In this PhD Thesis, supersymmetry and its formulation in the context of D=11
supergravity is discussed from several perspectives. The role of generalized
holonomy as a classification tool of supersymmetric supergravity solutions is
reviewed, with particular emphasis on how successive supercovariant derivatives
of the generalized curvature may be needed to properly define the generalized
holonomy algebra. The generalized curvature is also shown to contain the
supergravity equations of motion, even in the non-vanishing gravitino case. The
underlying gauge symmetry of D=11 supergravity is discussed and argued to
become manifest when its three-form field A_3 is expressed through a set of
one-form gauge fields, associated with the generators of a suitable family of
enlarged supersymmetry algebras. This family is related to osp(1|32) through
expansion, a method to obtain new Lie (super)algebras of increasing dimensions
from given ones. The analysis of the underlying gauge symmetry of D=11
supergravity leads naturally to enlarged supersymmetry algebras and superspaces
making, thus, natural to consider actions for objects moving in such spaces. In
particular, a string moving in tensorial space is discussed, describing the
excitations of a state preserving 30 out of 32 supersymmetries (hence composed
of two preons, hypothetical constituents of M Theory preserving 31
supersymmetries). A G-frame method is also discussed to study hypothetical
preonic solutions of supergravity.Comment: PhD Thesis (Advisor: J.A. de Azcarraga). 16+201 page

Varela, Oscar

English

arXiv

SUMMARY
Eleven-dimensional M Theory is conjectured to contain and unify quantum gravity and the rest of fundamental interactions. Despite its current lack of a precise formulation, hints and information about M Theory can be retrieved from its low energy limit, eleven-dimensional supergravity. In fact, this Thesis aims to make progress towards the understanding of M Theory by the study of eleven dimensional supergravity. Two important questions in M Theory yet to be solved are 1) how many supersymmetries can an M Theory background preserve and 2) what is the symmetry of M Theory. Both questions are addressed in this Thesis at the D=11 supergravity level.
The former question can be studied for purely bosonic solutions by means of generalized holonomy, namely, the holonomy of the supersymmetric connection defining the supersymmetry variation of the gravitino. The role of generalized holonomy in this context is reviewed, and successive covariant derivatives of the curvature of the supersymmetry connection are shown to be necessary to properly define the generalized holonomy algebra.
As for the later, the underlying gauge group structure of D=11 supergravity becomes manifest when its three-form field A_3 is expressed through a set of one-form gauge fields, associated with the generators of a family of enlarged supersymmetry algebras (which are fermionic extensions of the M Theory superalgebra). The problem can be tackled by trivializing a standard supersymmetry algebra Chevalley-Eilenberg four-cocycle (associated to dA_3) on the enlarged superalgebras. The family of trivializing superalgebras are related to osp(1|32) through "expansion", a method to obtain new Lie (super)algebras of increasing dimensions from given ones, that is also discussed.
This analysis of the underlying gauge symmetry of D=11 supergravity leads naturally to enlarged supersymmetry algebras and superspaces making, thus, natural to consider actions for objects moving in such spaces. In particular, a string moving in tensorial space is discussed, describing the excitations of a state preserving 30 out of 32 supersymmetries (hence composed of two preons, hypothetical constituents of M Theory preserving 31 supersymmetries). A G-frame method is also discussed to study hypothetical preonic solutions of supergravity.
__________________________________________________________________________________________________
RESUMEN
La Teoría M se postula como unificadora de la gravedad cuántica y el resto de interacciones fundamentales. Pese a carecer actualmente de formulación precisa, se puede obtener información sobre ella a partir de su límite de baja energía, la supergravedad once-dimensional. De hecho, esta Tesis pretende contribuir al progreso en el entendimiento de la Teoría M mediante el estudio de la supergravedad en D=11. Dos cuestiones importantes en Teoría M pendientes de resolver son 1) cuántas supersimetrías puede preservar una solución de Teoría M y 2) cuál es la simetría de la Teoría M. Se discuten estas cuestiones desde el punto de vista de la supergravedad en D=11.
La primera cuestión se puede estudiar mediante la holonomía generalizada, es decir, la holonomía de la conexión supersimétrica que define la variación del gravitino bajo supersimetría. Se revisa el papel de la holonomía generalizada y se demuestra que las derivadas covariantes sucesivas de la curvatura de la conexión supersimétrica son necesarias para definir convenientemente el álgebra de holonomía generalizada.
En cuanto a la segunda cuestión, el grupo de gauge subyacente a la supergravedad en D=11 queda al descubierto al expresar la tres-forma A_3 por medio de un conjunto de uno-formas de gauge asociadas a una familia de álgebras de supersimetría agrandadas. El problema se reduce a la trivialización sobre las álgebras agrandadas de un cuatro-cociclo (asociado a dA_3) del álgebra de supersimetría estándar. Las superálgebras de trivialización están relacionadas con osp(1|32) a través de cierta "expansión", un método, explicado en detalle, para obtener a partir de una (super)álgebras otras nuevas, de dimensiones crecientes.
Este análisis conduce de forma natural a álgebras de supersimetría y superespacios agrandados, haciendo, pues, posible el planteamiento de acciones para objetos móviles en esos espacios. En particular, se discute una cuerda moviéndose en el espacio tensorial, que describe las excitaciones de un estado que preserva 30 supersimetrías de 32 (y compuesto, por tanto, de dos preones, los constituyentes hipotéticos de la Teoría M). También se discute un método de "G-frame" para el estudio de hipotéticas soluciones preónicas de supergravedad

Symmetry and holonomy in M Theory

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