵′ , 𝐴′ = 𝐵
1
2
𝜖𝜇𝜈
𝜌𝜎
√−𝑔 𝜖𝛼𝛽𝐹𝛽,𝜌𝜎
𝐵′
= 𝑒−𝜙𝐺𝛼𝛽[𝐹𝛽,𝜇𝜈
𝐵 − 𝐶𝛽𝛾𝐹𝜇𝜈
𝐴,𝛾 − 𝐿ˆ 𝑖𝑗𝑌𝛽
𝑖𝐹𝜇𝜈
𝑗 ] − 1
2 𝑎 𝜖𝜇𝜈
𝜌𝜎
√−𝑔 𝐹𝜌𝜎
𝐴,𝛼
𝑊𝑖 = 𝑊1
𝑖 + 𝑖𝑊2
𝑖 = −𝑌2
𝑖 + 𝑈𝑌1
𝑖
𝐺𝛼𝛽 =
𝑇2 − ∑𝑖 (𝑊2
𝑖)2
2𝑈2
𝑈2
( 1 𝑈1
𝑈1 |𝑈|2) , 𝐵 = 𝑇1 − ∑𝑖 𝑊1
𝑖𝑊2
𝑖
2𝑈2
ℒscalar
het = − 1
2 𝜕𝑧𝑖 𝜕𝑧‾𝑗 𝐾(𝑧𝑘, 𝑧‾𝑘)𝜕𝜇𝑧𝑖𝜕𝜇𝑧‾𝑗
𝐾 = log [𝑆2 (𝑇2𝑈2 − 1
2 ∑
𝑖
(𝑊2
𝑖)2)]
𝐺𝛼𝛽 = 𝑇2
𝑈2
( 1 𝑈1
𝑈1 |𝑈|2) , 𝐵 = 𝑇1
𝑆 = 𝑎 − ∑𝑖 𝑊1
𝑖𝑊2
𝑖
2𝑈2
+ 𝑖 (𝑒−𝜙 − ∑𝑖 (𝑊2
𝑖)2
2𝑈2
)
𝑆′ = 𝑇 , 𝑇′ = 𝑆 , 𝑈 = 𝑈′ , 𝑊𝑖 = 𝑊′𝑖
(
𝑚1
𝑚2
𝑛1
𝑛2
𝑞𝑖 )
→
(
𝑚1
𝑚2
𝑛˜ 2
−𝑛˜1
𝑞𝑖 )
,
(
𝑚˜ 1
𝑚˜ 2
𝑛˜1
𝑛˜ 2
𝑞˜ 𝑖 )
→
(
𝑚˜ 1
𝑚˜ 2
−𝑛2
𝑛1
𝑞˜ 𝑖 )
𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏) = ∑
𝑛∈𝑍
𝑞1
2(𝑛−𝑎
2)2
𝑒2𝜋𝑖(𝑣−𝑏
2)(𝑛−𝑎
2)
𝜗 [𝑎 + 2
𝑏 ] (𝑣 ∣ 𝜏) = 𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏) , 𝜗 [ 𝑎
𝑏 + 2] (𝑣 ∣ 𝜏) = 𝑒𝑖𝜋𝑎𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏),
𝜗 [−𝑎
−𝑏] (𝑣 ∣ 𝜏) = 𝜗 [𝑎
𝑏] (−𝑣 ∣ 𝜏) , 𝜗 [𝑎
𝑏] (−𝑣 ∣ 𝜏) = 𝑒𝑖𝜋𝑎𝑏𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏) (𝑎, 𝑏 ∈ 𝑍).
𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏 + 1) = 𝑒−𝑖𝜋
4 𝑎(𝑎−2)𝜗 [ 𝑎
𝑎 + 𝑏 − 1] (𝑣 ∣ 𝜏),
𝜗 [𝑎
𝑏] (𝑣
𝜏 | − 1
𝜏) = √−𝑖𝜏𝑒𝑖𝜋
2 𝑎𝑏+𝑖𝜋𝑣2
𝜏 𝜗 [ 𝑏
−𝑎] (𝑣 ∣ 𝜏).
𝐴 = 𝐵′ , 𝐴′ = 𝐵
1
2
𝜖𝜇𝜈
𝜌𝜎
√−𝑔 𝜖𝛼𝛽𝐹𝛽,𝜌𝜎
𝐵′
= 𝑒−𝜙𝐺𝛼𝛽[𝐹𝛽,𝜇𝜈
𝐵 − 𝐶𝛽𝛾𝐹𝜇𝜈
𝐴,𝛾 − 𝐿ˆ 𝑖𝑗𝑌𝛽
𝑖𝐹𝜇𝜈
𝑗 ] − 1
2 𝑎 𝜖𝜇𝜈
𝜌𝜎
√−𝑔 𝐹𝜌𝜎
𝐴,𝛼
𝑊𝑖 = 𝑊1
𝑖 + 𝑖𝑊2
𝑖 = −𝑌2
𝑖 + 𝑈𝑌1
𝑖
𝐺𝛼𝛽 =
𝑇2 − ∑𝑖 (𝑊2
𝑖)2
2𝑈2
𝑈2
( 1 𝑈1
𝑈1 |𝑈|2) , 𝐵 = 𝑇1 − ∑𝑖 𝑊1
𝑖𝑊2
𝑖
2𝑈2
ℒscalar
het = − 1
2 𝜕𝑧𝑖 𝜕𝑧‾𝑗 𝐾(𝑧𝑘, 𝑧‾𝑘)𝜕𝜇𝑧𝑖𝜕𝜇𝑧‾𝑗
𝐾 = log [𝑆2 (𝑇2𝑈2 − 1
2 ∑
𝑖
(𝑊2
𝑖)2)]
𝐺𝛼𝛽 = 𝑇2
𝑈2
( 1 𝑈1
𝑈1 |𝑈|2) , 𝐵 = 𝑇1
𝑆 = 𝑎 − ∑𝑖 𝑊1
𝑖𝑊2
𝑖
2𝑈2
+ 𝑖 (𝑒−𝜙 − ∑𝑖 (𝑊2
𝑖)2
2𝑈2
)
𝑆′ = 𝑇 , 𝑇′ = 𝑆 , 𝑈 = 𝑈′ , 𝑊𝑖 = 𝑊′𝑖
(
𝑚1
𝑚2
𝑛1
𝑛2
𝑞𝑖 )
→
(
𝑚1
𝑚2
𝑛˜ 2
−𝑛˜1
𝑞𝑖 )
,
(
𝑚˜ 1
𝑚˜ 2
𝑛˜1
𝑛˜ 2
𝑞˜ 𝑖 )
→
(
𝑚˜ 1
𝑚˜ 2
−𝑛2
𝑛1
𝑞˜ 𝑖 )
𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏) = ∑
𝑛∈𝑍
𝑞1
2(𝑛−𝑎
2)2
𝑒2𝜋𝑖(𝑣−𝑏
2)(𝑛−𝑎
2)
𝜗 [𝑎 + 2
𝑏 ] (𝑣 ∣ 𝜏) = 𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏) , 𝜗 [ 𝑎
𝑏 + 2] (𝑣 ∣ 𝜏) = 𝑒𝑖𝜋𝑎𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏),
𝜗 [−𝑎
−𝑏] (𝑣 ∣ 𝜏) = 𝜗 [𝑎
𝑏] (−𝑣 ∣ 𝜏) , 𝜗 [𝑎
𝑏] (−𝑣 ∣ 𝜏) = 𝑒𝑖𝜋𝑎𝑏𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏) (𝑎, 𝑏 ∈ 𝑍).
𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏 + 1) = 𝑒−𝑖𝜋
4 𝑎(𝑎−2)𝜗 [ 𝑎
𝑎 + 𝑏 − 1] (𝑣 ∣ 𝜏),
𝜗 [𝑎
𝑏] (𝑣
𝜏 | − 1
𝜏) = √−𝑖𝜏𝑒𝑖𝜋
2 𝑎𝑏+𝑖𝜋𝑣2
𝜏 𝜗 [ 𝑏
−𝑎] (𝑣 ∣ 𝜏).
pág. 65
𝜗1(𝑣 ∣ 𝜏) = 2𝑞1
8sin [𝜋𝑣] ∏
∞
𝑛=1
(1 − 𝑞𝑛)(1 − 𝑞𝑛𝑒2𝜋𝑖𝑣)(1 − 𝑞𝑛𝑒−2𝜋𝑖𝑣)
𝜗2(𝑣 ∣ 𝜏) = 2𝑞1
8cos [𝜋𝑣] ∏
∞
𝑛=1
(1 − 𝑞𝑛)(1 + 𝑞𝑛𝑒2𝜋𝑖𝑣)(1 + 𝑞𝑛𝑒−2𝜋𝑖𝑣)
𝜗3(𝑣 ∣ 𝜏) = ∏
∞
𝑛=1
(1 − 𝑞𝑛)(1 + 𝑞𝑛−1/2𝑒2𝜋𝑖𝑣)(1 + 𝑞𝑛−1/2𝑒−2𝜋𝑖𝑣)
𝜗4(𝑣 ∣ 𝜏) = ∏
∞
𝑛=1
(1 − 𝑞𝑛)(1 − 𝑞𝑛−1/2𝑒2𝜋𝑖𝑣)(1 − 𝑞𝑛−1/2𝑒−2𝜋𝑖𝑣)
𝜂(𝜏) = 𝑞 1
24 ∏
∞
𝑛=1
(1 − 𝑞𝑛).
𝜕
𝜕𝑣 𝜗1(𝑣)|𝑣=0
≡ 𝜗1
′ = 2𝜋𝜂3(𝜏)
𝜂 (− 1
𝜏) = √−𝑖𝜏𝜂(𝜏)
𝜗 [𝑎
𝑏] (𝑣 + 𝜖1
2 𝜏 + 𝜖2
2 | 𝜏) = 𝑒−𝑖𝜋𝜏
4 𝜖1
2−𝑖𝜋𝜖1
2 (2𝑣−𝑏)−𝑖𝜋
2 𝜖1𝜖2 𝜗 [𝑎 − 𝜖1
𝑏 − 𝜖2] (𝑣 ∣ 𝜏)
𝜗2(0 ∣ 𝜏)𝜗3(0 ∣ 𝜏)𝜗4(0 ∣ 𝜏) = 2𝜂3
𝜗2
4(𝑣 ∣ 𝜏) − 𝜗1
4(𝑣 ∣ 𝜏) = 𝜗3
4(𝑣 ∣ 𝜏) − 𝜗4
4(𝑣 ∣ 𝜏)
𝜗2(2𝜏) = 1
√2 √𝜗3
2(𝜏) − 𝜗4
2(𝜏) , 𝜗3(2𝜏) = 1
√2 √𝜗3
2(𝜏) + 𝜗4
2(𝜏)
𝜗4(2𝜏) = √𝜗3(𝜏)𝜗4(𝜏) , 𝜂(2𝜏) = √𝜗2(𝜏)𝜂(𝜏)
2
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 ∏
4
𝑖=1
𝜗 [𝑎
𝑏] (𝑣𝑖) = − ∏
4
𝑖=1
𝜗1(𝑣𝑖
′),
𝑣1
′ = 1
2 (−𝑣1 + 𝑣2 + 𝑣3 + 𝑣4) , 𝑣2
′ = 1
2 (𝑣1 − 𝑣2 + 𝑣3 + 𝑣4)
𝑣3
′ = 1
2 (𝑣1 + 𝑣2 − 𝑣3 + 𝑣4) , 𝑣4
′ = 1
2 (𝑣1 + 𝑣2 + 𝑣3 − 𝑣4)
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 ∏
4
𝑖=1
𝜗 [𝑎 + ℎ𝑖
𝑏 + 𝑔𝑖
] (𝑣𝑖) = − ∏
4
𝑖=1
𝜗 [1 − ℎ𝑖
1 − 𝑔𝑖
] (𝑣𝑖
′)
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏 ∏
4
𝑖=1
𝜗 [𝑎
𝑏] (𝑣𝑖) = − ∏
4
𝑖=1
𝜗1(𝑣𝑖
′) + ∏
4
𝑖=1
𝜗1(𝑣𝑖)
𝜗1(𝑣 ∣ 𝜏) = 2𝑞1
8sin [𝜋𝑣] ∏
∞
𝑛=1
(1 − 𝑞𝑛)(1 − 𝑞𝑛𝑒2𝜋𝑖𝑣)(1 − 𝑞𝑛𝑒−2𝜋𝑖𝑣)
𝜗2(𝑣 ∣ 𝜏) = 2𝑞1
8cos [𝜋𝑣] ∏
∞
𝑛=1
(1 − 𝑞𝑛)(1 + 𝑞𝑛𝑒2𝜋𝑖𝑣)(1 + 𝑞𝑛𝑒−2𝜋𝑖𝑣)
𝜗3(𝑣 ∣ 𝜏) = ∏
∞
𝑛=1
(1 − 𝑞𝑛)(1 + 𝑞𝑛−1/2𝑒2𝜋𝑖𝑣)(1 + 𝑞𝑛−1/2𝑒−2𝜋𝑖𝑣)
𝜗4(𝑣 ∣ 𝜏) = ∏
∞
𝑛=1
(1 − 𝑞𝑛)(1 − 𝑞𝑛−1/2𝑒2𝜋𝑖𝑣)(1 − 𝑞𝑛−1/2𝑒−2𝜋𝑖𝑣)
𝜂(𝜏) = 𝑞 1
24 ∏
∞
𝑛=1
(1 − 𝑞𝑛).
𝜕
𝜕𝑣 𝜗1(𝑣)|𝑣=0
≡ 𝜗1
′ = 2𝜋𝜂3(𝜏)
𝜂 (− 1
𝜏) = √−𝑖𝜏𝜂(𝜏)
𝜗 [𝑎
𝑏] (𝑣 + 𝜖1
2 𝜏 + 𝜖2
2 | 𝜏) = 𝑒−𝑖𝜋𝜏
4 𝜖1
2−𝑖𝜋𝜖1
2 (2𝑣−𝑏)−𝑖𝜋
2 𝜖1𝜖2 𝜗 [𝑎 − 𝜖1
𝑏 − 𝜖2] (𝑣 ∣ 𝜏)
𝜗2(0 ∣ 𝜏)𝜗3(0 ∣ 𝜏)𝜗4(0 ∣ 𝜏) = 2𝜂3
𝜗2
4(𝑣 ∣ 𝜏) − 𝜗1
4(𝑣 ∣ 𝜏) = 𝜗3
4(𝑣 ∣ 𝜏) − 𝜗4
4(𝑣 ∣ 𝜏)
𝜗2(2𝜏) = 1
√2 √𝜗3
2(𝜏) − 𝜗4
2(𝜏) , 𝜗3(2𝜏) = 1
√2 √𝜗3
2(𝜏) + 𝜗4
2(𝜏)
𝜗4(2𝜏) = √𝜗3(𝜏)𝜗4(𝜏) , 𝜂(2𝜏) = √𝜗2(𝜏)𝜂(𝜏)
2
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 ∏
4
𝑖=1
𝜗 [𝑎
𝑏] (𝑣𝑖) = − ∏
4
𝑖=1
𝜗1(𝑣𝑖
′),
𝑣1
′ = 1
2 (−𝑣1 + 𝑣2 + 𝑣3 + 𝑣4) , 𝑣2
′ = 1
2 (𝑣1 − 𝑣2 + 𝑣3 + 𝑣4)
𝑣3
′ = 1
2 (𝑣1 + 𝑣2 − 𝑣3 + 𝑣4) , 𝑣4
′ = 1
2 (𝑣1 + 𝑣2 + 𝑣3 − 𝑣4)
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 ∏
4
𝑖=1
𝜗 [𝑎 + ℎ𝑖
𝑏 + 𝑔𝑖
] (𝑣𝑖) = − ∏
4
𝑖=1
𝜗 [1 − ℎ𝑖
1 − 𝑔𝑖
] (𝑣𝑖
′)
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏 ∏
4
𝑖=1
𝜗 [𝑎
𝑏] (𝑣𝑖) = − ∏
4
𝑖=1
𝜗1(𝑣𝑖
′) + ∏
4
𝑖=1
𝜗1(𝑣𝑖)
pág. 66
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏 ∏
4
𝑖=1
𝜗 [𝑎 + ℎ𝑖
𝑏 + 𝑔𝑖
] (𝑣𝑖) = − ∏
4
𝑖=1
𝜗 [1 − ℎ𝑖
1 − 𝑔𝑖
] (𝑣𝑖
′) + ∏
4
𝑖=1
𝜗 [1 + ℎ𝑖
1 + 𝑔𝑖
] (𝑣𝑖)
[ 1
(2𝜋𝑖)2
𝜕2
𝜕𝑣2 − 1
𝑖𝜋
𝜕
𝜕𝜏] 𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏) = 0
1
4𝜋𝑖
𝜗2
′′
𝜗2
= 𝜕𝜏log 𝜗2 = 𝑖𝜋
12 (𝐸2 + 𝜗3
4 + 𝜗4
4)
1
4𝜋𝑖
𝜗3
′′
𝜗3
= 𝜕𝜏log 𝜗3 = 𝑖𝜋
12 (𝐸2 + 𝜗2
4 − 𝜗4
4)
1
4𝜋𝑖
𝜗4
′′
𝜗4
= 𝜕𝜏log 𝜗4 = 𝑖𝜋
12 (𝐸2 − 𝜗2
4 − 𝜗3
4)
𝒫(𝑧) = 4𝜋𝑖𝜕𝜏log 𝜂(𝜏) − 𝜕𝑧
2log 𝜗1(𝑧) = 1
𝑧2 + 𝒪(𝑧2)
𝒫(−𝑧) = 𝒫(𝑧) , 𝒫(𝑧 + 1) = 𝒫(𝑧 + 𝜏) = 𝒫(𝑧)
𝒫(𝑧, 𝜏 + 1) = 𝒫(𝑧, 𝜏) , 𝒫 (𝑧
𝜏 , − 1
𝜏) = 𝜏2𝒫(𝑧, 𝜏)
∫ 𝑑2𝑧
𝜏2
𝒫(𝑧, 𝜏) = 4𝜋𝑖𝜕𝜏log (√𝜏2𝜂)
∫ 𝑑2𝑧
𝜏2
|𝒫(𝑧, 𝜏)|2 = |4𝜋𝑖𝜕𝜏log (√𝜏2𝜂)|2
∫ 𝑑2𝑧
𝜏2
𝒫(𝑧‾, 𝜏‾) [𝜕𝑧log 𝜗1(𝑧) + 2𝜋𝑖 𝐼𝑚𝑧
𝜏2
]
2
= 4𝜋𝑖𝜕𝜏log (𝜂√𝜏2)
∫ 𝑑2𝑧
𝜏2
𝜕𝑧
2log 𝜗1(𝑧) = − 𝜋
𝜏2
𝑓˜(𝑘) ≡ 1
2𝜋 ∫
+∞
−∞
𝑓(𝑥)𝑒𝑖𝑘𝑥𝑑𝑥
∑
𝑛∈𝑍
𝑓(2𝜋𝑛) = ∑
𝑛∈𝑍
𝑓˜(𝑛).
∑
𝑛∈𝑍
𝑒−𝜋𝑎𝑛2+𝜋𝑏𝑛 = 1
√𝑎 ∑
𝑛∈𝑍
𝑒−𝜋
𝑎(𝑛+𝑖𝑏
2)
2
,
∑
𝑛∈𝑍
𝑛𝑒−𝜋𝑎𝑛2+𝜋𝑏𝑛 = − 𝑖
√𝑎 ∑
𝑛∈𝑍
(𝑛 + 𝑖 𝑏
2)
𝑎 𝑒−𝜋
𝑎(𝑛+𝑖𝑏
2)
2
,
∑
𝑛∈𝑍
𝑛2𝑒−𝜋𝑎𝑛2+𝜋𝑏𝑛 = 1
√𝑎 ∑
𝑛∈𝑍
[ 1
2𝜋𝑎 − (𝑛 + 𝑖 𝑏
2)
2
𝑎2 ] 𝑒−𝜋
𝑎(𝑛+𝑖𝑏
2)
2
.
∑
𝑚𝑖∈𝑍
𝑒−𝜋𝑚𝑖𝑚𝑗𝐴𝑖𝑗+𝜋𝐵𝑖𝑚𝑖 = (det𝐴)−1
2 ∑
𝑚𝑖∈𝑍
𝑒−𝜋(𝑚𝑘+𝑖𝐵𝑘/2)(𝐴−1)𝑘𝑙(𝑚𝑙+𝑖𝐵𝑙/2)
1
2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏 ∏
4
𝑖=1
𝜗 [𝑎 + ℎ𝑖
𝑏 + 𝑔𝑖
] (𝑣𝑖) = − ∏
4
𝑖=1
𝜗 [1 − ℎ𝑖
1 − 𝑔𝑖
] (𝑣𝑖
′) + ∏
4
𝑖=1
𝜗 [1 + ℎ𝑖
1 + 𝑔𝑖
] (𝑣𝑖)
[ 1
(2𝜋𝑖)2
𝜕2
𝜕𝑣2 − 1
𝑖𝜋
𝜕
𝜕𝜏] 𝜗 [𝑎
𝑏] (𝑣 ∣ 𝜏) = 0
1
4𝜋𝑖
𝜗2
′′
𝜗2
= 𝜕𝜏log 𝜗2 = 𝑖𝜋
12 (𝐸2 + 𝜗3
4 + 𝜗4
4)
1
4𝜋𝑖
𝜗3
′′
𝜗3
= 𝜕𝜏log 𝜗3 = 𝑖𝜋
12 (𝐸2 + 𝜗2
4 − 𝜗4
4)
1
4𝜋𝑖
𝜗4
′′
𝜗4
= 𝜕𝜏log 𝜗4 = 𝑖𝜋
12 (𝐸2 − 𝜗2
4 − 𝜗3
4)
𝒫(𝑧) = 4𝜋𝑖𝜕𝜏log 𝜂(𝜏) − 𝜕𝑧
2log 𝜗1(𝑧) = 1
𝑧2 + 𝒪(𝑧2)
𝒫(−𝑧) = 𝒫(𝑧) , 𝒫(𝑧 + 1) = 𝒫(𝑧 + 𝜏) = 𝒫(𝑧)
𝒫(𝑧, 𝜏 + 1) = 𝒫(𝑧, 𝜏) , 𝒫 (𝑧
𝜏 , − 1
𝜏) = 𝜏2𝒫(𝑧, 𝜏)
∫ 𝑑2𝑧
𝜏2
𝒫(𝑧, 𝜏) = 4𝜋𝑖𝜕𝜏log (√𝜏2𝜂)
∫ 𝑑2𝑧
𝜏2
|𝒫(𝑧, 𝜏)|2 = |4𝜋𝑖𝜕𝜏log (√𝜏2𝜂)|2
∫ 𝑑2𝑧
𝜏2
𝒫(𝑧‾, 𝜏‾) [𝜕𝑧log 𝜗1(𝑧) + 2𝜋𝑖 𝐼𝑚𝑧
𝜏2
]
2
= 4𝜋𝑖𝜕𝜏log (𝜂√𝜏2)
∫ 𝑑2𝑧
𝜏2
𝜕𝑧
2log 𝜗1(𝑧) = − 𝜋
𝜏2
𝑓˜(𝑘) ≡ 1
2𝜋 ∫
+∞
−∞
𝑓(𝑥)𝑒𝑖𝑘𝑥𝑑𝑥
∑
𝑛∈𝑍
𝑓(2𝜋𝑛) = ∑
𝑛∈𝑍
𝑓˜(𝑛).
∑
𝑛∈𝑍
𝑒−𝜋𝑎𝑛2+𝜋𝑏𝑛 = 1
√𝑎 ∑
𝑛∈𝑍
𝑒−𝜋
𝑎(𝑛+𝑖𝑏
2)
2
,
∑
𝑛∈𝑍
𝑛𝑒−𝜋𝑎𝑛2+𝜋𝑏𝑛 = − 𝑖
√𝑎 ∑
𝑛∈𝑍
(𝑛 + 𝑖 𝑏
2)
𝑎 𝑒−𝜋
𝑎(𝑛+𝑖𝑏
2)
2
,
∑
𝑛∈𝑍
𝑛2𝑒−𝜋𝑎𝑛2+𝜋𝑏𝑛 = 1
√𝑎 ∑
𝑛∈𝑍
[ 1
2𝜋𝑎 − (𝑛 + 𝑖 𝑏
2)
2
𝑎2 ] 𝑒−𝜋
𝑎(𝑛+𝑖𝑏
2)
2
.
∑
𝑚𝑖∈𝑍
𝑒−𝜋𝑚𝑖𝑚𝑗𝐴𝑖𝑗+𝜋𝐵𝑖𝑚𝑖 = (det𝐴)−1
2 ∑
𝑚𝑖∈𝑍
𝑒−𝜋(𝑚𝑘+𝑖𝐵𝑘/2)(𝐴−1)𝑘𝑙(𝑚𝑙+𝑖𝐵𝑙/2)
pág. 67
𝑆𝑝,𝑞 = 1
4𝜋 ∫ 𝑑2𝜎√det𝑔𝑔𝑎𝑏𝐺𝛼𝛽𝜕𝑎𝑋𝛼𝜕𝑏𝑋𝛽 + 1
4𝜋 ∫ 𝑑2𝜎𝜖𝑎𝑏𝐵𝛼𝛽𝜕𝑎𝑋𝛼𝜕𝑏𝑋𝛽
+ 1
4𝜋 ∫ 𝑑2𝜎√det𝑔 ∑
𝐼
𝜓𝐼(∇‾ + 𝑌𝛼
𝐼(∇‾ 𝑋𝛼)𝜓‾𝐼
𝑍𝑝,𝑝+16(𝐺, 𝐵, 𝑌) = √detG
𝜏2
𝑝/2𝜂𝑝𝜂‾𝑝+16
× ∑
𝑚𝛼,𝑛𝛼∈𝑍
exp [− 𝜋
𝜏2
(𝐺 + 𝐵)𝛼𝛽(𝑚𝛼 + 𝜏𝑛𝛼)(𝑚𝛽 + 𝜏‾𝑛𝛽)]
× 1
2 ∑
1
𝑎,𝑏=0
∏
16
𝐼=1
𝑒𝑖𝜋(𝑚𝛼𝑌𝛼
𝐼𝑌𝛽
𝐼 𝑛𝛽−𝑏𝑛𝛼𝑌𝛼
𝐼)𝜗‾ [ 𝑎 − 2𝑛𝛼𝑌𝛼
𝐼
𝑏 − 2𝑚𝛽𝑌𝛽
𝐼]
= √detG
𝜏2
𝑝/2𝜂𝑝𝜂‾𝑝+16 ∑
𝑚𝛼,𝑛𝛼∈𝑍
exp [− 𝜋
𝜏2
(𝐺 + 𝐵)𝛼𝛽(𝑚𝛼 + 𝜏𝑛𝛼)(𝑚𝛽 + 𝜏‾𝑛𝛽)]
× exp [−𝑖𝜋 ∑
𝐼
𝑛𝛼(𝑚𝛽 + 𝜏‾𝑛𝛽)𝑌𝛼
𝐼𝑌𝛽
𝐼] 1
2 ∑
1
𝑎,𝑏=0
∏
16
𝐼=1
𝜗‾ [𝑎
𝑏] (𝑌𝛾
𝐼(𝑚𝛾 + 𝜏‾𝑛𝛾) ∣ 𝜏‾)
𝜏 → 𝜏 + 1 , 𝑍𝑝,𝑝+16 → 𝑒4𝜋𝑖/3𝑍𝑝,𝑝+16
Γ𝑝,𝑝+16(𝐺, 𝐵, 𝑌) = ∑
𝑚𝛼,𝑛𝛼,𝑄𝐼
𝑞𝑃𝐿
2/2𝑞‾𝑃𝑅
2/2
𝑀 = (
𝐺−1 𝐺−1𝐶 𝐺−1𝑌𝑡
𝐶𝑡𝐺−1 𝐺 + 𝐶𝑡𝐺−1𝐶 + 𝑌𝑡𝑌 𝐶𝑡𝐺−1𝑌𝑡 + 𝑌𝑡
𝑌𝐺−1 𝑌𝐺−1𝐶 + 𝑌 𝟏16 + 𝑌𝐺−1𝑌𝑡
)
𝐶𝛼𝛽 = 𝐵𝛼𝛽 − 1
2 𝑌𝛼
𝐼𝑌𝛽
𝐼
𝐿 = (
0 𝟏𝑝 0
𝟏𝑝 0 0
0 0 𝟏16
)
𝑀𝑇𝐿𝑀 = 𝑀𝐿𝑀 = 𝐿 , 𝑀−1 = 𝐿𝑀𝐿
1
2 𝑃𝐿
2 = 1
4 (𝑚𝛼, 𝑛𝛼, 𝑄𝐼) ⋅ (𝑀 − 𝐿) ⋅ (
𝑚𝛼
𝑛𝛼
𝑄𝐼
)
1
2 𝑃𝑅
2 = 1
4 (𝑚𝛼, 𝑛𝛼, 𝑄𝐼) ⋅ (𝑀 + 𝐿) ⋅ (
𝑚𝛼
𝑛𝛼
𝑄𝐼
) .
1
2 𝑃𝑅
2 − 1
2 𝑃𝐿
2 = 𝑚𝛼𝑛𝛼 − 1
2 𝑄𝐼𝑄𝐼
Γ𝑝,𝑝+16(𝐺, 𝐵, 𝑌 = 0) = Γ𝑝,𝑝(𝐺, 𝐵)Γ‾O(32)/𝑍2
Γ𝑝,𝑝+16(𝐺, 𝐵, 𝑌 = 𝑌˜ ) = Γ𝑝,𝑝(𝐺′, 𝐵′)Γ‾𝐸8×𝐸8
𝑆𝑝,𝑞 = 1
4𝜋 ∫ 𝑑2𝜎√det𝑔𝑔𝑎𝑏𝐺𝛼𝛽𝜕𝑎𝑋𝛼𝜕𝑏𝑋𝛽 + 1
4𝜋 ∫ 𝑑2𝜎𝜖𝑎𝑏𝐵𝛼𝛽𝜕𝑎𝑋𝛼𝜕𝑏𝑋𝛽
+ 1
4𝜋 ∫ 𝑑2𝜎√det𝑔 ∑
𝐼
𝜓𝐼(∇‾ + 𝑌𝛼
𝐼(∇‾ 𝑋𝛼)𝜓‾𝐼
𝑍𝑝,𝑝+16(𝐺, 𝐵, 𝑌) = √detG
𝜏2
𝑝/2𝜂𝑝𝜂‾𝑝+16
× ∑
𝑚𝛼,𝑛𝛼∈𝑍
exp [− 𝜋
𝜏2
(𝐺 + 𝐵)𝛼𝛽(𝑚𝛼 + 𝜏𝑛𝛼)(𝑚𝛽 + 𝜏‾𝑛𝛽)]
× 1
2 ∑
1
𝑎,𝑏=0
∏
16
𝐼=1
𝑒𝑖𝜋(𝑚𝛼𝑌𝛼
𝐼𝑌𝛽
𝐼 𝑛𝛽−𝑏𝑛𝛼𝑌𝛼
𝐼)𝜗‾ [ 𝑎 − 2𝑛𝛼𝑌𝛼
𝐼
𝑏 − 2𝑚𝛽𝑌𝛽
𝐼]
= √detG
𝜏2
𝑝/2𝜂𝑝𝜂‾𝑝+16 ∑
𝑚𝛼,𝑛𝛼∈𝑍
exp [− 𝜋
𝜏2
(𝐺 + 𝐵)𝛼𝛽(𝑚𝛼 + 𝜏𝑛𝛼)(𝑚𝛽 + 𝜏‾𝑛𝛽)]
× exp [−𝑖𝜋 ∑
𝐼
𝑛𝛼(𝑚𝛽 + 𝜏‾𝑛𝛽)𝑌𝛼
𝐼𝑌𝛽
𝐼] 1
2 ∑
1
𝑎,𝑏=0
∏
16
𝐼=1
𝜗‾ [𝑎
𝑏] (𝑌𝛾
𝐼(𝑚𝛾 + 𝜏‾𝑛𝛾) ∣ 𝜏‾)
𝜏 → 𝜏 + 1 , 𝑍𝑝,𝑝+16 → 𝑒4𝜋𝑖/3𝑍𝑝,𝑝+16
Γ𝑝,𝑝+16(𝐺, 𝐵, 𝑌) = ∑
𝑚𝛼,𝑛𝛼,𝑄𝐼
𝑞𝑃𝐿
2/2𝑞‾𝑃𝑅
2/2
𝑀 = (
𝐺−1 𝐺−1𝐶 𝐺−1𝑌𝑡
𝐶𝑡𝐺−1 𝐺 + 𝐶𝑡𝐺−1𝐶 + 𝑌𝑡𝑌 𝐶𝑡𝐺−1𝑌𝑡 + 𝑌𝑡
𝑌𝐺−1 𝑌𝐺−1𝐶 + 𝑌 𝟏16 + 𝑌𝐺−1𝑌𝑡
)
𝐶𝛼𝛽 = 𝐵𝛼𝛽 − 1
2 𝑌𝛼
𝐼𝑌𝛽
𝐼
𝐿 = (
0 𝟏𝑝 0
𝟏𝑝 0 0
0 0 𝟏16
)
𝑀𝑇𝐿𝑀 = 𝑀𝐿𝑀 = 𝐿 , 𝑀−1 = 𝐿𝑀𝐿
1
2 𝑃𝐿
2 = 1
4 (𝑚𝛼, 𝑛𝛼, 𝑄𝐼) ⋅ (𝑀 − 𝐿) ⋅ (
𝑚𝛼
𝑛𝛼
𝑄𝐼
)
1
2 𝑃𝑅
2 = 1
4 (𝑚𝛼, 𝑛𝛼, 𝑄𝐼) ⋅ (𝑀 + 𝐿) ⋅ (
𝑚𝛼
𝑛𝛼
𝑄𝐼
) .
1
2 𝑃𝑅
2 − 1
2 𝑃𝐿
2 = 𝑚𝛼𝑛𝛼 − 1
2 𝑄𝐼𝑄𝐼
Γ𝑝,𝑝+16(𝐺, 𝐵, 𝑌 = 0) = Γ𝑝,𝑝(𝐺, 𝐵)Γ‾O(32)/𝑍2
Γ𝑝,𝑝+16(𝐺, 𝐵, 𝑌 = 𝑌˜ ) = Γ𝑝,𝑝(𝐺′, 𝐵′)Γ‾𝐸8×𝐸8
pág. 68
(
𝑚𝛼
𝑛𝛼
𝑄𝐼
) → Ω ⋅ (
𝑚𝛼
𝑛𝛼
𝑄𝐼
) , 𝑀 → Ω𝑀Ω𝑇
𝑍𝑑,𝑑+16
𝑁 (𝜖) [ℎ
𝑔] =
Γ𝑝,𝑝+16(𝜖) [ℎ
𝑔]
𝜂𝑝𝜂‾𝑝+16 =
∑𝜆∈𝐿+𝜖ℎ
𝑁
𝑒2𝜋𝑖𝑔𝜖⋅𝜆
𝑁 𝑞𝑝𝐿
2/2𝑞‾𝑝𝑅
2 /2
𝜂𝑝𝜂‾𝑝+16
𝑍𝑁(−𝜖) [ℎ
𝑔] = 𝑍𝑁(𝜖) [ℎ
𝑔] , 𝑍𝑁(𝜖) [−ℎ
−𝑔] = 𝑍𝑁(𝜖) [ℎ
𝑔]
𝑍𝑁(𝜖) [ℎ + 1
𝑔 ] = exp [− 𝑖𝜋𝑔𝜖2
𝑁 ] 𝑍𝑁(𝜖) [ℎ
𝑔] , 𝑍𝑁(𝜖) [ℎ
𝑔 + 1] = 𝑍𝑁(𝜖) [ℎ
𝑔]
𝑍𝑁(𝜖 + 𝑁𝜖′) [ℎ
𝑔] = exp [2𝜋𝑖𝑔ℎ𝜖 ⋅ 𝜖′
𝑁 ] 𝑍𝑁(𝜖) [ℎ
𝑔]
𝜏 → 𝜏 + 1 ∶ 𝑍𝑁(𝜖) [ℎ
𝑔] → exp [4𝜋𝑖
3 + 𝑖𝜋ℎ2𝜖2
𝑁2 ] 𝑍𝑁(𝜖) [ℎ
ℎ + 𝑔]
𝜏 → − 1
𝜏 ∶ 𝑍𝑁(𝜖) [ℎ
𝑔] → exp [− 2𝜋𝑖ℎ𝑔𝜖2
𝑁2 ] 𝑍𝑁(𝜖) [𝑔
−ℎ]
𝑍𝑁(𝜖, Ω𝑀Ω𝑇) [ℎ
𝑔] = 𝑍𝑁(Ω ⋅ 𝜖, 𝑀) [ℎ
𝑔]
𝑒ˆ𝜇ˆ
𝑟ˆ = (𝑒𝜇
𝑟 𝐴𝜇
𝛽𝐸𝛽
𝑎
0 𝐸𝛼
𝑎 ) , 𝑒ˆ𝑟ˆ
𝜇ˆ = (𝑒𝑟
𝜇 −𝑒𝑟
𝜈𝐴𝜈
𝛼
0 𝐸𝑎
𝛼 )
𝐺ˆ𝜇ˆ 𝜈ˆ = (𝑔𝜇𝜈 + 𝐴𝜇
𝛼𝐺𝛼𝛽𝐴𝜈
𝛽 𝐺𝛼𝛽𝐴𝜇
𝛽
𝐺𝛼𝛽𝐴𝜈
𝛽 𝐺𝛼𝛽
) , 𝐺ˆ𝜇ˆ 𝜈ˆ = ( 𝑔𝜇𝜈 −𝐴𝜇𝛼
−𝐴𝜈𝛼 𝐺𝛼𝛽 + 𝐴𝜌
𝛼𝐴𝛽,𝜌) .
𝛼′𝐷−2𝑆𝐷
heterotic = ∫ 𝑑𝐷𝑥√−det𝑔𝑒−𝜙 [𝑅 + 𝜕𝜇𝜙𝜕𝜇𝜙 + 1
4 𝜕𝜇𝐺𝛼𝛽𝜕𝜇𝐺𝛼𝛽 +
− 1
4 𝐺𝛼𝛽𝐹𝜇𝜈
𝐴𝛼
𝐹𝐴
𝛽,𝜇𝜈]
𝜙 = Φˆ − 1
2 log (det𝐺𝛼𝛽)
𝐹𝜇𝜈
𝐴𝛼
= 𝜕𝜇𝐴𝜈
𝛼 − 𝜕𝜈𝐴𝜇
𝛼
− 1
12 ∫ 𝑑10𝑥√−det𝐺ˆ 𝑒−Φˆ 𝐻ˆ 𝜇ˆ 𝜈ˆ 𝜌ˆ 𝐻ˆ𝜇ˆ 𝜈ˆ 𝜌ˆ
= − ∫ 𝑑𝐷𝑥√−det𝑔𝑒−𝜙 [1
4 𝐻𝜇𝛼𝛽𝐻𝜇𝛼𝛽 + 1
4 𝐻𝜇𝜈𝛼𝐻𝜇𝜈𝛼 + 1
12 𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌]
𝐻𝜇𝛼𝛽 = 𝑒𝜇
𝑟𝑒ˆ𝑟ˆ
𝜇ˆ 𝐻ˆ𝜇ˆ 𝛼𝛽 = 𝐻ˆ𝜇𝛼𝛽
𝐻𝜇𝜈𝛼 = 𝑒𝜇
𝑟𝑒𝜈
𝑠𝑒ˆ𝑟
𝜇ˆ 𝑒ˆ𝑠
𝜈ˆ 𝐻ˆ𝜇ˆ 𝜈ˆ 𝛼 = 𝐻ˆ𝜇𝜈𝛼 − 𝐴𝜇
𝛽𝐻ˆ𝜈𝛼𝛽 + 𝐴𝜈
𝛽𝐻ˆ𝜇𝛼𝛽
𝐻𝜇𝜈𝜌 = 𝑒𝜇
𝑟𝑒𝜈
𝑠𝑒𝑡𝑒ˆ𝑟
𝜇ˆ 𝑒ˆ𝑠
𝜈ˆ 𝑒ˆ𝑡
𝜌ˆ 𝐻ˆ𝜇ˆ 𝜈ˆ 𝜌ˆ = 𝐻ˆ𝜇𝜈𝜌 + [−𝐴𝜇
𝛼𝐻ˆ𝛼𝜈𝜌 + 𝐴𝜇
𝛼𝐴𝜈
𝛽𝐻ˆ𝛼𝛽𝜌 + cyclic ]
(
𝑚𝛼
𝑛𝛼
𝑄𝐼
) → Ω ⋅ (
𝑚𝛼
𝑛𝛼
𝑄𝐼
) , 𝑀 → Ω𝑀Ω𝑇
𝑍𝑑,𝑑+16
𝑁 (𝜖) [ℎ
𝑔] =
Γ𝑝,𝑝+16(𝜖) [ℎ
𝑔]
𝜂𝑝𝜂‾𝑝+16 =
∑𝜆∈𝐿+𝜖ℎ
𝑁
𝑒2𝜋𝑖𝑔𝜖⋅𝜆
𝑁 𝑞𝑝𝐿
2/2𝑞‾𝑝𝑅
2 /2
𝜂𝑝𝜂‾𝑝+16
𝑍𝑁(−𝜖) [ℎ
𝑔] = 𝑍𝑁(𝜖) [ℎ
𝑔] , 𝑍𝑁(𝜖) [−ℎ
−𝑔] = 𝑍𝑁(𝜖) [ℎ
𝑔]
𝑍𝑁(𝜖) [ℎ + 1
𝑔 ] = exp [− 𝑖𝜋𝑔𝜖2
𝑁 ] 𝑍𝑁(𝜖) [ℎ
𝑔] , 𝑍𝑁(𝜖) [ℎ
𝑔 + 1] = 𝑍𝑁(𝜖) [ℎ
𝑔]
𝑍𝑁(𝜖 + 𝑁𝜖′) [ℎ
𝑔] = exp [2𝜋𝑖𝑔ℎ𝜖 ⋅ 𝜖′
𝑁 ] 𝑍𝑁(𝜖) [ℎ
𝑔]
𝜏 → 𝜏 + 1 ∶ 𝑍𝑁(𝜖) [ℎ
𝑔] → exp [4𝜋𝑖
3 + 𝑖𝜋ℎ2𝜖2
𝑁2 ] 𝑍𝑁(𝜖) [ℎ
ℎ + 𝑔]
𝜏 → − 1
𝜏 ∶ 𝑍𝑁(𝜖) [ℎ
𝑔] → exp [− 2𝜋𝑖ℎ𝑔𝜖2
𝑁2 ] 𝑍𝑁(𝜖) [𝑔
−ℎ]
𝑍𝑁(𝜖, Ω𝑀Ω𝑇) [ℎ
𝑔] = 𝑍𝑁(Ω ⋅ 𝜖, 𝑀) [ℎ
𝑔]
𝑒ˆ𝜇ˆ
𝑟ˆ = (𝑒𝜇
𝑟 𝐴𝜇
𝛽𝐸𝛽
𝑎
0 𝐸𝛼
𝑎 ) , 𝑒ˆ𝑟ˆ
𝜇ˆ = (𝑒𝑟
𝜇 −𝑒𝑟
𝜈𝐴𝜈
𝛼
0 𝐸𝑎
𝛼 )
𝐺ˆ𝜇ˆ 𝜈ˆ = (𝑔𝜇𝜈 + 𝐴𝜇
𝛼𝐺𝛼𝛽𝐴𝜈
𝛽 𝐺𝛼𝛽𝐴𝜇
𝛽
𝐺𝛼𝛽𝐴𝜈
𝛽 𝐺𝛼𝛽
) , 𝐺ˆ𝜇ˆ 𝜈ˆ = ( 𝑔𝜇𝜈 −𝐴𝜇𝛼
−𝐴𝜈𝛼 𝐺𝛼𝛽 + 𝐴𝜌
𝛼𝐴𝛽,𝜌) .
