Deforming field theories with $U(1)\times U(1)$ global symmetry and their gravity duals

TL;DR

This paper constructs gravity duals for marginal deformations of super Yang-Mills theories with $U(1) imes U(1)$ symmetry, revealing a method to generate solutions using an $SL(2,R)$ transformation related to star product deformations.

Contribution

It provides a general approach to find gravity duals of $eta$ deformations for theories with $U(1) imes U(1)$ symmetry, extending to D3 branes at toric singularities.

Findings

Gravity duals for $eta$ deformations are constructed using $SL(2,R)$ transformations.

The method applies to theories with $U(1) imes U(1)$ symmetry, including those on D3 branes at toric singularities.

Deformations can be interpreted as arising from a star product in the field theory.

Abstract

We find the gravity dual of a marginal deformation of ${\cal N}=4$ super Yang Mills, and discuss some of its properties. This deformation is intimately connected with an $SL(2,R)$ symmetry of the gravity theory. The $SL(2,R)$ transformation enables us to find the solutions in a simple way. These field theory deformations, sometimes called $\beta$ deformations, can be viewed as arising from a star product. Our method works for any theory that has a gravity dual with a $U(1)\times U(1)$ global symmetry which is realized geometrically. These include the field theories that live on D3 branes at the conifold or other toric singularities, as well as their cascading versions.