Marginal deformations of 3d $\mathcal{N}=2$ CFTs from AdS$_4$ backgrounds in generalised geometry

TL;DR

This paper investigates exactly marginal deformations of 3d $ ext{N}=2$ CFTs dual to AdS$_4$ backgrounds in eleven-dimensional supergravity, using generalised geometry to identify conditions for deformations to be exactly marginal.

Contribution

It provides a general framework for understanding marginal deformations in terms of generalised geometry and derives explicit conditions for their exact marginality.

Findings

Deformation corresponds to turning on a four-form flux related to a holomorphic function.

Explicit examples include S$^7$, Q$^{1,1,1}$, and M$^{1,1,1}$.

Conditions for deformations to extend to all orders are identified.

Abstract

We study exactly marginal deformations of 3d $\mathcal{N}=2$ CFTs dual to AdS$_4$ solutions in eleven-dimensional supergravity using generalised geometry. Focussing on Sasaki-Einstein backgrounds, we find that marginal deformations correspond to turning on a four-form flux on the internal space at first order. Viewing this as the deformation of a generalised structure, we derive a general expression for the four-form flux in terms of a holomorphic function. We discuss the explicit examples of S$^7$, Q$^{1,1,1}$ and M$^{1,1,1}$ and, using an obstruction analysis, find the conditions for the first-order deformations to extend all orders, thus identifying which marginal deformations are exactly marginal. We also show how the all-orders $\gamma$-deformation of Lunin and Maldacena can be encoded as a tri-vector deformation in generalised geometry and outline how to recover the supergravity solution from the generalised metric.