On skew tau-functions in higher spin theory

TL;DR

This paper explores skew tau-functions in higher spin theory, revealing their properties, relations to Toda recursion, and potential as a bridge between knot theory and integrability after quantization.

Contribution

It introduces skew tau-functions in higher spin theory, highlighting their unique properties, simple matrix model representations, and their role in connecting knot and integrability theories.

Findings

Skew tau-functions satisfy Toda recursion.

They have simple matrix model and Schur function representations.

Potential to connect knot theory and integrability after quantization.

Abstract

Recent studies of higher spin theory in three dimensions concentrate on Wilson loops in Chern-Simons theory, which in the classical limit reduce to peculiar corner matrix elements between the highest and lowest weight states in a given representation of SL(N). Despite these "skew" tau-functions can seem very different from conventional ones, which are the matrix elements between the two highest weight states, they also satisfy the Toda recursion between different fundamental representations. Moreover, in the most popular examples they possess simple representations in terms of matrix models and Schur functions. We provide a brief introduction to this new interesting field, which, after quantization, can serve as an additional bridge between knot and integrability theories.