Communications in Number Theory and Physics

Volume 14 (2020)

Number 2

Quantum Langlands dualities of boundary conditions, $D$-modules, and conformal blocks

Pages: 199 – 313



Edward Frenkel (Department of Mathematics, University of California at Berkeley)

Davide Gaiotto (Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada)


We review and extend the vertex algebra framework linking gauge theory constructions and a quantum deformation of the Geometric Langlands Program. The relevant vertex algebras are associated to junctions of two boundary conditions in a 4d gauge theory and can be constructed from the basic ones by following certain standard procedures. Conformal blocks of modules over these vertex algebras give rise to twisted $D$-modules on the moduli stacks of $G$-bundles on Riemann surfaces which have applications to the Langlands Program. In particular, we construct a series of vertex algebras for every simple Lie group $G$ which we expect to yield $D$-module kernels of various quantum Geometric Langlands dualities. We pay particular attention to the full duality group of gauge theory, which enables us to extend the standard qGL duality to a larger duality groupoid. We also discuss various subtleties related to the spin and gerbe structures and present a detailed analysis for the $U(1)$ and $SU(2)$ gauge theories.

E.F. was supported by NSF grant DMS-1601934. D.G. is supported by the NSERC Discovery Grant program and by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.

Received 14 June 2018

Accepted 22 October 2019

Published 30 March 2020