Schur Quantization and Complex Chern-Simons theory

TL;DR

This paper introduces a canonical Schur quantization method for K-theoretic Coulomb branches of 4D supersymmetric theories, enabling the quantization of complex phase spaces like character varieties and applications to Chern-Simons theory.

Contribution

It presents a novel Schur quantization approach for K-theoretic Coulomb branches, linking supersymmetric QFT to complex symplectic geometry and quantum topology.

Findings

Quantization of character varieties for complex gauge groups.

Application to Chern-Simons gauge theory.

New quantum deformation of the Lorentz group.

Abstract

Any four-dimensional Supersymmetric Quantum Field Theory with eight supercharges can be associated to a certain complex symplectic manifold called the "K-theoretic Coulomb branch" of the theory. The collection of K-theoretic Coulomb branches include many complex phase spaces of great interest, including in particular the "character varieties" of complex flat connections on a Riemann surface. The SQFT definition endows K-theoretic Coulomb branches with a variety of canonical structures, including a deformation quantization. In this paper we introduce a canonical "Schur" quantization of K-theoretic Coulomb branches. It is defined by a variant of the Gelfand-Naimark-Segal construction, applied to protected Schur correlation functions of half-BPS line defects. Schur quantization produces an actual quantization of the complex phase space. As a concrete application, we apply this construction to character varieties in order to quantize Chern-Simons gauge theory with a complex gauge group. Other applications include the definition of a new quantum deformation of the Lorentz group, and the solution of certain spectral problems via dualities.