Towards U(N|M) knot invariant from ABJM theory

TL;DR

This paper explores the U(N|M) supermatrix Chern-Simons theory to derive knot invariants, establishing a determinantal formula linking U(N|M) and U(1|1) averages, and introduces a new model related to torus knots.

Contribution

It derives a determinantal formula for U(N|M) character expectations and introduces a supermatrix model connected to torus knot invariants.

Findings

U(N|M) character expectation values expressed via U(1|1) averages.

A new supermatrix model related to torus knot Chern-Simons theory.

Spectral curve and Rosso-Jones-type formula derived for the new model.

Abstract

We study U(N|M) character expectation value with the supermatrix Chern-Simons theory, known as the ABJM matrix model, with emphasis on its connection to the knot invariant. This average just gives the half BPS circular Wilson loop expectation value in ABJM theory, which shall correspond to the unknot invariant. We derive the determinantal formula, which gives U(N|M) character expectation values in terms of U(1|1) averages for a particular type of character representations. This means that the U(1|1) character expectation value is a building block for all the U(N|M) averages, and in particular, by an appropriate limit, for the U(N) invariants. In addition to the original model, we introduce another supermatrix model obtained through the symplectic transform, which is motivated by the torus knot Chern-Simons matrix model. We obtain the Rosso-Jones-type formula and the spectral curve for this case.