𝛼′𝐷−2𝑆𝐷
heterotic = ∫ 𝑑𝐷𝑥√−det𝑔𝑒−𝜙 [𝑅 + 𝜕𝜇𝜙𝜕𝜇𝜙 + 1
4 𝜕𝜇𝐺𝛼𝛽𝜕𝜇𝐺𝛼𝛽 +
− 1
4 𝐺𝛼𝛽𝐹𝜇𝜈
𝐴𝛼
𝐹𝐴
𝛽,𝜇𝜈]
𝜙 = Φˆ − 1
2 log (det𝐺𝛼𝛽)
𝐹𝜇𝜈
𝐴𝛼
= 𝜕𝜇𝐴𝜈
𝛼 − 𝜕𝜈𝐴𝜇
𝛼
− 1
12 ∫ 𝑑10𝑥√−det𝐺ˆ 𝑒−Φˆ 𝐻ˆ 𝜇ˆ 𝜈ˆ 𝜌ˆ 𝐻ˆ𝜇ˆ 𝜈ˆ 𝜌ˆ
= − ∫ 𝑑𝐷𝑥√−det𝑔𝑒−𝜙 [1
4 𝐻𝜇𝛼𝛽𝐻𝜇𝛼𝛽 + 1
4 𝐻𝜇𝜈𝛼𝐻𝜇𝜈𝛼 + 1
12 𝐻𝜇𝜈𝜌𝐻𝜇𝜈𝜌]
𝐻𝜇𝛼𝛽 = 𝑒𝜇
𝑟𝑒ˆ𝑟ˆ
𝜇ˆ 𝐻ˆ𝜇ˆ 𝛼𝛽 = 𝐻ˆ𝜇𝛼𝛽
𝐻𝜇𝜈𝛼 = 𝑒𝜇
𝑟𝑒𝜈
𝑠𝑒ˆ𝑟
𝜇ˆ 𝑒ˆ𝑠
𝜈ˆ 𝐻ˆ𝜇ˆ 𝜈ˆ 𝛼 = 𝐻ˆ𝜇𝜈𝛼 − 𝐴𝜇
𝛽𝐻ˆ𝜈𝛼𝛽 + 𝐴𝜈
𝛽𝐻ˆ𝜇𝛼𝛽
𝐻𝜇𝜈𝜌 = 𝑒𝜇
𝑟𝑒𝜈
𝑠𝑒𝑡𝑒ˆ𝑟
𝜇ˆ 𝑒ˆ𝑠
𝜈ˆ 𝑒ˆ𝑡
𝜌ˆ 𝐻ˆ𝜇ˆ 𝜈ˆ 𝜌ˆ = 𝐻ˆ𝜇𝜈𝜌 + [−𝐴𝜇
𝛼𝐻ˆ𝛼𝜈𝜌 + 𝐴𝜇
𝛼𝐴𝜈
𝛽𝐻ˆ𝛼𝛽𝜌 + cyclic ]
pág. 69
∫ 𝑑10𝑥√−det𝐺ˆ 𝑒−Φˆ ∑
16
𝐼=1
𝐹ˆ𝜇ˆ 𝜈ˆ
𝐼 𝐹𝐼,𝜇ˆ 𝜈ˆ = ∫ 𝑑𝐷𝑥√−det𝑔𝑒−𝜙 ∑
16
𝐼=1
[𝐹˜𝜇𝜈
𝐼 𝐹˜𝐼,𝜇𝜈 + 2𝐹˜𝜇𝛼
𝐼 𝐹˜𝐼,𝜇𝛼]
𝑌𝛼
𝐼 = 𝐴ˆ𝛼
𝐼 , 𝐴𝜇
𝐼 = 𝐴ˆ𝜇
𝐼 − 𝑌𝛼
𝐼𝐴𝜇
𝑎 , 𝐹˜𝜇𝜈
𝐼 = 𝐹𝜇𝜈
𝐼 + 𝑌𝛼
𝐼𝐹𝜇𝜈
𝐴,𝛼
𝐹˜𝜇𝛼
𝐼 = 𝜕𝜇𝑌𝛼
𝐼 , 𝐹𝜇𝜈
𝐼 = 𝜕𝜇𝐴𝜈
𝐼 − 𝜕𝜈𝐴𝜇
𝐼
𝐻ˆ𝜇𝛼𝛽 = 𝜕𝜇𝐵ˆ𝛼𝛽 + 1
2 ∑
𝐼
[𝑌𝛼
𝐼𝜕𝜇𝑌𝛽
𝐼 − 𝑌𝛽
𝐼𝜕𝜇𝑌𝛼
𝐼]
𝐶𝛼𝛽 ≡ 𝐵ˆ𝛼𝛽 − 1
2 ∑
𝐼
𝑌𝛼
𝐼𝑌𝛽
𝐽
𝐻𝜇𝛼𝛽 = 𝜕𝜇𝐶𝛼𝛽 + ∑
𝐼
𝑌𝛼
𝐼𝜕𝜇𝑌𝛽
𝐼
𝐻ˆ𝜇𝜈𝛼 = 𝜕𝜇𝐵ˆ𝜈𝛼 − 𝜕𝜈𝐵ˆ𝜇𝛼 + 1
2 ∑
𝐼
[𝐴ˆ𝜈
𝐼 𝜕𝜇𝑌𝛼
𝐼 − 𝐴ˆ𝜇
𝐼 𝜕𝜈𝑌𝛼
𝐼 − 𝑌𝛼
𝐼𝐹ˆ𝜇𝜈
𝐼 ]
𝐵𝜇,𝛼 ≡ 𝐵ˆ𝜇𝛼 + 𝐵𝛼𝛽𝐴𝜇
𝛽 + 1
2 ∑
𝐼
𝑌𝛼
𝐼𝐴𝜇
𝐼
𝐹𝛼,𝜇𝜈
𝐵 = 𝜕𝜇𝐵𝛼,𝜈 − 𝜕𝜈𝐵𝛼,𝜇
𝐻𝜇𝜈𝛼 = 𝐹𝛼𝜇𝜈
𝐵 − 𝐶𝛼𝛽𝐹𝜇𝜈
𝐴,𝛽 − ∑
𝐼
𝑌𝛼
𝐼𝐹𝜇𝜈
𝐼
𝐵𝜇𝜈 = 𝐵ˆ𝜇𝜈 + 1
2 [𝐴𝜇
𝛼𝐵𝜈𝛼 + ∑
𝐼
𝐴𝜇
𝐼 𝐴𝜈
𝛼𝑌𝛼
𝐼 − (𝜇 ↔ 𝜈)] − 𝐴𝜇
𝛼𝐴𝜈
𝛽𝐵𝛼𝛽
𝐻𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 − 1
2 [𝐵𝜇𝛼𝐹𝜈𝜌
𝐴,𝛼 + 𝐴𝜇
𝛼𝐹𝑎,𝜈𝜌
𝐵 + ∑
𝐼
𝐴𝜇
𝐼 𝐹𝜈𝜌
𝐼 ] + cyclic
≡ 𝜕𝜇𝐵𝜈𝜌 − 1
2 𝐿𝑖𝑗𝐴𝜇
𝑖 𝐹𝜈𝜌
𝑗 + cyclic
𝑆𝐷
heterotic = ∫ 𝑑𝐷𝑥√−det𝑔𝑒−𝜙 [𝑅 + 𝜕𝜇𝜙𝜕𝜇𝜙 − 1
12 𝐻˜ 𝜇𝜈𝜌𝐻˜𝜇𝜈𝜌 −
− 1
4 (𝑀−1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝑆𝐶 = − 1
2 ⋅ 4! ∫ 𝑑𝑑𝑥√−𝐺𝐹ˆ 2
𝐹ˆ𝜇𝜈𝜌𝜎 = 𝜕𝜇𝐶ˆ𝜈𝜌𝜎 − 𝜕𝜎𝐶ˆ𝜇𝜈𝜌 + 𝜕𝜌𝐶ˆ𝜎𝜇𝜈 − 𝜕𝜈𝐶ˆ𝜌𝜎𝜇
𝐶𝛼𝛽𝛾 = 𝐶ˆ𝛼𝛽𝛾, 𝐶𝜇𝛼𝛽 = 𝐶ˆ𝜇𝛼𝛽 − 𝐶𝛼𝛽𝛾𝐴𝜇
𝛾
𝐶𝜇𝜈𝛼 = 𝐶ˆ𝜇𝜈𝛼 + 𝐶ˆ𝜇𝛼𝛽𝐴𝜈
𝛽 − 𝐶ˆ𝜈𝛼𝛽𝐴𝜇
𝛽 + 𝐶𝛼𝛽𝛾𝐴𝜇
𝛽𝐴𝜈
𝛾
𝐶𝜇𝜈𝜌 = 𝐶ˆ𝜇𝜈𝜌 + (−𝐶ˆ𝜈𝜌𝛼𝐴𝜇
𝛼 + 𝐶ˆ𝛼𝛽𝜌𝐴𝜇
𝛼𝐴𝜈
𝛽 + cyclic ) − 𝐶𝛼𝛽𝛾𝐴𝜇
𝛼𝐴𝜈
𝛽𝐴𝜌
𝛾
∫ 𝑑10𝑥√−det𝐺ˆ 𝑒−Φˆ ∑
16
𝐼=1
𝐹ˆ𝜇ˆ 𝜈ˆ
𝐼 𝐹𝐼,𝜇ˆ 𝜈ˆ = ∫ 𝑑𝐷𝑥√−det𝑔𝑒−𝜙 ∑
16
𝐼=1
[𝐹˜𝜇𝜈
𝐼 𝐹˜𝐼,𝜇𝜈 + 2𝐹˜𝜇𝛼
𝐼 𝐹˜𝐼,𝜇𝛼]
𝑌𝛼
𝐼 = 𝐴ˆ𝛼
𝐼 , 𝐴𝜇
𝐼 = 𝐴ˆ𝜇
𝐼 − 𝑌𝛼
𝐼𝐴𝜇
𝑎 , 𝐹˜𝜇𝜈
𝐼 = 𝐹𝜇𝜈
𝐼 + 𝑌𝛼
𝐼𝐹𝜇𝜈
𝐴,𝛼
𝐹˜𝜇𝛼
𝐼 = 𝜕𝜇𝑌𝛼
𝐼 , 𝐹𝜇𝜈
𝐼 = 𝜕𝜇𝐴𝜈
𝐼 − 𝜕𝜈𝐴𝜇
𝐼
𝐻ˆ𝜇𝛼𝛽 = 𝜕𝜇𝐵ˆ𝛼𝛽 + 1
2 ∑
𝐼
[𝑌𝛼
𝐼𝜕𝜇𝑌𝛽
𝐼 − 𝑌𝛽
𝐼𝜕𝜇𝑌𝛼
𝐼]
𝐶𝛼𝛽 ≡ 𝐵ˆ𝛼𝛽 − 1
2 ∑
𝐼
𝑌𝛼
𝐼𝑌𝛽
𝐽
𝐻𝜇𝛼𝛽 = 𝜕𝜇𝐶𝛼𝛽 + ∑
𝐼
𝑌𝛼
𝐼𝜕𝜇𝑌𝛽
𝐼
𝐻ˆ𝜇𝜈𝛼 = 𝜕𝜇𝐵ˆ𝜈𝛼 − 𝜕𝜈𝐵ˆ𝜇𝛼 + 1
2 ∑
𝐼
[𝐴ˆ𝜈
𝐼 𝜕𝜇𝑌𝛼
𝐼 − 𝐴ˆ𝜇
𝐼 𝜕𝜈𝑌𝛼
𝐼 − 𝑌𝛼
𝐼𝐹ˆ𝜇𝜈
𝐼 ]
𝐵𝜇,𝛼 ≡ 𝐵ˆ𝜇𝛼 + 𝐵𝛼𝛽𝐴𝜇
𝛽 + 1
2 ∑
𝐼
𝑌𝛼
𝐼𝐴𝜇
𝐼
𝐹𝛼,𝜇𝜈
𝐵 = 𝜕𝜇𝐵𝛼,𝜈 − 𝜕𝜈𝐵𝛼,𝜇
𝐻𝜇𝜈𝛼 = 𝐹𝛼𝜇𝜈
𝐵 − 𝐶𝛼𝛽𝐹𝜇𝜈
𝐴,𝛽 − ∑
𝐼
𝑌𝛼
𝐼𝐹𝜇𝜈
𝐼
𝐵𝜇𝜈 = 𝐵ˆ𝜇𝜈 + 1
2 [𝐴𝜇
𝛼𝐵𝜈𝛼 + ∑
𝐼
𝐴𝜇
𝐼 𝐴𝜈
𝛼𝑌𝛼
𝐼 − (𝜇 ↔ 𝜈)] − 𝐴𝜇
𝛼𝐴𝜈
𝛽𝐵𝛼𝛽
𝐻𝜇𝜈𝜌 = 𝜕𝜇𝐵𝜈𝜌 − 1
2 [𝐵𝜇𝛼𝐹𝜈𝜌
𝐴,𝛼 + 𝐴𝜇
𝛼𝐹𝑎,𝜈𝜌
𝐵 + ∑
𝐼
𝐴𝜇
𝐼 𝐹𝜈𝜌
𝐼 ] + cyclic
≡ 𝜕𝜇𝐵𝜈𝜌 − 1
2 𝐿𝑖𝑗𝐴𝜇
𝑖 𝐹𝜈𝜌
𝑗 + cyclic
𝑆𝐷
heterotic = ∫ 𝑑𝐷𝑥√−det𝑔𝑒−𝜙 [𝑅 + 𝜕𝜇𝜙𝜕𝜇𝜙 − 1
12 𝐻˜ 𝜇𝜈𝜌𝐻˜𝜇𝜈𝜌 −
− 1
4 (𝑀−1)𝑖𝑗𝐹𝜇𝜈
𝑖 𝐹𝑗𝜇𝜈 + 1
8 Tr(𝜕𝜇𝑀𝜕𝜇𝑀−1)]
𝑆𝐶 = − 1
2 ⋅ 4! ∫ 𝑑𝑑𝑥√−𝐺𝐹ˆ 2
𝐹ˆ𝜇𝜈𝜌𝜎 = 𝜕𝜇𝐶ˆ𝜈𝜌𝜎 − 𝜕𝜎𝐶ˆ𝜇𝜈𝜌 + 𝜕𝜌𝐶ˆ𝜎𝜇𝜈 − 𝜕𝜈𝐶ˆ𝜌𝜎𝜇
𝐶𝛼𝛽𝛾 = 𝐶ˆ𝛼𝛽𝛾, 𝐶𝜇𝛼𝛽 = 𝐶ˆ𝜇𝛼𝛽 − 𝐶𝛼𝛽𝛾𝐴𝜇
𝛾
𝐶𝜇𝜈𝛼 = 𝐶ˆ𝜇𝜈𝛼 + 𝐶ˆ𝜇𝛼𝛽𝐴𝜈
𝛽 − 𝐶ˆ𝜈𝛼𝛽𝐴𝜇
𝛽 + 𝐶𝛼𝛽𝛾𝐴𝜇
𝛽𝐴𝜈
𝛾
𝐶𝜇𝜈𝜌 = 𝐶ˆ𝜇𝜈𝜌 + (−𝐶ˆ𝜈𝜌𝛼𝐴𝜇
𝛼 + 𝐶ˆ𝛼𝛽𝜌𝐴𝜇
𝛼𝐴𝜈
𝛽 + cyclic ) − 𝐶𝛼𝛽𝛾𝐴𝜇
𝛼𝐴𝜈
𝛽𝐴𝜌
𝛾
pág. 70
𝑆𝐶 = − 1
2 ⋅ 4! ∫ 𝑑𝐷𝑥√−𝑔√det𝐺𝛼𝛽[𝐹𝜇𝜈𝜌𝜎𝐹𝜇𝜈𝜌𝜎 + 4𝐹𝜇𝜈𝜌𝛼𝐹𝜇𝜈𝜌𝛼 + 6𝐹𝜇𝜈𝛼𝛽𝐹𝜇𝜈𝛼𝛽 + 4𝐹𝜇𝛼𝛽𝛾𝐹𝜇𝛼𝛽𝛾]
𝐹𝜇𝛼𝛽𝛾 = 𝜕𝜇𝐶𝛼𝛽𝛾, 𝐹𝜇𝜈𝛼𝛽 = 𝜕𝜇𝐶𝜈𝛼𝛽 − 𝜕𝜈𝐶𝜇𝛼𝛽 + 𝐶𝛼𝛽𝛾𝐹𝜇𝜈
𝛾
𝐹𝜇𝜈𝜌𝛼 = 𝜕𝜇𝐶𝜈𝜌𝛼 + 𝐶𝜇𝛼𝛽𝐹𝜈𝜌
𝛽 + cyclic
𝐹𝜇𝜈𝜌𝜎 = (𝜕𝜇𝐶𝜈𝜌𝜎 + 3 perm. ) + (𝐶𝜌𝜎𝛼𝐹𝜇𝜈
𝛼 + 5 perm. )
ℒ𝑁=1 = − 1
2𝜅2 𝑅 + 𝐺𝑖𝑗‾𝐷𝜇𝜙𝑖𝐷𝜇𝜙‾ 𝑗‾ + 𝑉(𝜙, 𝜙‾) + ∑
𝑎
1
4𝑔𝑎
2 [𝐹𝜇𝜈𝐹𝜇𝜈]𝑎 + 𝜃𝑎
4 [𝐹𝜇𝜈𝐹˜𝜇𝜈]𝑎
𝐺𝑖𝑗‾ = 𝜕𝑖𝜕𝑗‾𝐾(𝜙, 𝜙‾)
1
𝑔𝑎
2 = Re𝑓𝑎(𝜙) , 𝜃𝑎 = −Im𝑓𝑎(𝜙)
𝑉(𝜙, 𝜙‾) = 𝑒𝜅2𝐾(𝐷𝑖𝑊𝐺𝑖𝑖‾𝐷‾𝑖‾𝑊‾ − 3𝜅2|𝑊|2)
𝐷𝑖𝑊 = 𝜕𝑊
𝜕𝜙𝑖 + 𝜅2 𝜕𝐾
𝜕𝜙𝑖 𝑊
𝐾 → 𝐾 + Λ(𝜙) + Λ‾ (𝜙‾) , 𝑊 → 𝑊𝑒−Λ , 𝑓𝑎 → 𝑓𝑎
𝐾 = −log [𝑖(𝑍‾𝐼𝐹𝐼 − 𝑍𝐼𝐹‾𝐼)]
𝐾 = −log [2 (𝑓(𝑇𝑖) + 𝑓‾(𝑇‾ 𝑖)) − (𝑇𝑖 − 𝑇‾ 𝑖)(𝑓𝑖 − 𝑓‾𝑖)]
𝑅𝑖𝑗‾𝑘𝑙‾ = 𝐺𝑖𝑗‾𝐺𝑘𝑙‾ + 𝐺𝑖𝑙‾𝐺𝑘𝑗‾ − 𝑒−2𝐾𝑊𝑖𝑘𝑚𝐺𝑚𝑚‾ 𝑊‾ 𝑚‾ 𝑗‾𝑙‾
ℒvectors = − 1
4 Ξ𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈 − 𝜃𝐼𝐽
4 𝐹𝜇𝜈
𝐼 𝐹˜ 𝐽,𝜇𝜈
Ξ𝐼𝐽 = 𝑖
4 [𝑁𝐼𝐽 − 𝑁‾𝐼𝐽] , 𝜃𝐼𝐽 = 1
4 [𝑁𝐼𝐽 + 𝑁‾𝐼𝐽]
𝑁𝐼𝐽 = 𝐹‾𝐼𝐽 + 2𝑖 Im𝐹𝐼𝐾Im𝐹𝐽𝐿𝑍𝐾𝑍𝐿
Im𝐹𝑀𝑁𝑍𝑀𝑍𝑁
𝑀𝐵𝑃𝑆
2 = |𝑒𝐼𝑍𝐼 + 𝑞𝐼𝐹𝐼|2
Im(𝑍𝐼𝐹‾𝐼)
𝑀𝐵𝑃𝑆
2 = 1
4Im𝑆 (𝛼𝑡 + 𝑆𝛽𝑡)𝑀+(𝛼 + 𝑆‾𝛽) + 1
2 √(𝛼𝑡𝑀+𝛼)(𝛽𝑡𝑀+𝛽) − (𝛼𝑡𝑀+𝛽)2,
{𝑄𝛼
𝐼 , 𝑄𝛽
𝐽 } = 𝜖𝛼𝛽𝑍𝐼𝐽 , {𝑄‾𝛼˙
𝐼 , 𝑄𝛽˙
𝐽} = 𝜖𝛼˙ 𝛽˙ 𝑍‾𝐼𝐽 , {𝑄𝛼
𝐼 , 𝑄‾𝛼˙
𝐽 } = 𝛿𝐼𝐽2𝜎𝛼𝛼˙
𝜇 𝑃𝜇
{𝑄𝛼
𝐼 , 𝑄‾𝛼˙
𝐽 } = 2𝑀𝛿𝛼𝛼˙ 𝛿𝐼𝐽 , {𝑄𝛼
𝐼 , 𝑄𝛽
𝐽 } = {𝑄‾𝛼˙
𝐼 , 𝑄‾𝛽˙
𝐽} = 0
𝑆𝐶 = − 1
2 ⋅ 4! ∫ 𝑑𝐷𝑥√−𝑔√det𝐺𝛼𝛽[𝐹𝜇𝜈𝜌𝜎𝐹𝜇𝜈𝜌𝜎 + 4𝐹𝜇𝜈𝜌𝛼𝐹𝜇𝜈𝜌𝛼 + 6𝐹𝜇𝜈𝛼𝛽𝐹𝜇𝜈𝛼𝛽 + 4𝐹𝜇𝛼𝛽𝛾𝐹𝜇𝛼𝛽𝛾]
𝐹𝜇𝛼𝛽𝛾 = 𝜕𝜇𝐶𝛼𝛽𝛾, 𝐹𝜇𝜈𝛼𝛽 = 𝜕𝜇𝐶𝜈𝛼𝛽 − 𝜕𝜈𝐶𝜇𝛼𝛽 + 𝐶𝛼𝛽𝛾𝐹𝜇𝜈
𝛾
𝐹𝜇𝜈𝜌𝛼 = 𝜕𝜇𝐶𝜈𝜌𝛼 + 𝐶𝜇𝛼𝛽𝐹𝜈𝜌
𝛽 + cyclic
𝐹𝜇𝜈𝜌𝜎 = (𝜕𝜇𝐶𝜈𝜌𝜎 + 3 perm. ) + (𝐶𝜌𝜎𝛼𝐹𝜇𝜈
𝛼 + 5 perm. )
ℒ𝑁=1 = − 1
2𝜅2 𝑅 + 𝐺𝑖𝑗‾𝐷𝜇𝜙𝑖𝐷𝜇𝜙‾ 𝑗‾ + 𝑉(𝜙, 𝜙‾) + ∑
𝑎
1
4𝑔𝑎
2 [𝐹𝜇𝜈𝐹𝜇𝜈]𝑎 + 𝜃𝑎
4 [𝐹𝜇𝜈𝐹˜𝜇𝜈]𝑎
𝐺𝑖𝑗‾ = 𝜕𝑖𝜕𝑗‾𝐾(𝜙, 𝜙‾)
1
𝑔𝑎
2 = Re𝑓𝑎(𝜙) , 𝜃𝑎 = −Im𝑓𝑎(𝜙)
𝑉(𝜙, 𝜙‾) = 𝑒𝜅2𝐾(𝐷𝑖𝑊𝐺𝑖𝑖‾𝐷‾𝑖‾𝑊‾ − 3𝜅2|𝑊|2)
𝐷𝑖𝑊 = 𝜕𝑊
𝜕𝜙𝑖 + 𝜅2 𝜕𝐾
𝜕𝜙𝑖 𝑊
𝐾 → 𝐾 + Λ(𝜙) + Λ‾ (𝜙‾) , 𝑊 → 𝑊𝑒−Λ , 𝑓𝑎 → 𝑓𝑎
𝐾 = −log [𝑖(𝑍‾𝐼𝐹𝐼 − 𝑍𝐼𝐹‾𝐼)]
𝐾 = −log [2 (𝑓(𝑇𝑖) + 𝑓‾(𝑇‾ 𝑖)) − (𝑇𝑖 − 𝑇‾ 𝑖)(𝑓𝑖 − 𝑓‾𝑖)]
𝑅𝑖𝑗‾𝑘𝑙‾ = 𝐺𝑖𝑗‾𝐺𝑘𝑙‾ + 𝐺𝑖𝑙‾𝐺𝑘𝑗‾ − 𝑒−2𝐾𝑊𝑖𝑘𝑚𝐺𝑚𝑚‾ 𝑊‾ 𝑚‾ 𝑗‾𝑙‾
ℒvectors = − 1
4 Ξ𝐼𝐽𝐹𝜇𝜈
𝐼 𝐹𝐽,𝜇𝜈 − 𝜃𝐼𝐽
4 𝐹𝜇𝜈
𝐼 𝐹˜ 𝐽,𝜇𝜈
Ξ𝐼𝐽 = 𝑖
4 [𝑁𝐼𝐽 − 𝑁‾𝐼𝐽] , 𝜃𝐼𝐽 = 1
4 [𝑁𝐼𝐽 + 𝑁‾𝐼𝐽]
𝑁𝐼𝐽 = 𝐹‾𝐼𝐽 + 2𝑖 Im𝐹𝐼𝐾Im𝐹𝐽𝐿𝑍𝐾𝑍𝐿
Im𝐹𝑀𝑁𝑍𝑀𝑍𝑁
𝑀𝐵𝑃𝑆
2 = |𝑒𝐼𝑍𝐼 + 𝑞𝐼𝐹𝐼|2
Im(𝑍𝐼𝐹‾𝐼)
𝑀𝐵𝑃𝑆
2 = 1
4Im𝑆 (𝛼𝑡 + 𝑆𝛽𝑡)𝑀+(𝛼 + 𝑆‾𝛽) + 1
2 √(𝛼𝑡𝑀+𝛼)(𝛽𝑡𝑀+𝛽) − (𝛼𝑡𝑀+𝛽)2,
{𝑄𝛼
𝐼 , 𝑄𝛽
𝐽 } = 𝜖𝛼𝛽𝑍𝐼𝐽 , {𝑄‾𝛼˙
𝐼 , 𝑄𝛽˙
𝐽} = 𝜖𝛼˙ 𝛽˙ 𝑍‾𝐼𝐽 , {𝑄𝛼
𝐼 , 𝑄‾𝛼˙
𝐽 } = 𝛿𝐼𝐽2𝜎𝛼𝛼˙
𝜇 𝑃𝜇
{𝑄𝛼
𝐼 , 𝑄‾𝛼˙
𝐽 } = 2𝑀𝛿𝛼𝛼˙ 𝛿𝐼𝐽 , {𝑄𝛼
𝐼 , 𝑄𝛽
𝐽 } = {𝑄‾𝛼˙
𝐼 , 𝑄‾𝛽˙
𝐽} = 0
pág. 71
𝐴𝛼
𝐼 = 1
√2𝑀 𝑄𝛼
𝐼 , 𝐴𝛼
†𝐼 = 1
√2𝑀 𝑄‾𝛼˙
𝐼
{𝑄𝛼
𝐼 , 𝑄‾𝛼˙
𝐽 } = 2 (2𝐸 0
0 0) 𝛿𝐼𝐽
{𝑄𝛼
𝑎𝑚, 𝑄‾𝛼˙
𝑏𝑛} = 2𝑀𝛿𝛼𝛼˙ 𝛿𝑎𝑏𝛿𝑚𝑛 , {𝑄𝛼
𝑎𝑚, 𝑄𝛽
𝑏𝑛} = 𝑍𝑛𝜖𝛼𝛽𝜖𝑎𝑏𝛿𝑚𝑛
𝐴𝛼
𝑚 = 1
√2 [𝑄𝛼
1𝑚 + 𝜖𝛼𝛽𝑄𝛽
2𝑚] , 𝐵𝛼
𝑚 = 1
√2 [𝑄𝛼
1𝑚 − 𝜖𝛼𝛽𝑄𝛽
2𝑚],
{𝐴𝛼
𝑚, 𝐴𝛽
𝑛} = {𝐴𝛼
𝑚, 𝐵𝛽
𝑛} = {𝐵𝛼
𝑚, 𝐵𝛽
𝑛} = 0,
{𝐴𝛼
𝑚, 𝐴𝛽
†𝑛} = 𝛿𝛼𝛽𝛿𝑚𝑛(2𝑀 + 𝑍𝑛), {𝐵𝛼
𝑚, 𝐵𝛽
†𝑛} = 𝛿𝛼𝛽𝛿𝑚𝑛(2𝑀 − 𝑍𝑛)
𝑀 ≥ max [𝑍𝑛
2 ]
𝐵2𝑛(𝑅) = Tr𝑅[(−1)2𝜆𝜆2𝑛]
𝑍𝑅(𝑦) = str𝑦2𝜆
𝑍[𝑗] = {(−)2𝑗 (𝑦2𝑗+1 − 𝑦−2𝑗−1
𝑦 − 1/𝑦 ) massive
(−)2𝑗(𝑦2𝑗 + 𝑦−2𝑗) supermassive
.
𝑍𝑟⊗𝑟˜ = 𝑍𝑟𝑍𝑟˜
𝐵𝑛(𝑅) = (𝑦2 𝑑
𝑑𝑦2)
𝑛
𝑍𝑅 (𝑦)|
𝑦=1
𝑍𝑚(𝑦) = 𝑍[𝑗](𝑦)(1 − 𝑦)𝑚(1 − 1/𝑦)𝑚
𝐿𝑗 ∶ [𝑗] ⊗ ([1] + 4[1/2] + 5[0])
𝑆𝑗 ∶ [𝑗] ⊗ (2[1/2] + 4[0])
𝑀𝜆
0 : ± (𝜆 + 1/2) + 2(±𝜆) + ±(𝜆 − 1/2)
𝐵0( any rep ) = 0
𝐵2(𝑀𝜆
0) = (−1)2𝜆+1 , 𝐵2(𝑆𝑗) = (−1)2𝑗+1𝐷𝑗 , 𝐵2(𝐿𝑗) = 0.
Lj ∶ [𝑗] ⊗ (42[0] + 48[1/2] + 27[1] + 8[3/2] + [2])
𝑠 = 2 supermassive long : 42[0] + 48[1/2] + 27[1] + 8[3/2] + [2]
𝐼𝑗: [𝑗] ⊗ (14[0] + 14[1/2] + 6[1] + [3/2])
𝐼3/2: 14[0] + 14[1/2] + 6[1] + [3/2]
𝑆𝑗 ∶ [𝑗] ⊗ (5[0] + 4[1/2] + [1]),
𝐴𝛼
𝐼 = 1
√2𝑀 𝑄𝛼
𝐼 , 𝐴𝛼
†𝐼 = 1
√2𝑀 𝑄‾𝛼˙
𝐼
{𝑄𝛼
𝐼 , 𝑄‾𝛼˙
𝐽 } = 2 (2𝐸 0
0 0) 𝛿𝐼𝐽
{𝑄𝛼
𝑎𝑚, 𝑄‾𝛼˙
𝑏𝑛} = 2𝑀𝛿𝛼𝛼˙ 𝛿𝑎𝑏𝛿𝑚𝑛 , {𝑄𝛼
𝑎𝑚, 𝑄𝛽
𝑏𝑛} = 𝑍𝑛𝜖𝛼𝛽𝜖𝑎𝑏𝛿𝑚𝑛
𝐴𝛼
𝑚 = 1
√2 [𝑄𝛼
1𝑚 + 𝜖𝛼𝛽𝑄𝛽
2𝑚] , 𝐵𝛼
𝑚 = 1
√2 [𝑄𝛼
1𝑚 − 𝜖𝛼𝛽𝑄𝛽
2𝑚],
{𝐴𝛼
𝑚, 𝐴𝛽
𝑛} = {𝐴𝛼
𝑚, 𝐵𝛽
𝑛} = {𝐵𝛼
𝑚, 𝐵𝛽
𝑛} = 0,
{𝐴𝛼
𝑚, 𝐴𝛽
†𝑛} = 𝛿𝛼𝛽𝛿𝑚𝑛(2𝑀 + 𝑍𝑛), {𝐵𝛼
𝑚, 𝐵𝛽
†𝑛} = 𝛿𝛼𝛽𝛿𝑚𝑛(2𝑀 − 𝑍𝑛)
𝑀 ≥ max [𝑍𝑛
2 ]
𝐵2𝑛(𝑅) = Tr𝑅[(−1)2𝜆𝜆2𝑛]
𝑍𝑅(𝑦) = str𝑦2𝜆
𝑍[𝑗] = {(−)2𝑗 (𝑦2𝑗+1 − 𝑦−2𝑗−1
𝑦 − 1/𝑦 ) massive
(−)2𝑗(𝑦2𝑗 + 𝑦−2𝑗) supermassive
.
𝑍𝑟⊗𝑟˜ = 𝑍𝑟𝑍𝑟˜
𝐵𝑛(𝑅) = (𝑦2 𝑑
𝑑𝑦2)
𝑛
𝑍𝑅 (𝑦)|
𝑦=1
𝑍𝑚(𝑦) = 𝑍[𝑗](𝑦)(1 − 𝑦)𝑚(1 − 1/𝑦)𝑚
𝐿𝑗 ∶ [𝑗] ⊗ ([1] + 4[1/2] + 5[0])
𝑆𝑗 ∶ [𝑗] ⊗ (2[1/2] + 4[0])
𝑀𝜆
0 : ± (𝜆 + 1/2) + 2(±𝜆) + ±(𝜆 − 1/2)
𝐵0( any rep ) = 0
𝐵2(𝑀𝜆
0) = (−1)2𝜆+1 , 𝐵2(𝑆𝑗) = (−1)2𝑗+1𝐷𝑗 , 𝐵2(𝐿𝑗) = 0.
Lj ∶ [𝑗] ⊗ (42[0] + 48[1/2] + 27[1] + 8[3/2] + [2])
𝑠 = 2 supermassive long : 42[0] + 48[1/2] + 27[1] + 8[3/2] + [2]
𝐼𝑗: [𝑗] ⊗ (14[0] + 14[1/2] + 6[1] + [3/2])
𝐼3/2: 14[0] + 14[1/2] + 6[1] + [3/2]
𝑆𝑗 ∶ [𝑗] ⊗ (5[0] + 4[1/2] + [1]),
pág. 72
𝑆1: 5[0] + 4[1/2] + [1]
𝑀𝜆
0: [±(𝜆 + 1)] + 4[±(𝜆 + 1/2)] + 6[±(𝜆)] + 4[±(𝜆 − 1/2)] + [±(𝜆 − 1)],
𝑀1
0 ∶ [±2] + 4[±3/2] + 6[±1] + 4[±1/2] + 2[0]
𝐿𝑗 → 2𝐼𝑗 + 𝐼𝑗+1/2 + 𝐼𝑗−1/2
𝐼𝑗 → 2𝑆𝑗 + 𝑆𝑗+1/2 + 𝑆𝑗−1/2
𝑆𝑗 → ∑
𝑗
𝜆=0
𝑀𝜆
0 , 𝑗 − 𝜆 ∈ ℤ
𝐵𝑛( any rep ) = 0 for 𝑛 = 0,2.
𝐵4(𝐿𝑗) = 𝐵4(𝐼𝑗) = 0, 𝐵4(𝑆𝑗) = (−1)2𝑗 3
2 𝐷𝑗
𝐵4(𝑀𝜆
0) = (−1)2𝜆3, 𝐵4(𝑉0) = 3
2
𝐵6(𝐿𝑗) = 0 , 𝐵6(𝐼𝑗) = (−1)2𝑗+1 45
4 𝐷𝑗 , 𝐵6(𝑆𝑗) = (−1)2𝑗 15
8 𝐷𝑗
3
𝐵6(𝑀𝜆
0) = (−1)2𝜆 15
4 (1 + 12𝜆2) , 𝐵6(𝑉0) = 15
8
𝐵8(𝐿𝑗) = (−1)2𝑗 315
4 𝐷𝑗 , 𝐵8(𝐼𝑗) = (−1)2𝑗+1 105
16 𝐷𝑗(1 + 𝐷𝑗
2)
𝐵8(𝑆𝑗) = (−1)2𝑗 21
64 𝐷𝑗(1 + 2𝐷𝑗
4)
𝐵8(𝑀𝜆
0) = (−1)2𝜆 21
16 (1 + 80𝜆2 + 160𝜆4) , 𝐵8(𝑉0) = 63
32
(𝜆 ± 2) + 8 (𝜆 ± 3
2) + 28(𝜆 ± 1) + 56 (𝜆 ± 1
2) + 70(𝜆).
(±2) + 8 (± 3
2) + 28(±1) + 56 (± 1
2) + 70(0)
[𝑗] ⊗ ([2] + 8[3/2] + 27[1] + 48[1/2] + 42[0])
𝑆𝑗 → ∑
𝑗
𝜆=0
𝑀0
𝜆
𝐼1
𝑗: [𝑗] ⊗ ([5/2] + 10[2] + 44[3/2] + 110[1] + 165[1/2] + 132[0])
𝐼2
𝑗: [𝑗] ⊗ ([3] + 12[5/2] + 65[2] + 208[3/2] + 429[1] +
+572[1/2] + 429[0])
𝐼3
𝑗: [𝑗] ⊗ ([7/2] + 14[3] + 90[5/2] + 350[2] + 910[3/2] +
+1638[1] + 2002[1/2] + 1430[0])
𝑆1: 5[0] + 4[1/2] + [1]
𝑀𝜆
0: [±(𝜆 + 1)] + 4[±(𝜆 + 1/2)] + 6[±(𝜆)] + 4[±(𝜆 − 1/2)] + [±(𝜆 − 1)],
𝑀1
0 ∶ [±2] + 4[±3/2] + 6[±1] + 4[±1/2] + 2[0]
𝐿𝑗 → 2𝐼𝑗 + 𝐼𝑗+1/2 + 𝐼𝑗−1/2
𝐼𝑗 → 2𝑆𝑗 + 𝑆𝑗+1/2 + 𝑆𝑗−1/2
𝑆𝑗 → ∑
𝑗
𝜆=0
𝑀𝜆
0 , 𝑗 − 𝜆 ∈ ℤ
𝐵𝑛( any rep ) = 0 for 𝑛 = 0,2.
𝐵4(𝐿𝑗) = 𝐵4(𝐼𝑗) = 0, 𝐵4(𝑆𝑗) = (−1)2𝑗 3
2 𝐷𝑗
𝐵4(𝑀𝜆
0) = (−1)2𝜆3, 𝐵4(𝑉0) = 3
2
𝐵6(𝐿𝑗) = 0 , 𝐵6(𝐼𝑗) = (−1)2𝑗+1 45
4 𝐷𝑗 , 𝐵6(𝑆𝑗) = (−1)2𝑗 15
8 𝐷𝑗
3
𝐵6(𝑀𝜆
0) = (−1)2𝜆 15
4 (1 + 12𝜆2) , 𝐵6(𝑉0) = 15
8
𝐵8(𝐿𝑗) = (−1)2𝑗 315
4 𝐷𝑗 , 𝐵8(𝐼𝑗) = (−1)2𝑗+1 105
16 𝐷𝑗(1 + 𝐷𝑗
2)
𝐵8(𝑆𝑗) = (−1)2𝑗 21
64 𝐷𝑗(1 + 2𝐷𝑗
4)
𝐵8(𝑀𝜆
0) = (−1)2𝜆 21
16 (1 + 80𝜆2 + 160𝜆4) , 𝐵8(𝑉0) = 63
32
(𝜆 ± 2) + 8 (𝜆 ± 3
2) + 28(𝜆 ± 1) + 56 (𝜆 ± 1
2) + 70(𝜆).
(±2) + 8 (± 3
2) + 28(±1) + 56 (± 1
2) + 70(0)
[𝑗] ⊗ ([2] + 8[3/2] + 27[1] + 48[1/2] + 42[0])
𝑆𝑗 → ∑
𝑗
𝜆=0
𝑀0
𝜆
𝐼1
𝑗: [𝑗] ⊗ ([5/2] + 10[2] + 44[3/2] + 110[1] + 165[1/2] + 132[0])
𝐼2
𝑗: [𝑗] ⊗ ([3] + 12[5/2] + 65[2] + 208[3/2] + 429[1] +
+572[1/2] + 429[0])
𝐼3
𝑗: [𝑗] ⊗ ([7/2] + 14[3] + 90[5/2] + 350[2] + 910[3/2] +
+1638[1] + 2002[1/2] + 1430[0])
pág. 73
[𝑗] ⊗ ([4] + 16[7/2] + 119[3] + 544[5/2] + 1700[2] + 3808[3/2] + 6188[1] + 7072[1/2] + 4862[0])
𝐿𝑗 → 𝐼3
𝑗+1
2 + 2𝐼3
𝑗 + 𝐼3
𝑗−1
2
𝐼3
𝑗 → 𝐼2
𝑗+1
2 + 2𝐼2
𝑗 + 𝐼2
𝑗−1
2
𝐼2
𝑗 → 𝐼1
𝑗+1
2 + 2𝐼1
𝑗 + 𝐼1
𝑗−1
2
𝐼1
𝑗 → 𝑆𝑗+1
2 + 2𝑆𝑗 + 𝑆𝑗−1
2
𝐵8(𝑀0
𝜆) = (−1)2𝜆315
𝐵10(𝑀0
𝜆) = (−1)2𝜆 4725
2 (6𝜆2 + 1)
𝐵12(𝑀0
𝜆) = (−1)2𝜆 10395
16 (240𝜆4 + 240𝜆2 + 19)
𝐵14(𝑀0
𝜆) = (−1)2𝜆 45045
16 (336𝜆6 + 840𝜆4 + 399𝜆2 + 20)
𝐵16(𝑀0
𝜆) = (−1)2𝜆 135135
256 (7680𝜆8 + 35840𝜆6 + 42560𝜆4 + 12800𝜆2 + 457)
𝐵8(𝑆𝑗) = (−1)2𝑗 ⋅ 315
2 𝐷𝑗,
𝐵10(𝑆𝑗) = (−1)2𝑗 ⋅ 4725
8 𝐷𝑗(𝐷𝑗
2 + 1),
𝐵12(𝑆𝑗) = (−1)2𝑗 ⋅ 10395
32 𝐷𝑗(3𝐷𝑗
4 + 10𝐷𝑗
2 + 6),
𝐵14(𝑆𝑗) = (−1)2𝑗 ⋅ 45045
128 𝐷𝑗(3𝐷𝑗
6 + 21𝐷𝑗
4 + 42𝐷𝑗
2 + 14),
𝐵16(𝑆𝑗) = (−1)2𝑗 ⋅ 45045
512 𝐷𝑗(10𝐷𝑗
8 + 120𝐷𝑗
6 + 504𝐷𝑗
4 + 560𝐷𝑗
2 + 177),
𝐵8(𝐼1
𝑗) = 0
𝐵10(𝐼1
𝑗) = (−1)2𝑗+1 ⋅ 14175
4 𝐷𝑗
𝐵12(𝐼1
𝑗) = (−1)2𝑗+1 ⋅ 155925
16 𝐷𝑗(2𝐷𝑗
2 + 3)
𝐵14(𝐼1
𝑗) = (−1)2𝑗+1 ⋅ 2837835
64 𝐷𝑗(𝐷𝑗
2 + 1)(𝐷𝑗
2 + 4)
𝐵16(𝐼1
𝑗) = (−1)2𝑗+1 ⋅ 2027025
128 𝐷𝑗(4𝐷𝑗
6 + 42𝐷𝑗
4 + 112𝐷𝑗
2 + 57)
𝐵8(𝐼3
𝑗) = 𝐵10(𝐼3
𝑗) = 𝐵12(𝐼3
𝑗) = 0
[𝑗] ⊗ ([4] + 16[7/2] + 119[3] + 544[5/2] + 1700[2] + 3808[3/2] + 6188[1] + 7072[1/2] + 4862[0])
𝐿𝑗 → 𝐼3
𝑗+1
2 + 2𝐼3
𝑗 + 𝐼3
𝑗−1
2
𝐼3
𝑗 → 𝐼2
𝑗+1
2 + 2𝐼2
𝑗 + 𝐼2
𝑗−1
2
𝐼2
𝑗 → 𝐼1
𝑗+1
2 + 2𝐼1
𝑗 + 𝐼1
𝑗−1
2
𝐼1
𝑗 → 𝑆𝑗+1
2 + 2𝑆𝑗 + 𝑆𝑗−1
2
𝐵8(𝑀0
𝜆) = (−1)2𝜆315
𝐵10(𝑀0
𝜆) = (−1)2𝜆 4725
2 (6𝜆2 + 1)
𝐵12(𝑀0
𝜆) = (−1)2𝜆 10395
16 (240𝜆4 + 240𝜆2 + 19)
𝐵14(𝑀0
𝜆) = (−1)2𝜆 45045
16 (336𝜆6 + 840𝜆4 + 399𝜆2 + 20)
𝐵16(𝑀0
𝜆) = (−1)2𝜆 135135
256 (7680𝜆8 + 35840𝜆6 + 42560𝜆4 + 12800𝜆2 + 457)
𝐵8(𝑆𝑗) = (−1)2𝑗 ⋅ 315
2 𝐷𝑗,
𝐵10(𝑆𝑗) = (−1)2𝑗 ⋅ 4725
8 𝐷𝑗(𝐷𝑗
2 + 1),
𝐵12(𝑆𝑗) = (−1)2𝑗 ⋅ 10395
32 𝐷𝑗(3𝐷𝑗
4 + 10𝐷𝑗
2 + 6),
𝐵14(𝑆𝑗) = (−1)2𝑗 ⋅ 45045
128 𝐷𝑗(3𝐷𝑗
6 + 21𝐷𝑗
4 + 42𝐷𝑗
2 + 14),
𝐵16(𝑆𝑗) = (−1)2𝑗 ⋅ 45045
512 𝐷𝑗(10𝐷𝑗
8 + 120𝐷𝑗
6 + 504𝐷𝑗
4 + 560𝐷𝑗
2 + 177),
𝐵8(𝐼1
𝑗) = 0
𝐵10(𝐼1
𝑗) = (−1)2𝑗+1 ⋅ 14175
4 𝐷𝑗
𝐵12(𝐼1
𝑗) = (−1)2𝑗+1 ⋅ 155925
16 𝐷𝑗(2𝐷𝑗
2 + 3)
𝐵14(𝐼1
𝑗) = (−1)2𝑗+1 ⋅ 2837835
64 𝐷𝑗(𝐷𝑗
2 + 1)(𝐷𝑗
2 + 4)
𝐵16(𝐼1
𝑗) = (−1)2𝑗+1 ⋅ 2027025
128 𝐷𝑗(4𝐷𝑗
6 + 42𝐷𝑗
4 + 112𝐷𝑗
2 + 57)
𝐵8(𝐼3
𝑗) = 𝐵10(𝐼3
𝑗) = 𝐵12(𝐼3
𝑗) = 0
pág. 74
𝐵14(𝐼3
𝑗) = (−1)2𝑗+1 ⋅ 42567525
8 𝐷𝑗,
𝐵16(𝐼3
𝑗) = (−1)2𝑗+1 ⋅ 212837625
8 𝐷𝑗(2𝐷𝑗
2 + 5),
𝐵8(𝐿𝑗 ) = 𝐵10(𝐿𝑗) = 𝐵12(𝐿𝑗) = 𝐵14(𝐿𝑗) = 0,
𝐵16(𝐿𝑗) = (−1)2𝑗 ⋅ 638512875
2 𝐷𝑗.
