Tau-functions, twistor theory, and quantum field theory

TL;DR

This paper explores the geometric origins of tau functions in integrable systems through twistor theory and connects quantum field theoretic tau functions to geometric quantization, providing new insights into their structure.

Contribution

It introduces a geometric quantization framework for tau functions using twistor correspondences and relates quantum field theory formulations to geometric data.

Findings

Tau functions can be derived from twistor geometry.

Quantum field theoretic tau functions arise from geometric quantization.

The approach unifies integrable systems and quantum field theories through geometry.

Abstract

This article is concerned with obtaining the standard tau function descriptions of integrable equations (in particular, here the KdV and Ernst equations are considered) from the geometry of their twistor correspondences. In particular, we will see that the quantum field theoretic formulae for tau functions can be understood as arising from geometric quantization of the twistor data. En route we give a geometric quantization formulation of Chern-Simons and WZW quantum field theories using the Quillen determinant line bundle construction and ingredients from Segal's conformal field theory. The $\tau$-functions are then seen to be amplitudes associated with gauge group actions on certain coherent states within these theories that can be obtained from the twistor description.