Applications of Supersymmetry: Exact Results, Gauge/Gravity Duality and Condensed Matter

Abstract

The study of supersymmetry has led us to a better understanding of field theories, specially in the strong coupling regime. In this thesis we have tried to show this through several examples. These are:

- The first of these examples has been the application of localization techniques in supersymmetric theories. Specifically, we have used the partition function of N=2 supersymmetric Chern-Simons theory with gauge group U(N) and 2Nf flavors. To regularize the theory, it is necessary to make the computation in a three sphere whose radius, R, serves as an IR regulator which can be taken to infinity at the end of the computation.

Once we have the exact partition function in terms of a matrix integral, we can solve the integral by means of a saddle-point approximation. This approximation becomes exact in the large N limit. The saddle-point equation can be solved exactly and in the decompactification limit it shows different phases depending on the value of the 't Hooft coupling.

We have also computed the free energy and the vacuum expectation value of a Wilson loop for a big circle of the three sphere. Both of them show discontinuities in their derivatives, in particular, the discontinuity in the free energy appears in the third derivative and thus, both phase transitions are third order.

- Other application that we have seen consists of the use of the gauge/gravity duality. In particular, starting from the gravity dual to N=1 super Yang-Mills, proposed by Maldacena and Núñez, we have reviewed how to add flavors (quarks) to this theory, without mass first and with mass later. We have also seen how to extract information about the field theory from these gravity duals, we have paid special attention to how the beta-function of the field theory dual is obtained from the gravity background proposed by Conde, Gaillard and Ramallo, dual to N=1 super Yang-Mills field theory with Nf massive flavors and a quartic superpotential.

The main result from the point of view of the field theory is that, in the case Nf=2N, the beta-function shows a non-trivial UV fixed point, which hints on possible IR fixed point as proposed by Seiberg in the conformal window picture. No evidence of non-trivial fixed points is found for Nf different from 2N.

- Again, in the context of the gauge/gravity duality, we have studied how to generate new supergravity solutions applying T-duality and how this affects the G-structures that describe the supersymmetry of these solutions. We have applied T-duality to the IIB supergravity solution of Klebanov and Witten with flavors. The supersymmetry of these backgrounds can be described by an SU(3)-structure and an SU(2)-structure before and after T-dualizing, respectively.

- Finally, we have presented an N=1 supersymmetric model that exhibits a superconducting phase transition. This model is based on a quartic Kähler potential for a chiral multiplet and no superpotential. The main difference with standard superconductivity is that the phase transition becomes first order rather than second order. Another difference is that, as it is typical in supersymmetric theories, the dependence on the cut-off is softened in our model.

El estudio de supersimetría nos ha permitido un mejor entendimiento de las teorías de campos, especialmente en el régimen de acoplamiento fuerte. En esta tesis hemos tratado de mostrar esto a través de varios ejemplos. A saber:

- Primero hemos mostrado cómo calcular de manera exacta con técnicas de localización la función de partición de la teoría de Chern-Simons supersimétrica N=2 con grupo gauge U(N) y 2Nf sabores en una tres esfera. Una vez que tenemos la función de partición exacta en términos de una integral de matrices, podemos resolverla por medio de la aproximación de punto silla. Esta aproximación se vuelve exacta cuando tomamos el límite de N grande. En el límite de descompactificación calculamos la energía libre y el valor de expectación de un lazo de Wilson correspondiente a un círculo máximo de la tres-esfera, mostrando transiciones de fase de tercer orden.

- Otra aplicación ha consistido en el uso de la dualidad gravedad/gauge. Hemos revisado como construir una solución de supergravedad dual a la teoría N=1 super Yang-Mills con Nf sabores masivos y un superpotencial cuártico. El principal resultado desde el punto de vista de la teoría de campos que corresponde a dicha solución, consiste en que, en el caso Nf=2N, la función beta exhibe un punto fijo UV no trivial que da lugar a indicios acerca de un punto fijo IR adicional tal y como propone Seiberg en su propuesta de la ventana conforme.

- También en el contexto de la dualidad gravedad/gauge hemos estudiado cómo generar nuevas soluciones de supergravedad por medio de la aplicación de T-dualidad a la solución de supergravedad de Klebanov y Witten con sabores. La supersimetría de estas soluciones pueden ser descritas en términos de una SU(3) estructura antes de T-dualizar, o una SU(2)-estructura, después de T-dualizar.

- Finalmente, hemos presentado un modelo supersimétrico que exhibe una transición de fase superconductora basado en un potencial de Kähler cuártico para un multiplete quiral y sin superpotencial. Encontramos dos diferencias con superconductores usuales: nuestro modelo exhibe una transición de fase de primer orden, en lugar de ser de segundo orden, y la dependencia con la energía de corte es más suave.