Blowup formulae in Donaldson-Witten theory and integrable hierarchies

TL;DR

This paper explores blowup formulae in Donaldson-Witten theory with gauge group SU(N), revealing connections to hyperelliptic functions, contact terms, and integrable hierarchies like KdV and multisoliton solutions.

Contribution

It demonstrates that the blowup function is a hyperelliptic sigma-function and provides explicit expansions, linking Donaldson-Witten theory to integrable hierarchies and tau-functions.

Findings

Blowup function is a hyperelliptic sigma-function.

Correlation functions are governed by the KdV hierarchy.

For manifolds of simple type, the blowup function is a multisoliton tau-function.

Abstract

We investigate blowup formulae in Donaldson-Witten theory with gauge group SU(N), using the theory of hyperelliptic Kleinian functions. We find that the blowup function is a hyperelliptic sigma-function and we describe an explicit procedure to expand it in terms of the Casimirs of the gauge group up to arbitrary order. As a corollary, we obtain a new expression for the contact terms and we show that the correlation functions involving the exceptional divisor are governed by the KdV hierarchy. We also show that, for manifolds of simple type, the blowup function becomes a tau-function for a multisoliton solution.