Twisted compactifications of 3d N = 4 theories and conformal blocks

TL;DR

This work proposes a framework in which 3d N=4 theories compactified on a Riemann surface are associated with non-unitary vertex operator algebras whose conformal blocks reproduce spaces of supersymmetric ground states, linking to Geometric Langlands via duality kernels. Central to the construction are the H- and C-twists, realized respectively by symplectic-boson and fermionic-current VOAs, and the coset/BRST formalisms that define the A_H[T] and A_C[T] algebras. The paper develops extensive free-hypermultiplet and Abelian examples, and then non-Abelian unitary quivers, to reveal IR symmetry enhancements in the VOAs that mirror the known IR enhancements of the gauge theories, including T[SU(N)] and related theories. The results provide concrete VOA-based descriptions of ground-state spaces, D-module/sheaf structures over moduli spaces of bundles and flat connections, and a path toward a geometric Langlands interpretation of 3d gauge theories via conformal blocks of these VOAs.

Abstract

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.