Sigma model instantons and singular tau function

TL;DR

This paper demonstrates that the instanton contributions in the 2D sigma model can be expressed as a formal singular tau function of the two-sided two-component KP hierarchy, which can be regularized to yield meaningful physical observables.

Contribution

It introduces a novel representation of the instanton series as a formal singular tau function within the KP hierarchy framework, enabling regularization for physical relevance.

Findings

The instanton series corresponds to a formal singular tau function.

Regularization of the singular tau function produces finite, meaningful observables.

The approach extends the family of tau functions to handle divergences.

Abstract

The generating series for the instanton contribution to Green functions of the $2D$ sigma model was found in the works of Schwarz, Fateev and Frolov. We show that this series can be written as a formal tau function of the two-sided two-component KP hierarchy. We call it formal singular tau function because this tau function is a sum where each term is the infrared and ultraviolet divergent one exactly as the series found by the mentioned authors. However one can regularize this singluar tau function and to obtain regular observables. This is because observables contains ratious of mentioned divergent expressions. Thus, we enladge the families of tau functions to work with.