  Three-dimensional N = 4 supersymmetric quantum field theories admit two
topological twists, the Rozansky-Witten twist and its mirror. Either twist can
be used to define a supersymmetric compactification on a Riemann surface and a
corre- sponding space of supersymmetric ground states. These spaces of ground
states can play an interesting role in the Geometric Langlands program. We
propose a description of these spaces as conformal blocks for certain
non-unitary Vertex Operator Algebras and test our conjecture in some important
examples. The two VOAs can be constructed respectively from a UV Lagrangian
description of the N = 4 theory or of its mirror. We further conjecture that
the VOAs associated to an N = 4 SQFT inherit properties of the theory which
only emerge in the IR, such as enhanced global symmetries. Thus knowledge of
the VOAs should allow one to compute the spaces of supersymmetric ground states
for a theory coupled to supersymmetric background connections for the full
symmetry group of the IR SCFT. In particular, we propose a conformal field
theory description of the spaces of ground states for the T[SU(N)] theories.
These theories play a role of S-duality kernel in maximally supersymmetric
SU(N) gauge theory and thus the corresponding spaces of supersymmetric ground
states should provide a kernel for the Geometric Langlands duality for special
unitary groups.
