Wilson Loop Invariants from $W_N$ Conformal Blocks

TL;DR

This paper develops a method to compute knot and link invariants, specifically HOMFLY polynomials, using $W_N$ conformal blocks in 2D CFT, extending the toolkit for topological invariants in quantum field theory.

Contribution

It introduces calculations of crossing and braiding matrices for $W_N$ conformal blocks with mixed representations, enabling new ways to derive knot invariants.

Findings

Computed crossing and braiding matrices for $W_N$ conformal blocks.

Derived HOMFLY invariants for specific representations.

Outlined potential generalizations to higher-representations.

Abstract

Knot and link polynomials are topological invariants calculated from the expectation value of loop operators in topological field theories. In 3D Chern-Simons theory, these invariants can be found from crossing and braiding matrices of four-point conformal blocks of the boundary 2D CFT. We calculate crossing and braiding matrices for $W_N$ conformal blocks with one component in the fundamental representation and another in a rectangular representation of $SU(N)$, which can be used to obtain HOMFLY knot and link invariants for these cases. We also discuss how our approach can be generalized to invariants in higher-representations of $W_N$ algebra.