Knot Invariants and New Weight Systems from General 3D TFTs

TL;DR

This paper explores Wilson loops in general 3D topological field theories, establishing their role in producing knot invariants and extending weight systems via advanced algebraic structures.

Contribution

It introduces a framework connecting Wilson loop expectation values to extended graph complexes and generalizes Lie algebra weight systems for knots.

Findings

Wilson loops yield knot invariants in general 3D TFTs

Established isomorphism between graph complexes and Chevalley-Eilenberg complexes

Constructed new knot invariants from holomorphic vector bundles over hyperKahler manifolds

Abstract

We introduce and study the Wilson loops in a general 3D topological field theories (TFTs), and show that the expectation value of Wilson loops also gives knot invariants as in Chern-Simons theory. We study the TFTs within the Batalin-Vilkovisky (BV) and Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) framework, and the Ward identities of these theories imply that the expectation value of the Wilson loop is a pairing of two dual constructions of (co)cycles of certain extended graph complex (extended from Kontsevich's graph complex to accommodate the Wilson loop). We also prove that there is an isomorphism between the same complex and certain extended Chevalley-Eilenberg complex of Hamiltonian vector fields. This isomorphism allows us to generalize the Lie algebra weight system for knots to weight systems associated with any homological vector field and its representations. As an example we construct knot invariants using holomorphic vector bundle over hyperKahler manifolds.