𝐹𝑑(−1/𝜏) = 𝜏𝑑𝐹𝑑(𝜏) 𝐹𝑑(𝜏 + 1) = 𝐹𝑑(𝜏)
𝐸2 = 12
𝑖𝜋 𝜕𝜏log 𝜂 = 1 − 24 ∑
∞
𝑛=1
𝑛𝑞𝑛
1 − 𝑞𝑛
𝐸4 = 1
2 (𝜗2
8 + 𝜗3
8 + 𝜗4
8) = 1 + 240 ∑
∞
𝑛=1
𝑛3𝑞𝑛
1 − 𝑞𝑛
𝐸6 = 1
2 (𝜗2
4 + 𝜗3
4)(𝜗3
4 + 𝜗4
4)(𝜗4
4 − 𝜗2
4) = 1 − 504 ∑
∞
𝑛=1
𝑛5𝑞𝑛
1 − 𝑞𝑛 .
𝐻2 ≡ 1 − 𝐸2
24 = ∑
∞
𝑛=1
𝑛𝑞𝑛
1 − 𝑞𝑛 ≡ ∑
∞
𝑛=1
𝑑2(𝑛)𝑞𝑛
𝐻4 ≡ 𝐸4 − 1
240 = ∑
∞
𝑛=1
𝑛3𝑞𝑛
1 − 𝑞𝑛 ≡ ∑
∞
𝑛=1
𝑑4(𝑛)𝑞𝑛
𝐻6 ≡ 1 − 𝐸6
504 = ∑
∞
𝑛=1
𝑛5𝑞𝑛
1 − 𝑞𝑛 ≡ ∑
∞
𝑛=1
𝑑6(𝑛)𝑞𝑛
𝑑2𝑘(𝑁) = ∑
𝑛∣𝑁
𝑛2𝑘−1 , 𝑘 = 1,2,3
𝐸ˆ2 = 𝐸2 − 3
𝜋𝜏2
𝑗 = 𝐸4
3
𝜂24 = 1
𝑞 + 744 + ⋯ , 𝜂24 = 1
26 ⋅ 33 [𝐸4
3 − 𝐸6
2].
𝐹𝑑+2 = ( 𝑖
𝜋 𝜕𝜏 + 𝑑/2
𝜋𝜏2
) 𝐹𝑑 ≡ 𝐷𝑑𝐹𝑑
𝐷𝑑1+𝑑2 (𝐹𝑑1 𝐹𝑑2 ) = 𝐹𝑑2 (𝐷𝑑1 𝐹𝑑1 ) + 𝐹𝑑1 (𝐷𝑑2 𝐹𝑑2 )
𝐷2𝐸ˆ2 = 1
6 𝐸4 − 1
6 𝐸ˆ2
2 , 𝐷4𝐸4 = 2
3 𝐸6 − 2
3 𝐸ˆ2𝐸4 , 𝐷6𝐸6 = 𝐸4
2 − 𝐸ˆ2𝐸6
𝐵14(𝐼3
𝑗) = (−1)2𝑗+1 ⋅ 42567525
8 𝐷𝑗,
𝐵16(𝐼3
𝑗) = (−1)2𝑗+1 ⋅ 212837625
8 𝐷𝑗(2𝐷𝑗
2 + 5),
𝐵8(𝐿𝑗 ) = 𝐵10(𝐿𝑗) = 𝐵12(𝐿𝑗) = 𝐵14(𝐿𝑗) = 0,
𝐵16(𝐿𝑗) = (−1)2𝑗 ⋅ 638512875
2 𝐷𝑗.
𝐹𝑑(−1/𝜏) = 𝜏𝑑𝐹𝑑(𝜏) 𝐹𝑑(𝜏 + 1) = 𝐹𝑑(𝜏)
𝐸2 = 12
𝑖𝜋 𝜕𝜏log 𝜂 = 1 − 24 ∑
∞
𝑛=1
𝑛𝑞𝑛
1 − 𝑞𝑛
𝐸4 = 1
2 (𝜗2
8 + 𝜗3
8 + 𝜗4
8) = 1 + 240 ∑
∞
𝑛=1
𝑛3𝑞𝑛
1 − 𝑞𝑛
𝐸6 = 1
2 (𝜗2
4 + 𝜗3
4)(𝜗3
4 + 𝜗4
4)(𝜗4
4 − 𝜗2
4) = 1 − 504 ∑
∞
𝑛=1
𝑛5𝑞𝑛
1 − 𝑞𝑛 .
𝐻2 ≡ 1 − 𝐸2
24 = ∑
∞
𝑛=1
𝑛𝑞𝑛
1 − 𝑞𝑛 ≡ ∑
∞
𝑛=1
𝑑2(𝑛)𝑞𝑛
𝐻4 ≡ 𝐸4 − 1
240 = ∑
∞
𝑛=1
𝑛3𝑞𝑛
1 − 𝑞𝑛 ≡ ∑
∞
𝑛=1
𝑑4(𝑛)𝑞𝑛
𝐻6 ≡ 1 − 𝐸6
504 = ∑
∞
𝑛=1
𝑛5𝑞𝑛
1 − 𝑞𝑛 ≡ ∑
∞
𝑛=1
𝑑6(𝑛)𝑞𝑛
𝑑2𝑘(𝑁) = ∑
𝑛∣𝑁
𝑛2𝑘−1 , 𝑘 = 1,2,3
𝐸ˆ2 = 𝐸2 − 3
𝜋𝜏2
𝑗 = 𝐸4
3
𝜂24 = 1
𝑞 + 744 + ⋯ , 𝜂24 = 1
26 ⋅ 33 [𝐸4
3 − 𝐸6
2].
𝐹𝑑+2 = ( 𝑖
𝜋 𝜕𝜏 + 𝑑/2
𝜋𝜏2
) 𝐹𝑑 ≡ 𝐷𝑑𝐹𝑑
𝐷𝑑1+𝑑2 (𝐹𝑑1 𝐹𝑑2 ) = 𝐹𝑑2 (𝐷𝑑1 𝐹𝑑1 ) + 𝐹𝑑1 (𝐷𝑑2 𝐹𝑑2 )
𝐷2𝐸ˆ2 = 1
6 𝐸4 − 1
6 𝐸ˆ2
2 , 𝐷4𝐸4 = 2
3 𝐸6 − 2
3 𝐸ˆ2𝐸4 , 𝐷6𝐸6 = 𝐸4
2 − 𝐸ˆ2𝐸6
pág. 75
𝜗1
′′′
𝜗1
′ = −𝜋2𝐸2, 𝜗1
(5)
𝜗1
′ = −𝜋2𝐸2(4𝜋𝑖𝜕𝜏log 𝐸2 − 𝜋2𝐸2)
−3 𝜗1
(5)
𝜗1
′ + 5 (𝜗1
′′′
𝜗1
′ )
2
= 2𝜋4𝐸4
−15 𝜗1
(7)
𝜗1
′ − 350
3 (𝜗1
′′′
𝜗1
′ )
3
+ 105 𝜗1
(5)𝜗1
′′′
𝜗1
′2 = 80𝜋6
3 𝐸6
1
2 ∑
4
𝑖=2
𝜗𝑖
′′𝜗𝑖
7
(2𝜋𝑖)2 = 1
12 (𝐸2𝐸4 − 𝐸6)
𝜉(𝑣) = ∏
∞
𝑛=1
(1 − 𝑞𝑛)2
(1 − 𝑞𝑛𝑒2𝜋𝑖𝑣)(1 − 𝑞𝑛𝑒−2𝜋𝑖𝑣) = sin 𝜋𝑣
𝜋
𝜗1
′
𝜗1(𝑣) 𝜉(𝑣) = 𝜉(−𝑣)
𝜉(0) = 1 , 𝜉(2)(0) = − 1
3 (𝜋2 + 𝜗1
′′′
𝜗1
′ ) = − 𝜋2
3 (1 − 𝐸2)
𝜉(4)(0) = 𝜋4
5 + 2𝜋2
3
𝜗1
′′′
𝜗1
′ + 2
3 (𝜗1
′′′
𝜗1
′ )
2
− 1
5
𝜗1
(5)
𝜗1
′ = 𝜋4
15 (3 − 10𝐸2 + 2𝐸4 + 5𝐸2
2)
𝜉(6)(0) = − 𝜋6
7 − 𝜋4 𝜗1
′′′
𝜗1
′ − 10𝜋2
3 (𝜗1
′′′
𝜗1
′ )
2
+ 𝜋2 𝜗1
(5)
𝜗1
′ +
− 10
3 (𝜗1
′′′
𝜗1
′ )
3
+ 2 𝜗1
(5)𝜗1
′′′
𝜗1
′2 − 1
7
𝜗1
(7)
𝜗1
′
= 𝜋6
63 (−9 + 63𝐸2 − 105𝐸2
2 − 42𝐸4 + 16𝐸6 +
+42𝐸2𝐸4 + 35𝐸2
3)
𝑍(𝑣, 𝑣‾) = Str[𝑞𝐿0 𝑞‾𝐿‾0 𝑒2𝜋𝑖𝑣𝜆𝑅−2𝜋𝑖𝑣‾𝜆𝐿 ]
𝑍𝐷=4
heterotic = 1
𝜏2𝜂2𝜂‾2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗 [𝑎
𝑏]
𝜂 𝐶int [𝑎
𝑏]
𝑍𝐷=4
heterotic (𝑣, 𝑣‾) = 𝜉(𝑣)𝜉‾(𝑣‾)
𝜏2𝜂2𝜂‾2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗 [𝑎
𝑏] (𝑣)
𝜂 𝐶int [𝑎
𝑏]
𝑍𝐷=4
heterotic (𝑣, 𝑣‾) = 𝜉(𝑣)𝜉‾(𝑣‾)
𝜏2𝜂2𝜂‾2
𝜗 [1
1] (𝑣/2)
𝜂 𝐶int [1
1] (𝑣/2)
𝐶int [1
1] (𝑣) = Tr𝑅 [(−1)𝐹int
𝑒2𝜋𝑖𝑣𝐽0 𝑞𝐿0
int−3/8𝑞‾𝐿‾0
int−11/12]
𝑄 = 1
2𝜋𝑖
𝜕
𝜕𝑣 , 𝑄‾ = − 1
2𝜋𝑖
𝜕
𝜕𝑣‾
𝐵2𝑛 ≡ Str[𝜆2𝑛] = (𝑄 + 𝑄‾)2𝑛𝑍𝐷=4
heterotic (𝑣, 𝑣‾)|𝑣=𝑣‾=0
𝜗1
′′′
𝜗1
′ = −𝜋2𝐸2, 𝜗1
(5)
𝜗1
′ = −𝜋2𝐸2(4𝜋𝑖𝜕𝜏log 𝐸2 − 𝜋2𝐸2)
−3 𝜗1
(5)
𝜗1
′ + 5 (𝜗1
′′′
𝜗1
′ )
2
= 2𝜋4𝐸4
−15 𝜗1
(7)
𝜗1
′ − 350
3 (𝜗1
′′′
𝜗1
′ )
3
+ 105 𝜗1
(5)𝜗1
′′′
𝜗1
′2 = 80𝜋6
3 𝐸6
1
2 ∑
4
𝑖=2
𝜗𝑖
′′𝜗𝑖
7
(2𝜋𝑖)2 = 1
12 (𝐸2𝐸4 − 𝐸6)
𝜉(𝑣) = ∏
∞
𝑛=1
(1 − 𝑞𝑛)2
(1 − 𝑞𝑛𝑒2𝜋𝑖𝑣)(1 − 𝑞𝑛𝑒−2𝜋𝑖𝑣) = sin 𝜋𝑣
𝜋
𝜗1
′
𝜗1(𝑣) 𝜉(𝑣) = 𝜉(−𝑣)
𝜉(0) = 1 , 𝜉(2)(0) = − 1
3 (𝜋2 + 𝜗1
′′′
𝜗1
′ ) = − 𝜋2
3 (1 − 𝐸2)
𝜉(4)(0) = 𝜋4
5 + 2𝜋2
3
𝜗1
′′′
𝜗1
′ + 2
3 (𝜗1
′′′
𝜗1
′ )
2
− 1
5
𝜗1
(5)
𝜗1
′ = 𝜋4
15 (3 − 10𝐸2 + 2𝐸4 + 5𝐸2
2)
𝜉(6)(0) = − 𝜋6
7 − 𝜋4 𝜗1
′′′
𝜗1
′ − 10𝜋2
3 (𝜗1
′′′
𝜗1
′ )
2
+ 𝜋2 𝜗1
(5)
𝜗1
′ +
− 10
3 (𝜗1
′′′
𝜗1
′ )
3
+ 2 𝜗1
(5)𝜗1
′′′
𝜗1
′2 − 1
7
𝜗1
(7)
𝜗1
′
= 𝜋6
63 (−9 + 63𝐸2 − 105𝐸2
2 − 42𝐸4 + 16𝐸6 +
+42𝐸2𝐸4 + 35𝐸2
3)
𝑍(𝑣, 𝑣‾) = Str[𝑞𝐿0 𝑞‾𝐿‾0 𝑒2𝜋𝑖𝑣𝜆𝑅−2𝜋𝑖𝑣‾𝜆𝐿 ]
𝑍𝐷=4
heterotic = 1
𝜏2𝜂2𝜂‾2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗 [𝑎
𝑏]
𝜂 𝐶int [𝑎
𝑏]
𝑍𝐷=4
heterotic (𝑣, 𝑣‾) = 𝜉(𝑣)𝜉‾(𝑣‾)
𝜏2𝜂2𝜂‾2 ∑
1
𝑎,𝑏=0
(−1)𝑎+𝑏+𝑎𝑏 𝜗 [𝑎
𝑏] (𝑣)
𝜂 𝐶int [𝑎
𝑏]
𝑍𝐷=4
heterotic (𝑣, 𝑣‾) = 𝜉(𝑣)𝜉‾(𝑣‾)
𝜏2𝜂2𝜂‾2
𝜗 [1
1] (𝑣/2)
𝜂 𝐶int [1
1] (𝑣/2)
𝐶int [1
1] (𝑣) = Tr𝑅 [(−1)𝐹int
𝑒2𝜋𝑖𝑣𝐽0 𝑞𝐿0
int−3/8𝑞‾𝐿‾0
int−11/12]
𝑄 = 1
2𝜋𝑖
𝜕
𝜕𝑣 , 𝑄‾ = − 1
2𝜋𝑖
𝜕
𝜕𝑣‾
𝐵2𝑛 ≡ Str[𝜆2𝑛] = (𝑄 + 𝑄‾)2𝑛𝑍𝐷=4
heterotic (𝑣, 𝑣‾)|𝑣=𝑣‾=0
pág. 76
𝑍𝑁=4
heterotic (𝑣, 𝑣‾) = 𝜗1
4(𝑣/2)
𝜂12𝜂‾24 𝜉(𝑣)𝜉‾(𝑣‾) Γ6,22
𝜏2
𝐵4 = ⟨(𝑄 + 𝑄‾)4⟩ = ⟨𝑄4⟩ = 3
2
1
𝜂‾24
𝐵6 = ⟨(𝑄 + 𝑄‾)6⟩ = ⟨𝑄6 + 15𝑄4𝑄‾ 2⟩ = 15
8
2 − 𝐸‾2
𝜂‾24
𝑍𝑁=8
𝐼𝐼 (𝑣, 𝑣‾) = Str[𝑞𝐿0 𝑞‾𝐿‾0 𝑒2𝜋𝑖𝑣𝜆𝑅−2𝜋𝑖𝑣‾𝜆𝐿 ]
= 1
4 ∑
1
𝛼,𝛽=0
∑
1
𝛼‾ ,𝛽‾ =0
(−1)𝛼+𝛽+𝛼𝛽 𝜗 [𝛼
𝛽] (𝑣)𝜗3 [𝛼
𝛽] (0)
𝜂4
× (−1)𝛼‾ +𝛽‾ +𝛼‾ 𝛽‾ 𝜗‾ [𝛼
𝛽‾] 𝜗‾3 [𝛼‾
𝛽‾] (0)
𝜂‾4
𝜉(𝑣)𝜉‾(𝑣‾)
Im𝜏|𝜂|4
Γ6,6
|𝜂|12 = Γ6,6
Im𝜏
𝜗1
4(𝑣/2)
𝜂12
𝜗‾1
4(𝑣‾/2)
𝜂‾12 𝜉(𝑣)𝜉‾(𝑣‾)
𝐵8 = ⟨(𝑄 + 𝑄‾)8⟩ = 70⟨𝑄4𝑄‾ 4⟩ = 315
2
Γ6,6
Im𝜏
𝑀2 = 1
4 𝑝𝐿
2 , 𝑚⃗⃗ ⋅ 𝑛⃗ = 0
𝐵10= ⟨(𝑄 + 𝑄‾)10⟩ = 210⟨𝑄6𝑄‾ 4 + 𝑄4𝑄‾6⟩
= − 4725
8𝜋2
Γ6,6
Im𝜏 (𝜗1
′′′
𝜗1
′ + 3𝜉′′ + 𝑐𝑐) = 4725
4
Γ6,6
Im𝜏
𝐵12 = ⟨495(𝑄4𝑄‾ 8 + 𝑄8𝑄‾ 4) + 924𝑄6𝑄‾ 6⟩ = [10395
2 + 31185
64 (𝐸4 + 𝐸‾4)] Γ6,6
Im𝜏
= [10395 ⋅ 19
32 + 10395 ⋅ 45
4 (𝐸4 − 1
240 + 𝑐𝑐)] Γ6,6
Im𝜏
𝐼2
𝑗: ∑
𝑗
(−1)2𝑗𝐷𝑗 = 𝑑4(𝑁)
𝐵14 = ⟨(𝑄 + 𝑄‾)14⟩ =
= [45045
32 20 + 14189175
16 (2 𝐸4 − 1
240 + 1 − 𝐸6
504 + 𝑐𝑐)] Γ6,6
Im𝜏
𝐼2
𝑗: ∑
𝑗
(−1)2𝑗𝐷𝑗
3 = 𝑑6(𝑁)
𝑍𝐼𝐼 = 1
8 ∑
1
𝑔,ℎ=0
∑
1
𝛼,𝛽=0
∑
1
𝛼‾ ,𝛽‾ =0
(−1)𝛼+𝛽+𝛼𝛽 𝜗2 [𝛼
𝛽]
𝜂2
𝜗 [𝛼 + ℎ
𝛽 + 𝑔]
𝜂
𝜗 [𝛼 − ℎ
𝛽 − 𝑔]
𝜂 ×
(−1)𝛼‾ +𝛽‾ +𝛼‾ 𝛽‾ 𝜗‾
𝜂‾2 [𝛼‾𝛽‾
𝜂‾2 ]
𝜗‾ [𝛼‾ + ℎ
𝛽‾ + 𝑔]
𝜂‾
𝜗‾ [𝛼‾ − ℎ
𝛽‾ − 𝑔]
𝜂‾
1
Im𝜏|𝜂|4
Γ2,2
|𝜂|4 𝑍4,4 [ℎ
𝑔]
𝑍𝑁=4
heterotic (𝑣, 𝑣‾) = 𝜗1
4(𝑣/2)
𝜂12𝜂‾24 𝜉(𝑣)𝜉‾(𝑣‾) Γ6,22
𝜏2
𝐵4 = ⟨(𝑄 + 𝑄‾)4⟩ = ⟨𝑄4⟩ = 3
2
1
𝜂‾24
𝐵6 = ⟨(𝑄 + 𝑄‾)6⟩ = ⟨𝑄6 + 15𝑄4𝑄‾ 2⟩ = 15
8
2 − 𝐸‾2
𝜂‾24
𝑍𝑁=8
𝐼𝐼 (𝑣, 𝑣‾) = Str[𝑞𝐿0 𝑞‾𝐿‾0 𝑒2𝜋𝑖𝑣𝜆𝑅−2𝜋𝑖𝑣‾𝜆𝐿 ]
= 1
4 ∑
1
𝛼,𝛽=0
∑
1
𝛼‾ ,𝛽‾ =0
(−1)𝛼+𝛽+𝛼𝛽 𝜗 [𝛼
𝛽] (𝑣)𝜗3 [𝛼
𝛽] (0)
𝜂4
× (−1)𝛼‾ +𝛽‾ +𝛼‾ 𝛽‾ 𝜗‾ [𝛼
𝛽‾] 𝜗‾3 [𝛼‾
𝛽‾] (0)
𝜂‾4
𝜉(𝑣)𝜉‾(𝑣‾)
Im𝜏|𝜂|4
Γ6,6
|𝜂|12 = Γ6,6
Im𝜏
𝜗1
4(𝑣/2)
𝜂12
𝜗‾1
4(𝑣‾/2)
𝜂‾12 𝜉(𝑣)𝜉‾(𝑣‾)
𝐵8 = ⟨(𝑄 + 𝑄‾)8⟩ = 70⟨𝑄4𝑄‾ 4⟩ = 315
2
Γ6,6
Im𝜏
𝑀2 = 1
4 𝑝𝐿
2 , 𝑚⃗⃗ ⋅ 𝑛⃗ = 0
𝐵10= ⟨(𝑄 + 𝑄‾)10⟩ = 210⟨𝑄6𝑄‾ 4 + 𝑄4𝑄‾6⟩
= − 4725
8𝜋2
Γ6,6
Im𝜏 (𝜗1
′′′
𝜗1
′ + 3𝜉′′ + 𝑐𝑐) = 4725
4
Γ6,6
Im𝜏
𝐵12 = ⟨495(𝑄4𝑄‾ 8 + 𝑄8𝑄‾ 4) + 924𝑄6𝑄‾ 6⟩ = [10395
2 + 31185
64 (𝐸4 + 𝐸‾4)] Γ6,6
Im𝜏
= [10395 ⋅ 19
32 + 10395 ⋅ 45
4 (𝐸4 − 1
240 + 𝑐𝑐)] Γ6,6
Im𝜏
𝐼2
𝑗: ∑
𝑗
(−1)2𝑗𝐷𝑗 = 𝑑4(𝑁)
𝐵14 = ⟨(𝑄 + 𝑄‾)14⟩ =
= [45045
32 20 + 14189175
16 (2 𝐸4 − 1
240 + 1 − 𝐸6
504 + 𝑐𝑐)] Γ6,6
Im𝜏
𝐼2
𝑗: ∑
𝑗
(−1)2𝑗𝐷𝑗
3 = 𝑑6(𝑁)
𝑍𝐼𝐼 = 1
8 ∑
1
𝑔,ℎ=0
∑
1
𝛼,𝛽=0
∑
1
𝛼‾ ,𝛽‾ =0
(−1)𝛼+𝛽+𝛼𝛽 𝜗2 [𝛼
𝛽]
𝜂2
𝜗 [𝛼 + ℎ
𝛽 + 𝑔]
𝜂
𝜗 [𝛼 − ℎ
𝛽 − 𝑔]
𝜂 ×
(−1)𝛼‾ +𝛽‾ +𝛼‾ 𝛽‾ 𝜗‾
𝜂‾2 [𝛼‾𝛽‾
𝜂‾2 ]
𝜗‾ [𝛼‾ + ℎ
𝛽‾ + 𝑔]
𝜂‾
𝜗‾ [𝛼‾ − ℎ
𝛽‾ − 𝑔]
𝜂‾
1
Im𝜏|𝜂|4
Γ2,2
|𝜂|4 𝑍4,4 [ℎ
𝑔]
pág. 77
𝑍4,4 [0
0] = Γ4,4
|𝜂|8 , 𝑍4,4 [0
1] = 16 |𝜂|4
|𝜗2|4 = |𝜗3𝜗4|4
|𝜂|8
𝑍4,4 [1
0] = 16 |𝜂|4
|𝜗4|4 = |𝜗2𝜗3|4
|𝜂|8 , 𝑍4,4 [1
1] = 16 |𝜂|4
|𝜗3|4 = |𝜗2𝜗4|4
|𝜂|8
𝑍𝐼𝐼(𝑣, 𝑣‾) = 1
4 ∑
𝛼𝛽𝛼‾ 𝛽‾
(−1)𝛼+𝛽+𝛼𝛽+𝛼‾ +𝛽‾ +𝛼‾ 𝛽‾ 𝜗 [𝛼
𝛽] (𝑣)𝜗 [𝛼
𝛽] (0)
𝜂6
×
𝜗‾ [𝛼‾
𝛽‾] (𝑣‾)𝜗‾ [𝛼‾
𝛽‾] (0)
𝜂‾6 𝜉(𝑣)𝜉‾(𝑣‾)𝐶 [ 𝛼 𝛼‾
𝛽‾] Γ2,2
𝜏2
= 𝜗1
2(𝑣/2)𝜗‾1
2(𝑣‾/2)
𝜂6𝜂‾6 𝜉(𝑣)𝜉‾(𝑣‾)𝐶 [1 1
1 1] (𝑣/2, 𝑣‾/2) Γ2,2
𝜏2
𝐶 [1 1
1 1] (𝑣, 0) = 8 ∑
4
𝑖=2
𝜗𝑖
2(𝑣)
𝜗𝑖
2(0)
⟨𝜆4⟩ =⟩(𝑄 + 𝑄‾)4⟩ = 6⟨𝑄2𝑄‾2 + 𝑄2𝑄‾ 4⟩ = 36 Γ2,2
𝜏2
.
⟨𝜆6⟩ =⟩(𝑄 + 𝑄‾)6⟩ = 15⟨𝑄4𝑄‾ 2 + 𝑄2𝑄‾ 4⟩ = 90 Γ2,2
𝜏2
,
𝜕𝑣
2𝐶 [1 1
1 1] (𝑣, 0)|𝑣=0
= −16𝜋2𝐸2
𝐿gauge = − 1
8 Im ∫ 𝑑4𝑥√−det𝑔𝐅𝜇𝜈
𝑖 𝑁𝑖𝑗𝐅𝑗,𝜇𝜈
𝐅𝜇𝜈 = 𝐹𝜇𝜈 + 𝑖⋆𝐹𝜇𝜈 , ⋆𝐹𝜇𝜈 = 1
2
𝜖𝜇𝜈 𝜌𝜎
√−𝑔 𝐹𝜌𝜎
𝐿gauge = − 1
4 ∫ 𝑑4𝑥[√−𝑔𝐹𝜇𝜈
𝑖 𝑁2
𝑖𝑗𝐹𝑗,𝜇𝜈 + 𝐹𝜇𝜈
𝑖 𝑁1
𝑖𝑗⋆𝐹𝑗,𝜇𝜈]
𝐆𝜇𝜈
𝑖 = 𝑁𝑖𝑗𝐅𝜇𝜈
𝑗 = 𝑁1𝐹 − 𝑁2 ⋆𝐹 + 𝑖(𝑁2𝐹 + 𝑁1 ⋆𝐹)
Im∇𝜇 (𝐆𝜇𝜈
𝑖
𝐅𝜇𝜈
𝑖 ) = (0
0)
(𝐆′ 𝜇𝜈
𝐅′ 𝜇𝜈
) = (𝐴 𝐵
𝐶 𝐷) (𝐆𝜇𝜈
𝐅𝜇𝜈
)
𝑁′ = (𝐴𝑁 + 𝐵)(𝐶𝑁 + 𝐷)−1
𝐹′ = 𝐶(𝑁1𝐹 − 𝑁2 ⋆𝐹) + 𝐷𝐹 , ⋆𝐹′ = 𝐶(𝑁2𝐹 + 𝑁1 ⋆𝐹) + 𝐷⋆𝐹
𝑍4,4 [0
0] = Γ4,4
|𝜂|8 , 𝑍4,4 [0
1] = 16 |𝜂|4
|𝜗2|4 = |𝜗3𝜗4|4
|𝜂|8
𝑍4,4 [1
0] = 16 |𝜂|4
|𝜗4|4 = |𝜗2𝜗3|4
|𝜂|8 , 𝑍4,4 [1
1] = 16 |𝜂|4
|𝜗3|4 = |𝜗2𝜗4|4
|𝜂|8
𝑍𝐼𝐼(𝑣, 𝑣‾) = 1
4 ∑
𝛼𝛽𝛼‾ 𝛽‾
(−1)𝛼+𝛽+𝛼𝛽+𝛼‾ +𝛽‾ +𝛼‾ 𝛽‾ 𝜗 [𝛼
𝛽] (𝑣)𝜗 [𝛼
𝛽] (0)
𝜂6
×
𝜗‾ [𝛼‾
𝛽‾] (𝑣‾)𝜗‾ [𝛼‾
𝛽‾] (0)
𝜂‾6 𝜉(𝑣)𝜉‾(𝑣‾)𝐶 [ 𝛼 𝛼‾
𝛽‾] Γ2,2
𝜏2
= 𝜗1
2(𝑣/2)𝜗‾1
2(𝑣‾/2)
𝜂6𝜂‾6 𝜉(𝑣)𝜉‾(𝑣‾)𝐶 [1 1
1 1] (𝑣/2, 𝑣‾/2) Γ2,2
𝜏2
𝐶 [1 1
1 1] (𝑣, 0) = 8 ∑
4
𝑖=2
𝜗𝑖
2(𝑣)
𝜗𝑖
2(0)
⟨𝜆4⟩ =⟩(𝑄 + 𝑄‾)4⟩ = 6⟨𝑄2𝑄‾2 + 𝑄2𝑄‾ 4⟩ = 36 Γ2,2
𝜏2
.
⟨𝜆6⟩ =⟩(𝑄 + 𝑄‾)6⟩ = 15⟨𝑄4𝑄‾ 2 + 𝑄2𝑄‾ 4⟩ = 90 Γ2,2
𝜏2
,
𝜕𝑣
2𝐶 [1 1
1 1] (𝑣, 0)|𝑣=0
= −16𝜋2𝐸2
𝐿gauge = − 1
8 Im ∫ 𝑑4𝑥√−det𝑔𝐅𝜇𝜈
𝑖 𝑁𝑖𝑗𝐅𝑗,𝜇𝜈
𝐅𝜇𝜈 = 𝐹𝜇𝜈 + 𝑖⋆𝐹𝜇𝜈 , ⋆𝐹𝜇𝜈 = 1
2
𝜖𝜇𝜈 𝜌𝜎
√−𝑔 𝐹𝜌𝜎
𝐿gauge = − 1
4 ∫ 𝑑4𝑥[√−𝑔𝐹𝜇𝜈
𝑖 𝑁2
𝑖𝑗𝐹𝑗,𝜇𝜈 + 𝐹𝜇𝜈
𝑖 𝑁1
𝑖𝑗⋆𝐹𝑗,𝜇𝜈]
𝐆𝜇𝜈
𝑖 = 𝑁𝑖𝑗𝐅𝜇𝜈
𝑗 = 𝑁1𝐹 − 𝑁2 ⋆𝐹 + 𝑖(𝑁2𝐹 + 𝑁1 ⋆𝐹)
Im∇𝜇 (𝐆𝜇𝜈
𝑖
𝐅𝜇𝜈
𝑖 ) = (0
0)
(𝐆′ 𝜇𝜈
𝐅′ 𝜇𝜈
) = (𝐴 𝐵
𝐶 𝐷) (𝐆𝜇𝜈
𝐅𝜇𝜈
)
𝑁′ = (𝐴𝑁 + 𝐵)(𝐶𝑁 + 𝐷)−1
𝐹′ = 𝐶(𝑁1𝐹 − 𝑁2 ⋆𝐹) + 𝐷𝐹 , ⋆𝐹′ = 𝐶(𝑁2𝐹 + 𝑁1 ⋆𝐹) + 𝐷⋆𝐹
pág. 78
𝐹′ = 𝑁1𝐹 − 𝑁2
⋆𝐹 , ⋆𝐹′ = 𝑁2𝐹 + 𝑁1
⋆𝐹 , 𝑁′ = − 1
𝑁 .
(𝐴 𝐵
𝐶 𝐷) = (𝟏 − 𝑒 −𝑒
𝑒 𝟏 − 𝑒) , 𝑒 = (
1 0 …
0 0 …
⋅ ⋅
) , .
𝑁00
′ = − 1
𝑁00
, 𝑁0𝑖
′ = 𝑁0𝑖
𝑁00
, 𝑁𝑖0
′ = 𝑁𝑖0
𝑁00
, 𝑁𝑖𝑗
′ = 𝑁𝑖𝑗 − 𝑁𝑖0𝑁0𝑗
𝑁00
.
(𝐴 𝐵
𝐶 𝐷) = (𝟏 − 𝑒1 𝑒2
−𝑒2 𝟏 − 𝑒1
) ,
𝑒1 = (
1 0 0 …
0 1 0 …
0 0 0 .
. . . .
) , 𝑒2 = (
0 1 0 …
−1 0 0 …
0 0 0 .
. . . .
) .
𝑁𝛼𝛽
′ = − 𝑁𝛼𝛽
det𝑁𝛼𝛽
𝑁𝛼𝑖
′ = − 𝑁𝛼𝛽𝜖𝛽𝛾𝑁𝛾𝑖
det𝑁𝛼𝛽
, 𝑁𝑖𝛼
′ = 𝑁𝑖𝛽 𝜖𝛽𝛾𝑁𝛼𝛾
det𝑁𝛼𝛽
𝑁𝑖𝑗
′ = 𝑁𝑖𝑗 + 𝑁𝑖𝛼𝜖𝛼𝛽𝑁𝛽𝛾𝜖𝛾𝛿𝑁𝛿𝑗
det𝑁𝛼𝛽
𝑁 = 𝑆1𝐿 + 𝑖𝑆2𝑀−1 , 𝑆 = 𝑆1 + 𝑖𝑆2
𝑁′ = −𝑁−1 = − 𝑆1
|𝑆|2 𝐿 + 𝑖 𝑆2
|𝑆|2 𝑀 = − 𝑆1
|𝑆|2 𝐿 + 𝑖 𝑆2
|𝑆|2 𝐿𝑀−1𝐿
𝑁′ = −𝐿𝑁−1𝐿 = − 𝑆1
|𝑆|2 𝐿 + 𝑖 𝑆2
|𝑆|2 𝑀−1
(𝐴 𝐵
𝐶 𝐷) = (𝑎𝟏28 𝑏𝐿
𝑐𝐿 𝑑𝟏28
) , 𝑎𝑑 − 𝑏𝑐 = 1
Supergravedad cuántica perturbativa en campos cuánticos relativistas. Modelo Matemático.
Espacio de Moduli, amplitudes y gauge fixing. Invariante BRST y tensores. Invariancia
difeomorfista. Invariancia de gauge. Operadores de Vértice.
𝐼 = 𝐼𝑋 + 𝐼gh
𝐼gh = 1
2𝜋 ∫ d2𝜎√𝑔𝑏𝑖𝑗𝐷𝑖𝑐𝑗
{𝑄𝐵, 𝑏𝑖𝑗} = 𝑇𝑖𝑗
𝛿𝐼 = 1
4𝜋 ∫ d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑇𝑖𝑗
[𝑄𝐵, 𝑔𝑖𝑗] = 𝛿𝑔𝑖𝑗, {𝑄𝐵, 𝛿𝑔𝑖𝑗} = 0
𝐼 → 𝐼̂ = 𝐼 + 1
4𝜋 ∫ d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗
𝐹′ = 𝑁1𝐹 − 𝑁2
⋆𝐹 , ⋆𝐹′ = 𝑁2𝐹 + 𝑁1
⋆𝐹 , 𝑁′ = − 1
𝑁 .
(𝐴 𝐵
𝐶 𝐷) = (𝟏 − 𝑒 −𝑒
𝑒 𝟏 − 𝑒) , 𝑒 = (
1 0 …
0 0 …
⋅ ⋅
) , .
𝑁00
′ = − 1
𝑁00
, 𝑁0𝑖
′ = 𝑁0𝑖
𝑁00
, 𝑁𝑖0
′ = 𝑁𝑖0
𝑁00
, 𝑁𝑖𝑗
′ = 𝑁𝑖𝑗 − 𝑁𝑖0𝑁0𝑗
𝑁00
.
(𝐴 𝐵
𝐶 𝐷) = (𝟏 − 𝑒1 𝑒2
−𝑒2 𝟏 − 𝑒1
) ,
𝑒1 = (
1 0 0 …
0 1 0 …
0 0 0 .
. . . .
) , 𝑒2 = (
0 1 0 …
−1 0 0 …
0 0 0 .
. . . .
) .
𝑁𝛼𝛽
′ = − 𝑁𝛼𝛽
det𝑁𝛼𝛽
𝑁𝛼𝑖
′ = − 𝑁𝛼𝛽𝜖𝛽𝛾𝑁𝛾𝑖
det𝑁𝛼𝛽
, 𝑁𝑖𝛼
′ = 𝑁𝑖𝛽 𝜖𝛽𝛾𝑁𝛼𝛾
det𝑁𝛼𝛽
𝑁𝑖𝑗
′ = 𝑁𝑖𝑗 + 𝑁𝑖𝛼𝜖𝛼𝛽𝑁𝛽𝛾𝜖𝛾𝛿𝑁𝛿𝑗
det𝑁𝛼𝛽
𝑁 = 𝑆1𝐿 + 𝑖𝑆2𝑀−1 , 𝑆 = 𝑆1 + 𝑖𝑆2
𝑁′ = −𝑁−1 = − 𝑆1
|𝑆|2 𝐿 + 𝑖 𝑆2
|𝑆|2 𝑀 = − 𝑆1
|𝑆|2 𝐿 + 𝑖 𝑆2
|𝑆|2 𝐿𝑀−1𝐿
𝑁′ = −𝐿𝑁−1𝐿 = − 𝑆1
|𝑆|2 𝐿 + 𝑖 𝑆2
|𝑆|2 𝑀−1
(𝐴 𝐵
𝐶 𝐷) = (𝑎𝟏28 𝑏𝐿
𝑐𝐿 𝑑𝟏28
) , 𝑎𝑑 − 𝑏𝑐 = 1
Supergravedad cuántica perturbativa en campos cuánticos relativistas. Modelo Matemático.
Espacio de Moduli, amplitudes y gauge fixing. Invariante BRST y tensores. Invariancia
difeomorfista. Invariancia de gauge. Operadores de Vértice.
𝐼 = 𝐼𝑋 + 𝐼gh
𝐼gh = 1
2𝜋 ∫ d2𝜎√𝑔𝑏𝑖𝑗𝐷𝑖𝑐𝑗
{𝑄𝐵, 𝑏𝑖𝑗} = 𝑇𝑖𝑗
𝛿𝐼 = 1
4𝜋 ∫ d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑇𝑖𝑗
[𝑄𝐵, 𝑔𝑖𝑗] = 𝛿𝑔𝑖𝑗, {𝑄𝐵, 𝛿𝑔𝑖𝑗} = 0
𝐼 → 𝐼̂ = 𝐼 + 1
4𝜋 ∫ d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗
pág. 79
𝐹(𝑔 ∣ 𝛿𝑔) = ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ (𝑋, 𝑏, 𝑐; 𝑔, 𝛿𝑔))= ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼)exp (− 1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗)
[𝑄𝐵, 𝐹(𝑔 ∣ 𝛿𝑔)} = 0
exp (− 1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗) = ∑
∞
𝑛=0
1
𝑛! (− 1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗)
𝑛
𝑄𝐵 = ∫
Σ
d2𝜎 ∑
𝑖,𝑗=1,2
√𝑔𝛿𝑔𝑖𝑗
𝛿
𝛿𝑔𝑖𝑗
d = ∑
𝑠
𝑖=1
d𝑥𝑖 𝜕
𝜕𝑥𝑖 ,
𝐹(𝑥1 … ∣ ⋯ d𝑥𝑠) = ∑
𝑖1<⋯<𝑖𝑝
𝐹𝑖1…𝑖𝑝 (𝑥1 … 𝑥𝑠)d𝑥𝑖1 … d𝑥𝑖𝑝
𝑔𝑖𝑗 → 𝑔𝑖𝑗 + 𝜖(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)
𝛿𝑔𝑖𝑗 → 𝛿𝑔𝑖𝑗 + 𝜖(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)
∫
Σ
d2𝜎√𝑔(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖) 𝛿
𝛿(𝛿𝑔𝑖𝑗) 𝐹(𝑔 ∣ 𝛿𝑔) = 0
∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) ∫
Σ
d2𝜎√𝑔(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)𝑏𝑖𝑗 = 0
𝐷𝑖𝑏𝑖𝑗 = 0.
𝑐𝑖 → 𝑐𝑖 + 𝜖𝑣𝑖,
∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) → ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) (1 + 𝜖
2𝜋 ∫
Σ
d2𝜎√𝑔𝑏𝑖𝑗𝐷𝑖𝑣𝑗)
𝑏𝑖𝑗 = ∑
6𝑔−6
𝛼=1
u𝛼b𝛼,𝑖𝑗 + ∑
𝜆
𝑤𝜆b𝜆𝑖𝑗
′ .
𝐼̂ = ⋯ + 1
4𝜋 ∑
6𝑔−6
𝛼=1
∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗u𝛼b𝛼
𝑖𝑗.
∏
6𝑔−6
𝛼=1
∫ d𝑢𝛼exp (u𝛼
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗 b𝛼
𝑖𝑗) = ∏
6𝑔−6
𝛼=1
1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗 b𝛼
𝑖𝑗.
𝑣𝛼 = 1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗 b𝛼
𝑖𝑗
𝐹(𝑔 ∣ 𝛿𝑔) = ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ (𝑋, 𝑏, 𝑐; 𝑔, 𝛿𝑔))= ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼)exp (− 1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗)
[𝑄𝐵, 𝐹(𝑔 ∣ 𝛿𝑔)} = 0
exp (− 1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗) = ∑
∞
𝑛=0
1
𝑛! (− 1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗)
𝑛
𝑄𝐵 = ∫
Σ
d2𝜎 ∑
𝑖,𝑗=1,2
√𝑔𝛿𝑔𝑖𝑗
𝛿
𝛿𝑔𝑖𝑗
d = ∑
𝑠
𝑖=1
d𝑥𝑖 𝜕
𝜕𝑥𝑖 ,
𝐹(𝑥1 … ∣ ⋯ d𝑥𝑠) = ∑
𝑖1<⋯<𝑖𝑝
𝐹𝑖1…𝑖𝑝 (𝑥1 … 𝑥𝑠)d𝑥𝑖1 … d𝑥𝑖𝑝
𝑔𝑖𝑗 → 𝑔𝑖𝑗 + 𝜖(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)
𝛿𝑔𝑖𝑗 → 𝛿𝑔𝑖𝑗 + 𝜖(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)
∫
Σ
d2𝜎√𝑔(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖) 𝛿
𝛿(𝛿𝑔𝑖𝑗) 𝐹(𝑔 ∣ 𝛿𝑔) = 0
∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) ∫
Σ
d2𝜎√𝑔(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)𝑏𝑖𝑗 = 0
𝐷𝑖𝑏𝑖𝑗 = 0.
𝑐𝑖 → 𝑐𝑖 + 𝜖𝑣𝑖,
∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) → ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) (1 + 𝜖
2𝜋 ∫
Σ
d2𝜎√𝑔𝑏𝑖𝑗𝐷𝑖𝑣𝑗)
𝑏𝑖𝑗 = ∑
6𝑔−6
𝛼=1
u𝛼b𝛼,𝑖𝑗 + ∑
𝜆
𝑤𝜆b𝜆𝑖𝑗
′ .
𝐼̂ = ⋯ + 1
4𝜋 ∑
6𝑔−6
𝛼=1
∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗u𝛼b𝛼
𝑖𝑗.
∏
6𝑔−6
𝛼=1
∫ d𝑢𝛼exp (u𝛼
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗 b𝛼
𝑖𝑗) = ∏
6𝑔−6
𝛼=1
1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗 b𝛼
𝑖𝑗.
𝑣𝛼 = 1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗 b𝛼
𝑖𝑗
pág. 80
∏
6𝑔−6
𝛼=1
𝛿(𝑣𝛼)
𝑐𝑖 = ∑
𝜆
𝛾𝜆𝑐𝜆
𝑖
∏
𝜆
∫ 𝐷𝛾𝜆exp (𝑚𝜆𝛾𝜆𝑤𝜆) = ∏
𝜆
𝑚𝜆 ⋅ ∏
𝜆
𝛿(𝑤𝜆)
𝑍g = ∫
ℳg
𝐹(𝑔 ∣ 𝛿𝑔)
𝛿𝑔𝑖𝑗 = ∑
p
𝑠=1
𝜕𝑔𝑖𝑗
𝜕𝑚𝑠
d𝑚𝑠.
