Holographic duals of lower dimensional theories

Abstract

This thesis describes some results in AdS/CFT duality for quantum field theories in 2+1 and 1+1 dimensions.

In first place, a one-parameter family of supergravity solutions is described, obtained as a deformation of a D2-branes system. These solutions are dual to a family of field theories in 2+1 dimensions. All the backgrounds in this family present a geometry that ends smoothly. But, except in a particular case, there is no confinement. The reason for this is that the string theory description of these solution is not valid and they are only regular solutions in the context of M-theory. In this description quarks (i.e. fields in the fundamental representation) are represented by membranes wrapped around a circle that can end smoothly in a cigar-like tip. This allows a disconnected configuration. Consequently, in the field theory there exists a mass gap due to the end of the geometry at a finite value of the radial coordinate, but there is no confinement. This unexpected lack of confinement is not the only remarkable feature of this family of solutions. One of the dual theories has a renormalization group (RG) flow that ends in a fixed point, that is, a conformal theory. Varying slightly the parameter that labels the solutions we end in solutions dual to theories whose RG flow passes near the fixed point, thus exhibiting a quasiconformal behaviour. All these theories also flow to the gapped phase at lower energies.

The existence of a quasiconformal regime in a RG flow implies that there is an approximated conformal symmetry. When the flow leaves the quasiconformal region this approximated symmetry is broken and this breaking can be either spontaneous or explicit. By the Goldstone’s theorem, when a symmetry is spontaneously broken a massless scalar Goldstone boson must appear for each of the generators of the symmetry. If the symmetry is not exact, the Goldstone boson is not massless but light. In the case of the conformal symmetry, the Goldsonte boson associated to its spontaneous breaking is a dilaton, a light pseudo-dilaton for the approximated symmetry. In chapter 3 we compute the spectra of spin-0 and spin-2 states. The result is that we did not find such pseudo-dilaton. The reason for this is that the spontaneous breaking of the symmetry is of the same order than the explicit breaking, Nevertheless, it is possible to simulate a spontaneous breaking of the conformal symmetry by putting a hard-wall cut-off in the infrared. By tuning the energy scale at which we impose the cut-off we can explore different energy regions of the flow, finding the quasiconformal behaviour in the appropriate solutions.

Finally, we consider a new system, consisting in D1-branes in a space that has as transverse manifold a space with special holonomy. This implies that the maximal supersymmetry corresponding to the D1-branes in flat space is partially broken and reduces to a number of supercharges that depends on the particular holonomy of the space considered. The internal manifolds that we study are tri-Sasaki, Sasaki-Einstein and weak-G2 , that reduce supersymmetry from N=(8, 8) to N=(0, 3), (0 ,2) and (0, 1) respectively. Furthermore, we introduce D5-branes sharing the gauge directions with the D1-branes, which in the gauge theory translates into the presence of matter in the fundamental representation. For this reason they are called flavour branes. In these flavoured supersymmetric theories we have found renormalization group flows that connect the brane-like ultraviolet with a fixed point.

En la presente tesis se exponen una serie de resultados en el marco de la dualidad AdS/CFT concernientes a teorías de campos en 2+1 y 1+1 dimensiones.

En primer lugar se detalla una familia uniparamétrica de soluciones de supergravedad duales a teorías 2+1-dimensionales basadas en conjuntos de N D2-branas con D4-branas fraccionales con una variedad transversa de holonomía Spin(7). Estas teorías tienen la particularidad de no ser confinantes a pesar de tener un mass gap. Además la presencia de un punto fijo en el flujo del grupo de renormalización de una de las teorías hace que exista un comportamiento cuasiconforme para las que tienen un valor del parámetro cercano.

Se calcula el espectro de modos de spin 0 y spin 2 para toda la familia de soluciones descrita y se estudia el comportamiento cuasiconforme de las teorías que lo presentan en función de la escala de energía usando una hard-wall en el infrarrojo.

Finalmente, se encuentran ecuaciones BPS que describen soluciones duales a teorías de supersimetría reducida (N=(0,3), N=(0,2) y N=(0,1)) en 1+1 dimensiones con grados de libertad de sabor. El sistema se construye considerando D1-branas con un espacio transverso de holonomía especial (Sp(2), SU(4) y Spin(7) respectivamente) y añadiendo D5-branas que comparten con las D1-branas las direcciones de la teoría gauge. En todos los sistemas considerados se encuentra una solución que fluye desde el ultravioleta de tipo D1-brana hasta una solución AdS3 en el infrarrojo.