Integrability in non-perturbative QFT

TL;DR

This paper explores the deep connections between non-perturbative quantum field theories and integrable systems, revealing hidden symmetries and relations among correlation functions, with applications to various models including knot invariants.

Contribution

It extends the understanding of integrability in non-perturbative QFT, linking partition functions and correlation functions to tau-functions of integrable systems, and explores their applications to knot theory.

Findings

Partition functions exhibit hidden symmetries and integrability.

Knot polynomials can be organized into tau-functions.

Hurwitz partition functions encompass various knot invariants.

Abstract

Exact non-perturbative partition functions of coupling constants and external fields exhibit huge hidden symmetry, reflecting the possibility to change integration variables in the functional integral. In many cases this implies also some non-linear relations between correlation functions, typical for the tau-functions of integrable systems. To a variety of old examples, from matrix models to Seiberg-Witten theory and AdS/CFT correspondence, now adds the Chern-Simons theory of knot invariants. Some knot polynomials are already shown to combine into tau-functions, the search for entire set of relations is still in progress. It is already known, that generic knot polynomials fit into the set of Hurwitz partition functions -- and this provides one more stimulus for studying this increasingly important class of deformations of the ordinary KP/Toda tau-functions.