Chern-Simons Theory with Wilson Lines and Boundary in the BV-BFV Formalism

TL;DR

This paper explores the BV-BFV formalism for Chern-Simons theory with Wilson lines, revealing connections to boundary actions, representation theory, and conformal blocks in WZW models.

Contribution

It extends the BV-BFV formalism to include Wilson lines ending on boundaries, linking boundary actions to known models and representation theory.

Findings

Quantized boundary actions match known operators in representation theory.

Boundary state space aligns with conformal blocks of WZW models.

Wilson lines influence boundary dynamics and state spaces.

Abstract

We consider the Chern-Simons theory with Wilson lines in 3D and in 1D in the BV-BFV formalism of Cattaneo-Mnev-Reshetikhin. In particular, we allow for Wilson lines to end on the boundary of the space-time manifold. In the toy model of 1D Chern-Simons theory, the quantized BFV boundary action coincides with the Kostant cubic Dirac operator which plays an important role in representation theory. In the case of 3D Chern-Simons theory, the boundary action turns out to be the odd (degree 1) version of the BF model with source terms for the B field at the points where the Wilson lines meet the boundary. The boundary space of states arising as the cohomology of the quantized BFV action coincides with the space of conformal blocks of the corresponding WZW model.