1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗 = 1
4𝜋 ∑
p
𝑠=1
d𝑚𝑠 ∫
Σ
d2𝜎 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝑚𝑠
𝑏𝑖𝑗
𝐹(𝑔 ∣ 𝛿𝑔) = ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) = ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼 − 1
4𝜋 ∑
p
𝑠=1
d𝑚𝑠 ∫
Σ
d2𝜎 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝑚𝑠
𝑏𝑖𝑗)
d𝑚1, … , d𝑚p
𝐹top (𝑚1 … ∣ ⋯ d𝑚𝔭) = (−1)𝔭(𝔭+1)/2 d𝑚1 … d𝑚𝔭 ∫ 𝒟(𝑋, 𝑏, 𝑐)𝑒−𝐼 ∏
𝔭
𝑠=1
1
4𝜋 ∫
Σ
d2𝜎 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝑚𝑠
𝑏𝑖𝑗
Ψ𝑠 = 1
4𝜋 ∫
Σ
d2𝜎 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝑚𝑠
𝑏𝑖𝑗.
(−1)𝔭(𝔭+1)/2 ∫ 𝒟(𝑚1 … ∣ ⋯ d𝑚𝔭)d𝑚1 … d𝑚𝔭 ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼)Ψ1 … Ψ𝔭.
(−1)𝔭(𝔭+1)/2 ∫ 𝒟(𝑚1 … ∣ ⋯ d𝑚𝔭)d𝑚1 … d𝑚𝔭 ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼)𝛿(Ψ1) … 𝛿(Ψ𝔭).
∫ 𝒟(𝑚1, … , 𝑚𝔭) ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼)𝛿(Ψ1) … 𝛿(Ψ𝔭)
𝐹Ω(𝑔 ∣ 𝛿𝑔) = ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ (𝑋, 𝑏, 𝑐, 𝑔, 𝛿𝑔))Ω.
d𝐹Ω + 𝐹𝑄𝐵Ω = 0
∏
6𝑔−6
𝛼=1
𝛿(𝑣𝛼)
𝑐𝑖 = ∑
𝜆
𝛾𝜆𝑐𝜆
𝑖
∏
𝜆
∫ 𝐷𝛾𝜆exp (𝑚𝜆𝛾𝜆𝑤𝜆) = ∏
𝜆
𝑚𝜆 ⋅ ∏
𝜆
𝛿(𝑤𝜆)
𝑍g = ∫
ℳg
𝐹(𝑔 ∣ 𝛿𝑔)
𝛿𝑔𝑖𝑗 = ∑
p
𝑠=1
𝜕𝑔𝑖𝑗
𝜕𝑚𝑠
d𝑚𝑠.
1
4𝜋 ∫
Σ
d2𝜎√𝑔𝛿𝑔𝑖𝑗𝑏𝑖𝑗 = 1
4𝜋 ∑
p
𝑠=1
d𝑚𝑠 ∫
Σ
d2𝜎 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝑚𝑠
𝑏𝑖𝑗
𝐹(𝑔 ∣ 𝛿𝑔) = ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) = ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼 − 1
4𝜋 ∑
p
𝑠=1
d𝑚𝑠 ∫
Σ
d2𝜎 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝑚𝑠
𝑏𝑖𝑗)
d𝑚1, … , d𝑚p
𝐹top (𝑚1 … ∣ ⋯ d𝑚𝔭) = (−1)𝔭(𝔭+1)/2 d𝑚1 … d𝑚𝔭 ∫ 𝒟(𝑋, 𝑏, 𝑐)𝑒−𝐼 ∏
𝔭
𝑠=1
1
4𝜋 ∫
Σ
d2𝜎 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝑚𝑠
𝑏𝑖𝑗
Ψ𝑠 = 1
4𝜋 ∫
Σ
d2𝜎 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝑚𝑠
𝑏𝑖𝑗.
(−1)𝔭(𝔭+1)/2 ∫ 𝒟(𝑚1 … ∣ ⋯ d𝑚𝔭)d𝑚1 … d𝑚𝔭 ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼)Ψ1 … Ψ𝔭.
(−1)𝔭(𝔭+1)/2 ∫ 𝒟(𝑚1 … ∣ ⋯ d𝑚𝔭)d𝑚1 … d𝑚𝔭 ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼)𝛿(Ψ1) … 𝛿(Ψ𝔭).
∫ 𝒟(𝑚1, … , 𝑚𝔭) ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼)𝛿(Ψ1) … 𝛿(Ψ𝔭)
𝐹Ω(𝑔 ∣ 𝛿𝑔) = ∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ (𝑋, 𝑏, 𝑐, 𝑔, 𝛿𝑔))Ω.
d𝐹Ω + 𝐹𝑄𝐵Ω = 0
pág. 81
Ω = ∏
𝑛
𝑠=1
𝒱𝑠(𝑋, 𝑏, 𝑐; 𝑝𝑠)
𝑏𝑛𝒱𝑠 = 𝑏̃𝑛𝒱𝑠 = 0, 𝑛 ≥ 0
𝑏𝑛 = 1
2𝜋𝑖 ∮ d𝑧𝑧𝑛+1𝑏𝑧𝑧, 𝑏̃𝑛 = 1
2𝜋𝑖 ∮ d𝑧̃ 𝑧̃ 𝑛+1𝑏𝑧̃ 𝑧̃
𝑏𝑛𝒲𝑠 = 𝑏̃𝑛𝒲𝑠 = 0, 𝑛 ≥ 0.
𝛿𝑐𝑖 = 𝑐𝑗𝜕𝑗𝑐𝑖
𝛿𝑋 = 𝑐𝑗𝜕𝑗𝑋
𝛿𝑔𝑖𝑗 = 𝐷𝑖𝑐𝑗 + 𝐷𝑗𝑐𝑖 − 𝑔𝑖𝑗𝐷𝑘𝑐𝑘
𝑣𝑖(𝑝1) = ⋯ = 𝑣𝑖(𝑝𝔫) = 0
ℳg,n = 𝒥/𝒟𝑝1,…,𝑝n
∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) ∏
n
𝑠=1
𝒱𝑠(𝑋, 𝑏, 𝑐; 𝑝𝑠) ∫
Σ
d2𝜎√𝑔(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)𝑏𝑖𝑗 = 0
⟨𝒱1 … 𝒱n⟩g = ∫
ℳg,n
𝐹𝒱1,…,𝒱n
∫
ℳg,n
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = 0
d𝐹𝒲1,𝒱2,…,𝒱n + 𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = 0
∫
ℳg,n
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = − ∫
ℳg,n
d𝐹𝒲1,𝒱2,…,𝒱n
• Deligne-Mumford – Riemann.
𝑧 = 𝐳, 𝑧̃ = 𝐳̃
𝑧 = {𝑤 + 𝐳 if |𝑤| < 𝜖
𝑤 if |𝑤| > 1 − 𝜖
𝑧 = 𝑤 + 𝑓(|𝑤|)𝐳
𝑧̃ = 𝑤̃ + 𝑓(|𝑤|)𝐳̃
𝜕𝐳𝑔𝐼𝐽 = 𝐷𝐼𝑣𝐽 + 𝐷𝐽𝑣𝐼
𝜕𝐳̃ 𝑔𝐼𝐽 = 𝐷𝐼𝑣̃𝐽 + 𝐷𝐽𝑣̃𝐼,
(𝑣𝑤, 𝑣𝑤̃ ) = −(𝜕𝐳|𝑧,𝑧̃ 𝑤, 𝜕𝐳|𝑧,𝑧̃ 𝑤̃ )
Ω = ∏
𝑛
𝑠=1
𝒱𝑠(𝑋, 𝑏, 𝑐; 𝑝𝑠)
𝑏𝑛𝒱𝑠 = 𝑏̃𝑛𝒱𝑠 = 0, 𝑛 ≥ 0
𝑏𝑛 = 1
2𝜋𝑖 ∮ d𝑧𝑧𝑛+1𝑏𝑧𝑧, 𝑏̃𝑛 = 1
2𝜋𝑖 ∮ d𝑧̃ 𝑧̃ 𝑛+1𝑏𝑧̃ 𝑧̃
𝑏𝑛𝒲𝑠 = 𝑏̃𝑛𝒲𝑠 = 0, 𝑛 ≥ 0.
𝛿𝑐𝑖 = 𝑐𝑗𝜕𝑗𝑐𝑖
𝛿𝑋 = 𝑐𝑗𝜕𝑗𝑋
𝛿𝑔𝑖𝑗 = 𝐷𝑖𝑐𝑗 + 𝐷𝑗𝑐𝑖 − 𝑔𝑖𝑗𝐷𝑘𝑐𝑘
𝑣𝑖(𝑝1) = ⋯ = 𝑣𝑖(𝑝𝔫) = 0
ℳg,n = 𝒥/𝒟𝑝1,…,𝑝n
∫ 𝒟(𝑋, 𝑏, 𝑐)exp (−𝐼̂ ) ∏
n
𝑠=1
𝒱𝑠(𝑋, 𝑏, 𝑐; 𝑝𝑠) ∫
Σ
d2𝜎√𝑔(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)𝑏𝑖𝑗 = 0
⟨𝒱1 … 𝒱n⟩g = ∫
ℳg,n
𝐹𝒱1,…,𝒱n
∫
ℳg,n
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = 0
d𝐹𝒲1,𝒱2,…,𝒱n + 𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = 0
∫
ℳg,n
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = − ∫
ℳg,n
d𝐹𝒲1,𝒱2,…,𝒱n
• Deligne-Mumford – Riemann.
𝑧 = 𝐳, 𝑧̃ = 𝐳̃
𝑧 = {𝑤 + 𝐳 if |𝑤| < 𝜖
𝑤 if |𝑤| > 1 − 𝜖
𝑧 = 𝑤 + 𝑓(|𝑤|)𝐳
𝑧̃ = 𝑤̃ + 𝑓(|𝑤|)𝐳̃
𝜕𝐳𝑔𝐼𝐽 = 𝐷𝐼𝑣𝐽 + 𝐷𝐽𝑣𝐼
𝜕𝐳̃ 𝑔𝐼𝐽 = 𝐷𝐼𝑣̃𝐽 + 𝐷𝐽𝑣̃𝐼,
(𝑣𝑤, 𝑣𝑤̃ ) = −(𝜕𝐳|𝑧,𝑧̃ 𝑤, 𝜕𝐳|𝑧,𝑧̃ 𝑤̃ )
pág. 82
(𝑣𝑤, 𝑣𝑤̃ ) = (1,0)
(𝑣̃ 𝑤, 𝑣̃ 𝑤̃ ) = (0,1)
d𝐳 d𝐳̃Ψ𝐳Ψ𝐳̃
Ψ𝐳 = 1
2𝜋 ∫
Σ
d𝑤 d𝑤̃ √𝑔𝑏𝑖𝑗𝐷𝑖𝑣𝑗, Ψ𝐳̃ = 1
2𝜋 ∫
Σ
d𝑤 d𝑤̃ √𝑔𝑏𝑖𝑗𝐷𝑖𝑣̃𝑗
Ψ𝐳Ψ𝐳̃ 𝑐̃ 𝑐𝑉(0)
∫
Σ
d𝐳 d𝐳̃𝑉(𝐳, 𝐳̃ )
Σ → ℳg,n
↓ 𝜋
ℳg,n−1
𝒱 = 𝛿(𝑐̃ )𝛿(𝑐)𝑉
𝛿𝐼 = −𝑖
4𝜋 ∫
Σ
d𝑧̃ d𝑧(𝛿𝐽𝑧˜
𝑧𝑇𝑧𝑧 − 𝛿𝐽𝑧
𝑧˜ 𝑇𝑧̃ 𝑧˜ )
𝐼̂ = 𝐼 + −𝑖
4𝜋 ∫
Σ
d𝑧̃ d𝑧(𝛿𝐽𝑧̃
𝑧𝑏𝑧𝑧 − 𝛿𝐽𝑧
𝑧̃ 𝑏𝑧̃ 𝑧̃ )
Espacios de Supermoduli - Riemann. Variables holomórficas y antiholomórficas y deformaciones.
𝐷𝜃 = 𝜕
𝜕𝜃 + 𝜃 𝜕
𝜕𝑧 .
Φ[𝑛] = 𝑢 + 𝜃𝑣,
𝜈𝑓 = 𝑓(𝑧)(𝜕𝜃 − 𝜃𝜕𝑧)
𝑉𝑔 = 𝑔(𝑧)𝜕𝑧 + 𝑔′(𝑧)
2 𝜃𝜕𝜃
𝐺𝑟 = 𝑧𝑟+1/2(𝜕𝜃 − 𝜃𝜕𝑧), 𝑟 ∈ ℤ + 1/2
𝐿𝑛 = −𝑧𝑛+1𝜕𝑧 − 1
2 (𝑛 + 1)𝑧𝑛𝜃𝜕𝜃, 𝑛 ∈ ℤ
𝑟 ≥ −1/2, 𝑛 ≥ −1
[𝐿𝑚, 𝐿𝑛] = (𝑚 − 𝑛)𝐿𝑚+𝑛
{𝐺𝑟, 𝐺𝑠} = 2𝐿𝑟+𝑠
[𝐿𝑚, 𝐺𝑟] = (𝑚
2 − 𝑟) 𝐺𝑚+𝑟
𝛿𝒥𝑧˜
𝑧 = ℎ𝑧˜
𝑧 + 𝜃𝜒𝑧˜
𝜃
𝛿𝒥𝜃
𝑧˜ = 𝑒𝜃
𝑧˜ + 𝜃ℎ𝑧
𝑧˜
(𝑣𝑤, 𝑣𝑤̃ ) = (1,0)
(𝑣̃ 𝑤, 𝑣̃ 𝑤̃ ) = (0,1)
d𝐳 d𝐳̃Ψ𝐳Ψ𝐳̃
Ψ𝐳 = 1
2𝜋 ∫
Σ
d𝑤 d𝑤̃ √𝑔𝑏𝑖𝑗𝐷𝑖𝑣𝑗, Ψ𝐳̃ = 1
2𝜋 ∫
Σ
d𝑤 d𝑤̃ √𝑔𝑏𝑖𝑗𝐷𝑖𝑣̃𝑗
Ψ𝐳Ψ𝐳̃ 𝑐̃ 𝑐𝑉(0)
∫
Σ
d𝐳 d𝐳̃𝑉(𝐳, 𝐳̃ )
Σ → ℳg,n
↓ 𝜋
ℳg,n−1
𝒱 = 𝛿(𝑐̃ )𝛿(𝑐)𝑉
𝛿𝐼 = −𝑖
4𝜋 ∫
Σ
d𝑧̃ d𝑧(𝛿𝐽𝑧˜
𝑧𝑇𝑧𝑧 − 𝛿𝐽𝑧
𝑧˜ 𝑇𝑧̃ 𝑧˜ )
𝐼̂ = 𝐼 + −𝑖
4𝜋 ∫
Σ
d𝑧̃ d𝑧(𝛿𝐽𝑧̃
𝑧𝑏𝑧𝑧 − 𝛿𝐽𝑧
𝑧̃ 𝑏𝑧̃ 𝑧̃ )
Espacios de Supermoduli - Riemann. Variables holomórficas y antiholomórficas y deformaciones.
𝐷𝜃 = 𝜕
𝜕𝜃 + 𝜃 𝜕
𝜕𝑧 .
Φ[𝑛] = 𝑢 + 𝜃𝑣,
𝜈𝑓 = 𝑓(𝑧)(𝜕𝜃 − 𝜃𝜕𝑧)
𝑉𝑔 = 𝑔(𝑧)𝜕𝑧 + 𝑔′(𝑧)
2 𝜃𝜕𝜃
𝐺𝑟 = 𝑧𝑟+1/2(𝜕𝜃 − 𝜃𝜕𝑧), 𝑟 ∈ ℤ + 1/2
𝐿𝑛 = −𝑧𝑛+1𝜕𝑧 − 1
2 (𝑛 + 1)𝑧𝑛𝜃𝜕𝜃, 𝑛 ∈ ℤ
𝑟 ≥ −1/2, 𝑛 ≥ −1
[𝐿𝑚, 𝐿𝑛] = (𝑚 − 𝑛)𝐿𝑚+𝑛
{𝐺𝑟, 𝐺𝑠} = 2𝐿𝑟+𝑠
[𝐿𝑚, 𝐺𝑟] = (𝑚
2 − 𝑟) 𝐺𝑚+𝑟
𝛿𝒥𝑧˜
𝑧 = ℎ𝑧˜
𝑧 + 𝜃𝜒𝑧˜
𝜃
𝛿𝒥𝜃
𝑧˜ = 𝑒𝜃
𝑧˜ + 𝜃ℎ𝑧
𝑧˜
pág. 83
𝑞𝑧˜ 𝜕𝑧˜ + (𝑞𝑧𝜕𝑧 + 1
2 𝐷𝜃𝑞𝑧𝐷𝜃) + 𝑞𝜃𝐷𝜃
𝛿𝐷𝜃 = [𝐷𝜃, 𝑞𝜃𝐷𝜃] = (𝐷𝜃𝑞𝜃)𝐷𝜃 − 2𝑞𝜃𝜕𝑧
𝛿𝒥𝑧˜
𝑧 = 𝜕𝑧˜ 𝑞𝑧
𝛿𝒥𝜃
𝑧˜ = 𝐷𝜃𝑞𝑧˜ (3.10)
Propagadores, indentidades y números fantasma.
𝛿𝐼 = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)(𝛿𝒥𝑧̃
𝑧𝒮𝑧𝜃 + 𝛿𝒥𝜃
𝑧̃ 𝑇𝑧̃ 𝑧̃ )
𝒟(𝑧̃ , 𝑧 ∣ 𝜃) = −𝑖[ d𝑧̃ ; d𝑧 ∣ d𝜃]
𝒮𝑧𝜃 = 𝑆𝑧𝜃 + 𝜃𝑇𝑧𝑧
𝜕𝑧̃ 𝒮𝑧𝜃 = 0 = 𝐷𝜃𝑇𝑧̃ 𝑧˜ = 𝜕𝑧𝑇𝑧̃ 𝑧˜ ,
𝒮𝑧𝜃(𝑧 ∣ 𝜃) = 1
2 ∑
𝑟∈ℤ+1/2
𝑧−𝑟−3/2𝐺𝑟 + 𝜃 ∑
𝑛∈ℤ
𝑧−𝑛−2𝐿𝑛
𝑇𝑧̃ 𝑧̃ = ∑
𝑛∈ℤ
𝑧̃ −𝑛−2𝐿̃ 𝑛.
𝐶𝑧 = 𝑐𝑧 + 𝜃𝛾𝜃
𝐵𝑧𝜃 = 𝛽𝑧𝜃 + 𝜃𝑏𝑧𝑧
𝐵̃𝑧̃ 𝑧 = 𝑏̃𝑧̃ 𝑧̃ + 𝜃𝑓̃𝑧̃ 𝑧˜𝜃, 𝐶̃ 𝑧̃ = 𝑐̃ 𝑧̃ + 𝜃𝑔̃𝜃
𝑧̃
𝐼gh = 1
2𝜋 ∫ 𝒟(𝑧̃ , 𝑧 ∣ 𝜃)(𝐵𝑧𝜃𝜕𝑧̃ 𝐶𝑧 + 𝐵̃𝑧̃ 𝑧̃ 𝐷𝜃𝐶̃ 𝑧̃ )
𝜕𝑧̃ 𝐵𝑧𝜃 = 0 = 𝐷𝜃𝐵̃𝑧̃ 𝑧̃
[𝑄𝐵, 𝐵𝑧𝜃] = 𝒮𝑧𝜃
{𝑄𝐵, 𝐵̃𝑧̃ 𝑧} = 𝑇𝑧̃ 𝑧˜
{𝑄𝐵, 𝛿𝒥𝑧˜
𝑧˜ } = [𝑄𝐵, 𝛿𝒥𝜃
𝑧˜ ] = 0
𝐼 → 𝐼̂ = 𝐼 + 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)(𝛿𝒥𝑧˜
𝑧𝐵𝑧𝜃 − 𝛿𝒥𝜃
𝑧˜ 𝐵̃𝑧̃ 𝑧˜ )
𝐹(𝒥, 𝛿𝒥) = ∫ 𝒟(𝑋, 𝐵, 𝐶, 𝐵̃ , 𝐶̃ )exp (−𝐼̂ )
[𝑄𝐵, 𝐹(𝒥, 𝛿𝒥)} = 0
d𝐹(𝒥, 𝛿𝒥) = 0
𝑞𝑧˜ 𝜕𝑧˜ + (𝑞𝑧𝜕𝑧 + 1
2 𝐷𝜃𝑞𝑧𝐷𝜃) + 𝑞𝜃𝐷𝜃
𝛿𝐷𝜃 = [𝐷𝜃, 𝑞𝜃𝐷𝜃] = (𝐷𝜃𝑞𝜃)𝐷𝜃 − 2𝑞𝜃𝜕𝑧
𝛿𝒥𝑧˜
𝑧 = 𝜕𝑧˜ 𝑞𝑧
𝛿𝒥𝜃
𝑧˜ = 𝐷𝜃𝑞𝑧˜ (3.10)
Propagadores, indentidades y números fantasma.
𝛿𝐼 = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)(𝛿𝒥𝑧̃
𝑧𝒮𝑧𝜃 + 𝛿𝒥𝜃
𝑧̃ 𝑇𝑧̃ 𝑧̃ )
𝒟(𝑧̃ , 𝑧 ∣ 𝜃) = −𝑖[ d𝑧̃ ; d𝑧 ∣ d𝜃]
𝒮𝑧𝜃 = 𝑆𝑧𝜃 + 𝜃𝑇𝑧𝑧
𝜕𝑧̃ 𝒮𝑧𝜃 = 0 = 𝐷𝜃𝑇𝑧̃ 𝑧˜ = 𝜕𝑧𝑇𝑧̃ 𝑧˜ ,
𝒮𝑧𝜃(𝑧 ∣ 𝜃) = 1
2 ∑
𝑟∈ℤ+1/2
𝑧−𝑟−3/2𝐺𝑟 + 𝜃 ∑
𝑛∈ℤ
𝑧−𝑛−2𝐿𝑛
𝑇𝑧̃ 𝑧̃ = ∑
𝑛∈ℤ
𝑧̃ −𝑛−2𝐿̃ 𝑛.
𝐶𝑧 = 𝑐𝑧 + 𝜃𝛾𝜃
𝐵𝑧𝜃 = 𝛽𝑧𝜃 + 𝜃𝑏𝑧𝑧
𝐵̃𝑧̃ 𝑧 = 𝑏̃𝑧̃ 𝑧̃ + 𝜃𝑓̃𝑧̃ 𝑧˜𝜃, 𝐶̃ 𝑧̃ = 𝑐̃ 𝑧̃ + 𝜃𝑔̃𝜃
𝑧̃
𝐼gh = 1
2𝜋 ∫ 𝒟(𝑧̃ , 𝑧 ∣ 𝜃)(𝐵𝑧𝜃𝜕𝑧̃ 𝐶𝑧 + 𝐵̃𝑧̃ 𝑧̃ 𝐷𝜃𝐶̃ 𝑧̃ )
𝜕𝑧̃ 𝐵𝑧𝜃 = 0 = 𝐷𝜃𝐵̃𝑧̃ 𝑧̃
[𝑄𝐵, 𝐵𝑧𝜃] = 𝒮𝑧𝜃
{𝑄𝐵, 𝐵̃𝑧̃ 𝑧} = 𝑇𝑧̃ 𝑧˜
{𝑄𝐵, 𝛿𝒥𝑧˜
𝑧˜ } = [𝑄𝐵, 𝛿𝒥𝜃
𝑧˜ ] = 0
𝐼 → 𝐼̂ = 𝐼 + 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)(𝛿𝒥𝑧˜
𝑧𝐵𝑧𝜃 − 𝛿𝒥𝜃
𝑧˜ 𝐵̃𝑧̃ 𝑧˜ )
𝐹(𝒥, 𝛿𝒥) = ∫ 𝒟(𝑋, 𝐵, 𝐶, 𝐵̃ , 𝐶̃ )exp (−𝐼̂ )
[𝑄𝐵, 𝐹(𝒥, 𝛿𝒥)} = 0
d𝐹(𝒥, 𝛿𝒥) = 0
pág. 84
Supermanifolds e integración bosónica.
𝐹(𝑥, 𝜆 d𝑥) = 𝜆𝑠𝐹(𝑥, d𝑥)
d = ∑
𝐼
d𝑥𝐼 𝜕
𝜕𝑥𝐼 , d2 = 0
∫ 𝒟(𝑥, d𝑥)𝐹(𝑥, d𝑥)
𝐹(𝑥, d𝑥) = 𝑓(𝑡1 … ∣ ⋯ 𝜃𝑛)d𝑡1 … d𝑡𝑚𝛿( d𝜃1) … 𝛿(d𝜃𝑛)
𝐹(𝑥, d𝑥) = 𝑓(𝑡1 … ∣ ⋯ 𝜃𝑛)𝛿(d𝑡1) … 𝛿(d𝜃𝑛)
= 𝑓(𝑡1 … ∣ ⋯ 𝜃𝑛)𝛿𝑚∣𝑛( d𝑡1 … ∣ ⋯ d𝜃𝑛)
𝜕𝑟
𝜕( d𝜃)𝑟 𝛿( d𝜃)
𝐹(𝑥, d𝑥) = 𝜃2 d𝑡𝛿′(d𝜃1)
𝐼𝛽𝛾 = 1
𝜋 ∫
Σ
d2𝑧𝛽𝜕𝑧˜ 𝛾
∫ 𝒟(d𝜃) 𝜕𝑟
𝜕( d𝜃)𝑟 𝛿( d𝜃) = 𝛿𝑟,0
∫ 𝒟(𝑢, 𝑣)exp (− ∑
𝑖,𝑗
𝑢𝑖𝑚𝑖𝑗𝑣𝑗) = 1
det𝑚
∫ 𝒟(𝑢, 𝑣)exp (− ∑
𝑖,𝑗
𝑢𝑖𝑚𝑖𝑗𝑣𝑗 − ∑
𝑘
(𝑟𝑘𝑢𝑘 + 𝑠𝑘𝑣𝑘)) = exp (∑𝑖,𝑗 𝑠𝑖(𝑚−1)𝑖𝑗𝑟𝑗)
det𝑚
∫ 𝒟(𝑢, 𝑣)exp (− ∑
𝑖,𝑗
𝑢𝑖𝑚𝑖𝑗𝑣𝑗 − ∑
𝑘
𝑠𝑘𝑣𝑘) = 1
det𝑚
𝐺(𝑣) = ∫ 𝒟(𝑢)exp (−(𝑢, 𝑚𝑣))
∫ 𝒟(𝑣)𝐴(𝑣)𝐺(𝑣) = ∫ 𝒟(𝑢, 𝑣)𝐴(𝑣)exp (−(𝑢, 𝑚𝑣))
∫ 𝒟(𝑣)exp (−(𝑠, 𝑣))𝐺(𝑣) = 1
det𝑚
∫ 𝒟(𝑢)exp (−(𝑢, 𝑚𝑣)) = 𝛿(𝑚𝑣)
Supermanifolds e integración bosónica.
𝐹(𝑥, 𝜆 d𝑥) = 𝜆𝑠𝐹(𝑥, d𝑥)
d = ∑
𝐼
d𝑥𝐼 𝜕
𝜕𝑥𝐼 , d2 = 0
∫ 𝒟(𝑥, d𝑥)𝐹(𝑥, d𝑥)
𝐹(𝑥, d𝑥) = 𝑓(𝑡1 … ∣ ⋯ 𝜃𝑛)d𝑡1 … d𝑡𝑚𝛿( d𝜃1) … 𝛿(d𝜃𝑛)
𝐹(𝑥, d𝑥) = 𝑓(𝑡1 … ∣ ⋯ 𝜃𝑛)𝛿(d𝑡1) … 𝛿(d𝜃𝑛)
= 𝑓(𝑡1 … ∣ ⋯ 𝜃𝑛)𝛿𝑚∣𝑛( d𝑡1 … ∣ ⋯ d𝜃𝑛)
𝜕𝑟
𝜕( d𝜃)𝑟 𝛿( d𝜃)
𝐹(𝑥, d𝑥) = 𝜃2 d𝑡𝛿′(d𝜃1)
𝐼𝛽𝛾 = 1
𝜋 ∫
Σ
d2𝑧𝛽𝜕𝑧˜ 𝛾
∫ 𝒟(d𝜃) 𝜕𝑟
𝜕( d𝜃)𝑟 𝛿( d𝜃) = 𝛿𝑟,0
∫ 𝒟(𝑢, 𝑣)exp (− ∑
𝑖,𝑗
𝑢𝑖𝑚𝑖𝑗𝑣𝑗) = 1
det𝑚
∫ 𝒟(𝑢, 𝑣)exp (− ∑
𝑖,𝑗
𝑢𝑖𝑚𝑖𝑗𝑣𝑗 − ∑
𝑘
(𝑟𝑘𝑢𝑘 + 𝑠𝑘𝑣𝑘)) = exp (∑𝑖,𝑗 𝑠𝑖(𝑚−1)𝑖𝑗𝑟𝑗)
det𝑚
∫ 𝒟(𝑢, 𝑣)exp (− ∑
𝑖,𝑗
𝑢𝑖𝑚𝑖𝑗𝑣𝑗 − ∑
𝑘
𝑠𝑘𝑣𝑘) = 1
det𝑚
𝐺(𝑣) = ∫ 𝒟(𝑢)exp (−(𝑢, 𝑚𝑣))
∫ 𝒟(𝑣)𝐴(𝑣)𝐺(𝑣) = ∫ 𝒟(𝑢, 𝑣)𝐴(𝑣)exp (−(𝑢, 𝑚𝑣))
∫ 𝒟(𝑣)exp (−(𝑠, 𝑣))𝐺(𝑣) = 1
det𝑚
∫ 𝒟(𝑢)exp (−(𝑢, 𝑚𝑣)) = 𝛿(𝑚𝑣)
pág. 85
∫ 𝒟(𝑣)𝛿(𝑚𝑣) = 1
det𝑚
𝐼̃𝛿𝒥𝐵 = 1
2𝜋 ∫ 𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝛿𝒥𝑧˜
𝑧𝐵𝑧𝜃
𝐵 = ∑
𝛼=1…∣⋯2𝑔−2
𝑢𝛼B𝛼 + ∑
𝜆
𝑤𝜆B𝜆
′ .
𝐶 = ∑
𝜆
𝛾𝜆C𝜆,
∫ d𝛾exp (−𝑚𝑤𝛾) = 𝛿(𝑚𝑤)
𝐼̃𝛿𝒥𝐵 = ∑
𝛼=1…∣⋯2𝑔−2
𝑢𝛼Ψ𝛼,
Ψ𝛼 = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝛿𝒥𝑧˜
𝑧 B𝑧𝜃𝛼
∏
𝛼=1…∣⋯2𝑔−2
∫ d𝑢𝛼exp (𝑢𝛼Ψ𝛼) = ∏
𝛼=1…∣⋯2𝑔−2
𝛿(Ψ𝛼) = 𝛿3𝑔−3∣2𝑔−2(Ψ1
′ … ∣ ⋯ Ψ2𝑔−2
′′ )
𝐼̃𝛿𝒥𝐵̃ = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝛿𝒥𝜃
𝑧̃ 𝐵̃𝑧̃ 𝑧̃
𝐼̃𝛿𝒥𝐵̃ = ∑
3𝑔−3
𝛼=1
𝑢̃ 𝛼Ψ̃𝛼
Ψ̃𝛼 = 1
2𝜋 ∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝛿𝒥𝜃
𝑧̃ B̃𝑧̃ 𝑧̃
∏
3𝑔−3
𝛼=1
∫ d𝔲̃𝛼exp (𝔲̃𝛼Ψ̃𝛼) = 𝛿3𝑔−3(Ψ̃1, … , Ψ̃3𝑔−3)
𝛿3𝑔−3(Ψ̃1 … Ψ̃3𝑔−3)𝛿3𝑔−3∣2𝑔−2(Ψ1
′ … ∣ ⋯ Ψ2𝑔−2
′′ )
𝛿𝒥𝑧˜
𝑧 → 𝛿𝒥𝑧˜
𝑧 + 𝜕𝑧˜ 𝑞𝑧, 𝛿𝒥𝜃
𝑧˜ → 𝛿𝒥𝜃
𝑧˜ + 𝐷𝜃𝑞𝑧˜
𝑚′ = 𝑓(𝑚 ∣ 𝜂1, 𝜂2)
𝜂′𝑖 = 𝜓𝑖(𝑚 ∣ 𝜂1, 𝜂2), 𝑖 = 1,2
𝑚′ = 𝑓0(𝑚) + 𝜂1𝜂2𝑓2(𝑚)
𝑚̃ ′ = 𝑓̃ (𝑚̃ )
𝑍g = ∫
Γ
𝒟(𝒥, 𝛿𝒥)𝐹(𝒥, 𝛿𝒥)
∫ 𝒟(𝑣)𝛿(𝑚𝑣) = 1
det𝑚
𝐼̃𝛿𝒥𝐵 = 1
2𝜋 ∫ 𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝛿𝒥𝑧˜
𝑧𝐵𝑧𝜃
𝐵 = ∑
𝛼=1…∣⋯2𝑔−2
𝑢𝛼B𝛼 + ∑
𝜆
𝑤𝜆B𝜆
′ .
𝐶 = ∑
𝜆
𝛾𝜆C𝜆,
∫ d𝛾exp (−𝑚𝑤𝛾) = 𝛿(𝑚𝑤)
𝐼̃𝛿𝒥𝐵 = ∑
𝛼=1…∣⋯2𝑔−2
𝑢𝛼Ψ𝛼,
Ψ𝛼 = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝛿𝒥𝑧˜
𝑧 B𝑧𝜃𝛼
∏
𝛼=1…∣⋯2𝑔−2
∫ d𝑢𝛼exp (𝑢𝛼Ψ𝛼) = ∏
𝛼=1…∣⋯2𝑔−2
𝛿(Ψ𝛼) = 𝛿3𝑔−3∣2𝑔−2(Ψ1
′ … ∣ ⋯ Ψ2𝑔−2
′′ )
𝐼̃𝛿𝒥𝐵̃ = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝛿𝒥𝜃
𝑧̃ 𝐵̃𝑧̃ 𝑧̃
𝐼̃𝛿𝒥𝐵̃ = ∑
3𝑔−3
𝛼=1
𝑢̃ 𝛼Ψ̃𝛼
Ψ̃𝛼 = 1
2𝜋 ∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝛿𝒥𝜃
𝑧̃ B̃𝑧̃ 𝑧̃
∏
3𝑔−3
𝛼=1
∫ d𝔲̃𝛼exp (𝔲̃𝛼Ψ̃𝛼) = 𝛿3𝑔−3(Ψ̃1, … , Ψ̃3𝑔−3)
𝛿3𝑔−3(Ψ̃1 … Ψ̃3𝑔−3)𝛿3𝑔−3∣2𝑔−2(Ψ1
′ … ∣ ⋯ Ψ2𝑔−2
′′ )
𝛿𝒥𝑧˜
𝑧 → 𝛿𝒥𝑧˜
𝑧 + 𝜕𝑧˜ 𝑞𝑧, 𝛿𝒥𝜃
𝑧˜ → 𝛿𝒥𝜃
𝑧˜ + 𝐷𝜃𝑞𝑧˜
𝑚′ = 𝑓(𝑚 ∣ 𝜂1, 𝜂2)
𝜂′𝑖 = 𝜓𝑖(𝑚 ∣ 𝜂1, 𝜂2), 𝑖 = 1,2
𝑚′ = 𝑓0(𝑚) + 𝜂1𝜂2𝑓2(𝑚)
𝑚̃ ′ = 𝑓̃ (𝑚̃ )
𝑍g = ∫
Γ
𝒟(𝒥, 𝛿𝒥)𝐹(𝒥, 𝛿𝒥)
pág. 86
𝛿𝒥𝑧˜
𝑧 = ∑
𝛼=1…∣⋯2𝔤−2
𝜕𝒥𝑧˜
𝑧
𝜕𝒎𝛼
d𝒎𝛼
𝛿𝒥𝜃
𝑧̃ = ∑
𝛽=1…3𝐠−3
𝜕𝒥𝜃
𝑧˜
𝜕𝑚̃𝛽
d𝑚̃𝛽
𝐼d𝒎, d𝑚̃ = ∑
𝛼=1…∣⋯2𝑔−2
d𝒎𝛼𝐵(𝛼) + ∑
𝛽=1…3𝑔−3
d𝑚̃𝛽𝐵̃ (𝛽)
𝐵(𝛼) = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃) 𝜕𝒥𝑧˜
𝑧
𝜕𝒎𝛼
𝐵𝑧𝜃
𝐵̃ (𝛽) = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃) 𝜕𝒥𝜃
𝑧˜
𝜕𝑚̃𝛽
𝐵̃𝑧̃ 𝑧˜
∏
𝛽=1…3𝑔−3
∫ 𝒟( d𝑚̃𝛽) exp(−d𝑚̃𝛽𝐵̃ (𝛽)) ⋅ ∏
𝛼=1…∣⋯2𝑔−2
𝒟( d𝒎𝛼) exp(−d𝒎𝛼𝐵(𝛼))
= ∏
𝛽=1…3𝑔−3
𝛿(𝐵̃ (𝛽)) ⋅ ∏
𝛼=1…∣⋯2𝑔−2
𝛿(𝐵(𝛼))
𝛿3𝑔−3(𝐵̃ (𝛽))𝛿3𝑔−3∣2𝑔−2(𝐵(𝛼))
Λ(𝑚̃ , 𝒎) = ∫ 𝒟(𝑋, 𝐵, 𝐶, 𝐵̃ , 𝐶̃ )exp (−𝐼)𝛿3𝑔−3(𝐵̃ (𝛽))𝛿3𝑔−3∣2𝑔−2(𝐵(𝛼))
𝒟(𝑥) = [d𝑚̃1 … d𝑚̃3𝑔−3; d𝑚1 … d𝑚3𝑔−3 ∣ d𝜂1 … d𝜂2𝑔−2]
Ξ(𝑥) = 𝒟(𝑥)Λ(𝑥)
Ξ(𝑥) = ∫ 𝒟(d𝑥)𝐹(𝒥, 𝛿𝒥)
𝑍g = ∫
Γ
Ξ(𝑥)
Sistema de Coordenadas – Moduli y transformaciones de gauge.
Ξ = [d𝑚̃ ; d𝑚 ∣ d𝜂1, d𝜂2](Υ0(𝑚̃ , 𝑚) + Υ2(𝑚̃ , 𝑚)𝜂1𝜂2)
𝑚′ = 𝑚 + 𝑎(𝑚)𝜂1𝜂2
𝜂′1 = 𝜂1
𝜂′2 = 𝜂2
Ξ = [d𝑚̃ ; d𝑚′ ∣ d𝜂′1, d𝜂′2](Υ0
′(𝑚̃ , 𝑚′) + Υ2
′(𝑚̃ , 𝑚′)𝜂′1𝜂′2)
Υ2
′(𝑚̃ , 𝑚′) = Υ2(𝑚̃ , 𝑚′) − 𝜕𝑚′ (𝑎(𝑚′)Υ0(𝑚̃ ; 𝑚′))
𝛿𝒥𝑧˜
𝑧 = ℎ𝑧˜
𝑧 + 𝜃𝜒𝑧˜
𝜃
𝜒𝑧˜
𝜃 → 𝜒𝑧˜
𝜃 + 𝜕𝑧˜ 𝑦𝜃
𝛿𝒥𝑧˜
𝑧 = ∑
𝛼=1…∣⋯2𝔤−2
𝜕𝒥𝑧˜
𝑧
𝜕𝒎𝛼
d𝒎𝛼
𝛿𝒥𝜃
𝑧̃ = ∑
𝛽=1…3𝐠−3
𝜕𝒥𝜃
𝑧˜
𝜕𝑚̃𝛽
d𝑚̃𝛽
𝐼d𝒎, d𝑚̃ = ∑
𝛼=1…∣⋯2𝑔−2
d𝒎𝛼𝐵(𝛼) + ∑
𝛽=1…3𝑔−3
d𝑚̃𝛽𝐵̃ (𝛽)
𝐵(𝛼) = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃) 𝜕𝒥𝑧˜
𝑧
𝜕𝒎𝛼
𝐵𝑧𝜃
𝐵̃ (𝛽) = 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃) 𝜕𝒥𝜃
𝑧˜
𝜕𝑚̃𝛽
𝐵̃𝑧̃ 𝑧˜
∏
𝛽=1…3𝑔−3
∫ 𝒟( d𝑚̃𝛽) exp(−d𝑚̃𝛽𝐵̃ (𝛽)) ⋅ ∏
𝛼=1…∣⋯2𝑔−2
𝒟( d𝒎𝛼) exp(−d𝒎𝛼𝐵(𝛼))
= ∏
𝛽=1…3𝑔−3
𝛿(𝐵̃ (𝛽)) ⋅ ∏
𝛼=1…∣⋯2𝑔−2
𝛿(𝐵(𝛼))
𝛿3𝑔−3(𝐵̃ (𝛽))𝛿3𝑔−3∣2𝑔−2(𝐵(𝛼))
Λ(𝑚̃ , 𝒎) = ∫ 𝒟(𝑋, 𝐵, 𝐶, 𝐵̃ , 𝐶̃ )exp (−𝐼)𝛿3𝑔−3(𝐵̃ (𝛽))𝛿3𝑔−3∣2𝑔−2(𝐵(𝛼))
𝒟(𝑥) = [d𝑚̃1 … d𝑚̃3𝑔−3; d𝑚1 … d𝑚3𝑔−3 ∣ d𝜂1 … d𝜂2𝑔−2]
Ξ(𝑥) = 𝒟(𝑥)Λ(𝑥)
Ξ(𝑥) = ∫ 𝒟(d𝑥)𝐹(𝒥, 𝛿𝒥)
𝑍g = ∫
Γ
Ξ(𝑥)
Sistema de Coordenadas – Moduli y transformaciones de gauge.
Ξ = [d𝑚̃ ; d𝑚 ∣ d𝜂1, d𝜂2](Υ0(𝑚̃ , 𝑚) + Υ2(𝑚̃ , 𝑚)𝜂1𝜂2)
𝑚′ = 𝑚 + 𝑎(𝑚)𝜂1𝜂2
𝜂′1 = 𝜂1
𝜂′2 = 𝜂2
Ξ = [d𝑚̃ ; d𝑚′ ∣ d𝜂′1, d𝜂′2](Υ0
′(𝑚̃ , 𝑚′) + Υ2
′(𝑚̃ , 𝑚′)𝜂′1𝜂′2)
Υ2
′(𝑚̃ , 𝑚′) = Υ2(𝑚̃ , 𝑚′) − 𝜕𝑚′ (𝑎(𝑚′)Υ0(𝑚̃ ; 𝑚′))
𝛿𝒥𝑧˜
𝑧 = ℎ𝑧˜
𝑧 + 𝜃𝜒𝑧˜
𝜃
𝜒𝑧˜
𝜃 → 𝜒𝑧˜
𝜃 + 𝜕𝑧˜ 𝑦𝜃
pág. 87
𝜒𝑧̃
𝜃 = ∑
2𝑔−2
𝜎=1
𝜂𝜎𝜒𝑧̃
(𝜎)𝜃
𝜕𝑧˜ 𝑦𝜃 = ∑
𝜎
𝑒𝜎𝜒𝑧˜
(𝜎)𝜃
𝜒𝑧˜
(𝜎)𝜃 → 𝜒𝑧˜
(𝜎)𝜃 + 𝜕𝑧˜ 𝑦(𝜎)𝜃
𝜒 → 𝜒 + 𝜕𝑧˜ 𝑦,
𝑦 = ∑
2𝑔−2
𝜎=1
𝜂𝜎𝑦(𝜎)
ℎ𝑧˜
𝑧 → ℎ𝑧˜
𝑧 + 𝑦𝜃𝜒𝑧˜
𝜃.
ℎ𝑧˜
𝑧 → ℎ𝑧˜
𝑧 + ∑
𝜎,𝜎′
𝜂𝜎𝜂𝜎′ 𝑦(𝜎)𝜃𝜒𝑧˜
(𝜎′)𝜃
𝐼𝜂 = ∑
2 g−2
𝜎=1
𝜂𝜎
2𝜋 ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝑆𝑧𝜃
𝐼d𝜂 = ∑
2 g−2
𝜎=1
d𝜂𝜎
2𝜋 ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃
exp (− 1
2𝜋 ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃(𝜂𝜎𝑆𝑧𝜃 + d𝜂𝜎𝛽𝑧𝜃))
𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃) ⋅ ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝑆𝑧𝜃
𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃) ⋅ 𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝑆𝑧𝜃)
𝜒𝑧˜
(𝜎)𝜃 → 𝜆𝜒𝑧˜
(𝜎)𝜃, 𝜂𝜎 → 𝜆−1𝜂𝜎, 𝜆 ∈ ℂ∗
(
𝜒𝑧˜
(1)
⋮
𝜒𝑧˜
(𝑠)𝜃
) → 𝑀 (
𝜒𝑧˜
(1)
⋮
𝜒𝑧˜
(𝑠)𝜃
)
(𝜂1 … 𝜂𝑠) → (𝜂1 … 𝜂𝑠)𝑀−1, ( d𝜂1 … d𝜂𝑠) → (d𝜂1 … d𝜂𝑠)𝑀−1
∏
𝑠
𝜎=1
(𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃) ⋅ ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝑆𝑧𝜃)
𝜒𝑧̃
𝜃 = ∑
2𝑔−2
𝜎=1
𝜂𝜎𝜒𝑧̃
(𝜎)𝜃
𝜕𝑧˜ 𝑦𝜃 = ∑
𝜎
𝑒𝜎𝜒𝑧˜
(𝜎)𝜃
𝜒𝑧˜
(𝜎)𝜃 → 𝜒𝑧˜
(𝜎)𝜃 + 𝜕𝑧˜ 𝑦(𝜎)𝜃
𝜒 → 𝜒 + 𝜕𝑧˜ 𝑦,
𝑦 = ∑
2𝑔−2
𝜎=1
𝜂𝜎𝑦(𝜎)
ℎ𝑧˜
𝑧 → ℎ𝑧˜
𝑧 + 𝑦𝜃𝜒𝑧˜
𝜃.
