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

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.