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

TL;DR

The paper develops a general, solution-generating approach to construct gravity duals of marginal beta-deformations of gauge theories with a geometric $U(1)\times U(1)$ symmetry. By applying an $SL(2,\mathbb{R})$ transformation to backgrounds with a two-torus, the authors realize the field-theory beta deformation as a star-product and derive explicit deformed gravity solutions, including the ${\cal N}=4$ SYM case and extensions to toric conifold theories and KS-like cascades. Rational deformation parameters give rise to orbifolds with discrete torsion and rich Coulomb branches, while nonzero sigma extends the moduli via an $SL(3,\mathbb{R})$ structure and yields a calculable Zamolodchikov metric. The framework also encompasses pp-wave limits and generalizations to $U(1)^3$ symmetries, establishing a versatile, broadly applicable bridge between marginal deformations in field theory and their gravity duals.

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 $β$ 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.