ℎ𝑧˜
𝑧 → ℎ𝑧˜
𝑧 + ∑
𝜎,𝜎′
𝜂𝜎𝜂𝜎′ 𝑦(𝜎)𝜃𝜒𝑧˜
(𝜎′)𝜃
𝐼𝜂 = ∑
2 g−2
𝜎=1
𝜂𝜎
2𝜋 ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝑆𝑧𝜃
𝐼d𝜂 = ∑
2 g−2
𝜎=1
d𝜂𝜎
2𝜋 ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃
exp (− 1
2𝜋 ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃(𝜂𝜎𝑆𝑧𝜃 + d𝜂𝜎𝛽𝑧𝜃))
𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃) ⋅ ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝑆𝑧𝜃
𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃) ⋅ 𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝑆𝑧𝜃)
𝜒𝑧˜
(𝜎)𝜃 → 𝜆𝜒𝑧˜
(𝜎)𝜃, 𝜂𝜎 → 𝜆−1𝜂𝜎, 𝜆 ∈ ℂ∗
(
𝜒𝑧˜
(1)
⋮
𝜒𝑧˜
(𝑠)𝜃
) → 𝑀 (
𝜒𝑧˜
(1)
⋮
𝜒𝑧˜
(𝑠)𝜃
)
(𝜂1 … 𝜂𝑠) → (𝜂1 … 𝜂𝑠)𝑀−1, ( d𝜂1 … d𝜂𝑠) → (d𝜂1 … d𝜂𝑠)𝑀−1
∏
𝑠
𝜎=1
(𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃) ⋅ ∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝑆𝑧𝜃)
pág. 88
Operador Picture-Changing.
𝜒𝑧̃
(𝜎)𝜃 = 𝛿𝑝𝜎 .
𝑦(𝑝) = 𝛿(𝛽(𝑝))𝑆𝑧𝜃(𝑝)
𝜒𝑧˜
(𝜎)𝜃 = 𝛿𝑝𝜎 .
∏
2𝑔−2
𝜎=1
𝑦(𝑝𝜎)
𝜕𝑧˜ 𝑦𝜃 = ∑
2𝔤−2
𝜎=1
𝑒𝜎𝛿𝑝𝜎
𝐻0 (Σred, 𝑇1/2(Σ𝜎𝑝𝜎)) ≠ 0
𝜒𝑧̃
(1)𝜃 = 𝛿2(𝑧), 𝜒𝑧̃
(2)𝜃 = 𝛿2(𝑧 − 𝜖)
𝜒̂𝑧˜
(2)𝜃 = − 1
𝜖 (𝜒𝑧˜
(2)𝜃 − 𝜒𝑧˜
(1)𝜃) = − 1
𝜖 (𝛿2(𝑧 − 𝜖) − 𝛿2(𝑧)).
𝜒̂𝑧̃
(2)𝜃 = 𝜕𝑧𝛿2(𝑧).
𝛿(𝜕𝑧𝛽(𝑧0))𝜕𝑧𝑆𝑧𝜃(𝑧0).
𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃)
𝜕𝑧𝛿(𝛽(𝑧)) = 𝜕𝑧𝛽(𝑧) ⋅ 𝛿′(𝛽(𝑧))
𝛿(𝑘) (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃)
𝑟 − 𝑠 = 2𝑔 − 2
𝛿3𝑔−3∣2𝑔−2(𝐵(𝛼)) = ∏
3𝑔−3
𝛼′=1
𝛿(𝑏(𝛼′)) ∏
2𝑔−2
𝛼′′=1
𝛿(𝛽(𝛼′′))
Métrica Neveu-Schwarz.
𝑏𝑛𝒱 = 𝑏̃𝑛𝒱 = 𝛽𝑟𝒱 = 0, 𝑛, 𝑟 ≥ 0
𝛽𝑟 = 1
2𝜋𝑖 ∮ d𝑧𝑧𝑟+1/2𝛽𝑧𝜃
𝐿𝑛𝒱 = 0, 𝑛 ≥ 0
𝐺𝑟𝒱 = 0, 𝑟 ≥ 1/2,
Operador Picture-Changing.
𝜒𝑧̃
(𝜎)𝜃 = 𝛿𝑝𝜎 .
𝑦(𝑝) = 𝛿(𝛽(𝑝))𝑆𝑧𝜃(𝑝)
𝜒𝑧˜
(𝜎)𝜃 = 𝛿𝑝𝜎 .
∏
2𝑔−2
𝜎=1
𝑦(𝑝𝜎)
𝜕𝑧˜ 𝑦𝜃 = ∑
2𝔤−2
𝜎=1
𝑒𝜎𝛿𝑝𝜎
𝐻0 (Σred, 𝑇1/2(Σ𝜎𝑝𝜎)) ≠ 0
𝜒𝑧̃
(1)𝜃 = 𝛿2(𝑧), 𝜒𝑧̃
(2)𝜃 = 𝛿2(𝑧 − 𝜖)
𝜒̂𝑧˜
(2)𝜃 = − 1
𝜖 (𝜒𝑧˜
(2)𝜃 − 𝜒𝑧˜
(1)𝜃) = − 1
𝜖 (𝛿2(𝑧 − 𝜖) − 𝛿2(𝑧)).
𝜒̂𝑧̃
(2)𝜃 = 𝜕𝑧𝛿2(𝑧).
𝛿(𝜕𝑧𝛽(𝑧0))𝜕𝑧𝑆𝑧𝜃(𝑧0).
𝛿 (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃)
𝜕𝑧𝛿(𝛽(𝑧)) = 𝜕𝑧𝛽(𝑧) ⋅ 𝛿′(𝛽(𝑧))
𝛿(𝑘) (∫
Σred
d2𝑧𝜒𝑧˜
(𝜎)𝜃𝛽𝑧𝜃)
𝑟 − 𝑠 = 2𝑔 − 2
𝛿3𝑔−3∣2𝑔−2(𝐵(𝛼)) = ∏
3𝑔−3
𝛼′=1
𝛿(𝑏(𝛼′)) ∏
2𝑔−2
𝛼′′=1
𝛿(𝛽(𝛼′′))
Métrica Neveu-Schwarz.
𝑏𝑛𝒱 = 𝑏̃𝑛𝒱 = 𝛽𝑟𝒱 = 0, 𝑛, 𝑟 ≥ 0
𝛽𝑟 = 1
2𝜋𝑖 ∮ d𝑧𝑧𝑟+1/2𝛽𝑧𝜃
𝐿𝑛𝒱 = 0, 𝑛 ≥ 0
𝐺𝑟𝒱 = 0, 𝑟 ≥ 1/2,
pág. 89
𝐿̃ 𝑛𝒱 = 0, 𝑛 ≥ 0.
𝛿(𝑏̃ )𝛿(𝑏)𝛿(𝛽)
𝒱 = 𝑐̃ 𝑐𝛿(𝛾)𝑉,
𝐿𝑛
𝑋𝑉 = 1
2 𝛿𝑛,0𝑉, 𝑛 ≥ 0
𝐺𝑟
𝑋𝑉 = 0, 𝑟 > 0
𝐿̃ 𝑛
𝑋𝑉 = 𝛿𝑛,0𝑉, 𝑛 ≥ 0
𝑄𝐵
∗ = ∑
𝑟∈ℤ+1/2
𝛾𝑟𝐺−𝑟
𝑋
𝛾(𝑧) = ∑
𝑟∈ℤ+1/2
𝑧−𝑟+1/2𝛾𝑟
𝐹𝒱1,…,𝒱𝑛 (𝒥, 𝛿𝒥) = ∫ 𝒟(𝑋, 𝐵, 𝐶, 𝐵̃ , 𝐶̃ )exp (−𝐼̂ ) ∏
𝑛
𝑖=1
𝒱𝑖(𝑝𝑖)
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n + d𝐹𝒲1,𝒱2,…,𝒱𝑛 = 0
𝑏̃𝑛𝒱 = 𝑏𝑛𝒱 = 𝛽𝑟𝒱 = 0, 𝑛, 𝑟 ≥ 0.
𝑏̃𝑛𝒲 = 𝑏𝑛𝒲 = 𝛽𝑟𝒲 = 0, 𝑛, 𝑟 ≥ 0.
𝐼̂ → 𝐼̂ + 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝜕𝑧˜ 𝑞𝑧𝐵𝑧𝜃
⟨𝒱1 … 𝒱n⟩ = ∫
Γ
𝐹𝒱1,…,𝒱n (𝒥, 𝛿𝒥)
∫
Γ
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = − ∫
Γ
d𝐹𝒲1,𝒱2,…,𝒱𝑛 = 0
𝐾𝒱 = ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝑉(𝑧̃ ; 𝑧 ∣ 𝜃)
( 1
2𝜋 ∫
Σ
d2𝑧𝑔𝑧𝜕𝑧˜ 𝑏𝑧𝑧) ⋅ 𝑐𝑧(0) = 𝑔𝑧(0)
𝛿 ( 1
2𝜋 ∫
Σ
d2𝑧𝑔𝑧𝜕𝑧̃ 𝑏𝑧𝑧) ⋅ 𝛿(𝑐𝑧(0)) = 𝑔𝑧(0)
𝛿 ( 1
2𝜋 ∫
Σ
d2𝑧𝑓𝜃𝜕𝑧˜ 𝛽𝑧𝜃) ⋅ 𝛿(𝛾𝜃(0)) = 1
𝑓𝜃(0)
𝛿(𝐵̃ (𝐳̃))𝛿(𝐵(𝐳))𝛿(𝐵(𝜽))
[d𝐳̃; d𝐳 ∣ d𝜽]𝛿(𝐵̃ (𝐳̃))𝛿(𝐵(𝐳))𝛿(𝐵(𝜽))𝑐̃ 𝑐𝛿(𝛾)𝑉
𝐿̃ 𝑛𝒱 = 0, 𝑛 ≥ 0.
𝛿(𝑏̃ )𝛿(𝑏)𝛿(𝛽)
𝒱 = 𝑐̃ 𝑐𝛿(𝛾)𝑉,
𝐿𝑛
𝑋𝑉 = 1
2 𝛿𝑛,0𝑉, 𝑛 ≥ 0
𝐺𝑟
𝑋𝑉 = 0, 𝑟 > 0
𝐿̃ 𝑛
𝑋𝑉 = 𝛿𝑛,0𝑉, 𝑛 ≥ 0
𝑄𝐵
∗ = ∑
𝑟∈ℤ+1/2
𝛾𝑟𝐺−𝑟
𝑋
𝛾(𝑧) = ∑
𝑟∈ℤ+1/2
𝑧−𝑟+1/2𝛾𝑟
𝐹𝒱1,…,𝒱𝑛 (𝒥, 𝛿𝒥) = ∫ 𝒟(𝑋, 𝐵, 𝐶, 𝐵̃ , 𝐶̃ )exp (−𝐼̂ ) ∏
𝑛
𝑖=1
𝒱𝑖(𝑝𝑖)
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n + d𝐹𝒲1,𝒱2,…,𝒱𝑛 = 0
𝑏̃𝑛𝒱 = 𝑏𝑛𝒱 = 𝛽𝑟𝒱 = 0, 𝑛, 𝑟 ≥ 0.
𝑏̃𝑛𝒲 = 𝑏𝑛𝒲 = 𝛽𝑟𝒲 = 0, 𝑛, 𝑟 ≥ 0.
𝐼̂ → 𝐼̂ + 1
2𝜋 ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝜕𝑧˜ 𝑞𝑧𝐵𝑧𝜃
⟨𝒱1 … 𝒱n⟩ = ∫
Γ
𝐹𝒱1,…,𝒱n (𝒥, 𝛿𝒥)
∫
Γ
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = − ∫
Γ
d𝐹𝒲1,𝒱2,…,𝒱𝑛 = 0
𝐾𝒱 = ∫
Σ
𝒟(𝑧̃ , 𝑧 ∣ 𝜃)𝑉(𝑧̃ ; 𝑧 ∣ 𝜃)
( 1
2𝜋 ∫
Σ
d2𝑧𝑔𝑧𝜕𝑧˜ 𝑏𝑧𝑧) ⋅ 𝑐𝑧(0) = 𝑔𝑧(0)
𝛿 ( 1
2𝜋 ∫
Σ
d2𝑧𝑔𝑧𝜕𝑧̃ 𝑏𝑧𝑧) ⋅ 𝛿(𝑐𝑧(0)) = 𝑔𝑧(0)
𝛿 ( 1
2𝜋 ∫
Σ
d2𝑧𝑓𝜃𝜕𝑧˜ 𝛽𝑧𝜃) ⋅ 𝛿(𝛾𝜃(0)) = 1
𝑓𝜃(0)
𝛿(𝐵̃ (𝐳̃))𝛿(𝐵(𝐳))𝛿(𝐵(𝜽))
[d𝐳̃; d𝐳 ∣ d𝜽]𝛿(𝐵̃ (𝐳̃))𝛿(𝐵(𝐳))𝛿(𝐵(𝜽))𝑐̃ 𝑐𝛿(𝛾)𝑉
pág. 90
[d𝐳̃; d𝐳 ∣ d𝜽]𝑉(𝐳̃; 𝐳 ∣ 𝜽).
∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝑉 =
{
∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝐷𝜃𝑊
∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝜕𝑧̃ 𝑊′
⟨∏
3
𝑖=1
𝑐̃ 𝑐𝑉𝑖(𝑧̃𝑖, 𝑧𝑖) ∏
𝑛
𝑗=4
∫ d2𝑧𝑗𝑉(𝑧̃𝑗, 𝑧𝑗)⟩
∫ d𝜃3𝑐̃ 𝑐𝑉3(𝑧̃3; 𝑧3 ∣ 𝜃3) = 𝑐̃ 𝑐𝐷𝜃𝑉3(𝑧̃3; 𝑧3 ∣ 0)
⟨𝑐̃ 𝑐𝛿(𝛾)𝑉1(𝑧̃1, 𝑧1 ∣ 0)𝑐̃ 𝑐𝛿(𝛾)𝑉2(𝑧̃2, 𝑧2 ∣ 0)𝑐̃ 𝑐𝐷𝜃𝑉3(𝑧̃3, 𝑧3 ∣ 0)⟩
⟨𝑐̃ 𝑐𝛿(𝛾)𝑉1(𝑧̃1, 𝑧1 ∣ 0)𝑐̃ 𝑐𝛿(𝛾)𝑉2(𝑧̃2, 𝑧2 ∣ 0)𝑐̃ 𝑐𝐷𝜃𝑉3(𝑧̃3, 𝑧3 ∣ 0) ∏
𝑛
𝑗=4
∫ [ d𝑧̃𝑗; d𝑧𝑗 ∣ d𝜃𝑗]𝑉𝑗(𝑧̃𝑗; 𝑧𝑗 ∣ 𝜃𝑗)⟩
𝜒𝑧̃
𝜃 = ∑
2𝑔−2+𝑛
𝜎=1
𝜂𝜎𝜒𝑧̃
(𝜎)𝜃
𝐻0 (Σred, 𝑇1/2 ⊗ 𝒪 ( ∑
2 g−2+n
𝜎=1
𝑝𝜎 − ∑
n
𝑖=1
𝑞𝑖)) ≠ 0
⟨∏
3
𝑖=1
𝑐̃ 𝑐𝒱(𝑠𝑖)(𝑧̃𝑖, 𝑧𝑖) ∏
𝑛
𝑗=4
∫ d2𝑧𝑗𝑉̂𝑗
(𝑠𝑗)(𝑧̃𝑗, 𝑧𝑗)⟩
∑
𝑖
𝑠𝑖 = −2
Métrica Ramond.
𝐷𝜃
∗ = 𝜕𝜃 + 𝜃𝑧𝜕𝑧.
𝜏|𝜁 = (log 𝑧)|𝜃
(𝐷𝜃
∗ )2 = 𝑧𝜕𝑧,
𝜈𝑓 = 𝑓(𝑧)(𝜕𝜃 − 𝜃𝑧𝜕𝑧)
𝑉𝑔 = 𝑧 (𝑔(𝑧)𝜕𝑧 + 𝑔′(𝑧)
2 𝜃𝜕𝜃)
[d𝐳̃; d𝐳 ∣ d𝜽]𝑉(𝐳̃; 𝐳 ∣ 𝜽).
∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝑉 =
{
∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝐷𝜃𝑊
∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝜕𝑧̃ 𝑊′
⟨∏
3
𝑖=1
𝑐̃ 𝑐𝑉𝑖(𝑧̃𝑖, 𝑧𝑖) ∏
𝑛
𝑗=4
∫ d2𝑧𝑗𝑉(𝑧̃𝑗, 𝑧𝑗)⟩
∫ d𝜃3𝑐̃ 𝑐𝑉3(𝑧̃3; 𝑧3 ∣ 𝜃3) = 𝑐̃ 𝑐𝐷𝜃𝑉3(𝑧̃3; 𝑧3 ∣ 0)
⟨𝑐̃ 𝑐𝛿(𝛾)𝑉1(𝑧̃1, 𝑧1 ∣ 0)𝑐̃ 𝑐𝛿(𝛾)𝑉2(𝑧̃2, 𝑧2 ∣ 0)𝑐̃ 𝑐𝐷𝜃𝑉3(𝑧̃3, 𝑧3 ∣ 0)⟩
⟨𝑐̃ 𝑐𝛿(𝛾)𝑉1(𝑧̃1, 𝑧1 ∣ 0)𝑐̃ 𝑐𝛿(𝛾)𝑉2(𝑧̃2, 𝑧2 ∣ 0)𝑐̃ 𝑐𝐷𝜃𝑉3(𝑧̃3, 𝑧3 ∣ 0) ∏
𝑛
𝑗=4
∫ [ d𝑧̃𝑗; d𝑧𝑗 ∣ d𝜃𝑗]𝑉𝑗(𝑧̃𝑗; 𝑧𝑗 ∣ 𝜃𝑗)⟩
𝜒𝑧̃
𝜃 = ∑
2𝑔−2+𝑛
𝜎=1
𝜂𝜎𝜒𝑧̃
(𝜎)𝜃
𝐻0 (Σred, 𝑇1/2 ⊗ 𝒪 ( ∑
2 g−2+n
𝜎=1
𝑝𝜎 − ∑
n
𝑖=1
𝑞𝑖)) ≠ 0
⟨∏
3
𝑖=1
𝑐̃ 𝑐𝒱(𝑠𝑖)(𝑧̃𝑖, 𝑧𝑖) ∏
𝑛
𝑗=4
∫ d2𝑧𝑗𝑉̂𝑗
(𝑠𝑗)(𝑧̃𝑗, 𝑧𝑗)⟩
∑
𝑖
𝑠𝑖 = −2
Métrica Ramond.
𝐷𝜃
∗ = 𝜕𝜃 + 𝜃𝑧𝜕𝑧.
𝜏|𝜁 = (log 𝑧)|𝜃
(𝐷𝜃
∗ )2 = 𝑧𝜕𝑧,
𝜈𝑓 = 𝑓(𝑧)(𝜕𝜃 − 𝜃𝑧𝜕𝑧)
𝑉𝑔 = 𝑧 (𝑔(𝑧)𝜕𝑧 + 𝑔′(𝑧)
2 𝜃𝜕𝜃)
pág. 91
𝐺𝑟 = 𝑧𝑟(𝜕𝜃 − 𝜃𝑧𝜕𝑧)
𝐿𝑛 = −𝑧𝑛+1𝜕𝑧 − 𝑛𝑧𝑛
2 𝜃𝜕𝜃
[𝐿𝑚, 𝐿𝑛] = (𝑚 − 𝑛)𝐿𝑚+𝑛
{𝐺𝑟, 𝐺𝑠} = 2𝐿𝑟+𝑠
[𝐿𝑚, 𝐺𝑟] = (𝑚
2 − 𝑟) 𝐺𝑚+𝑟
𝐺0|𝑧=0 = 𝜕𝜃
𝜃 → 𝜃 + 𝛼
𝐺𝑟𝒱 = 𝐿𝑛𝒱 = 0, 𝑟, 𝑛 ≥ 0
𝒱 = 𝑐̃ 𝑐Θ𝑉,
𝐿𝑛
𝑋𝑉 = 5
8 𝛿𝑛,0𝑉, 𝑛 ≥ 0
𝐺𝑟
𝑋𝑉 = 0, 𝑟 ≥ 0
𝛽(𝑧) = ∑
𝑟
𝑧−𝑟−3/2𝛽𝑟, 𝛾(𝑧) = ∑
𝑟
𝑧−𝑟+1/2𝛾𝑟
𝑄𝐵
∗ = ∑
𝑟∈ℤ
𝛾−𝑟𝐺𝑟
𝑋
𝛾𝑟Θ = 0, 𝑟 > 0
𝛽𝑟Θ = 0, 𝑟 ≥ 0
dim𝔐g,𝔫NS,𝔫R = 3 g − 3 + 𝔫NS + 𝔫R | 2 g − 2 + 𝔫NS + 1
2 𝔫R
𝛾𝑟Θ = 0, 𝑟 ≥ 0
𝛽𝑟Θ = 0, 𝑟 > 0.
𝐷𝜃
∗ = 𝜕𝜃 + 𝜃𝑤(𝑧)𝜕𝑧
𝑤(𝑧) = ∏
nR
𝑖=1
(𝑧 − 𝑧𝑖)
𝜈𝑓 = 𝑓(𝑧)(𝜕𝜃 − 𝜃𝑤(𝑧)𝜕𝑧)
𝑉𝑔 = 𝑤(𝑧) (𝑔(𝑧)𝜕𝑧 + 𝑔′(𝑧)
2 𝜃𝜕𝜃)
𝐷𝜃
∗ = 𝜕𝜃 + 𝜃𝑧(𝑧 − 𝑎)𝜕𝑧
𝜈𝑠 = 𝑧𝑠(𝜕𝜃 − 𝜃𝑧(𝑧 − 𝑎)𝜕𝑧)
𝑉𝑛 = 𝑧(𝑧 − 𝑎) (𝑧𝑛𝜕𝑧 + 𝑛𝑧𝑛−1
2 𝜃𝜕𝜃)
𝐺𝑟 = 𝑧𝑟(𝜕𝜃 − 𝜃𝑧𝜕𝑧)
𝐿𝑛 = −𝑧𝑛+1𝜕𝑧 − 𝑛𝑧𝑛
2 𝜃𝜕𝜃
[𝐿𝑚, 𝐿𝑛] = (𝑚 − 𝑛)𝐿𝑚+𝑛
{𝐺𝑟, 𝐺𝑠} = 2𝐿𝑟+𝑠
[𝐿𝑚, 𝐺𝑟] = (𝑚
2 − 𝑟) 𝐺𝑚+𝑟
𝐺0|𝑧=0 = 𝜕𝜃
𝜃 → 𝜃 + 𝛼
𝐺𝑟𝒱 = 𝐿𝑛𝒱 = 0, 𝑟, 𝑛 ≥ 0
𝒱 = 𝑐̃ 𝑐Θ𝑉,
𝐿𝑛
𝑋𝑉 = 5
8 𝛿𝑛,0𝑉, 𝑛 ≥ 0
𝐺𝑟
𝑋𝑉 = 0, 𝑟 ≥ 0
𝛽(𝑧) = ∑
𝑟
𝑧−𝑟−3/2𝛽𝑟, 𝛾(𝑧) = ∑
𝑟
𝑧−𝑟+1/2𝛾𝑟
𝑄𝐵
∗ = ∑
𝑟∈ℤ
𝛾−𝑟𝐺𝑟
𝑋
𝛾𝑟Θ = 0, 𝑟 > 0
𝛽𝑟Θ = 0, 𝑟 ≥ 0
dim𝔐g,𝔫NS,𝔫R = 3 g − 3 + 𝔫NS + 𝔫R | 2 g − 2 + 𝔫NS + 1
2 𝔫R
𝛾𝑟Θ = 0, 𝑟 ≥ 0
𝛽𝑟Θ = 0, 𝑟 > 0.
𝐷𝜃
∗ = 𝜕𝜃 + 𝜃𝑤(𝑧)𝜕𝑧
𝑤(𝑧) = ∏
nR
𝑖=1
(𝑧 − 𝑧𝑖)
𝜈𝑓 = 𝑓(𝑧)(𝜕𝜃 − 𝜃𝑤(𝑧)𝜕𝑧)
𝑉𝑔 = 𝑤(𝑧) (𝑔(𝑧)𝜕𝑧 + 𝑔′(𝑧)
2 𝜃𝜕𝜃)
𝐷𝜃
∗ = 𝜕𝜃 + 𝜃𝑧(𝑧 − 𝑎)𝜕𝑧
𝜈𝑠 = 𝑧𝑠(𝜕𝜃 − 𝜃𝑧(𝑧 − 𝑎)𝜕𝑧)
𝑉𝑛 = 𝑧(𝑧 − 𝑎) (𝑧𝑛𝜕𝑧 + 𝑛𝑧𝑛−1
2 𝜃𝜕𝜃)
pág. 92
𝑐Θ(𝑎)𝑐Θ(0) ∼ 𝑐𝜕𝑐𝛿(𝛾), 𝑎 → 0
Θ(𝑎)Θ(0) ∼ 𝛿(𝛾), 𝑎 → 0
𝐼𝑋 = 1
2𝜋𝑖 ∫
Σ
[d𝜏̃ ; d𝜏 ∣ d𝜁]𝐺𝐼𝐽(𝑋𝐾)𝜕𝜏̃ 𝑋𝐼𝐷𝜁𝑋𝐽
𝜏̃ = log 𝑧̃
𝜏 = log 𝑧
𝜁 = 𝜃
𝐼𝑋 = 1
2𝜋𝑖 ∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃] 1
𝑧 𝐺𝐼𝐽(𝑋𝐾)𝜕𝑧̃ 𝑋𝐼𝐷𝜃
∗ 𝑋𝐽
𝐼𝑋 = 1
2𝜋 ∫
Σred
d2𝑧 (𝐺𝐼𝐽𝜕𝑧̃ 𝑥𝐼𝜕𝑧𝑥𝐽 + 1
𝑧 𝐺𝐼𝐽𝜓𝐼 𝐷
𝐷𝑧̃ 𝜓𝐽)
𝜏 ≅ 𝜏 + 2𝜋𝑖
𝐼𝜓 = 1
2𝜋 ∫
Σred
d2𝜏𝐺𝐼𝐽(𝑥𝐾)𝜓𝐼 𝐷
𝐷𝜏̃ 𝜓𝐽
𝐼𝐵𝐶 = 1
2𝜋𝑖 ∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝐵𝜕𝑧̃ 𝐶
[𝑑𝑧 ∣ d𝜃] = [d𝑧∗ ∣ d𝜃∗]Ber−1 (𝜕𝑧𝑧∗ 𝜕𝑧𝜃∗
𝜕𝜃𝑧∗ 𝜕𝜃𝜃∗)
𝐶 = 𝐶∗ ⋅ Ber−2 (𝜕𝑧𝑧∗ 𝜕𝑧𝜃∗
𝜕𝜃𝑧∗ 𝜕𝜃𝜃∗)
𝐵 = 𝐵∗ ⋅ Ber3 (𝜕𝑧𝑧∗ 𝜕𝑧𝜃∗
𝜕𝜃𝑧∗ 𝜕𝜃𝜃∗)
ℛ2 ≅ 𝑇 ⊗ 𝒪(−𝑞1 − ⋯ − 𝑞𝔫R )
ℛ2 ≅ 𝑇 ⊗ 𝒪(−𝑞)
ℛ−2 ≅ 𝐾 ⊗ 𝒪(𝑞)
𝐶(𝑧 ∣ 𝜃) = 𝑐̂ (𝑧) + 𝜃𝛾̂ (𝑧)
𝐵 = 𝛽̂ + 𝜃𝑏̂
𝐼𝐵𝐶 = 1
2𝜋 ∫
Σred
d2𝑧(𝑏̂ 𝜕𝑧‾𝑐̂ + 𝛽̂ 𝜕𝑧‾𝛾̂ )
𝛾 ∼ 𝑧1/2, 𝛽 ∼ 𝑧−1/2, 𝑧 → 0
𝛾(𝑧) = ∑
𝑟∈ℤ
𝑧−𝑟+1/2𝛾𝑟, 𝛽(𝑧) = ∑
𝑟∈ℤ
𝑧−𝑟−3/2𝛽𝑟
𝑐Θ(𝑎)𝑐Θ(0) ∼ 𝑐𝜕𝑐𝛿(𝛾), 𝑎 → 0
Θ(𝑎)Θ(0) ∼ 𝛿(𝛾), 𝑎 → 0
𝐼𝑋 = 1
2𝜋𝑖 ∫
Σ
[d𝜏̃ ; d𝜏 ∣ d𝜁]𝐺𝐼𝐽(𝑋𝐾)𝜕𝜏̃ 𝑋𝐼𝐷𝜁𝑋𝐽
𝜏̃ = log 𝑧̃
𝜏 = log 𝑧
𝜁 = 𝜃
𝐼𝑋 = 1
2𝜋𝑖 ∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃] 1
𝑧 𝐺𝐼𝐽(𝑋𝐾)𝜕𝑧̃ 𝑋𝐼𝐷𝜃
∗ 𝑋𝐽
𝐼𝑋 = 1
2𝜋 ∫
Σred
d2𝑧 (𝐺𝐼𝐽𝜕𝑧̃ 𝑥𝐼𝜕𝑧𝑥𝐽 + 1
𝑧 𝐺𝐼𝐽𝜓𝐼 𝐷
𝐷𝑧̃ 𝜓𝐽)
𝜏 ≅ 𝜏 + 2𝜋𝑖
𝐼𝜓 = 1
2𝜋 ∫
Σred
d2𝜏𝐺𝐼𝐽(𝑥𝐾)𝜓𝐼 𝐷
𝐷𝜏̃ 𝜓𝐽
𝐼𝐵𝐶 = 1
2𝜋𝑖 ∫
Σ
[d𝑧̃ ; d𝑧 ∣ d𝜃]𝐵𝜕𝑧̃ 𝐶
[𝑑𝑧 ∣ d𝜃] = [d𝑧∗ ∣ d𝜃∗]Ber−1 (𝜕𝑧𝑧∗ 𝜕𝑧𝜃∗
𝜕𝜃𝑧∗ 𝜕𝜃𝜃∗)
𝐶 = 𝐶∗ ⋅ Ber−2 (𝜕𝑧𝑧∗ 𝜕𝑧𝜃∗
𝜕𝜃𝑧∗ 𝜕𝜃𝜃∗)
𝐵 = 𝐵∗ ⋅ Ber3 (𝜕𝑧𝑧∗ 𝜕𝑧𝜃∗
𝜕𝜃𝑧∗ 𝜕𝜃𝜃∗)
ℛ2 ≅ 𝑇 ⊗ 𝒪(−𝑞1 − ⋯ − 𝑞𝔫R )
ℛ2 ≅ 𝑇 ⊗ 𝒪(−𝑞)
ℛ−2 ≅ 𝐾 ⊗ 𝒪(𝑞)
𝐶(𝑧 ∣ 𝜃) = 𝑐̂ (𝑧) + 𝜃𝛾̂ (𝑧)
𝐵 = 𝛽̂ + 𝜃𝑏̂
𝐼𝐵𝐶 = 1
2𝜋 ∫
Σred
d2𝑧(𝑏̂ 𝜕𝑧‾𝑐̂ + 𝛽̂ 𝜕𝑧‾𝛾̂ )
𝛾 ∼ 𝑧1/2, 𝛽 ∼ 𝑧−1/2, 𝑧 → 0
𝛾(𝑧) = ∑
𝑟∈ℤ
𝑧−𝑟+1/2𝛾𝑟, 𝛽(𝑧) = ∑
𝑟∈ℤ
𝑧−𝑟−3/2𝛽𝑟
pág. 93
𝛾𝑟Θ−1/2 = 0, 𝑟 > 0
𝛽𝑟Θ−1/2 = 0, 𝑟 ≥ 0
𝛽𝑧𝜃′ (𝑧)𝛾𝜃′
(𝑤) ∼ − 1
𝑧 − 𝑤
⟨𝛽𝑧𝜃′ (𝑧)𝛾𝜃′
(𝑤)⟩Θ−1/2
= − 1
𝑧 − 𝑤 √𝑤
𝑧 .
𝐽𝛽𝛾(𝑤) = lim
𝑧→𝑤 (−𝛽(𝑧)𝛾(𝑤) − 1
𝑧 − 𝑤) ,
⟨𝐽𝛽𝛾(𝑤)⟩Θ−1/2
= − 1
2𝑤 ,
𝑇𝛽𝛾 =: 𝜕𝑧𝛽𝑧𝜃′ ⋅ 𝛾𝜃′
: − 3
2 𝜕𝑧(: 𝛽𝑧𝜃′ 𝛾𝜃′
: )
⟨𝑇𝛽𝛾(𝑤)⟩Θ−1/2
= 3
8𝑤2 ,
⟨𝛽𝑧𝜃′ (𝑧)𝛾𝜃′
(𝑤)⟩Θ−𝑡
= − 1
𝑧 − 𝑤 (𝑤
𝑧 )
𝑡
⟨𝐽𝛽𝛾(𝑤)⟩Θ−𝑡
= − 𝑡
𝑤
⟨𝑇𝛽𝛾(𝑤)⟩Θ−𝑡
= − 𝑡(𝑡 − 2)
2𝑤2
𝑏𝑛𝒱1 = 𝛽𝑟𝒱1 = 0, 𝑛, 𝑟 ≥ 0,
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱𝑠 + d𝐹𝒲1,𝒱2,…,𝒱𝑠 = 0
∫
Γ
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱𝑠 + ∫
Γ
d𝐹𝒲1,𝒱2,…,𝒱𝑠 = 0
Σ → 𝔐g,𝔫NS,𝔫R
↓ 𝜋
𝔐g,𝔫NS−1,𝔫R .
ℛ2 ≅ 𝑇 ⊗ 𝒪(−𝑞1 − ⋯ − 𝑞𝔫R )
degℛ = 1 − g − 1
2 𝔫R
degℛ̂ = 1 − g − nNS − 1
2 nR
𝜒𝑧˜
𝜃 = ∑
Δ
𝜎=1
𝜂𝜎𝛿𝑟𝜎
𝛾𝑟Θ−1/2 = 0, 𝑟 > 0
𝛽𝑟Θ−1/2 = 0, 𝑟 ≥ 0
𝛽𝑧𝜃′ (𝑧)𝛾𝜃′
(𝑤) ∼ − 1
𝑧 − 𝑤
⟨𝛽𝑧𝜃′ (𝑧)𝛾𝜃′
(𝑤)⟩Θ−1/2
= − 1
𝑧 − 𝑤 √𝑤
𝑧 .
𝐽𝛽𝛾(𝑤) = lim
𝑧→𝑤 (−𝛽(𝑧)𝛾(𝑤) − 1
𝑧 − 𝑤) ,
⟨𝐽𝛽𝛾(𝑤)⟩Θ−1/2
= − 1
2𝑤 ,
𝑇𝛽𝛾 =: 𝜕𝑧𝛽𝑧𝜃′ ⋅ 𝛾𝜃′
: − 3
2 𝜕𝑧(: 𝛽𝑧𝜃′ 𝛾𝜃′
: )
⟨𝑇𝛽𝛾(𝑤)⟩Θ−1/2
= 3
8𝑤2 ,
⟨𝛽𝑧𝜃′ (𝑧)𝛾𝜃′
(𝑤)⟩Θ−𝑡
= − 1
𝑧 − 𝑤 (𝑤
𝑧 )
𝑡
⟨𝐽𝛽𝛾(𝑤)⟩Θ−𝑡
= − 𝑡
𝑤
⟨𝑇𝛽𝛾(𝑤)⟩Θ−𝑡
= − 𝑡(𝑡 − 2)
2𝑤2
𝑏𝑛𝒱1 = 𝛽𝑟𝒱1 = 0, 𝑛, 𝑟 ≥ 0,
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱𝑠 + d𝐹𝒲1,𝒱2,…,𝒱𝑠 = 0
∫
Γ
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱𝑠 + ∫
Γ
d𝐹𝒲1,𝒱2,…,𝒱𝑠 = 0
Σ → 𝔐g,𝔫NS,𝔫R
↓ 𝜋
𝔐g,𝔫NS−1,𝔫R .
ℛ2 ≅ 𝑇 ⊗ 𝒪(−𝑞1 − ⋯ − 𝑞𝔫R )
degℛ = 1 − g − 1
2 𝔫R
degℛ̂ = 1 − g − nNS − 1
2 nR
𝜒𝑧˜
𝜃 = ∑
Δ
𝜎=1
𝜂𝜎𝛿𝑟𝜎
pág. 94
𝐻0 (Σ, ℛ̂ ⊗ 𝒪 (∑
Δ
𝜎=1
𝑟𝜎)) ≠ 0
⟨𝑐̃ 𝑐𝛿(𝛾)𝑉1(𝑧̃1; 𝑧1 ∣ 0)𝑐̃ 𝑐Θ−1/2𝑉2
′(𝑧̃2; 𝑧2)𝑐̃ 𝑐Θ−1/2𝑉3
′(𝑧̃3; 𝑧3)⟩.
⟨∏
3
𝑖=1
𝑐̃ 𝑐Θ−1/2𝑉𝑖′(𝑧̃𝑖; 𝑧𝑖) ∫ d2𝑧4Θ−1/2𝑉4
′(𝑧̃4; 𝑧4)⟩
Dualidad isotrópica.
𝜔(𝒱, 𝒲) = ⟨𝒱(0)𝒲(1)⟩.
𝜔(𝑄𝐵𝒰, 𝒱) + (−1)|𝒰|𝜔(𝒰, 𝑄𝐵𝒱) = 0.
⟨𝑐𝜕𝑐𝑈(0)𝑐𝑉(1)⟩ ≠ 0.
𝜔: ℋ𝑛
∗ ≅ ℋ3−𝑛.
𝜔: ℋ𝑛;𝑘
∗ ≅ ℋ1−𝑛;−2−𝑘
⟨𝑐𝜕𝑐𝛿(𝛾)𝑈(0)𝑐𝛿(𝛾)𝑉(1)⟩ ≠ 0
𝜔: ℋ𝑛;−3/2
∗ ≅ ℋ1−𝑛;−1/2
𝐷𝜃
∗ = 𝜕𝜃 + 𝑧(𝑧 − 1)𝜃𝜕𝑧
𝜈 = 𝜕𝜃 − 𝑧(𝑧 − 1)𝜃𝜕𝑧
Supermasa y deformaciones dimensionales – GSO – Propagador de una Partícula Supermasiva –
NS - Ramond en un espacio de Fock.
⟨𝑐𝜕𝑐Θ−1/2𝑢𝛼Σ𝛼(𝑝) ⋅ 𝑐Θ−3/2𝑣𝛽Σ𝛽(𝑞)⟩ = 𝑢𝛼𝑣𝛼(2𝜋)10𝛿10(𝑝 + 𝑞).
𝑝𝐼Γ𝐼𝛼𝛽𝑢𝛽 = 0
𝑣𝛽 ≅ 𝑣𝛽 + 𝑝𝐼Γ𝐼𝛽𝛾𝑤𝛾
1
𝑝2 + 𝑚2 = ∫
∞
0
d𝑠exp (−𝑠(𝑝2 + 𝑚2))
𝐻0 (Σ, ℛ̂ ⊗ 𝒪 (∑
Δ
𝜎=1
𝑟𝜎)) ≠ 0
⟨𝑐̃ 𝑐𝛿(𝛾)𝑉1(𝑧̃1; 𝑧1 ∣ 0)𝑐̃ 𝑐Θ−1/2𝑉2
′(𝑧̃2; 𝑧2)𝑐̃ 𝑐Θ−1/2𝑉3
′(𝑧̃3; 𝑧3)⟩.
⟨∏
3
𝑖=1
𝑐̃ 𝑐Θ−1/2𝑉𝑖′(𝑧̃𝑖; 𝑧𝑖) ∫ d2𝑧4Θ−1/2𝑉4
′(𝑧̃4; 𝑧4)⟩
Dualidad isotrópica.
𝜔(𝒱, 𝒲) = ⟨𝒱(0)𝒲(1)⟩.
𝜔(𝑄𝐵𝒰, 𝒱) + (−1)|𝒰|𝜔(𝒰, 𝑄𝐵𝒱) = 0.
⟨𝑐𝜕𝑐𝑈(0)𝑐𝑉(1)⟩ ≠ 0.
𝜔: ℋ𝑛
∗ ≅ ℋ3−𝑛.
𝜔: ℋ𝑛;𝑘
∗ ≅ ℋ1−𝑛;−2−𝑘
⟨𝑐𝜕𝑐𝛿(𝛾)𝑈(0)𝑐𝛿(𝛾)𝑉(1)⟩ ≠ 0
𝜔: ℋ𝑛;−3/2
∗ ≅ ℋ1−𝑛;−1/2
𝐷𝜃
∗ = 𝜕𝜃 + 𝑧(𝑧 − 1)𝜃𝜕𝑧
𝜈 = 𝜕𝜃 − 𝑧(𝑧 − 1)𝜃𝜕𝑧
Supermasa y deformaciones dimensionales – GSO – Propagador de una Partícula Supermasiva –
NS - Ramond en un espacio de Fock.
⟨𝑐𝜕𝑐Θ−1/2𝑢𝛼Σ𝛼(𝑝) ⋅ 𝑐Θ−3/2𝑣𝛽Σ𝛽(𝑞)⟩ = 𝑢𝛼𝑣𝛼(2𝜋)10𝛿10(𝑝 + 𝑞).
𝑝𝐼Γ𝐼𝛼𝛽𝑢𝛽 = 0
𝑣𝛽 ≅ 𝑣𝛽 + 𝑝𝐼Γ𝐼𝛽𝛾𝑤𝛾
1
𝑝2 + 𝑚2 = ∫
∞
0
d𝑠exp (−𝑠(𝑝2 + 𝑚2))
pág. 95
𝑥𝑦 = 𝑞
𝑥 = 𝑒𝜚, 𝑦 = 𝑞𝑒−𝜚
𝜚 = 𝑢 + 𝑖𝜑, 𝑢, 𝜑 ∈ ℝ
0 ≤ 𝑢 ≤ 𝑠, 0 ≤ 𝜑 ≤ 𝜋
ds2 = d𝑢2 + d𝜑2
𝐿0 = − 1
2𝜋 ∫
𝜋
0
d𝜑𝑇𝑢𝑢
𝑏0 = − 1
2𝜋 ∫
𝜋
0
d𝜑𝑏𝑢𝑢
𝑧 = −𝑖𝜚 = 𝜑 − 𝑖𝑢
𝑇𝑧𝑧(𝑢, 𝜑) = 𝑇𝑧̃ (𝑢, −𝜑) = 𝑇𝑧𝑧(𝑢, 𝜑 + 2𝜋)
𝑏𝑧𝑧(𝑢, 𝜑) = 𝑏𝑧̃ 𝑧˜ (𝑢, −𝜑) = 𝑏𝑧𝑧(𝑢, 𝜑 + 2𝜋)
𝐿0 = 1
2𝜋 ∫
2𝜋
0
d𝜑𝑇𝑧𝑧
𝑏0 = 1
2𝜋 ∫
2𝜋
0
d𝜑𝑏𝑧𝑧
0 ≤ 𝑡 ≤ 1, 0 ≤ 𝜑 ≤ 𝜋
ds2 = 𝑠2 d𝑡2 + d𝜑2
Ψ𝑠= 1
4𝜋 ∫
𝑆
d𝑡 d𝜑 𝜕(√𝑔𝑔𝑖𝑗)
d𝑠 𝑏𝑖𝑗 = 1
2𝜋 ∫
1
0
d𝑡 ∫
𝜋
0
d𝜑𝑠2𝑏𝑡𝑡
= 1
2𝜋 ∫
1
0
d𝑡 ∫
𝜋
0
d𝜑𝑏𝑢𝑢 = ∫
1
0
d𝑡𝑏0 = 𝑏0
∫
∞
0
d𝑠𝑏0exp (−𝑠𝐿0) = 𝑏0
𝐿0
𝑏0 ∫
1
0
d𝑞
𝑞 𝑞𝐿0
𝐿0 = 𝛼′
4 𝑝2 + 𝑁,
4
𝛼′
𝑏0
𝑝2 + 𝑀2
𝑍Σ;𝑠 = ⟨𝜓ℓ| 𝑏0
𝐿0
|𝜓𝑟⟩.
