Chern–Simons theory with Wilson lines and boundary in the BV–BFV formalism

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.