𝑥𝑦 = 𝑞
𝑥 = 𝑒𝜚, 𝑦 = 𝑞𝑒−𝜚
𝜚 = 𝑢 + 𝑖𝜑, 𝑢, 𝜑 ∈ ℝ
0 ≤ 𝑢 ≤ 𝑠, 0 ≤ 𝜑 ≤ 𝜋
ds2 = d𝑢2 + d𝜑2
𝐿0 = − 1
2𝜋 ∫
𝜋
0
d𝜑𝑇𝑢𝑢
𝑏0 = − 1
2𝜋 ∫
𝜋
0
d𝜑𝑏𝑢𝑢
𝑧 = −𝑖𝜚 = 𝜑 − 𝑖𝑢
𝑇𝑧𝑧(𝑢, 𝜑) = 𝑇𝑧̃ (𝑢, −𝜑) = 𝑇𝑧𝑧(𝑢, 𝜑 + 2𝜋)
𝑏𝑧𝑧(𝑢, 𝜑) = 𝑏𝑧̃ 𝑧˜ (𝑢, −𝜑) = 𝑏𝑧𝑧(𝑢, 𝜑 + 2𝜋)
𝐿0 = 1
2𝜋 ∫
2𝜋
0
d𝜑𝑇𝑧𝑧
𝑏0 = 1
2𝜋 ∫
2𝜋
0
d𝜑𝑏𝑧𝑧
0 ≤ 𝑡 ≤ 1, 0 ≤ 𝜑 ≤ 𝜋
ds2 = 𝑠2 d𝑡2 + d𝜑2
Ψ𝑠= 1
4𝜋 ∫
𝑆
d𝑡 d𝜑 𝜕(√𝑔𝑔𝑖𝑗)
d𝑠 𝑏𝑖𝑗 = 1
2𝜋 ∫
1
0
d𝑡 ∫
𝜋
0
d𝜑𝑠2𝑏𝑡𝑡
= 1
2𝜋 ∫
1
0
d𝑡 ∫
𝜋
0
d𝜑𝑏𝑢𝑢 = ∫
1
0
d𝑡𝑏0 = 𝑏0
∫
∞
0
d𝑠𝑏0exp (−𝑠𝐿0) = 𝑏0
𝐿0
𝑏0 ∫
1
0
d𝑞
𝑞 𝑞𝐿0
𝐿0 = 𝛼′
4 𝑝2 + 𝑁,
4
𝛼′
𝑏0
𝑝2 + 𝑀2
𝑍Σ;𝑠 = ⟨𝜓ℓ| 𝑏0
𝐿0
|𝜓𝑟⟩.
pág. 96
𝜃̃ = {𝜃 if 𝜑 = 0
−𝜃 if 𝜑 = 𝜋
𝜑 → 𝜑 + 2𝜋, 𝜃 → −𝜃
𝜒𝑧
𝜃̃ = {𝜒𝑧̃
𝜃 if 𝜑 = 0
−𝜒𝑧̃
𝜃 if 𝜑 = 𝜋
𝜒𝑧˜
𝜃(𝑢, 𝜑 + 2𝜋) = −𝜒𝑧˜
𝜃(𝑢, 𝜑)
𝜒𝑧˜
𝜃 → 𝜒𝑧˜
𝜃 + 𝜕𝑧˜ 𝑦𝜃,
𝑏0
𝐿0
ΠGSO.
∫
∞
0
d𝑠Tr𝒪𝑏0exp (−𝑠𝐿0)ΠGSO
𝛽𝑟|q⟩ = 0, 𝑟 > −q − 3/2
𝛾𝑟|q⟩ = 0, 𝑟 ≥ q + 3/2
𝛽𝑟|q⟩ = 𝛾𝑟|q⟩ = 0, 𝑟 > 0
det𝜕̃𝛽𝛾 = Tr(−1)𝐹exp (−𝑠𝐿0;𝛽∗𝛾∗ ) = 𝑞𝑓 ∏
𝑟=1
2,3
2,…
(1 − 𝑞𝑟)
1
det𝜕̃𝛽𝛾
= 𝑞−𝑓 ∏
𝑟=1
2,3
2,…
1
1 − 𝑞𝑟 .
Tr𝒪𝑞𝐿0
𝜃̃ = {𝜃 if 𝜑 = 0
𝜃 if 𝜑 = 𝜋
𝐺0 = 1
2𝜋 ∫
2𝜋
0
d𝜑𝑆𝑧𝜃(0, 𝜑)
𝛽0 = 1
2𝜋 ∫
2𝜋
0
d𝜑𝛽𝑧𝜃(𝑢, 𝜑)
𝜒𝑧˜
𝜃(𝑢, 𝜑 + 2𝜋) = 𝜒𝑧˜
𝜃(𝑢, 𝜑).
𝜒𝑧̃
𝜃 = 𝜂,
𝜒𝑧˜
𝜃 → 𝜒𝑧˜
𝜃 + 𝜕𝑧˜ 𝑦𝜃
• Deligne-Mumford:
𝜃̃ = {𝜃 if 𝜑 = 0
−𝜃 if 𝜑 = 𝜋
𝜑 → 𝜑 + 2𝜋, 𝜃 → −𝜃
𝜒𝑧
𝜃̃ = {𝜒𝑧̃
𝜃 if 𝜑 = 0
−𝜒𝑧̃
𝜃 if 𝜑 = 𝜋
𝜒𝑧˜
𝜃(𝑢, 𝜑 + 2𝜋) = −𝜒𝑧˜
𝜃(𝑢, 𝜑)
𝜒𝑧˜
𝜃 → 𝜒𝑧˜
𝜃 + 𝜕𝑧˜ 𝑦𝜃,
𝑏0
𝐿0
ΠGSO.
∫
∞
0
d𝑠Tr𝒪𝑏0exp (−𝑠𝐿0)ΠGSO
𝛽𝑟|q⟩ = 0, 𝑟 > −q − 3/2
𝛾𝑟|q⟩ = 0, 𝑟 ≥ q + 3/2
𝛽𝑟|q⟩ = 𝛾𝑟|q⟩ = 0, 𝑟 > 0
det𝜕̃𝛽𝛾 = Tr(−1)𝐹exp (−𝑠𝐿0;𝛽∗𝛾∗ ) = 𝑞𝑓 ∏
𝑟=1
2,3
2,…
(1 − 𝑞𝑟)
1
det𝜕̃𝛽𝛾
= 𝑞−𝑓 ∏
𝑟=1
2,3
2,…
1
1 − 𝑞𝑟 .
Tr𝒪𝑞𝐿0
𝜃̃ = {𝜃 if 𝜑 = 0
𝜃 if 𝜑 = 𝜋
𝐺0 = 1
2𝜋 ∫
2𝜋
0
d𝜑𝑆𝑧𝜃(0, 𝜑)
𝛽0 = 1
2𝜋 ∫
2𝜋
0
d𝜑𝛽𝑧𝜃(𝑢, 𝜑)
𝜒𝑧˜
𝜃(𝑢, 𝜑 + 2𝜋) = 𝜒𝑧˜
𝜃(𝑢, 𝜑).
𝜒𝑧̃
𝜃 = 𝜂,
𝜒𝑧˜
𝜃 → 𝜒𝑧˜
𝜃 + 𝜕𝑧˜ 𝑦𝜃
• Deligne-Mumford:
pág. 97
exp (− 1
2𝜋 ∫
2𝜋
0
d𝜑 ∫
𝑠
0
d𝑢(𝜂𝑆𝑧𝜃 + d𝜂𝛽𝑧𝜃)) = exp (−𝑠(𝜂𝐺0 + d𝜂𝛽0))
𝐺0𝛿(𝛽0)
𝑏0𝛿(𝛽0)ΠGSO𝐺0
𝐿0
𝑏0𝛿(𝛽0)ΠGSO
𝐺0
ds2 = d𝑢2 + d𝜑2, 0 ≤ 𝑢 ≤ 𝑠, 0 ≤ 𝜑 ≤ 2𝜋.
𝐿0 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑇𝑧𝑧
𝐿̃ 0 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑇𝑧𝑧̃
𝑏0 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑏𝑧𝑧
𝑏̃0 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑏𝑧˜𝑧˜
𝜑̂ = 𝜑 − 𝛼𝑓(𝑢),
ds2 = d𝑢2 + d(𝜑̂ + 𝛼𝑓(𝑢))2.
Ψ𝛼 = 1
4𝜋 ∫
2𝜋
0
d𝜑 ∫
𝑠
0
d𝑢 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝛼 𝑏𝑖𝑗
𝜕𝑔𝑖𝑗
𝜕𝛼 = 𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖
𝑣 = 𝑓(𝑢) 𝜕
𝜕𝜑 ,
Ψ𝛼 = 1
4𝜋 ∫
2𝜋
0
d𝜑 ∫
𝑠
0
d𝑢√𝑔𝑏𝑖𝑗(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)
Ψ𝛼 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑏𝑢𝜑(𝜑, 𝑠) = 𝑏0 − 𝑏̃0
exp (−𝑠(𝐿0 + 𝐿̃ 0)) exp (−𝑖𝛼(𝐿0 − 𝐿̃ 0))
2Ψ𝑠Ψ𝛼 ∫
∞
0
d𝑠 ∫
2𝜋
0
d𝛼exp (−𝑠(𝐿0 + 𝐿̃ 0)) exp (−𝑖𝛼(𝐿0 − 𝐿̃ 0))
= 4𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃0 ∫
∞
0
d𝑠exp (−𝑠(𝐿0 + 𝐿̃ 0)) = 2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃0
𝐿0
exp (− 1
2𝜋 ∫
2𝜋
0
d𝜑 ∫
𝑠
0
d𝑢(𝜂𝑆𝑧𝜃 + d𝜂𝛽𝑧𝜃)) = exp (−𝑠(𝜂𝐺0 + d𝜂𝛽0))
𝐺0𝛿(𝛽0)
𝑏0𝛿(𝛽0)ΠGSO𝐺0
𝐿0
𝑏0𝛿(𝛽0)ΠGSO
𝐺0
ds2 = d𝑢2 + d𝜑2, 0 ≤ 𝑢 ≤ 𝑠, 0 ≤ 𝜑 ≤ 2𝜋.
𝐿0 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑇𝑧𝑧
𝐿̃ 0 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑇𝑧𝑧̃
𝑏0 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑏𝑧𝑧
𝑏̃0 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑏𝑧˜𝑧˜
𝜑̂ = 𝜑 − 𝛼𝑓(𝑢),
ds2 = d𝑢2 + d(𝜑̂ + 𝛼𝑓(𝑢))2.
Ψ𝛼 = 1
4𝜋 ∫
2𝜋
0
d𝜑 ∫
𝑠
0
d𝑢 𝜕(√𝑔𝑔𝑖𝑗)
𝜕𝛼 𝑏𝑖𝑗
𝜕𝑔𝑖𝑗
𝜕𝛼 = 𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖
𝑣 = 𝑓(𝑢) 𝜕
𝜕𝜑 ,
Ψ𝛼 = 1
4𝜋 ∫
2𝜋
0
d𝜑 ∫
𝑠
0
d𝑢√𝑔𝑏𝑖𝑗(𝐷𝑖𝑣𝑗 + 𝐷𝑗𝑣𝑖)
Ψ𝛼 = − 1
2𝜋 ∫
2𝜋
0
d𝜑𝑏𝑢𝜑(𝜑, 𝑠) = 𝑏0 − 𝑏̃0
exp (−𝑠(𝐿0 + 𝐿̃ 0)) exp (−𝑖𝛼(𝐿0 − 𝐿̃ 0))
2Ψ𝑠Ψ𝛼 ∫
∞
0
d𝑠 ∫
2𝜋
0
d𝛼exp (−𝑠(𝐿0 + 𝐿̃ 0)) exp (−𝑖𝛼(𝐿0 − 𝐿̃ 0))
= 4𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃0 ∫
∞
0
d𝑠exp (−𝑠(𝐿0 + 𝐿̃ 0)) = 2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃0
𝐿0
pág. 98
𝑏̃0𝑏0 ∫
|𝑞|≤1
d2𝑞
|𝑞|2 𝑞𝐿0 𝑞̃ 𝐿̃0
2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃ 0 ΠGSO
𝐿0
2𝜋𝑏̃0𝑏0𝛿(𝛽0)𝛿𝐿0−𝐿̃0 𝐺0ΠGSO
𝐿0
= 2𝜋𝑏̃0𝑏0𝛿(𝛽0)𝛿𝐿0−𝐿̃ 0 ΠGSO
𝐺0
2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃0 ΠGSOΠ̃GSO
𝐿0
2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃ 0 𝛿(𝛽0)𝐺0ΠGSOΠ̃GSO
𝐿0
2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃ 0 𝛿(𝛽̃0)𝐺̃0ΠGSOΠ̃GSO
𝐿0
2𝜋𝑏̃0𝑏0𝛿(𝛽0)𝐺0𝛿(𝛽̃0)𝐺̃0𝛿𝐿0−𝐿̃ 0 ΠGSOΠ̃GSO
𝐿0
⟨𝜙𝛼𝛽(𝑝)𝜙𝛼′𝛽′ (−𝑝)⟩ = (Γ ⋅ 𝑝)𝛼𝛼′ (Γ ⋅ 𝑝)𝛽𝛽′
𝑝2 + constant.
Métrica de Feynman – singularidad y propagador de Lorentz – BRST – OPE.
1
𝑝2 + 𝑚2 = ∫
∞
0
dtexp (−𝔱(𝑝2 + 𝑚2))
1
𝑝2 + 𝑚2 − 𝑖𝜖 = 𝑖 ∫
∞
0
d𝜏exp (−𝑖𝜏(𝑝2 + 𝑚2) − 𝜖𝜏)
𝒟nonsep ≅ ℳ̂𝑔−1,𝑛+2.
𝒟sep ≅ ℳ̂g1,n1+1 × ℳ̂g2,𝑛2+1,
g1 + g2 = g, n1 + n2 = n.
(𝑥 − 𝑎)(𝑦 − 𝑏) = 𝑞.
𝑏̃0𝑏0 ∫
|𝑞|≤1
d2𝑞
|𝑞|2 𝑞𝐿0 𝑞‾𝐿̃0
∫ d𝑑𝑝
(2𝜋)𝑑
1
𝑝2
𝑏̃0𝑏0 ∫
|𝑞|≤1
d2𝑞
|𝑞|2 𝑞𝐿0 𝑞̃ 𝐿̃0
2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃ 0 ΠGSO
𝐿0
2𝜋𝑏̃0𝑏0𝛿(𝛽0)𝛿𝐿0−𝐿̃0 𝐺0ΠGSO
𝐿0
= 2𝜋𝑏̃0𝑏0𝛿(𝛽0)𝛿𝐿0−𝐿̃ 0 ΠGSO
𝐺0
2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃0 ΠGSOΠ̃GSO
𝐿0
2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃ 0 𝛿(𝛽0)𝐺0ΠGSOΠ̃GSO
𝐿0
2𝜋𝑏̃0𝑏0𝛿𝐿0−𝐿̃ 0 𝛿(𝛽̃0)𝐺̃0ΠGSOΠ̃GSO
𝐿0
2𝜋𝑏̃0𝑏0𝛿(𝛽0)𝐺0𝛿(𝛽̃0)𝐺̃0𝛿𝐿0−𝐿̃ 0 ΠGSOΠ̃GSO
𝐿0
⟨𝜙𝛼𝛽(𝑝)𝜙𝛼′𝛽′ (−𝑝)⟩ = (Γ ⋅ 𝑝)𝛼𝛼′ (Γ ⋅ 𝑝)𝛽𝛽′
𝑝2 + constant.
Métrica de Feynman – singularidad y propagador de Lorentz – BRST – OPE.
1
𝑝2 + 𝑚2 = ∫
∞
0
dtexp (−𝔱(𝑝2 + 𝑚2))
1
𝑝2 + 𝑚2 − 𝑖𝜖 = 𝑖 ∫
∞
0
d𝜏exp (−𝑖𝜏(𝑝2 + 𝑚2) − 𝜖𝜏)
𝒟nonsep ≅ ℳ̂𝑔−1,𝑛+2.
𝒟sep ≅ ℳ̂g1,n1+1 × ℳ̂g2,𝑛2+1,
g1 + g2 = g, n1 + n2 = n.
(𝑥 − 𝑎)(𝑦 − 𝑏) = 𝑞.
𝑏̃0𝑏0 ∫
|𝑞|≤1
d2𝑞
|𝑞|2 𝑞𝐿0 𝑞‾𝐿̃0
∫ d𝑑𝑝
(2𝜋)𝑑
1
𝑝2
pág. 99
𝑥𝑦= −𝜀2
𝑦𝜃 = 𝜀𝜓
𝑥𝜓= −𝜀𝜃
𝜃𝜓 = 0.
𝑞NS = −𝜀2
𝐷𝜃
∗ = 𝜕
𝜕𝜃 + 𝜃𝑥 𝜕
𝜕𝑥 , 𝐷𝜓
∗ = 𝜕
𝜕𝜓 + 𝜓𝑦 𝜕
𝜕𝑦 .
𝑥𝑦 = 𝑞R
𝜃 = ±√−1𝜓.
𝒟nonsep ≅ 𝔐̂g,nNS+2,nR
𝒟sep ≅ 𝔐̂ g1,nNS,1+1,nR,1 × 𝔐̂g2,nNS,2+1,nR,2
𝚷: 𝒟sep → 𝔐̂g1,nNS,1,nR,1+1 × 𝔐̂g2,nNS,2,nR,2+1
𝐺0 = 𝜕
𝜕𝜓 − 𝜓𝑦 𝜕
𝜕𝑦
𝒟sep ≅ 𝔐̂g1,nNS,1+1,nR,1 × 𝔐̂g2,nNS,2+1,nR,2 ,
R = 𝑏0Π0
{𝑄𝐵, R} = {𝑄𝐵, 𝑏0}Π0 = 𝐿0Π0 = 0
⊕𝑛∈ℤ ℋ𝑛 ⊗ ℋ𝑛+1
∗
R′ ∈⊕𝑛∈ℤ ℋ𝑛 ⊗ ℋ2−𝑛
R′ ∈ (ℋ ⊗ ℋ)2
R′ = ∑
𝑖
𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑄𝐵𝒳
𝒳 = ∑
𝑗
𝒮𝑗 ⊗ 𝒯𝑗, 𝒮𝑗, 𝒯𝑗 ∈ ℋ
d𝐹Ω + 𝐹𝑄𝐵Ω = 0.
d𝐹𝒱1⋯𝒱n𝒳 + 𝐹𝒱1⋯𝒱n𝑄𝐵𝒳 = 0
∫
ℳg−1,n+2
𝐹𝒱1⋯𝒱n𝑄𝐵𝒳 = − ∫
ℳg−1,n+2
d𝐹𝒱1⋯𝒱n𝒳
R̃′ = ∑
𝑖
(𝑐𝜕𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑐𝑈𝑖 ⊗ 𝑐𝜕𝑐𝑈𝑖) + 𝑄𝐵𝒴
𝑥𝑦= −𝜀2
𝑦𝜃 = 𝜀𝜓
𝑥𝜓= −𝜀𝜃
𝜃𝜓 = 0.
𝑞NS = −𝜀2
𝐷𝜃
∗ = 𝜕
𝜕𝜃 + 𝜃𝑥 𝜕
𝜕𝑥 , 𝐷𝜓
∗ = 𝜕
𝜕𝜓 + 𝜓𝑦 𝜕
𝜕𝑦 .
𝑥𝑦 = 𝑞R
𝜃 = ±√−1𝜓.
𝒟nonsep ≅ 𝔐̂g,nNS+2,nR
𝒟sep ≅ 𝔐̂ g1,nNS,1+1,nR,1 × 𝔐̂g2,nNS,2+1,nR,2
𝚷: 𝒟sep → 𝔐̂g1,nNS,1,nR,1+1 × 𝔐̂g2,nNS,2,nR,2+1
𝐺0 = 𝜕
𝜕𝜓 − 𝜓𝑦 𝜕
𝜕𝑦
𝒟sep ≅ 𝔐̂g1,nNS,1+1,nR,1 × 𝔐̂g2,nNS,2+1,nR,2 ,
R = 𝑏0Π0
{𝑄𝐵, R} = {𝑄𝐵, 𝑏0}Π0 = 𝐿0Π0 = 0
⊕𝑛∈ℤ ℋ𝑛 ⊗ ℋ𝑛+1
∗
R′ ∈⊕𝑛∈ℤ ℋ𝑛 ⊗ ℋ2−𝑛
R′ ∈ (ℋ ⊗ ℋ)2
R′ = ∑
𝑖
𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑄𝐵𝒳
𝒳 = ∑
𝑗
𝒮𝑗 ⊗ 𝒯𝑗, 𝒮𝑗, 𝒯𝑗 ∈ ℋ
d𝐹Ω + 𝐹𝑄𝐵Ω = 0.
d𝐹𝒱1⋯𝒱n𝒳 + 𝐹𝒱1⋯𝒱n𝑄𝐵𝒳 = 0
∫
ℳg−1,n+2
𝐹𝒱1⋯𝒱n𝑄𝐵𝒳 = − ∫
ℳg−1,n+2
d𝐹𝒱1⋯𝒱n𝒳
R̃′ = ∑
𝑖
(𝑐𝜕𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑐𝑈𝑖 ⊗ 𝑐𝜕𝑐𝑈𝑖) + 𝑄𝐵𝒴
pág. 100
(𝑐𝜕𝑐𝜕2𝑐 ⊗ 1 + 1 ⊗ 𝑐𝜕𝑐𝜕2𝑐)
Cohomología de Ramond.
R′ = ∑
𝑖
𝑐𝛿(𝛾)𝑈𝑖 ⊗ 𝑐𝛿(𝛾)𝑈𝑖 + 𝑄𝐵𝒳
R = 𝑏0𝛿(𝛽0)𝐺0Π0
R′ ∈ (ℋ ⊗ ℋ)1;−1/2⊗−1/2
R′ = ∑
𝑖
𝑐Θ−1/2Φ𝑖 ⊗ 𝑐Θ−1/2Φ𝑖
′ + 𝑄𝐵𝒳
∑
𝛼𝛽
(𝑝 ⋅ Γ)𝛼𝛽𝑐Θ−1/2Σ𝛼(𝑝) ⊗ 𝑐Θ−1/2Σ𝛽(−𝑝)
R′ = ∑
𝑖
𝑐̃ 𝑐𝑈𝑖 ⊗ 𝑐̃ 𝑐𝑈𝑖 + {𝑄𝐵, 𝒳}
R̃′ = ∑
𝑖
(𝑐̃ 𝑐(𝜕̃ 𝑐̃ + 𝜕𝑐)𝑈𝑖 ⊗ 𝑐̃ 𝑐𝑈𝑖 + 𝑐̃ 𝑐𝑈𝑖 ⊗ 𝑐̃ 𝑐(𝜕̃ 𝑐̃ + 𝜕𝑐)𝑈𝑖) + {𝑄𝐵, 𝒳}
∑
𝐼
(𝑐̃ 𝜕̃ 𝑐̃ 𝜕̃ 2𝑐̃ ⋅ 𝑐𝜕𝑋𝐼 ⊗ 𝑐𝜕𝑋𝐼 + 𝑐𝜕𝑋𝐼 ⊗ 𝑐̃ 𝜕̃ 𝑐̃ 𝜕̃ 2𝑐̃ ⋅ 𝑐𝜕𝑋𝐼) + 𝑧 ↔ 𝑧̃
(𝑥 − 𝑎)(𝑦 − 𝑏) = 𝑞
(𝑥, 𝑎) → (𝜆𝑥, 𝜆𝑎), (𝑦, 𝑏) → (𝜆̃ 𝑦, 𝜆̃ 𝑏), 𝑞 → 𝜆𝜆̃ 𝑞.
Ω ∼ d𝑎 d𝑏 d𝑞
𝑞2
Ω ∼ 𝑐(𝑎) ⊗ 𝑐(𝑏) d𝑞
𝑞2
𝒱ℓ(𝑎) ⊗ 𝒱𝑟(𝑏)d𝑞𝑞𝐿0−1
(𝑥 − 𝑎 − 𝛼𝜃)(𝑦 − 𝑏 − 𝛽𝜓) = −𝜀2
(𝑦 − 𝑏 − 𝛽𝜓)(𝜃 − 𝛼)= 𝜀(𝜓 − 𝛽)
(𝑥 − 𝑎 − 𝛼𝜃)(𝜓 − 𝛽)= −𝜀(𝜃 − 𝛼)
(𝜃 − 𝛼)(𝜓 − 𝛽) = 0
(𝑥, 𝑎, 𝛼)→ (𝜆𝑥, 𝜆𝑎, 𝜆1/2𝛼)
(𝑦, 𝑏, 𝛽)→ (𝜆̃ 𝑦, 𝜆̃ 𝑏, 𝜆̃1/2𝛽)
𝜀 → (𝜆𝜆̃ )1/2𝜀
Ω ∼ [d𝑎 ∣ d𝛼] ⊗ [d𝑏 ∣ d𝛽] d𝜀
𝜀2
(𝑐𝜕𝑐𝜕2𝑐 ⊗ 1 + 1 ⊗ 𝑐𝜕𝑐𝜕2𝑐)
Cohomología de Ramond.
R′ = ∑
𝑖
𝑐𝛿(𝛾)𝑈𝑖 ⊗ 𝑐𝛿(𝛾)𝑈𝑖 + 𝑄𝐵𝒳
R = 𝑏0𝛿(𝛽0)𝐺0Π0
R′ ∈ (ℋ ⊗ ℋ)1;−1/2⊗−1/2
R′ = ∑
𝑖
𝑐Θ−1/2Φ𝑖 ⊗ 𝑐Θ−1/2Φ𝑖
′ + 𝑄𝐵𝒳
∑
𝛼𝛽
(𝑝 ⋅ Γ)𝛼𝛽𝑐Θ−1/2Σ𝛼(𝑝) ⊗ 𝑐Θ−1/2Σ𝛽(−𝑝)
R′ = ∑
𝑖
𝑐̃ 𝑐𝑈𝑖 ⊗ 𝑐̃ 𝑐𝑈𝑖 + {𝑄𝐵, 𝒳}
R̃′ = ∑
𝑖
(𝑐̃ 𝑐(𝜕̃ 𝑐̃ + 𝜕𝑐)𝑈𝑖 ⊗ 𝑐̃ 𝑐𝑈𝑖 + 𝑐̃ 𝑐𝑈𝑖 ⊗ 𝑐̃ 𝑐(𝜕̃ 𝑐̃ + 𝜕𝑐)𝑈𝑖) + {𝑄𝐵, 𝒳}
∑
𝐼
(𝑐̃ 𝜕̃ 𝑐̃ 𝜕̃ 2𝑐̃ ⋅ 𝑐𝜕𝑋𝐼 ⊗ 𝑐𝜕𝑋𝐼 + 𝑐𝜕𝑋𝐼 ⊗ 𝑐̃ 𝜕̃ 𝑐̃ 𝜕̃ 2𝑐̃ ⋅ 𝑐𝜕𝑋𝐼) + 𝑧 ↔ 𝑧̃
(𝑥 − 𝑎)(𝑦 − 𝑏) = 𝑞
(𝑥, 𝑎) → (𝜆𝑥, 𝜆𝑎), (𝑦, 𝑏) → (𝜆̃ 𝑦, 𝜆̃ 𝑏), 𝑞 → 𝜆𝜆̃ 𝑞.
Ω ∼ d𝑎 d𝑏 d𝑞
𝑞2
Ω ∼ 𝑐(𝑎) ⊗ 𝑐(𝑏) d𝑞
𝑞2
𝒱ℓ(𝑎) ⊗ 𝒱𝑟(𝑏)d𝑞𝑞𝐿0−1
(𝑥 − 𝑎 − 𝛼𝜃)(𝑦 − 𝑏 − 𝛽𝜓) = −𝜀2
(𝑦 − 𝑏 − 𝛽𝜓)(𝜃 − 𝛼)= 𝜀(𝜓 − 𝛽)
(𝑥 − 𝑎 − 𝛼𝜃)(𝜓 − 𝛽)= −𝜀(𝜃 − 𝛼)
(𝜃 − 𝛼)(𝜓 − 𝛽) = 0
(𝑥, 𝑎, 𝛼)→ (𝜆𝑥, 𝜆𝑎, 𝜆1/2𝛼)
(𝑦, 𝑏, 𝛽)→ (𝜆̃ 𝑦, 𝜆̃ 𝑏, 𝜆̃1/2𝛽)
𝜀 → (𝜆𝜆̃ )1/2𝜀
Ω ∼ [d𝑎 ∣ d𝛼] ⊗ [d𝑏 ∣ d𝛽] d𝜀
𝜀2
pág. 101
Ω ∼ 𝑐𝛿(𝛾) ⊗ 𝑐𝛿(𝛾) d𝜀
𝜀2
𝒱ℓ(𝑎 ∣ 𝛼) ⊗ 𝒱𝑟(𝑏 ∣ 𝛽)d𝜀𝜀2𝐿0−1
d𝜀𝜀2𝐿0−1 ∼ d𝑞NS𝑞NS
𝐿0−1
(𝑥̂ − 𝑎̂ )(𝑦̂ − 𝑏̂ ) = 𝑞̂
𝑞̂ = 𝑞 𝜕𝑎̂
𝜕𝑎
𝜕𝑏̂
𝜕𝑏
Compactificación Deligne-Mumford.
𝑞̃ = 𝑞NS
𝑞̃ = 𝑞NS (1 + 𝒪(𝜂𝑖𝜂𝑗)) ,
𝑞̃ = 𝑞NS + 𝒪(𝜂𝑖𝜂𝑗).
Ξ = [d𝑞̃ ; d𝑞NS ∣ d𝜂1, d𝜂2]𝑞̃ −1.
∫ Ξ → ∫ Ξ + ∫ [d𝑞̃ ; d𝑞NS ∣ d𝜂1, d𝜂2]𝜂1𝜂2
𝜕
𝜕𝑞NS
1
𝑞̃ .
𝑞̃ → 𝑒𝜑̃ 𝑞̃ , 𝑞NS → 𝑒𝜑𝑞NS
𝑞̃ = 𝑢̃1 − 𝑢̃ 2
𝑞NS = 𝑢1 − 𝑢2 − 𝜁1𝜁2.
𝑞NS(1 + 𝒪(𝜂2)) = 𝑡1 + 𝑖𝑡2
𝑞̃NS(1 + 𝒪(𝜂̃ 2)) = 𝑡1 − 𝑖𝑡2
𝑡1 = 𝑞NS(1 + 𝒪(𝜂2))
Anomalías BRST para las partículas sin masa. Supersimetrías de gauge.
∫
Γ
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = − ∫
𝜕Γ
𝐹𝒲1,𝒱2,…,𝒱n
𝑏0 ∫
∞
0
d𝑠exp (−𝑠𝐿0)
2𝜋(𝑏0 − 𝑏̃0)𝛿𝐿0−𝐿̃0 exp (−𝑠(𝐿0 + 𝐿̃ 0)) .
∑
𝑖
(𝑐𝜕𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑐𝑈𝑖 ⊗ 𝑐𝜕𝑐𝑈𝑖)
𝒪 = ∑
𝑖
𝑎𝑖𝒴𝑖
Ω ∼ 𝑐𝛿(𝛾) ⊗ 𝑐𝛿(𝛾) d𝜀
𝜀2
𝒱ℓ(𝑎 ∣ 𝛼) ⊗ 𝒱𝑟(𝑏 ∣ 𝛽)d𝜀𝜀2𝐿0−1
d𝜀𝜀2𝐿0−1 ∼ d𝑞NS𝑞NS
𝐿0−1
(𝑥̂ − 𝑎̂ )(𝑦̂ − 𝑏̂ ) = 𝑞̂
𝑞̂ = 𝑞 𝜕𝑎̂
𝜕𝑎
𝜕𝑏̂
𝜕𝑏
Compactificación Deligne-Mumford.
𝑞̃ = 𝑞NS
𝑞̃ = 𝑞NS (1 + 𝒪(𝜂𝑖𝜂𝑗)) ,
𝑞̃ = 𝑞NS + 𝒪(𝜂𝑖𝜂𝑗).
Ξ = [d𝑞̃ ; d𝑞NS ∣ d𝜂1, d𝜂2]𝑞̃ −1.
∫ Ξ → ∫ Ξ + ∫ [d𝑞̃ ; d𝑞NS ∣ d𝜂1, d𝜂2]𝜂1𝜂2
𝜕
𝜕𝑞NS
1
𝑞̃ .
𝑞̃ → 𝑒𝜑̃ 𝑞̃ , 𝑞NS → 𝑒𝜑𝑞NS
𝑞̃ = 𝑢̃1 − 𝑢̃ 2
𝑞NS = 𝑢1 − 𝑢2 − 𝜁1𝜁2.
𝑞NS(1 + 𝒪(𝜂2)) = 𝑡1 + 𝑖𝑡2
𝑞̃NS(1 + 𝒪(𝜂̃ 2)) = 𝑡1 − 𝑖𝑡2
𝑡1 = 𝑞NS(1 + 𝒪(𝜂2))
Anomalías BRST para las partículas sin masa. Supersimetrías de gauge.
∫
Γ
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = − ∫
𝜕Γ
𝐹𝒲1,𝒱2,…,𝒱n
𝑏0 ∫
∞
0
d𝑠exp (−𝑠𝐿0)
2𝜋(𝑏0 − 𝑏̃0)𝛿𝐿0−𝐿̃0 exp (−𝑠(𝐿0 + 𝐿̃ 0)) .
∑
𝑖
(𝑐𝜕𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑐𝑈𝑖 ⊗ 𝑐𝜕𝑐𝑈𝑖)
𝒪 = ∑
𝑖
𝑎𝑖𝒴𝑖
pág. 102
∑
𝑖
𝑎𝑖⟨𝒴𝑖𝒱n⟩g𝑟
.
𝒬𝐵(𝒲1) = {𝑄𝐵, 𝒲1} − 𝑔st
2 gℓ 𝒪(𝒲1),
𝒬𝐵(𝒱n) = {𝑄𝐵, 𝒱n} + ∑
𝑖
𝑦𝑖𝑐𝜕𝑐𝑈𝑖,
Rompimiento de simetría de gauge - BRST.
𝜎 → 𝜎 − 𝑔st
2 gℓ 𝜆
∫ (𝐹𝐼𝐽𝐹𝐼𝐽 + (𝜕𝐼𝜎 + 𝐴𝐼)2)
𝜆 → 𝜆 − 𝑔st
2𝑔ℓ 𝜁.
𝒲1 = 𝜀𝐼 ⋅ (𝑐𝑧𝜕𝑧𝑋𝐼 − 𝑐̃ 𝑧̃ 𝜕𝑧˜ 𝑋𝐼)exp (𝑖𝑝 ⋅ 𝑋), 𝑝2 = 𝜀 ⋅ 𝑝 = 0,
𝑊 = 𝜀𝐼 d𝑋𝐼exp (𝑖𝑝 ⋅ 𝑋), 𝑝2 = 𝜀 ⋅ 𝑝 = 0.
𝒪(𝒲) = 𝒲open , 𝒲open = 𝑐𝑊open
∫ d𝐷𝑥 ( 1
𝑔st
2 𝐻𝐼𝐽𝐾
2 + 1
𝑔st
(𝜕𝐼𝐴𝐽 − 𝜕𝐽𝐴𝐼 + 𝐵𝐼𝐽)2) , 𝐻 = d𝐵
∫
ℝ4
d4𝑥𝑒−2𝜙(( d𝐴)2 + (d𝐵)2)
∫
ℝ4
𝐵 ∧ d𝐴
𝑥𝑦 = 𝑞,
𝒟 = ℳ̂ℓ × ℳ̂𝑟
𝑞 → 𝑒𝑓ℓ+𝑓𝑟 𝑞
𝒩 = ℒℓ ⊗ ℒ𝑟
𝒜𝑔(𝑝1, 𝜁1; … ; 𝑝𝑛, 𝜁𝑛) = ∫
ℳ̂
𝐹𝑝1,𝜁1;…;𝑝𝑛,𝜁𝑛 (𝑔 ∣ 𝛿𝑔)
𝒜𝑔, sing = ∑
𝑠
𝛼=1
∫ d2𝑞
𝑞‾𝑞 ∫
ℳ̂ ℓ
𝒢ℓ,𝛼 ∫
ℳ̂𝑟
𝒢𝑟,𝛼.
∫
ℳ̂𝑟
𝒢𝑟,𝛼 = 0, 𝛼 = 1, … , 𝑠.
∑
𝑖
𝑎𝑖⟨𝒴𝑖𝒱n⟩g𝑟
.
𝒬𝐵(𝒲1) = {𝑄𝐵, 𝒲1} − 𝑔st
2 gℓ 𝒪(𝒲1),
𝒬𝐵(𝒱n) = {𝑄𝐵, 𝒱n} + ∑
𝑖
𝑦𝑖𝑐𝜕𝑐𝑈𝑖,
Rompimiento de simetría de gauge - BRST.
𝜎 → 𝜎 − 𝑔st
2 gℓ 𝜆
∫ (𝐹𝐼𝐽𝐹𝐼𝐽 + (𝜕𝐼𝜎 + 𝐴𝐼)2)
𝜆 → 𝜆 − 𝑔st
2𝑔ℓ 𝜁.
𝒲1 = 𝜀𝐼 ⋅ (𝑐𝑧𝜕𝑧𝑋𝐼 − 𝑐̃ 𝑧̃ 𝜕𝑧˜ 𝑋𝐼)exp (𝑖𝑝 ⋅ 𝑋), 𝑝2 = 𝜀 ⋅ 𝑝 = 0,
𝑊 = 𝜀𝐼 d𝑋𝐼exp (𝑖𝑝 ⋅ 𝑋), 𝑝2 = 𝜀 ⋅ 𝑝 = 0.
𝒪(𝒲) = 𝒲open , 𝒲open = 𝑐𝑊open
∫ d𝐷𝑥 ( 1
𝑔st
2 𝐻𝐼𝐽𝐾
2 + 1
𝑔st
(𝜕𝐼𝐴𝐽 − 𝜕𝐽𝐴𝐼 + 𝐵𝐼𝐽)2) , 𝐻 = d𝐵
∫
ℝ4
d4𝑥𝑒−2𝜙(( d𝐴)2 + (d𝐵)2)
∫
ℝ4
𝐵 ∧ d𝐴
𝑥𝑦 = 𝑞,
𝒟 = ℳ̂ℓ × ℳ̂𝑟
𝑞 → 𝑒𝑓ℓ+𝑓𝑟 𝑞
𝒩 = ℒℓ ⊗ ℒ𝑟
𝒜𝑔(𝑝1, 𝜁1; … ; 𝑝𝑛, 𝜁𝑛) = ∫
ℳ̂
𝐹𝑝1,𝜁1;…;𝑝𝑛,𝜁𝑛 (𝑔 ∣ 𝛿𝑔)
𝒜𝑔, sing = ∑
𝑠
𝛼=1
∫ d2𝑞
𝑞‾𝑞 ∫
ℳ̂ ℓ
𝒢ℓ,𝛼 ∫
ℳ̂𝑟
𝒢𝑟,𝛼.
∫
ℳ̂𝑟
𝒢𝑟,𝛼 = 0, 𝛼 = 1, … , 𝑠.
pág. 103
∫
𝜖<|𝑞|<Λ
d2𝑞
𝑞‾𝑞
𝒜g,𝜖 = ∫
ℳ̂𝜖
𝐹(𝑔 ∣ 𝛿𝑔),
𝒜g → 𝒜g − 4𝜋 ∑
𝛼
∫
ℳ̂ ℓ×ℳ̂𝑟
ℎ𝒢ℓ,𝛼𝒢𝑟,𝛼
Δ𝑔𝑟 𝜙𝛼 = − 1
4𝜋 ∫
ℳ̂𝑟
ℎ𝑟𝒢𝑟,𝛼
𝒜g → 𝒜g + ∑
𝛼
Δg𝑟 𝜙𝛼 ∫
ℳ̂ ℓ
𝒢ℓ,𝛼
𝒜g → 𝒜g + ∑
𝛼
Δg𝑟 𝜙𝛼
𝜕
𝜕𝜙𝛼
𝒜gℓ
𝒜g → 𝒜g + ∑
gℓ+g𝑟=g
∑
𝛼
Δg𝑟 𝜙𝛼
𝜕
𝜕𝜙𝛼
𝒜gℓ
𝒜 → 𝒜 + ∑
𝛼
Δ𝜙𝛼
𝜕
𝜕𝜙𝛼
𝒜.
𝒜 → exp (∑
𝛼
Δ𝜙𝛼
𝜕
𝜕𝜙𝛼
) 𝒜.
∫
𝜖<|𝑞|<Λ
d2𝑞
𝑞‾𝑞
𝒜g,𝜖 = ∫
ℳ̂𝜖
𝐹(𝑔 ∣ 𝛿𝑔),
𝒜g → 𝒜g − 4𝜋 ∑
𝛼
∫
ℳ̂ ℓ×ℳ̂𝑟
ℎ𝒢ℓ,𝛼𝒢𝑟,𝛼
Δ𝑔𝑟 𝜙𝛼 = − 1
4𝜋 ∫
ℳ̂𝑟
ℎ𝑟𝒢𝑟,𝛼
𝒜g → 𝒜g + ∑
𝛼
Δg𝑟 𝜙𝛼 ∫
ℳ̂ ℓ
𝒢ℓ,𝛼
𝒜g → 𝒜g + ∑
𝛼
Δg𝑟 𝜙𝛼
𝜕
𝜕𝜙𝛼
𝒜gℓ
𝒜g → 𝒜g + ∑
gℓ+g𝑟=g
∑
𝛼
Δg𝑟 𝜙𝛼
𝜕
𝜕𝜙𝛼
𝒜gℓ
𝒜 → 𝒜 + ∑
𝛼
Δ𝜙𝛼
𝜕
𝜕𝜙𝛼
𝒜.
𝒜 → exp (∑
𝛼
Δ𝜙𝛼
𝜕
𝜕𝜙𝛼
) 𝒜.
pág. 104
𝒦 = ∑
𝛼
Δ𝜙𝛼
𝜕
𝜕𝜙𝛼
Figura 1. Superespacios producidos por supermembranas.
𝐹0 = ∑
𝛼
d2𝑞
𝑞‾𝑞 ∧ 𝒢ℓ,𝛼 ∧ 𝒢𝑟,𝛼
𝒜𝑔,𝜖 = ∫
ℳ̂𝜖
𝐹 − ∫
𝜕ℳ̂𝜖
Λ
𝜒𝑟,𝛼 = d2𝑞
𝑞‾𝑞 𝒢𝑟,𝛼
𝜒𝑟,𝛼 = d𝜆𝑟,𝛼
Λ = ∑
𝛼
𝒢ℓ,𝛼 ∧ 𝜆𝑟,𝛼
Δ𝜙𝛼 = − ∫
𝒳𝜖
Δ𝜆𝑟,𝛼
Λ → Λ + ∑
𝛼
𝒢ℓ,𝛼 ∧ Δ𝜆𝑟,𝛼
𝒜g → 𝒜g + ∑
𝛼
Δ𝜙𝛼 ∫
ℳ̂ ℓ
𝒢ℓ,𝛼.
𝜆𝑟,𝛼
(0) = d2𝑞
𝑞‾𝑞 𝛽𝑟,𝛼.
𝒦 = ∑
𝛼
Δ𝜙𝛼
𝜕
𝜕𝜙𝛼
Figura 1. Superespacios producidos por supermembranas.
𝐹0 = ∑
𝛼
d2𝑞
𝑞‾𝑞 ∧ 𝒢ℓ,𝛼 ∧ 𝒢𝑟,𝛼
𝒜𝑔,𝜖 = ∫
ℳ̂𝜖
𝐹 − ∫
𝜕ℳ̂𝜖
Λ
𝜒𝑟,𝛼 = d2𝑞
𝑞‾𝑞 𝒢𝑟,𝛼
𝜒𝑟,𝛼 = d𝜆𝑟,𝛼
Λ = ∑
𝛼
𝒢ℓ,𝛼 ∧ 𝜆𝑟,𝛼
Δ𝜙𝛼 = − ∫
𝒳𝜖
Δ𝜆𝑟,𝛼
Λ → Λ + ∑
𝛼
𝒢ℓ,𝛼 ∧ Δ𝜆𝑟,𝛼
𝒜g → 𝒜g + ∑
𝛼
Δ𝜙𝛼 ∫
ℳ̂ ℓ
𝒢ℓ,𝛼.
𝜆𝑟,𝛼
(0) = d2𝑞
𝑞‾𝑞 𝛽𝑟,𝛼.
pág. 105
𝜆𝑟,𝛼 = d2𝑞
𝑞‾𝑞 𝛽𝑟,𝛼 + d𝑞
𝑞 𝛾𝑟,𝛼 + d𝑞‾
𝑞‾ 𝛾̃𝑟,𝛼,
Renormalización de la función de onda.
𝒜g,sing = ∑
𝑠
𝛼=1
∫ d2𝑞
𝑞‾𝑞 ∫
ℳ̂ ℓ
𝒢ℓ,𝛼 ∫
ℳ̂𝑟
𝒢𝑟,𝛼
∫
ℳ̂ ℓ
𝒢ℓ,𝛼 = 0
𝑥𝑦 = 𝑞
ℬ ≅ ℳ̂ℓ × ℳ̂𝑟
∑
𝑖
(𝑐𝜕𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑐𝑈𝑖 ⊗ 𝑐𝜕𝑐𝑈𝑖)
𝒪(𝒲1) = ∑
𝑖
𝑎𝑖𝒱𝑖
∫
ℳ̂g,n
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = − ∫
𝜕ℳ̂g,n
𝐹𝒲1,𝒱2,…,𝒱n
d𝑞
𝑞 ∑
𝑖
𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖
d𝑞
𝑞 ∑ 𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + ∑
𝑖
(𝑐𝜕𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑐𝑈𝑖 ⊗ 𝑐𝜕𝑐𝑈𝑖)
𝑣(𝑥)𝜕𝑥 + 𝜕𝑣(𝑥)
𝜕𝑥 |𝑥=0
𝑞𝜕𝑞, 𝑣(0) = 0
𝒳𝑖 = d𝑞
𝑞 𝑐𝑈𝑖 + 𝑐𝜕𝑐𝑈𝑖
𝑎𝑖 = ∫
ℳℓ
∗
𝐹𝒲1,…,𝒳𝑖
d(|𝑞|2)
|𝑞|2 ∑
𝑖
𝑐̃ 𝑐𝑈𝑖 ⊗ 𝑐̃ 𝑐𝑈𝑖 + ∑
𝑖
(𝑐̃ 𝑐(𝜕̃ 𝑐̃ + 𝜕𝑐)𝑈𝑖 ⊗ 𝑐̃ 𝑐𝑈𝑖 + 𝑐̃ 𝑐𝑈𝑖 ⊗ 𝑐̃ 𝑐(𝜕̃ 𝑐̃ + 𝜕𝑐)𝑈𝑖)
Supersimetrías de espacio – tiempo.
𝑖
𝑝𝑖
2 − 𝑚𝑖
2
−𝑖𝑒𝑖𝜀 ⋅ (𝑝𝑖 + 𝑝𝑖
′), 𝑝𝑖
′ = 𝑝𝑖 + 𝑘
𝜆𝑟,𝛼 = d2𝑞
𝑞‾𝑞 𝛽𝑟,𝛼 + d𝑞
𝑞 𝛾𝑟,𝛼 + d𝑞‾
𝑞‾ 𝛾̃𝑟,𝛼,
Renormalización de la función de onda.
𝒜g,sing = ∑
𝑠
𝛼=1
∫ d2𝑞
𝑞‾𝑞 ∫
ℳ̂ ℓ
𝒢ℓ,𝛼 ∫
ℳ̂𝑟
𝒢𝑟,𝛼
∫
ℳ̂ ℓ
𝒢ℓ,𝛼 = 0
𝑥𝑦 = 𝑞
ℬ ≅ ℳ̂ℓ × ℳ̂𝑟
∑
𝑖
(𝑐𝜕𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑐𝑈𝑖 ⊗ 𝑐𝜕𝑐𝑈𝑖)
𝒪(𝒲1) = ∑
𝑖
𝑎𝑖𝒱𝑖
∫
ℳ̂g,n
𝐹{𝑄𝐵,𝒲1},𝒱2,…,𝒱n = − ∫
𝜕ℳ̂g,n
𝐹𝒲1,𝒱2,…,𝒱n
d𝑞
𝑞 ∑
𝑖
𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖
d𝑞
𝑞 ∑ 𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + ∑
𝑖
(𝑐𝜕𝑐𝑈𝑖 ⊗ 𝑐𝑈𝑖 + 𝑐𝑈𝑖 ⊗ 𝑐𝜕𝑐𝑈𝑖)
𝑣(𝑥)𝜕𝑥 + 𝜕𝑣(𝑥)
𝜕𝑥 |𝑥=0
𝑞𝜕𝑞, 𝑣(0) = 0
𝒳𝑖 = d𝑞
𝑞 𝑐𝑈𝑖 + 𝑐𝜕𝑐𝑈𝑖
𝑎𝑖 = ∫
ℳℓ
∗
𝐹𝒲1,…,𝒳𝑖
d(|𝑞|2)
|𝑞|2 ∑
𝑖
𝑐̃ 𝑐𝑈𝑖 ⊗ 𝑐̃ 𝑐𝑈𝑖 + ∑
𝑖
(𝑐̃ 𝑐(𝜕̃ 𝑐̃ + 𝜕𝑐)𝑈𝑖 ⊗ 𝑐̃ 𝑐𝑈𝑖 + 𝑐̃ 𝑐𝑈𝑖 ⊗ 𝑐̃ 𝑐(𝜕̃ 𝑐̃ + 𝜕𝑐)𝑈𝑖)
Supersimetrías de espacio – tiempo.
𝑖
𝑝𝑖
2 − 𝑚𝑖
2
−𝑖𝑒𝑖𝜀 ⋅ (𝑝𝑖 + 𝑝𝑖
′), 𝑝𝑖
′ = 𝑝𝑖 + 𝑘
pág. 106
−𝑖𝑒𝑖𝜀 ⋅ (𝑝𝑖 + 𝑝𝑖
′) 𝑖
(𝑝𝑖
′)2 − 𝑚𝑖
2 = 𝑒𝑖𝜀 ⋅ 𝑝𝑖
𝑘 ⋅ 𝑝𝑖
,
𝒜 ∼ ∑
𝑛
𝑖=1
𝑒𝑖𝜀 ⋅ 𝑝𝑖
𝑘 ⋅ 𝑝𝑖
𝒜′.
∑
𝑖
𝑒𝑖 = 0,
−𝑖𝑡𝑎,𝑖𝜀 ⋅ (𝑝 + 𝑝′)
𝒜 ∼ ∑
n
𝑖=1
𝑡𝑎,𝑖𝜀 ⋅ 𝑝𝑖
𝑘 ⋅ 𝑝𝑖
𝒜′
∑
𝑖
𝑡𝑎,𝑖𝒜′ = 0.
∑
𝑖≠𝑖′
𝑡𝑎,𝑖𝜀𝑎 ⋅ 𝑝𝑖
𝑘𝑎 ⋅ 𝑝𝑖
𝑡𝑏,𝑖′ 𝜀𝑏 ⋅ 𝑝𝑖′
𝑘𝑏 ⋅ 𝑝𝑖′
𝒜′
∑
𝑖≠𝑖′
𝑡𝑎,𝑖
𝑡𝑏,𝑖′ 𝜀𝑏 ⋅ 𝑝𝑖′
𝑘𝑏 ⋅ 𝑝𝑖′
𝒜′ = − ∑
𝑖
𝑡𝑏,𝑖𝜀𝑏 ⋅ 𝑝𝑖
𝑘𝑏 ⋅ 𝑝𝑖
𝑡𝑎,𝑖𝒜′
(∑
𝑖
𝑡𝑎,𝑖𝜀𝑎 ⋅ 𝑝𝑖
𝑘𝑎 ⋅ 𝑝𝑖
𝑡𝑏,𝑖𝜀𝑏 ⋅ 𝑝𝑖
(𝑘𝑎 + 𝑘𝑏) ⋅ 𝑝𝑖
+ 𝑎 ↔ 𝑏) 𝒜′.
(∑
𝑖
𝑡𝑏,𝑖𝜀𝑏 ⋅ 𝑝𝑖
𝑘𝑏 ⋅ 𝑝𝑖
𝑡𝑎,𝑖 + ∑
𝑖
[𝑡𝑎,𝑖, 𝑡𝑏,𝑖] 𝜀𝑏 ⋅ 𝑝𝑖
(𝑘𝑎 + 𝑘𝑏) ⋅ 𝑝𝑖
) 𝒜′.
Gravedad y supergravedad. Métrica de Ward.
∑
n
𝑖=1
𝑄𝛼,𝑖𝒜′ = 0
𝒱 = {𝑄𝐵, 𝒲} = 𝑐̃ 𝑐𝑖𝑘𝐽𝜀𝐼𝜕̃ 𝑋𝐽𝜕𝑋𝐼𝑒𝑖𝑘⋅𝑋
𝒱 = 𝑐̃ 𝑐𝜕̃ (𝜀𝐼𝜕𝑋𝐼𝑒𝑖𝑘⋅𝑋)
0 = {𝑄𝐵, 𝒲}
−𝑖𝑒𝑖𝜀 ⋅ (𝑝𝑖 + 𝑝𝑖
′) 𝑖
(𝑝𝑖
′)2 − 𝑚𝑖
2 = 𝑒𝑖𝜀 ⋅ 𝑝𝑖
𝑘 ⋅ 𝑝𝑖
,
𝒜 ∼ ∑
𝑛
𝑖=1
𝑒𝑖𝜀 ⋅ 𝑝𝑖
𝑘 ⋅ 𝑝𝑖
𝒜′.
∑
𝑖
𝑒𝑖 = 0,
−𝑖𝑡𝑎,𝑖𝜀 ⋅ (𝑝 + 𝑝′)
𝒜 ∼ ∑
n
𝑖=1
𝑡𝑎,𝑖𝜀 ⋅ 𝑝𝑖
𝑘 ⋅ 𝑝𝑖
𝒜′
∑
𝑖
𝑡𝑎,𝑖𝒜′ = 0.
∑
𝑖≠𝑖′
𝑡𝑎,𝑖𝜀𝑎 ⋅ 𝑝𝑖
𝑘𝑎 ⋅ 𝑝𝑖
𝑡𝑏,𝑖′ 𝜀𝑏 ⋅ 𝑝𝑖′
𝑘𝑏 ⋅ 𝑝𝑖′
𝒜′
∑
𝑖≠𝑖′
𝑡𝑎,𝑖
𝑡𝑏,𝑖′ 𝜀𝑏 ⋅ 𝑝𝑖′
𝑘𝑏 ⋅ 𝑝𝑖′
𝒜′ = − ∑
𝑖
𝑡𝑏,𝑖𝜀𝑏 ⋅ 𝑝𝑖
𝑘𝑏 ⋅ 𝑝𝑖
𝑡𝑎,𝑖𝒜′
(∑
𝑖
𝑡𝑎,𝑖𝜀𝑎 ⋅ 𝑝𝑖
𝑘𝑎 ⋅ 𝑝𝑖
𝑡𝑏,𝑖𝜀𝑏 ⋅ 𝑝𝑖
(𝑘𝑎 + 𝑘𝑏) ⋅ 𝑝𝑖
+ 𝑎 ↔ 𝑏) 𝒜′.
(∑
𝑖
𝑡𝑏,𝑖𝜀𝑏 ⋅ 𝑝𝑖
𝑘𝑏 ⋅ 𝑝𝑖
𝑡𝑎,𝑖 + ∑
𝑖
[𝑡𝑎,𝑖, 𝑡𝑏,𝑖] 𝜀𝑏 ⋅ 𝑝𝑖
(𝑘𝑎 + 𝑘𝑏) ⋅ 𝑝𝑖
) 𝒜′.
Gravedad y supergravedad. Métrica de Ward.
∑
n
𝑖=1
𝑄𝛼,𝑖𝒜′ = 0
𝒱 = {𝑄𝐵, 𝒲} = 𝑐̃ 𝑐𝑖𝑘𝐽𝜀𝐼𝜕̃ 𝑋𝐽𝜕𝑋𝐼𝑒𝑖𝑘⋅𝑋
𝒱 = 𝑐̃ 𝑐𝜕̃ (𝜀𝐼𝜕𝑋𝐼𝑒𝑖𝑘⋅𝑋)
0 = {𝑄𝐵, 𝒲}
pág. 107
0 = 𝜕̃ 𝐽, 𝐽 = 𝜀𝐼𝜕𝑋𝐼.
𝐽𝐼 = 𝜕𝑋𝐼.
1
2𝜋𝛼′ ∮ 𝛾𝐽𝐼 ⋅ 𝒱 = 𝑝𝐼𝒱
0 = ⟨∫
Σ′
d𝐽 ⋅ 𝒱1 … 𝒱𝑛⟩ = ∑
𝑛
𝑖=1
⟨𝒱1 … 𝒱𝑖−1(∮ 𝛾𝑖 𝐽 ⋅ 𝒱𝑖)𝒱𝑖+1 … 𝒱𝑛⟩.
0 = (∑
𝑛
𝑖=1
𝑝𝑖) ⟨𝒱1 … 𝒱𝑛⟩.
∑
n
𝑖=1
𝑝𝑖 = 0.
Spacetime Supersymmetry. Métrica de Ramond – BRST – Stokes - Ward. Conservación de
energía – momentum – stress.
𝒲(𝑘, 𝑢) = 𝑐Θ−1/2𝑢𝛼Σ𝛼𝑒𝑖𝑘⋅𝑋
(𝛾 ⋅ 𝑘)𝛼𝛽𝑢𝛽 = 0
𝒱(𝑘, 𝑢) = {𝑄𝐵, 𝒲(𝑘, 𝑢)} = 𝑖𝑐̃ 𝑐𝑘 ⋅ 𝜕̃ 𝑋Θ−1/2𝑢𝛼Σ𝛼𝑒𝑖𝑘⋅𝑋
𝒮𝛼 = 𝑐Θ−1/2Σ𝛼
d𝐹𝒮𝛼 𝒱1 … 𝒱n = 0
0 = ∫
Γ
d𝐹𝒮𝛼𝒱1…𝒱n = ∫
𝜕Γ
𝐹𝒮𝛼𝒱1…𝒱n
∑
𝑛
𝑖=1
⟨𝒱1 … 𝒱𝑖−1𝑄𝛼(𝒱𝑖)𝒱𝑖+1 … 𝒱𝑛⟩ = 0
Figura 2. Supermembranas.
𝒮𝛼 = 𝑐𝑆̂𝛼, 𝑆̂𝛼 = Θ−1/2Σ𝛼
0 = 𝜕̃ 𝐽, 𝐽 = 𝜀𝐼𝜕𝑋𝐼.
𝐽𝐼 = 𝜕𝑋𝐼.
1
2𝜋𝛼′ ∮ 𝛾𝐽𝐼 ⋅ 𝒱 = 𝑝𝐼𝒱
0 = ⟨∫
Σ′
d𝐽 ⋅ 𝒱1 … 𝒱𝑛⟩ = ∑
𝑛
𝑖=1
⟨𝒱1 … 𝒱𝑖−1(∮ 𝛾𝑖 𝐽 ⋅ 𝒱𝑖)𝒱𝑖+1 … 𝒱𝑛⟩.
0 = (∑
𝑛
𝑖=1
𝑝𝑖) ⟨𝒱1 … 𝒱𝑛⟩.
∑
n
𝑖=1
𝑝𝑖 = 0.
Spacetime Supersymmetry. Métrica de Ramond – BRST – Stokes - Ward. Conservación de
energía – momentum – stress.
𝒲(𝑘, 𝑢) = 𝑐Θ−1/2𝑢𝛼Σ𝛼𝑒𝑖𝑘⋅𝑋
(𝛾 ⋅ 𝑘)𝛼𝛽𝑢𝛽 = 0
𝒱(𝑘, 𝑢) = {𝑄𝐵, 𝒲(𝑘, 𝑢)} = 𝑖𝑐̃ 𝑐𝑘 ⋅ 𝜕̃ 𝑋Θ−1/2𝑢𝛼Σ𝛼𝑒𝑖𝑘⋅𝑋
𝒮𝛼 = 𝑐Θ−1/2Σ𝛼
d𝐹𝒮𝛼 𝒱1 … 𝒱n = 0
0 = ∫
Γ
d𝐹𝒮𝛼𝒱1…𝒱n = ∫
𝜕Γ
𝐹𝒮𝛼𝒱1…𝒱n
∑
𝑛
𝑖=1
⟨𝒱1 … 𝒱𝑖−1𝑄𝛼(𝒱𝑖)𝒱𝑖+1 … 𝒱𝑛⟩ = 0
Figura 2. Supermembranas.
𝒮𝛼 = 𝑐𝑆̂𝛼, 𝑆̂𝛼 = Θ−1/2Σ𝛼
pág. 108
1
2𝜋𝑖 ∮ |𝑧|=𝜖d𝑧𝑆̂𝛼 ⋅ 𝒱
0 = ∫
ℳ̂g,n+1
d𝐹𝒫𝐼𝒱1…𝒱n = ∫
𝜕ℳ̂g,n+1
𝐹𝒫𝐼𝒱1…𝒱n
0 = ∫
ℬ
d𝐹(𝒥, 𝛿𝒥) = ∫
𝜕ℬ
𝐹(𝒥, 𝛿𝒥)
𝑄𝛼(𝑄𝛽(𝒱)) + 𝑄𝛽(𝑄𝛼(𝒱)) + 𝒪(𝒱) = 0
𝒪 = −{𝑄𝛼, 𝑄𝛽}
𝑐̃ 𝜕̃ 𝑐̃ 𝜕̃ 2𝑐̃ 𝑐𝛿(𝛾)𝐷𝜃𝑋𝐼 ⊗ 𝑐𝛿(𝛾)𝐷𝜃𝑋𝐼 + ⋯
𝒪𝒮𝛼,𝒮𝛽 = 1
2𝜋𝑖 ∮ |𝑧|=𝜖d𝑧𝑆̂𝛼(𝑧)𝒮𝛽(0)
𝒱𝜙 = ∑
𝛼
𝑄𝛼(𝒱𝜓𝛼 )
𝒱𝜙 = 𝑐̃ 𝑐𝛿(𝛾)𝜕̃ 𝑋𝐼𝐷𝑋𝐼
𝒱𝜓𝛼 = Γ𝐼
𝛼𝛽𝑐̃ 𝑐𝜕̃ 𝑋𝐼Θ−1/2Σ𝛽
Figura 3. Supermembranas.
∑
𝛼
⟨𝒮𝛼𝒱𝜓𝛼 ⟩
1
2𝜋𝑖 ∮ |𝑧|=𝜖d𝑧𝑆̂𝛼 ⋅ 𝒱
0 = ∫
ℳ̂g,n+1
d𝐹𝒫𝐼𝒱1…𝒱n = ∫
𝜕ℳ̂g,n+1
𝐹𝒫𝐼𝒱1…𝒱n
0 = ∫
ℬ
d𝐹(𝒥, 𝛿𝒥) = ∫
𝜕ℬ
𝐹(𝒥, 𝛿𝒥)
𝑄𝛼(𝑄𝛽(𝒱)) + 𝑄𝛽(𝑄𝛼(𝒱)) + 𝒪(𝒱) = 0
𝒪 = −{𝑄𝛼, 𝑄𝛽}
𝑐̃ 𝜕̃ 𝑐̃ 𝜕̃ 2𝑐̃ 𝑐𝛿(𝛾)𝐷𝜃𝑋𝐼 ⊗ 𝑐𝛿(𝛾)𝐷𝜃𝑋𝐼 + ⋯
𝒪𝒮𝛼,𝒮𝛽 = 1
2𝜋𝑖 ∮ |𝑧|=𝜖d𝑧𝑆̂𝛼(𝑧)𝒮𝛽(0)
𝒱𝜙 = ∑
𝛼
𝑄𝛼(𝒱𝜓𝛼 )
𝒱𝜙 = 𝑐̃ 𝑐𝛿(𝛾)𝜕̃ 𝑋𝐼𝐷𝑋𝐼
𝒱𝜓𝛼 = Γ𝐼
𝛼𝛽𝑐̃ 𝑐𝜕̃ 𝑋𝐼Θ−1/2Σ𝛽
Figura 3. Supermembranas.
∑
𝛼
⟨𝒮𝛼𝒱𝜓𝛼 ⟩
pág. 109
0 = ∫
Γ
d𝐹𝒮𝒱 = ∫
𝜕Γ
𝐹𝒮𝒱
Supermembranas. Métrica de Ward – Chan – Paton. Función Delta.
𝒱NS,NS = ∑
𝛼
{𝑄𝛼
′ , 𝒱R,NS
𝛼 } = ∑
𝛽
{𝑄𝛽
′′, 𝒱NS,R
𝛽 }
𝑞 → 𝑒𝑓𝑞
𝑞 → 𝑒𝑓ℓ+𝑓𝑟 𝑞
𝑞Σℓ,D
𝑞Σℓ,ℝℙ2
→ 𝑒𝜅 𝑞Σℓ,D
𝑞Σℓ,ℝℙ2
, 𝜅 ∈ ℝ
0 = ∫
𝜕Γ̂
𝐹𝑆𝛼 𝒱1 … 𝒱n
𝒪(𝒮𝛼
′ ) + 𝒪(𝒮𝛼˜
′′) = 0
𝒮𝛼 = 𝒮𝛼
′ + 𝜙D(𝒮𝛼
′ )
⟨𝒱NS−NS⟩ + ⟨𝒱R−R⟩ = 0,
𝒱NS−NS = ∑
𝛼
{𝑄𝛼
′ , 𝒱R−NS
𝛼 }
0 = ∫
Γ
d𝐹𝒮𝒱 = ∫
𝜕Γ
𝐹𝒮𝒱
Supermembranas. Métrica de Ward – Chan – Paton. Función Delta.
𝒱NS,NS = ∑
𝛼
{𝑄𝛼
′ , 𝒱R,NS
𝛼 } = ∑
𝛽
{𝑄𝛽
′′, 𝒱NS,R
𝛽 }
𝑞 → 𝑒𝑓𝑞
𝑞 → 𝑒𝑓ℓ+𝑓𝑟 𝑞
𝑞Σℓ,D
𝑞Σℓ,ℝℙ2
→ 𝑒𝜅 𝑞Σℓ,D
𝑞Σℓ,ℝℙ2
, 𝜅 ∈ ℝ
0 = ∫
𝜕Γ̂
𝐹𝑆𝛼 𝒱1 … 𝒱n
𝒪(𝒮𝛼
′ ) + 𝒪(𝒮𝛼˜
′′) = 0
𝒮𝛼 = 𝒮𝛼
′ + 𝜙D(𝒮𝛼
′ )
⟨𝒱NS−NS⟩ + ⟨𝒱R−R⟩ = 0,
𝒱NS−NS = ∑
𝛼
{𝑄𝛼
′ , 𝒱R−NS
𝛼 }
pág. 110
𝒱NS−NS + 𝒱R−R = ∑
𝛼
{𝑄𝛼
′ + 𝑄𝛼
′′, 𝒱R−NS
𝛼 }.
𝒱NS−NS + 𝒱R−R = ∑
𝛼
{𝑄𝛼
′ + 𝑄𝛼
′′, 𝒱R−NS
𝛼 + 𝒱NS−R
𝛼 }
𝒴𝐼 = 𝑐𝛿(𝛾)𝐷𝜃𝑋𝐼exp (𝑖𝑘 ⋅ 𝑋)
𝒵𝛼 = 𝑐Θ−1/2Σ𝛼exp (𝑖𝑘 ⋅ 𝑋)
𝒴̃ 𝐼 = 𝑐̃ 𝛿(𝛾̃ )𝐷𝜃̃ ′ 𝑋𝐼exp (𝑖𝑘 ⋅ 𝑋)
𝒵̃𝛼 = 𝑐̃ Θ̃−1/2Σ̃𝛼exp (𝑖𝑘 ⋅ 𝑋)
𝒵∗
𝛼= 𝑐Θ−3/2Σ𝛼exp (𝑖𝑘 ⋅ 𝑋)
𝒵̃∗
𝛼 = 𝑐̃ Θ̃−3/2Σ̃ 𝛼exp (𝑖𝑘 ⋅ 𝑋̃ )
𝑦 ⋅ 𝒵∗
𝛼 = (Γ ⋅ 𝑘)𝛼𝛽𝒵𝛽 + {𝑄𝐵,⋅}
𝒮̂𝛼(𝑧)𝒵𝛽(𝑤) ∼ 1
𝑧 − 𝑤 Γ𝛼𝛽
𝐼 𝒴𝐼
{𝑄𝛼, 𝒵𝛽} = Γ𝛼𝛽
𝐼 𝒴𝐼
𝒮̂𝛼(𝑧)𝒴𝐼(𝑤) ∼ 1
𝑧 − 𝑤 Γ𝐼𝛼𝛽𝒵∗
𝛽
{𝑄𝛼, 𝒴𝐼} = (Γ𝐼Γ ⋅ 𝑘)𝛼 𝛽𝒵𝛽
𝒮𝛼(𝑧)𝒴𝐼(𝑤) ∼ Γ𝐼𝛼𝛽𝜕𝑐𝒵∗
𝛽
𝜈̃ = 𝜕𝜃̃ − 𝜃̃ 𝑧̃ 𝜕𝑧̃
𝜈̃ 2 = −𝑧̃ 𝜕𝑧˜
𝜃̃ → 𝜃̃ + 𝛼,
𝒱NS−NS + 𝒱R−R = ∑
𝛼
{𝑄𝛼
′ + 𝑄𝛼
′′, 𝒱R−NS
𝛼 }.
𝒱NS−NS + 𝒱R−R = ∑
𝛼
{𝑄𝛼
′ + 𝑄𝛼
′′, 𝒱R−NS
𝛼 + 𝒱NS−R
𝛼 }
𝒴𝐼 = 𝑐𝛿(𝛾)𝐷𝜃𝑋𝐼exp (𝑖𝑘 ⋅ 𝑋)
𝒵𝛼 = 𝑐Θ−1/2Σ𝛼exp (𝑖𝑘 ⋅ 𝑋)
𝒴̃ 𝐼 = 𝑐̃ 𝛿(𝛾̃ )𝐷𝜃̃ ′ 𝑋𝐼exp (𝑖𝑘 ⋅ 𝑋)
𝒵̃𝛼 = 𝑐̃ Θ̃−1/2Σ̃𝛼exp (𝑖𝑘 ⋅ 𝑋)
𝒵∗
𝛼= 𝑐Θ−3/2Σ𝛼exp (𝑖𝑘 ⋅ 𝑋)
𝒵̃∗
𝛼 = 𝑐̃ Θ̃−3/2Σ̃ 𝛼exp (𝑖𝑘 ⋅ 𝑋̃ )
𝑦 ⋅ 𝒵∗
𝛼 = (Γ ⋅ 𝑘)𝛼𝛽𝒵𝛽 + {𝑄𝐵,⋅}
𝒮̂𝛼(𝑧)𝒵𝛽(𝑤) ∼ 1
𝑧 − 𝑤 Γ𝛼𝛽
𝐼 𝒴𝐼
{𝑄𝛼, 𝒵𝛽} = Γ𝛼𝛽
𝐼 𝒴𝐼
𝒮̂𝛼(𝑧)𝒴𝐼(𝑤) ∼ 1
𝑧 − 𝑤 Γ𝐼𝛼𝛽𝒵∗
𝛽
{𝑄𝛼, 𝒴𝐼} = (Γ𝐼Γ ⋅ 𝑘)𝛼 𝛽𝒵𝛽
𝒮𝛼(𝑧)𝒴𝐼(𝑤) ∼ Γ𝐼𝛼𝛽𝜕𝑐𝒵∗
𝛽
𝜈̃ = 𝜕𝜃̃ − 𝜃̃ 𝑧̃ 𝜕𝑧̃
𝜈̃ 2 = −𝑧̃ 𝜕𝑧˜
𝜃̃ → 𝜃̃ + 𝛼,
pág. 111
Figura 4. Supermembranas.
0 = ∫
𝜕Γ̂
𝐹𝒮𝛼 𝒱NS/R
𝛼
⟨𝒱NS−NS⟩Σ = 0
⟨𝒱NS−NS⟩Σ + ⟨𝑉R−R⟩Σ = 0.
𝒱R−R = 𝜕˜𝑐˜𝒵̃∗
𝛼𝒵𝛼 + 𝒵̃𝛼𝜕𝑐𝒵∗
𝛼
⟨𝒱NS−NS⟩D + ⟨𝒱NS−NS⟩ℝℙ2 = 0.
𝐼1 ∼ ∫ [ d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢1(𝑚1 … ; 𝑞Σℓ,D ∣ ⋯ 𝜂𝑠) d𝑞Σℓ,D
𝑞Σℓ,D
𝐼2 ∼ ∫ [ d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢2(𝑚1 … ; 𝑞Σℓ,ℝℙ2 ∣ ⋯ 𝜂𝑠) d𝑞Σℓ,ℝℙ2
𝑞Σℓ,ℝℙ2
𝐼1,log= ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢1(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠)
𝐼2,log = ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢2(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠)
𝒢1(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠)= 𝒢0(𝑚1 … ∣ ⋯ 𝜂𝑠)⟨𝒱NS−NS⟩D
𝒢2(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠) = 𝒢0(𝑚1 … ∣ ⋯ 𝜂𝑠)⟨𝒱NS−NS⟩ℝℙ2
𝒜𝒱1…𝒱n;𝒱NS−NS = ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢0
𝐼1,log + 𝐼2,log = 0
𝐼1→ 𝐼1 − ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]ℎ𝒢1(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠)
= 𝐼1 − ⟨𝒱NS−NS⟩D ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]ℎ𝒢0(𝑚1, … ∣ ⋯ 𝜂𝑠)
Figura 4. Supermembranas.
0 = ∫
𝜕Γ̂
𝐹𝒮𝛼 𝒱NS/R
𝛼
⟨𝒱NS−NS⟩Σ = 0
⟨𝒱NS−NS⟩Σ + ⟨𝑉R−R⟩Σ = 0.
𝒱R−R = 𝜕˜𝑐˜𝒵̃∗
𝛼𝒵𝛼 + 𝒵̃𝛼𝜕𝑐𝒵∗
𝛼
⟨𝒱NS−NS⟩D + ⟨𝒱NS−NS⟩ℝℙ2 = 0.
𝐼1 ∼ ∫ [ d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢1(𝑚1 … ; 𝑞Σℓ,D ∣ ⋯ 𝜂𝑠) d𝑞Σℓ,D
𝑞Σℓ,D
𝐼2 ∼ ∫ [ d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢2(𝑚1 … ; 𝑞Σℓ,ℝℙ2 ∣ ⋯ 𝜂𝑠) d𝑞Σℓ,ℝℙ2
𝑞Σℓ,ℝℙ2
𝐼1,log= ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢1(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠)
𝐼2,log = ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢2(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠)
𝒢1(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠)= 𝒢0(𝑚1 … ∣ ⋯ 𝜂𝑠)⟨𝒱NS−NS⟩D
𝒢2(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠) = 𝒢0(𝑚1 … ∣ ⋯ 𝜂𝑠)⟨𝒱NS−NS⟩ℝℙ2
𝒜𝒱1…𝒱n;𝒱NS−NS = ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢0
𝐼1,log + 𝐼2,log = 0
𝐼1→ 𝐼1 − ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]ℎ𝒢1(𝑚1 … ; 0 ∣ ⋯ 𝜂𝑠)
= 𝐼1 − ⟨𝒱NS−NS⟩D ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]ℎ𝒢0(𝑚1, … ∣ ⋯ 𝜂𝑠)
pág. 112
𝐼2 → 𝐼2 − ⟨𝒱NS−NS⟩ℝℙ2 ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]ℎ𝒢0(𝑚1, … ∣ ⋯ 𝜂𝑠)
𝐼2 → 𝐼2 + 𝜅⟨𝒱NS−NS⟩ℝℙ2 ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢0(𝑚1, … ∣ ⋯ 𝜂𝑠)
𝒜𝒱1…𝒱n → 𝒜𝒱1…𝒱n + 𝜅⟨𝒱NS−NS⟩ℝℙ2 𝒜𝒱1…𝒱n;𝒱NS−NS
Figura 5. Supermembranas.
⟨𝛿(𝛾(𝑧1))𝛿(𝛽(𝑧2))⟩ ≠ 0
⟨𝛿′(𝛾(𝑧1))𝛿(𝛽(𝑧2))⟩ = 0
⟨𝛿′(𝛾(𝑧1))𝛿(𝛽(𝑧2))𝛾(𝑧3)⟩ ≠ 0
⟨𝑏(𝑧1)𝑐(𝑧2)⟩ ≠ 0.
⟨𝑏𝑐𝛿(𝛽)𝛿(𝛾)⟩ ≠ 0
⟨𝑏𝑐𝛿(𝛽)𝛿′(𝛾)𝛾⟩ ≠ 0
∏
𝑛
𝑖=2
∮ 𝜕ℓΣ[d𝑧 ∣ d𝜃]𝜀𝐼
(𝑖)𝐷𝜃𝑋𝐼exp (𝑖𝑘(𝑖) ⋅ 𝑋)
𝒱1 = 𝑐𝛿(𝛾)𝜀(1) ⋅ 𝐷𝜃𝑋exp (𝑖𝑘(1) ⋅ 𝑋)
(𝑏0 + 𝑏̃0) ∫
∞
𝑠0
d𝑠exp (−𝑠(𝐿0 + 𝐿̃ 0)) 𝛿(𝛽0 + 𝛽̃0)(𝐺0 + 𝐺̃0)
𝑐𝛿(𝛾)𝜀(1) ⋅ 𝐷𝜃𝑋exp (𝑖𝑘(1) ⋅ 𝑋)(𝑏0 + 𝑏̃0) ∫
∞
𝑠0
d𝑠exp (−𝑠(𝐿0 + 𝐿̃ 0)) 𝛿(𝛽0 + 𝛽̃0)(𝐺0 + 𝐺̃0)
𝐺0 = 𝐺0
𝑋 + 𝐺0
gh
𝐺̃0 = 𝐺̃0
𝑋 + 𝐺̃0
gh
𝒲1 = 𝑐𝛿′(𝛾)exp (𝑖𝑘(1) ⋅ 𝑋)
𝐼2 → 𝐼2 − ⟨𝒱NS−NS⟩ℝℙ2 ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]ℎ𝒢0(𝑚1, … ∣ ⋯ 𝜂𝑠)
𝐼2 → 𝐼2 + 𝜅⟨𝒱NS−NS⟩ℝℙ2 ∫
ℳℓ
[d𝑚1 … ∣ ⋯ d𝜂𝑠]𝒢0(𝑚1, … ∣ ⋯ 𝜂𝑠)
𝒜𝒱1…𝒱n → 𝒜𝒱1…𝒱n + 𝜅⟨𝒱NS−NS⟩ℝℙ2 𝒜𝒱1…𝒱n;𝒱NS−NS
Figura 5. Supermembranas.
⟨𝛿(𝛾(𝑧1))𝛿(𝛽(𝑧2))⟩ ≠ 0
⟨𝛿′(𝛾(𝑧1))𝛿(𝛽(𝑧2))⟩ = 0
⟨𝛿′(𝛾(𝑧1))𝛿(𝛽(𝑧2))𝛾(𝑧3)⟩ ≠ 0
⟨𝑏(𝑧1)𝑐(𝑧2)⟩ ≠ 0.
⟨𝑏𝑐𝛿(𝛽)𝛿(𝛾)⟩ ≠ 0
⟨𝑏𝑐𝛿(𝛽)𝛿′(𝛾)𝛾⟩ ≠ 0
∏
𝑛
𝑖=2
∮ 𝜕ℓΣ[d𝑧 ∣ d𝜃]𝜀𝐼
(𝑖)𝐷𝜃𝑋𝐼exp (𝑖𝑘(𝑖) ⋅ 𝑋)
𝒱1 = 𝑐𝛿(𝛾)𝜀(1) ⋅ 𝐷𝜃𝑋exp (𝑖𝑘(1) ⋅ 𝑋)
(𝑏0 + 𝑏̃0) ∫
∞
𝑠0
d𝑠exp (−𝑠(𝐿0 + 𝐿̃ 0)) 𝛿(𝛽0 + 𝛽̃0)(𝐺0 + 𝐺̃0)
𝑐𝛿(𝛾)𝜀(1) ⋅ 𝐷𝜃𝑋exp (𝑖𝑘(1) ⋅ 𝑋)(𝑏0 + 𝑏̃0) ∫
∞
𝑠0
d𝑠exp (−𝑠(𝐿0 + 𝐿̃ 0)) 𝛿(𝛽0 + 𝛽̃0)(𝐺0 + 𝐺̃0)
𝐺0 = 𝐺0
𝑋 + 𝐺0
gh
𝐺̃0 = 𝐺̃0
𝑋 + 𝐺̃0
gh
𝒲1 = 𝑐𝛿′(𝛾)exp (𝑖𝑘(1) ⋅ 𝑋)
pág. 113
𝑐𝛿′(𝛾)exp (𝑖𝑘(1) ⋅ 𝑋)exp (−𝑠(𝐿0 + 𝐿̃ 0)) 𝛿(𝛽0 + 𝛽̃0)(𝐺0 + 𝐺̃0)
𝐼𝛽𝛾 = 1
2𝜋 ∫
Σ
d2𝑧𝛽𝜕𝑧˜ 𝛾
𝐼𝛽∗𝛾∗ = 1
2𝜋 ∫
Σ
d2𝑧𝛽∗𝜕𝑧˜ 𝛾∗
∫ 𝒟𝛽∗𝒟𝛾∗exp (−𝐼𝛽∗𝛾∗) = det𝑀
∫ 𝒟𝛽𝒟𝛾exp (−𝐼𝛽𝛾) = 1
det𝑀
⟨𝒪⟩𝑁 = ⟨𝒪⟩
⟨1⟩
∫ d𝑛𝑥exp (− 1
2 (𝑥, 𝑁𝑥)) = 1
√det𝑁
𝑁 = ( 0 𝑀
𝑀𝑡 0 )
√det𝑁 = det𝑀,
𝑁̃ = ( 0 𝑀
−𝑀𝑡 0 )
⟨𝛾(𝑢)𝛽(𝑤)⟩𝑁 = ⟨𝛾∗(𝑢)𝛽∗(𝑤)⟩𝑁 = 𝑆(𝑢, 𝑤),
𝜕𝑧̃
2𝜋 𝑆(𝑧, 𝑧′) = 𝛿2(𝑧, 𝑧′)
⟨𝛾∗(𝑢1)𝛾∗(𝑢2) … 𝛾∗(𝑢𝑠)𝛽∗(𝑤𝑠)𝛽∗(𝑤𝑠−1) … 𝛽∗(𝑤1)⟩𝑁 = ∑
𝜋
(−1)𝜋 ∏
𝑠
𝑖=1
𝑆(𝑢𝑖, 𝑤𝜋(𝑖)),
⟨∏
𝑠
𝑖=1
𝛾(𝑢𝑖) ∏
𝑠
𝑗=1
𝛽(𝑤𝑗)⟩
𝑁
= ∑
𝜋
∏
𝑠
𝑖=1
𝑆(𝑢𝑖, 𝑤𝜋(𝑖)).
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))⟩ = (det𝑀)𝑆(𝑢, 𝑤)
∫ d𝜏∗ d𝜎∗exp (−𝜎∗𝛾∗(𝑢) − 𝛽∗(𝑤)𝜏∗) = 𝛾∗(𝑢)𝛽∗(𝑤) = 𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))⟩ = ∫ 𝒟𝛽∗𝒟𝛾∗ d𝜏∗ d𝜎∗exp (−𝐼𝛽∗𝛾∗ − 𝜎∗𝛾∗(𝑢) − 𝛽∗(𝑤)𝜏∗)
𝐼̂𝛽̂ ∗𝛾̂ ∗ = 𝐼𝛽∗𝛾∗ + 𝜎∗𝛾∗(𝑢) + 𝛽∗(𝑤)𝜏∗
𝛽̂ ∗ = (𝛽∗(𝑧)𝜎∗), 𝛾̂ ∗ = (𝛾∗(𝑧)
𝜏∗ )
𝑐𝛿′(𝛾)exp (𝑖𝑘(1) ⋅ 𝑋)exp (−𝑠(𝐿0 + 𝐿̃ 0)) 𝛿(𝛽0 + 𝛽̃0)(𝐺0 + 𝐺̃0)
𝐼𝛽𝛾 = 1
2𝜋 ∫
Σ
d2𝑧𝛽𝜕𝑧˜ 𝛾
𝐼𝛽∗𝛾∗ = 1
2𝜋 ∫
Σ
d2𝑧𝛽∗𝜕𝑧˜ 𝛾∗
∫ 𝒟𝛽∗𝒟𝛾∗exp (−𝐼𝛽∗𝛾∗) = det𝑀
∫ 𝒟𝛽𝒟𝛾exp (−𝐼𝛽𝛾) = 1
det𝑀
⟨𝒪⟩𝑁 = ⟨𝒪⟩
⟨1⟩
∫ d𝑛𝑥exp (− 1
2 (𝑥, 𝑁𝑥)) = 1
√det𝑁
𝑁 = ( 0 𝑀
𝑀𝑡 0 )
√det𝑁 = det𝑀,
𝑁̃ = ( 0 𝑀
−𝑀𝑡 0 )
⟨𝛾(𝑢)𝛽(𝑤)⟩𝑁 = ⟨𝛾∗(𝑢)𝛽∗(𝑤)⟩𝑁 = 𝑆(𝑢, 𝑤),
𝜕𝑧̃
2𝜋 𝑆(𝑧, 𝑧′) = 𝛿2(𝑧, 𝑧′)
⟨𝛾∗(𝑢1)𝛾∗(𝑢2) … 𝛾∗(𝑢𝑠)𝛽∗(𝑤𝑠)𝛽∗(𝑤𝑠−1) … 𝛽∗(𝑤1)⟩𝑁 = ∑
𝜋
(−1)𝜋 ∏
𝑠
𝑖=1
𝑆(𝑢𝑖, 𝑤𝜋(𝑖)),
⟨∏
𝑠
𝑖=1
𝛾(𝑢𝑖) ∏
𝑠
𝑗=1
𝛽(𝑤𝑗)⟩
𝑁
= ∑
𝜋
∏
𝑠
𝑖=1
𝑆(𝑢𝑖, 𝑤𝜋(𝑖)).
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))⟩ = (det𝑀)𝑆(𝑢, 𝑤)
∫ d𝜏∗ d𝜎∗exp (−𝜎∗𝛾∗(𝑢) − 𝛽∗(𝑤)𝜏∗) = 𝛾∗(𝑢)𝛽∗(𝑤) = 𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))⟩ = ∫ 𝒟𝛽∗𝒟𝛾∗ d𝜏∗ d𝜎∗exp (−𝐼𝛽∗𝛾∗ − 𝜎∗𝛾∗(𝑢) − 𝛽∗(𝑤)𝜏∗)
𝐼̂𝛽̂ ∗𝛾̂ ∗ = 𝐼𝛽∗𝛾∗ + 𝜎∗𝛾∗(𝑢) + 𝛽∗(𝑤)𝜏∗
𝛽̂ ∗ = (𝛽∗(𝑧)𝜎∗), 𝛾̂ ∗ = (𝛾∗(𝑧)
𝜏∗ )
pág. 114
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))⟩ = det𝑀̂
det𝑀̂ = (det𝑀)𝑆(𝑢, 𝑤)
𝛿(𝛾(𝑢))𝛿(𝛽(𝑤)) = ∫ d𝜏 d𝜎exp (−𝜎𝛾(𝑢) − 𝛽(𝑤)𝜏)
𝐼̂𝛽̂ 𝛾̂ = 𝐼𝛽𝛾 + 𝜎𝛾(𝑢) + 𝛽(𝑤)𝜏,
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤))⟩ = ∫ 𝒟𝛽̂ 𝒟𝛾̂ exp (−𝐼̂𝛽̂ 𝛾̂ ) = 1
det𝑀̂
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤))⟩𝑁 = 1
𝑆(𝑢, 𝑤)
𝛿(𝛾(𝑢))𝛿(𝛽(𝑤)) ∼ 𝑢 − 𝑤
𝛾∗(𝑢)= 0
𝜕𝑧˜ 𝛾∗
2𝜋 + 𝛿2(𝑧, 𝑤)𝜏∗ = 0
∫ 𝒟𝛽∗𝒟𝛾∗ ∏
𝑠
𝑖=1
d𝜏𝑖
∗ d𝜎𝑖∗exp (−𝐼𝛽∗𝛾∗ − ∑
𝑖
𝜎𝑖∗𝛾∗(𝑢𝑖) − ∑
𝑗
𝛽∗(𝑤𝑗)𝜏𝑗
∗)
⟨∏
𝑠
𝑖=1
𝛿(𝛾(𝑢𝑖)) ∏
𝑠
𝑗=1
𝛿 (𝛽(𝑤𝑗))⟩
𝑁
= 1
∑𝜋 (−1)𝜋 ∏𝑠
𝑖=1 𝑆(𝑢𝑖, 𝑤𝜋(𝑖))
𝛿(𝛾(𝑢1))𝛿(𝛾(𝑢2)) ∼ − 1
𝑢1 − 𝑢2
𝛿(𝛾)𝛿(𝜕𝛾)(𝑢2)
Θ−𝑡 = 𝛿(𝛾)𝛿(𝜕𝛾) … 𝛿(𝜕𝑡−1𝛾)
𝛿(𝛾(𝑢1))𝛿(𝛽)𝛿(𝜕𝛽)(𝑢2) ∼ (𝑢1 − 𝑢2)2𝛿(𝛽(𝑢2))
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))𝛾∗(𝑢′)𝛽∗(𝑤′)⟩ = det𝑀(𝑆(𝑢, 𝑤)𝑆(𝑢′, 𝑤′) − 𝑆(𝑢, 𝑤′)𝑆(𝑢′, 𝑤))
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))𝛾∗(𝑢′)𝛽∗(𝑤′)⟩ = ∫ 𝒟𝛽̂ ∗𝒟𝛾̂ ∗exp (−𝐼̂𝛽̂ ∗𝛾̂ ∗ )𝛾∗(𝑢′)𝛽∗(𝑤′)
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))𝛾∗(𝑢′)𝛽∗(𝑤′)⟩ = (det𝑀̂ )𝑆̂ (𝑢′, 𝑤′),
𝑆̂ (𝑢′, 𝑤′) = 1
𝑆(𝑢, 𝑤) (𝑆(𝑢, 𝑤)𝑆(𝑢′, 𝑤′) − 𝑆(𝑢, 𝑤′)𝑆(𝑢′, 𝑤)).
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤))𝛾(𝑢′)𝛽(𝑤′)⟩ = 1
det𝑀̂ 𝑆̂ (𝑢′, 𝑤′).
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤))𝛾(𝑢′)𝛽(𝑤′)⟩𝑁 = 1
𝑆(𝑢, 𝑤)2 (𝑆(𝑢, 𝑤)𝑆(𝑢′, 𝑤′) − 𝑆(𝑢, 𝑤′)𝑆(𝑢′, 𝑤)).
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))⟩ = det𝑀̂
det𝑀̂ = (det𝑀)𝑆(𝑢, 𝑤)
𝛿(𝛾(𝑢))𝛿(𝛽(𝑤)) = ∫ d𝜏 d𝜎exp (−𝜎𝛾(𝑢) − 𝛽(𝑤)𝜏)
𝐼̂𝛽̂ 𝛾̂ = 𝐼𝛽𝛾 + 𝜎𝛾(𝑢) + 𝛽(𝑤)𝜏,
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤))⟩ = ∫ 𝒟𝛽̂ 𝒟𝛾̂ exp (−𝐼̂𝛽̂ 𝛾̂ ) = 1
det𝑀̂
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤))⟩𝑁 = 1
𝑆(𝑢, 𝑤)
𝛿(𝛾(𝑢))𝛿(𝛽(𝑤)) ∼ 𝑢 − 𝑤
𝛾∗(𝑢)= 0
𝜕𝑧˜ 𝛾∗
2𝜋 + 𝛿2(𝑧, 𝑤)𝜏∗ = 0
∫ 𝒟𝛽∗𝒟𝛾∗ ∏
𝑠
𝑖=1
d𝜏𝑖
∗ d𝜎𝑖∗exp (−𝐼𝛽∗𝛾∗ − ∑
𝑖
𝜎𝑖∗𝛾∗(𝑢𝑖) − ∑
𝑗
𝛽∗(𝑤𝑗)𝜏𝑗
∗)
⟨∏
𝑠
𝑖=1
𝛿(𝛾(𝑢𝑖)) ∏
𝑠
𝑗=1
𝛿 (𝛽(𝑤𝑗))⟩
𝑁
= 1
∑𝜋 (−1)𝜋 ∏𝑠
𝑖=1 𝑆(𝑢𝑖, 𝑤𝜋(𝑖))
𝛿(𝛾(𝑢1))𝛿(𝛾(𝑢2)) ∼ − 1
𝑢1 − 𝑢2
𝛿(𝛾)𝛿(𝜕𝛾)(𝑢2)
Θ−𝑡 = 𝛿(𝛾)𝛿(𝜕𝛾) … 𝛿(𝜕𝑡−1𝛾)
𝛿(𝛾(𝑢1))𝛿(𝛽)𝛿(𝜕𝛽)(𝑢2) ∼ (𝑢1 − 𝑢2)2𝛿(𝛽(𝑢2))
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))𝛾∗(𝑢′)𝛽∗(𝑤′)⟩ = det𝑀(𝑆(𝑢, 𝑤)𝑆(𝑢′, 𝑤′) − 𝑆(𝑢, 𝑤′)𝑆(𝑢′, 𝑤))
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))𝛾∗(𝑢′)𝛽∗(𝑤′)⟩ = ∫ 𝒟𝛽̂ ∗𝒟𝛾̂ ∗exp (−𝐼̂𝛽̂ ∗𝛾̂ ∗ )𝛾∗(𝑢′)𝛽∗(𝑤′)
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤))𝛾∗(𝑢′)𝛽∗(𝑤′)⟩ = (det𝑀̂ )𝑆̂ (𝑢′, 𝑤′),
𝑆̂ (𝑢′, 𝑤′) = 1
𝑆(𝑢, 𝑤) (𝑆(𝑢, 𝑤)𝑆(𝑢′, 𝑤′) − 𝑆(𝑢, 𝑤′)𝑆(𝑢′, 𝑤)).
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤))𝛾(𝑢′)𝛽(𝑤′)⟩ = 1
det𝑀̂ 𝑆̂ (𝑢′, 𝑤′).
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤))𝛾(𝑢′)𝛽(𝑤′)⟩𝑁 = 1
𝑆(𝑢, 𝑤)2 (𝑆(𝑢, 𝑤)𝑆(𝑢′, 𝑤′) − 𝑆(𝑢, 𝑤′)𝑆(𝑢′, 𝑤)).
pág. 115
𝛾(𝑢′)𝛿(𝛾(𝑢)) ∼ (𝑢′ − 𝑢)𝜕𝛾 ⋅ 𝛿(𝛾)(𝑢).
𝛽(𝑤′)𝛿(𝛾(𝑢)) ∼ 1
𝑤′ − 𝑢 𝛿′(𝛾(𝑢)).
𝛾(𝑢′)𝛿′(𝛾(𝑢)) ∼ −𝛿(𝛾(𝑢))
𝛾(𝑢′)𝛿(𝛽(𝑤))∼ 1
𝑢′ − 𝑤 𝛿′(𝛽(𝑤))
𝛽(𝑤′)𝛿(𝛽(𝑤)) ∼ (𝑤′ − 𝑤)𝜕𝛽 ⋅ 𝛿(𝛽)(𝑤)
⟨𝛾(𝑢)𝛽(𝑤)𝛾(𝑢′)𝛽(𝑤′)⟩ = 𝑆(𝑢, 𝑤)𝑆(𝑢′, 𝑤′) + 𝑆(𝑢, 𝑤′)𝑆(𝑢′, 𝑤)
𝛿(𝛾(𝑢))𝛿(𝛾(0)) = 𝛿 (𝛾(0) + 𝑢𝛾′(0) + 𝑢2
2 𝛾′′(0) + ⋯ ) 𝛿(𝛾(0))
𝛿 (𝛾(0) + 𝑢𝛾′(0) + 𝑢2
2 𝛾′′(0) + ⋯ ) 𝛿(𝛾(0))= 𝛿 (𝑢𝛾′(0) + 𝑢2
2 𝛾′′(0) + ⋯ ) 𝛿(𝛾(0))= 1
𝑢 𝛿 (𝛾′(0) + 𝑢
2 𝛾′′(0) + ⋯ ) 𝛿(𝛾(0))
𝛿 (𝛾′(0) + 𝑢
2 𝛾′′(0) + ⋯ ) = 𝛿(𝛾′(0)) + 𝑢
2 𝛾′′(0)𝛿′(𝛾′(0)) + ⋯
𝛿(𝛾(𝑢))𝛿(𝛾(0)) ∼ 1
𝑢 𝛿(𝛾′)𝛿(𝛾)(0) + 1
2 𝛾′′𝛿′(𝛾′)𝛿(𝛾)(0) + ⋯
𝛾𝑓 = ∫
Σ
𝑓(𝑧)𝛾(𝑧), 𝛽𝑔 = ∫
Σ
𝑔(𝑧)𝛽(𝑧)
⟨𝛿(𝛾𝑓
∗)𝛿(𝛽𝑔
∗)⟩𝑁 = ⟨𝛾𝑓
∗𝛽𝑔
∗⟩𝑁 = ∫
Σ×Σ
𝑓(𝑧)𝑆(𝑧, 𝑧′)𝑔(𝑧′)
⟨𝛿(𝛾𝑓)𝛿(𝛽𝑔)⟩𝑁 = 1
∫Σ×Σ 𝑓(𝑧)𝑆(𝑧, 𝑧′)𝑔(𝑧′)
⟨𝛿(𝛾(𝑢))𝛿(𝜕𝛽(𝑤))⟩𝑁 = 1
𝜕𝑤𝑆(𝑢, 𝑤)
𝐻1(Σ, 𝐾 ⊗ ℒ−1) = 0
𝑀ty𝑖(𝑧) = 0, 𝑖 = 1, … , 𝑡
⟨𝛽∗(𝑤1) … 𝛽∗(𝑤𝑡)⟩ = det′𝑀det𝑁.
⟨𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡))⟩ = det′𝑀det𝑁.
⟨𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))⟩ = 1
det′𝑀
1
det𝑁
𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡)) = ∫ d𝜏1
∗ … d𝜏𝑡
∗exp (− ∑
𝑖
𝛽(𝑤𝑖)𝜏𝑖
∗)
𝛾(𝑢′)𝛿(𝛾(𝑢)) ∼ (𝑢′ − 𝑢)𝜕𝛾 ⋅ 𝛿(𝛾)(𝑢).
𝛽(𝑤′)𝛿(𝛾(𝑢)) ∼ 1
𝑤′ − 𝑢 𝛿′(𝛾(𝑢)).
𝛾(𝑢′)𝛿′(𝛾(𝑢)) ∼ −𝛿(𝛾(𝑢))
𝛾(𝑢′)𝛿(𝛽(𝑤))∼ 1
𝑢′ − 𝑤 𝛿′(𝛽(𝑤))
𝛽(𝑤′)𝛿(𝛽(𝑤)) ∼ (𝑤′ − 𝑤)𝜕𝛽 ⋅ 𝛿(𝛽)(𝑤)
⟨𝛾(𝑢)𝛽(𝑤)𝛾(𝑢′)𝛽(𝑤′)⟩ = 𝑆(𝑢, 𝑤)𝑆(𝑢′, 𝑤′) + 𝑆(𝑢, 𝑤′)𝑆(𝑢′, 𝑤)
𝛿(𝛾(𝑢))𝛿(𝛾(0)) = 𝛿 (𝛾(0) + 𝑢𝛾′(0) + 𝑢2
2 𝛾′′(0) + ⋯ ) 𝛿(𝛾(0))
𝛿 (𝛾(0) + 𝑢𝛾′(0) + 𝑢2
2 𝛾′′(0) + ⋯ ) 𝛿(𝛾(0))= 𝛿 (𝑢𝛾′(0) + 𝑢2
2 𝛾′′(0) + ⋯ ) 𝛿(𝛾(0))= 1
𝑢 𝛿 (𝛾′(0) + 𝑢
2 𝛾′′(0) + ⋯ ) 𝛿(𝛾(0))
𝛿 (𝛾′(0) + 𝑢
2 𝛾′′(0) + ⋯ ) = 𝛿(𝛾′(0)) + 𝑢
2 𝛾′′(0)𝛿′(𝛾′(0)) + ⋯
𝛿(𝛾(𝑢))𝛿(𝛾(0)) ∼ 1
𝑢 𝛿(𝛾′)𝛿(𝛾)(0) + 1
2 𝛾′′𝛿′(𝛾′)𝛿(𝛾)(0) + ⋯
𝛾𝑓 = ∫
Σ
𝑓(𝑧)𝛾(𝑧), 𝛽𝑔 = ∫
Σ
𝑔(𝑧)𝛽(𝑧)
⟨𝛿(𝛾𝑓
∗)𝛿(𝛽𝑔
∗)⟩𝑁 = ⟨𝛾𝑓
∗𝛽𝑔
∗⟩𝑁 = ∫
Σ×Σ
𝑓(𝑧)𝑆(𝑧, 𝑧′)𝑔(𝑧′)
⟨𝛿(𝛾𝑓)𝛿(𝛽𝑔)⟩𝑁 = 1
∫Σ×Σ 𝑓(𝑧)𝑆(𝑧, 𝑧′)𝑔(𝑧′)
⟨𝛿(𝛾(𝑢))𝛿(𝜕𝛽(𝑤))⟩𝑁 = 1
𝜕𝑤𝑆(𝑢, 𝑤)
𝐻1(Σ, 𝐾 ⊗ ℒ−1) = 0
𝑀ty𝑖(𝑧) = 0, 𝑖 = 1, … , 𝑡
⟨𝛽∗(𝑤1) … 𝛽∗(𝑤𝑡)⟩ = det′𝑀det𝑁.
⟨𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡))⟩ = det′𝑀det𝑁.
⟨𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))⟩ = 1
det′𝑀
1
det𝑁
𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡)) = ∫ d𝜏1
∗ … d𝜏𝑡
∗exp (− ∑
𝑖
𝛽(𝑤𝑖)𝜏𝑖
∗)
pág. 116
𝐼̂𝛽∗𝛾̂ ∗ = 𝐼𝛽∗𝛾∗ + ∑
𝑡
𝑖=1
𝛽∗(𝑤𝑖)𝜏𝑖
∗
⟨𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡))⟩ = ∫ 𝒟𝛽∗𝒟𝛾̂ ∗exp (−𝐼̂𝛽∗𝛾̂ ∗ )
𝐼̂𝛽𝛾̂ = 𝐼𝛽𝛾 + ∑
𝑡
𝑖=1
𝛽(𝑤𝑖)𝜏𝑖,
⟨𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))⟩ = ∫ 𝒟𝛽𝒟𝛾̂ exp (−𝐼̂𝛽𝛾̂ )
⟨𝛾∗(𝑢)𝛽∗(𝑤1) … 𝛽∗(𝑤𝑡+1)⟩.
𝑆(𝑢, 𝑤) → 𝑆(𝑢, 𝑤) + ∑
𝑖
ℎ𝑖(𝑢)y𝑖(𝑤)
𝑁(𝑡+1) =
(
y1(𝑤1) y1(𝑤2) … y1(𝑤𝑡+1)
y2(𝑤1) y2(𝑤2) … y2(𝑤𝑡+1)
⋱
y𝑡(𝑤1) y𝑡(𝑤2) … y𝑡(𝑤𝑡+1)
𝑆(𝑢, 𝑤1) 𝑆(𝑢, 𝑤2) … 𝑆(𝑢, 𝑤𝑡+1))
⟨𝛾∗(𝑢)𝛽∗(𝑤1) … 𝛽∗(𝑤𝑡+1)⟩ = det′𝑀det𝑁(𝑡+1)
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡+1))⟩ = det′𝑀det𝑁(𝑡+1)
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡+1))⟩ = 1
det′𝑀
1
det𝑁(𝑡+1)
⟨𝛾∗(𝑢)𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡))𝛽∗(𝑤𝑡+1)⟩
⟨𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡))⟩ = det𝑁(𝑡+1)
det𝑁(𝑡)
⟨𝛾(𝑢)𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))𝛽(𝑤𝑡+1)⟩
⟨𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))⟩ = det𝑁(𝑡+1)
det𝑁(𝑡)
⟨𝛾(𝑢)𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))𝛽(𝑤𝑡+1)⟩ = det𝑁(𝑡+1)
det′𝑀(det𝑁(𝑡))2
𝑁(𝑡+𝑠) =
(
y1(𝑤1) y1(𝑤2) … y1(𝑤𝑡+𝑠)
y2(𝑤1) y2(𝑤2) … y2(𝑤𝑡+𝑠)
⋱
y𝑡(𝑤1) y𝑡(𝑤2) … y𝑡(𝑤𝑡+𝑠)
𝑆(𝑢1, 𝑤1) 𝑆(𝑢1, 𝑤2) … 𝑆(𝑢1, 𝑤𝑡+𝑠)
⋱
𝑆(𝑢𝑠, 𝑤1) 𝑆(𝑢𝑠, 𝑤2) … 𝑆(𝑢𝑠, 𝑤𝑡+𝑠))
⟨𝛾∗(𝑢1) … 𝛾∗(𝑢𝑠)𝛽∗(𝑢1) … 𝛽∗(𝑢𝑡+𝑠)⟩ = det′𝑀det𝑁(𝑡+𝑠)
𝐼̂𝛽∗𝛾̂ ∗ = 𝐼𝛽∗𝛾∗ + ∑
𝑡
𝑖=1
𝛽∗(𝑤𝑖)𝜏𝑖
∗
⟨𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡))⟩ = ∫ 𝒟𝛽∗𝒟𝛾̂ ∗exp (−𝐼̂𝛽∗𝛾̂ ∗ )
𝐼̂𝛽𝛾̂ = 𝐼𝛽𝛾 + ∑
𝑡
𝑖=1
𝛽(𝑤𝑖)𝜏𝑖,
⟨𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))⟩ = ∫ 𝒟𝛽𝒟𝛾̂ exp (−𝐼̂𝛽𝛾̂ )
⟨𝛾∗(𝑢)𝛽∗(𝑤1) … 𝛽∗(𝑤𝑡+1)⟩.
𝑆(𝑢, 𝑤) → 𝑆(𝑢, 𝑤) + ∑
𝑖
ℎ𝑖(𝑢)y𝑖(𝑤)
𝑁(𝑡+1) =
(
y1(𝑤1) y1(𝑤2) … y1(𝑤𝑡+1)
y2(𝑤1) y2(𝑤2) … y2(𝑤𝑡+1)
⋱
y𝑡(𝑤1) y𝑡(𝑤2) … y𝑡(𝑤𝑡+1)
𝑆(𝑢, 𝑤1) 𝑆(𝑢, 𝑤2) … 𝑆(𝑢, 𝑤𝑡+1))
⟨𝛾∗(𝑢)𝛽∗(𝑤1) … 𝛽∗(𝑤𝑡+1)⟩ = det′𝑀det𝑁(𝑡+1)
⟨𝛿(𝛾∗(𝑢))𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡+1))⟩ = det′𝑀det𝑁(𝑡+1)
⟨𝛿(𝛾(𝑢))𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡+1))⟩ = 1
det′𝑀
1
det𝑁(𝑡+1)
⟨𝛾∗(𝑢)𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡))𝛽∗(𝑤𝑡+1)⟩
⟨𝛿(𝛽∗(𝑤1)) … 𝛿(𝛽∗(𝑤𝑡))⟩ = det𝑁(𝑡+1)
det𝑁(𝑡)
⟨𝛾(𝑢)𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))𝛽(𝑤𝑡+1)⟩
⟨𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))⟩ = det𝑁(𝑡+1)
det𝑁(𝑡)
⟨𝛾(𝑢)𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡))𝛽(𝑤𝑡+1)⟩ = det𝑁(𝑡+1)
det′𝑀(det𝑁(𝑡))2
𝑁(𝑡+𝑠) =
(
y1(𝑤1) y1(𝑤2) … y1(𝑤𝑡+𝑠)
y2(𝑤1) y2(𝑤2) … y2(𝑤𝑡+𝑠)
⋱
y𝑡(𝑤1) y𝑡(𝑤2) … y𝑡(𝑤𝑡+𝑠)
𝑆(𝑢1, 𝑤1) 𝑆(𝑢1, 𝑤2) … 𝑆(𝑢1, 𝑤𝑡+𝑠)
⋱
𝑆(𝑢𝑠, 𝑤1) 𝑆(𝑢𝑠, 𝑤2) … 𝑆(𝑢𝑠, 𝑤𝑡+𝑠))
⟨𝛾∗(𝑢1) … 𝛾∗(𝑢𝑠)𝛽∗(𝑢1) … 𝛽∗(𝑢𝑡+𝑠)⟩ = det′𝑀det𝑁(𝑡+𝑠)
pág. 117
⟨𝛿(𝛾(𝑢1)) … 𝛿(𝛾∗(𝑢𝑠))𝛿(𝛽∗(𝑢1)) … 𝛿(𝛽∗(𝑢𝑡+𝑠))⟩ = 1
det′𝑀
1
det𝑁(𝑡+𝑠)
⟨𝛿(𝛾(𝑢1))𝛿(𝛾(𝑢2)) … 𝛿(𝛾(𝑢𝑠))𝛾(𝑢𝑠+1)𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡+𝑠))𝛽(𝑤𝑡+𝑠+1)⟩ = det𝑁(𝑡+𝑠+1)
det′𝑀(det𝑁(𝑡+𝑠))2
⟨𝛿(𝛾(𝑢1))𝛿(𝛾(𝑢2)) … 𝛿(𝛾(𝑢𝑠))𝛾(𝑢𝑠+1)𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡+𝑠))𝛽(𝑤𝑡+𝑠+1)⟩
⟨𝛿(𝛾(𝑢1))𝛿(𝛾(𝑢2)) … 𝛿(𝛾(𝑢𝑠))𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡+𝑠))⟩ = det𝑁(𝑡+𝑠+1)
det𝑁(𝑡+𝑠)
ℒ2 ≅ 𝐾−1 ⊗𝑖=1
𝑛R 𝒪(−𝑝𝑖)
Θ−𝑡 = 𝛿(𝛾)𝛿(𝜕𝛾) … 𝛿(𝜕𝑡−1𝛾)
ℒ2 ≅ 𝐾−1 ⊗ 𝒪(−𝑝)
𝛾̂ (𝑧)(𝑧−1𝜕𝑧)1/2
𝑠 = 1
𝑧1/2 𝑠⋄
𝛾⋄(𝑧)(𝜕𝑧)1/2.
⟨𝛾(𝑢)𝛽(𝑤)⟩𝑁,𝛿 = det𝑁(𝑡+𝑠+1)
det𝑁(𝑡+𝑠)
ℒ̃ = ℒ ⊗𝑖=1
𝑠 𝒪(−𝑢𝑖) ⊗𝑗=1
𝑡+𝑠 𝒪(𝑤𝑗)
𝑉𝑎 = ∑
𝑛−𝑚
𝑖=1
𝑣𝑎,𝑖
𝜕
𝜕𝑓𝑖
𝐢𝑉𝑎 = ∑
𝑛−𝑚
𝑖=1
𝑣𝑎,𝑖
𝜕
𝜕 d𝑓𝑖
.
𝑉 = ∑
∞
𝑛=1
𝐿−𝑛𝑈𝑛
𝑏𝑛𝒲 = 0, 𝑛 ≥ 0,
𝑉 = 𝐿−1Φ0,
𝒲 = Φ0.
𝑉 = (𝐿−2 + 3
2 𝐿−1
2 ) Φ−1,
𝒲 = 𝑏𝑐Φ−1 + 3
2 𝐿−1Φ−1.
𝑉 = 𝐿−1Φ0 + (𝐿−2 + 3
2 𝐿−1
2 ) Φ−1
⟨𝛿(𝛾(𝑢1)) … 𝛿(𝛾∗(𝑢𝑠))𝛿(𝛽∗(𝑢1)) … 𝛿(𝛽∗(𝑢𝑡+𝑠))⟩ = 1
det′𝑀
1
det𝑁(𝑡+𝑠)
⟨𝛿(𝛾(𝑢1))𝛿(𝛾(𝑢2)) … 𝛿(𝛾(𝑢𝑠))𝛾(𝑢𝑠+1)𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡+𝑠))𝛽(𝑤𝑡+𝑠+1)⟩ = det𝑁(𝑡+𝑠+1)
det′𝑀(det𝑁(𝑡+𝑠))2
⟨𝛿(𝛾(𝑢1))𝛿(𝛾(𝑢2)) … 𝛿(𝛾(𝑢𝑠))𝛾(𝑢𝑠+1)𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡+𝑠))𝛽(𝑤𝑡+𝑠+1)⟩
⟨𝛿(𝛾(𝑢1))𝛿(𝛾(𝑢2)) … 𝛿(𝛾(𝑢𝑠))𝛿(𝛽(𝑤1)) … 𝛿(𝛽(𝑤𝑡+𝑠))⟩ = det𝑁(𝑡+𝑠+1)
det𝑁(𝑡+𝑠)
ℒ2 ≅ 𝐾−1 ⊗𝑖=1
𝑛R 𝒪(−𝑝𝑖)
Θ−𝑡 = 𝛿(𝛾)𝛿(𝜕𝛾) … 𝛿(𝜕𝑡−1𝛾)
ℒ2 ≅ 𝐾−1 ⊗ 𝒪(−𝑝)
𝛾̂ (𝑧)(𝑧−1𝜕𝑧)1/2
𝑠 = 1
𝑧1/2 𝑠⋄
𝛾⋄(𝑧)(𝜕𝑧)1/2.
⟨𝛾(𝑢)𝛽(𝑤)⟩𝑁,𝛿 = det𝑁(𝑡+𝑠+1)
det𝑁(𝑡+𝑠)
ℒ̃ = ℒ ⊗𝑖=1
𝑠 𝒪(−𝑢𝑖) ⊗𝑗=1
𝑡+𝑠 𝒪(𝑤𝑗)
𝑉𝑎 = ∑
𝑛−𝑚
𝑖=1
𝑣𝑎,𝑖
𝜕
𝜕𝑓𝑖
𝐢𝑉𝑎 = ∑
𝑛−𝑚
𝑖=1
𝑣𝑎,𝑖
𝜕
𝜕 d𝑓𝑖
.
𝑉 = ∑
∞
𝑛=1
𝐿−𝑛𝑈𝑛
𝑏𝑛𝒲 = 0, 𝑛 ≥ 0,
𝑉 = 𝐿−1Φ0,
𝒲 = Φ0.
𝑉 = (𝐿−2 + 3
2 𝐿−1
2 ) Φ−1,
𝒲 = 𝑏𝑐Φ−1 + 3
2 𝐿−1Φ−1.
𝑉 = 𝐿−1Φ0 + (𝐿−2 + 3
2 𝐿−1
2 ) Φ−1
pág. 118
𝒲 = Φ0 + (𝑏𝑐 + 3
2 𝐿−1) Φ−1
𝑄𝐵 = 𝑄> + 𝑄0
𝑄𝐵𝒲 = 𝒱,
𝑁∗
lc = ∑
𝑚≥1
1
𝑚 𝛼−𝑚
+ 𝛼𝑚
−
𝑄> = 𝑄>,1 + 𝑄>,0 + 𝑄>,−1
𝑄>,1 = −(2𝛼′)1/2𝑘+ ∑
𝑚≥1
𝛼𝑚
− 𝑐−𝑚
𝑅 = 1
(2𝛼′)1/2𝑘+ ∑
𝑚≥1
𝛼−𝑚
+ 𝑏𝑚
𝑆 = {𝑄>,1, 𝑅} = ∑
∞
𝑚=1
(𝑚𝑐−𝑚𝑏𝑚 − 𝛼−𝑚
+ 𝛼𝑚
− )
𝑉 = ∑
𝑛>0
𝐿−𝑛
𝑋 𝑊𝑛 + ∑
𝑟>0
𝐺−𝑟
𝑋 Λ𝑟
𝑉 = 𝐺−1/2
𝑋 Φ0
𝒲 = 𝑐𝛿′(𝛾)Φ0
𝛽𝑟| − 1⟩ = 𝛾𝑟| − 1⟩ = 0, 𝑟 > 0
𝒲 = −𝑐1𝛽−1/2| − 1⟩ ⊗ Φ0
𝑉 = (𝐺−3/2
𝑋 + 2𝐺−1/2
𝑋 𝐿−1
𝑋 )Φ−1
𝒲 = (𝛿(𝛾)𝐺−1/2
𝑋 − 𝑐𝛽𝛿(𝛾) + 𝑐𝛿′(𝛾)𝐿−1
𝑋 )Φ−1
𝒲 = (𝐺−1/2
𝑋 − 𝑐1𝛽−3/2 − 𝑐1𝛽−1/2𝐿−1
𝑋 )| − 1⟩ ⊗ Φ−1
𝑉 = (𝐿−1
𝑋 − 1
2 𝐺−1
𝑋 𝐺0
𝑋) Φ
𝛽𝑛Θ−1/2 = 0, 𝑛 ≥ 0
𝛾𝑛Θ−1/2 = 0, 𝑛 > 0
𝒲 = (1 − 𝐺0
gh𝐺0
𝑋) Θ−1/2Φ
𝒲 = (1 + 1
2 𝑐1𝛽−1𝐺0
𝑋) Θ−1/2Φ
𝑄𝐵 = 𝑄> + 𝑄0,
𝒲 = Φ0 + (𝑏𝑐 + 3
2 𝐿−1) Φ−1
𝑄𝐵 = 𝑄> + 𝑄0
𝑄𝐵𝒲 = 𝒱,
𝑁∗
lc = ∑
𝑚≥1
1
𝑚 𝛼−𝑚
+ 𝛼𝑚
−
𝑄> = 𝑄>,1 + 𝑄>,0 + 𝑄>,−1
𝑄>,1 = −(2𝛼′)1/2𝑘+ ∑
𝑚≥1
𝛼𝑚
− 𝑐−𝑚
𝑅 = 1
(2𝛼′)1/2𝑘+ ∑
𝑚≥1
𝛼−𝑚
+ 𝑏𝑚
𝑆 = {𝑄>,1, 𝑅} = ∑
∞
𝑚=1
(𝑚𝑐−𝑚𝑏𝑚 − 𝛼−𝑚
+ 𝛼𝑚
− )
𝑉 = ∑
𝑛>0
𝐿−𝑛
𝑋 𝑊𝑛 + ∑
𝑟>0
𝐺−𝑟
𝑋 Λ𝑟
𝑉 = 𝐺−1/2
𝑋 Φ0
𝒲 = 𝑐𝛿′(𝛾)Φ0
𝛽𝑟| − 1⟩ = 𝛾𝑟| − 1⟩ = 0, 𝑟 > 0
𝒲 = −𝑐1𝛽−1/2| − 1⟩ ⊗ Φ0
𝑉 = (𝐺−3/2
𝑋 + 2𝐺−1/2
𝑋 𝐿−1
𝑋 )Φ−1
𝒲 = (𝛿(𝛾)𝐺−1/2
𝑋 − 𝑐𝛽𝛿(𝛾) + 𝑐𝛿′(𝛾)𝐿−1
𝑋 )Φ−1
𝒲 = (𝐺−1/2
𝑋 − 𝑐1𝛽−3/2 − 𝑐1𝛽−1/2𝐿−1
𝑋 )| − 1⟩ ⊗ Φ−1
𝑉 = (𝐿−1
𝑋 − 1
2 𝐺−1
𝑋 𝐺0
𝑋) Φ
𝛽𝑛Θ−1/2 = 0, 𝑛 ≥ 0
𝛾𝑛Θ−1/2 = 0, 𝑛 > 0
𝒲 = (1 − 𝐺0
gh𝐺0
𝑋) Θ−1/2Φ
𝒲 = (1 + 1
2 𝑐1𝛽−1𝐺0
𝑋) Θ−1/2Φ
𝑄𝐵 = 𝑄> + 𝑄0,
pág. 119
𝑁∗
lc = ∑
𝑚≥1
1
𝑚 𝛼−𝑚
+ 𝛼𝑚
− − ∑
𝑟≥1/2
𝜓−𝑟
+ 𝜓𝑟
−
𝑄> = 𝑄>,1 + 𝑄>,0 + 𝑄>,−1
𝑄>,1 = −(2𝛼′)1/2𝑘+ ∑
𝑚≥1
𝛼𝑚
− 𝑐−𝑚 + (2𝛼′)1/2𝑘+ ∑
𝑟≥1/2
𝛾−𝑟𝜓𝑟
−
∏
𝑠≥1
𝛽−𝑠
𝑛𝑠 ∏
𝑟≥0
𝛾−𝑟
𝑚𝑟 Θ−1/2
𝑁∗
lc = ∑
𝑚≥1
1
𝑚 𝛼−𝑚
+ 𝛼𝑚
− − ∑
𝑟≥0
𝜓−𝑟
+ 𝜓𝑟
−
𝑄>,1 = −(2𝛼′)1/2𝑘+ ∑
𝑚≥1
𝛼𝑚
− 𝑐−𝑚 + (2𝛼′)1/2𝑘+ ∑
𝑟≥0
𝛾−𝑟𝜓𝑟
−
𝑧 ≅ 𝑧 + 1
𝜃 ≅ −𝜃
𝑧 ≅ 𝑧 + 𝜏
𝜃 ≅ 𝜃
𝑧̃ ≅ 𝑧̃ + 1 ≅ 𝑧̃ + 𝜏̃
𝑞NS = 𝑢1 − 𝑢2 − 𝜁1𝜁2
𝑢≅ 𝑢 + 1
𝜁1≅ −𝜁1
𝜁2≅ 𝜁2
𝑢̃ ≅ 𝑢̃ + 1
𝑢≅ 𝑢 + 𝜏
𝜁1≅ 𝜁1
𝜁2≅ 𝜁2
𝑢̃ ≅ 𝑢̃ + 𝜏‾
𝑞̃ = 𝑢̃
𝑞NS = 𝑢 − 𝜁1𝜁2
𝑢̃ = 𝑢 + 𝜁1𝜁2ℎ(𝑢̃ , 𝑢)
ℎ(𝑢̃ + 1, 𝑢 + 1) = −ℎ(𝑢̃ , 𝑢), ℎ(𝑢̃ + 𝜏‾, 𝑢 + 𝜏) = ℎ(𝑢̃ , 𝑢)
ℎ(0,0) = −1
ℎ𝜆 = 𝜆ℎ1 + (1 − 𝜆)ℎ2, 0 ≤ 𝜆 ≤ 1
Ω = [d𝑢̃ ; d𝑢 ∣ d𝜁1 d𝜁2]𝑃(𝑢̃ ),
𝑃(𝑢̃ + 1) = −𝑃(𝑢̃ ), 𝑃(𝑢̃ + 𝜏‾) = 𝑃(𝑢̃ )
𝑁∗
lc = ∑
𝑚≥1
1
𝑚 𝛼−𝑚
+ 𝛼𝑚
− − ∑
𝑟≥1/2
𝜓−𝑟
+ 𝜓𝑟
−
𝑄> = 𝑄>,1 + 𝑄>,0 + 𝑄>,−1
𝑄>,1 = −(2𝛼′)1/2𝑘+ ∑
𝑚≥1
𝛼𝑚
− 𝑐−𝑚 + (2𝛼′)1/2𝑘+ ∑
𝑟≥1/2
𝛾−𝑟𝜓𝑟
−
∏
𝑠≥1
𝛽−𝑠
𝑛𝑠 ∏
𝑟≥0
𝛾−𝑟
𝑚𝑟 Θ−1/2
𝑁∗
lc = ∑
𝑚≥1
1
𝑚 𝛼−𝑚
+ 𝛼𝑚
− − ∑
𝑟≥0
𝜓−𝑟
+ 𝜓𝑟
−
𝑄>,1 = −(2𝛼′)1/2𝑘+ ∑
𝑚≥1
𝛼𝑚
− 𝑐−𝑚 + (2𝛼′)1/2𝑘+ ∑
𝑟≥0
𝛾−𝑟𝜓𝑟
−
𝑧 ≅ 𝑧 + 1
𝜃 ≅ −𝜃
𝑧 ≅ 𝑧 + 𝜏
𝜃 ≅ 𝜃
𝑧̃ ≅ 𝑧̃ + 1 ≅ 𝑧̃ + 𝜏̃
𝑞NS = 𝑢1 − 𝑢2 − 𝜁1𝜁2
𝑢≅ 𝑢 + 1
𝜁1≅ −𝜁1
𝜁2≅ 𝜁2
𝑢̃ ≅ 𝑢̃ + 1
𝑢≅ 𝑢 + 𝜏
𝜁1≅ 𝜁1
𝜁2≅ 𝜁2
𝑢̃ ≅ 𝑢̃ + 𝜏‾
𝑞̃ = 𝑢̃
𝑞NS = 𝑢 − 𝜁1𝜁2
𝑢̃ = 𝑢 + 𝜁1𝜁2ℎ(𝑢̃ , 𝑢)
ℎ(𝑢̃ + 1, 𝑢 + 1) = −ℎ(𝑢̃ , 𝑢), ℎ(𝑢̃ + 𝜏‾, 𝑢 + 𝜏) = ℎ(𝑢̃ , 𝑢)
ℎ(0,0) = −1
ℎ𝜆 = 𝜆ℎ1 + (1 − 𝜆)ℎ2, 0 ≤ 𝜆 ≤ 1
Ω = [d𝑢̃ ; d𝑢 ∣ d𝜁1 d𝜁2]𝑃(𝑢̃ ),
𝑃(𝑢̃ + 1) = −𝑃(𝑢̃ ), 𝑃(𝑢̃ + 𝜏‾) = 𝑃(𝑢̃ )
pág. 120
𝑃(𝑢̃ ) = ∑
𝑛,𝑚∈ℤ
(−1)𝑛
𝑢̃ + 𝑛 + 𝑚𝜏‾
𝐼 = ∫
Γ
Ω
𝑢 = 𝑢̃ − 𝜁1𝜁2ℎ(𝑢̃ , 𝑢̃ )
[d𝑢̃ ; d𝑢 ∣ d𝜁1 d𝜁2] = (1 − 𝜁1𝜁2
𝜕ℎ
𝜕𝑢̃ ) [d𝑢̃ d𝑢̃ ∣ d𝜁1 d𝜁2]
𝐼 = ∫
Γ
[d𝑢̃ d𝑢̃ ∣ d𝜁1 d𝜁2] (1 − 𝜁1𝜁2
𝜕ℎ
𝜕𝑢̃ ) 𝑃(𝑢̃ )
𝐼 = − ∫
Σred
d𝑢̃ ∧ d𝑢̃ 𝜕ℎ
𝜕𝑢̃ 𝑃(𝑢̃ )
𝐼 = 4𝜋𝑖
𝑧 → −𝑧
𝜃 → ±√−1𝜃.
𝑢→ −𝑢
𝜁𝑖 → ±√−1𝜁𝑖, 𝑖 = 1,2
𝑢→ 𝑢
𝜁𝑖 → −𝜁𝑖, 𝑖 = 1,2
CONCLUSIONES.
Se concluye entonces, que la supergravedad cuántica, es un estado de la gravedad propiamente dicha,
en la que, el tejido temporo – espacial, no conserva su geometría inicial. La deformación y por ende, la
torsión del espacio – tiempo cuántico, no es local, de tal suerte, que la producción de multiespacios es
inminente. La supergravedad cuántica, ocurre en circunstancias extremadamente hostiles de la materia,
en las que, a escala microscópica, una partícula estrella u oscura, según sea el caso, se aniquila o colapsa,
provocando, inicialmente, un agujero negro masivo o supermasivo, según la emisión de radición que se
tenga por implícita, y simultáneamente, la formación de dimensiones en ℝ𝜂. En este punto, es
indispensable anotar, que una partícula, sea cual sea, puede interactuar en distintas dimensiones, al
mismo tiempo. En un escenario de supergravedad cuántica, la materia y la energía alcanzan densidades
exponenciales, lo que las funde, transformándolas finalmente, en materia y energía oscuras,
yuxtapuestas en un tejido espacio – tiempo pluridimensional y no local.
𝑃(𝑢̃ ) = ∑
𝑛,𝑚∈ℤ
(−1)𝑛
𝑢̃ + 𝑛 + 𝑚𝜏‾
𝐼 = ∫
Γ
Ω
𝑢 = 𝑢̃ − 𝜁1𝜁2ℎ(𝑢̃ , 𝑢̃ )
[d𝑢̃ ; d𝑢 ∣ d𝜁1 d𝜁2] = (1 − 𝜁1𝜁2
𝜕ℎ
𝜕𝑢̃ ) [d𝑢̃ d𝑢̃ ∣ d𝜁1 d𝜁2]
𝐼 = ∫
Γ
[d𝑢̃ d𝑢̃ ∣ d𝜁1 d𝜁2] (1 − 𝜁1𝜁2
𝜕ℎ
𝜕𝑢̃ ) 𝑃(𝑢̃ )
𝐼 = − ∫
Σred
d𝑢̃ ∧ d𝑢̃ 𝜕ℎ
𝜕𝑢̃ 𝑃(𝑢̃ )
𝐼 = 4𝜋𝑖
𝑧 → −𝑧
𝜃 → ±√−1𝜃.
𝑢→ −𝑢
𝜁𝑖 → ±√−1𝜁𝑖, 𝑖 = 1,2
𝑢→ 𝑢
𝜁𝑖 → −𝜁𝑖, 𝑖 = 1,2
CONCLUSIONES.
Se concluye entonces, que la supergravedad cuántica, es un estado de la gravedad propiamente dicha,
en la que, el tejido temporo – espacial, no conserva su geometría inicial. La deformación y por ende, la
torsión del espacio – tiempo cuántico, no es local, de tal suerte, que la producción de multiespacios es
inminente. La supergravedad cuántica, ocurre en circunstancias extremadamente hostiles de la materia,
en las que, a escala microscópica, una partícula estrella u oscura, según sea el caso, se aniquila o colapsa,
provocando, inicialmente, un agujero negro masivo o supermasivo, según la emisión de radición que se
tenga por implícita, y simultáneamente, la formación de dimensiones en ℝ𝜂. En este punto, es
indispensable anotar, que una partícula, sea cual sea, puede interactuar en distintas dimensiones, al
mismo tiempo. En un escenario de supergravedad cuántica, la materia y la energía alcanzan densidades
exponenciales, lo que las funde, transformándolas finalmente, en materia y energía oscuras,
yuxtapuestas en un tejido espacio – tiempo pluridimensional y no local.
pág. 121
REFERENCIAS BIBLIOGRÁFICAS.
Elias Kiritsis, INTRODUCTION TO SUPERSTRING THEORY, https://arxiv.org/abs/hep-
th/9709062v2.
Edward Witten, Perturbative Superstring Theory Revisited, arXiv:1209.5461v4 [hep-th] 5 Nov 2023.
Albuja Bustamante, M. I. (2024). Demostración del Espectro Hamiltoniano para un Campo de Yang-
Mills no Abeliano que Poseen una Brecha de Masa Finita con Respecto al Estado de
Vacío. Ciencia Latina Revista Científica Multidisciplinar, 8(1).
https://doi.org/10.37811/cl_rcm.v8i1.9738.
Manuel Ignacio Albuja Bustamante, Teoría de Campos: Reforzamiento Teórico – Matemático al
Modelo Estándar de Partículas, bajo la estructura ecuacional de Yang – Mills, Ciencia Latina
Revista Científica Multidisciplinar: Vol. 8 Núm. 2 (2024)
Manuel Ignacio Albuja Bustamante, La brecha de masa y la curvatura de los campos cuánticos, Ciencia
Latina Revista Científica Multidisciplinar: Vol. 8 Núm. 4 (2024)
Manuel Ignacio Albuja Bustamante, Aportes Matemáticos en Econometría Teórica, Ciencia Latina
Revista Científica Multidisciplinar: Vol. 8 Núm. 4 (2024)
Manuel Ignacio Albuja Bustamante, Formalización Matemática y en Física de Partículas, en Relación a
la Brecha de Masa y la Curvatura Geométrica de los Campos Cuánticos, Ciencia Latina Revista
Científica Multidisciplinar: Vol. 8 Núm. 5 (2024)
Manuel Ignacio Albuja Bustamante, Campos Cuánticos Relativistas: Aproximaciones Teórico –
Matemáticas Relativas a los Espacios Cuánticos Geométricamente Deformados o Perforados
por Partículas y Antipartículas Supermasivas y Masivas E Hiperpartículas y Suprapartículas,
Ciencia Latina Revista Científica Multidisciplinar: Vol. 9 Núm. 1 (2025)
Manuel Ignacio Albuja Bustamante, Teoría Cuántica de Campos Relativistas. Formalización Teórica,
Ciencia Latina Revista Científica Multidisciplinar: Vol. 9 Núm. 3 (2025)
Albuja Bustamante, M. I. (2025). Las partículas supermasivas: naturaleza fenomenológica de la
partícula oscura. Ciencia Latina Revista Científica Multidisciplinar, 9(3).
https://doi.org/10.37811/cl_rcm.v9i3.18172
REFERENCIAS BIBLIOGRÁFICAS.
Elias Kiritsis, INTRODUCTION TO SUPERSTRING THEORY, https://arxiv.org/abs/hep-
th/9709062v2.
Edward Witten, Perturbative Superstring Theory Revisited, arXiv:1209.5461v4 [hep-th] 5 Nov 2023.
Albuja Bustamante, M. I. (2024). Demostración del Espectro Hamiltoniano para un Campo de Yang-
Mills no Abeliano que Poseen una Brecha de Masa Finita con Respecto al Estado de
Vacío. Ciencia Latina Revista Científica Multidisciplinar, 8(1).
https://doi.org/10.37811/cl_rcm.v8i1.9738.
Manuel Ignacio Albuja Bustamante, Teoría de Campos: Reforzamiento Teórico – Matemático al
Modelo Estándar de Partículas, bajo la estructura ecuacional de Yang – Mills, Ciencia Latina
Revista Científica Multidisciplinar: Vol. 8 Núm. 2 (2024)
Manuel Ignacio Albuja Bustamante, La brecha de masa y la curvatura de los campos cuánticos, Ciencia
Latina Revista Científica Multidisciplinar: Vol. 8 Núm. 4 (2024)
Manuel Ignacio Albuja Bustamante, Aportes Matemáticos en Econometría Teórica, Ciencia Latina
Revista Científica Multidisciplinar: Vol. 8 Núm. 4 (2024)
Manuel Ignacio Albuja Bustamante, Formalización Matemática y en Física de Partículas, en Relación a
la Brecha de Masa y la Curvatura Geométrica de los Campos Cuánticos, Ciencia Latina Revista
Científica Multidisciplinar: Vol. 8 Núm. 5 (2024)
Manuel Ignacio Albuja Bustamante, Campos Cuánticos Relativistas: Aproximaciones Teórico –
Matemáticas Relativas a los Espacios Cuánticos Geométricamente Deformados o Perforados
por Partículas y Antipartículas Supermasivas y Masivas E Hiperpartículas y Suprapartículas,
Ciencia Latina Revista Científica Multidisciplinar: Vol. 9 Núm. 1 (2025)
Manuel Ignacio Albuja Bustamante, Teoría Cuántica de Campos Relativistas. Formalización Teórica,
Ciencia Latina Revista Científica Multidisciplinar: Vol. 9 Núm. 3 (2025)
Albuja Bustamante, M. I. (2025). Las partículas supermasivas: naturaleza fenomenológica de la
partícula oscura. Ciencia Latina Revista Científica Multidisciplinar, 9(3).
https://doi.org/10.37811/cl_rcm.v9i3.18172