Universal Functions for Topological Correlators

TL;DR

This paper derives universal functions describing topological correlators in twisted supersymmetric Yang-Mills theory, linking them to intersection theory, Seiberg-Witten geometry, and moduli spaces of sheaves, with explicit formulas and consistency checks.

Contribution

It provides explicit closed-form expressions for universal functions in topologically twisted $ ext{SU}(2)$ gauge theories with matter, connecting physical correlators to geometric invariants.

Findings

Universal functions encode intersection numbers of instanton moduli spaces.

Explicit formulas derived using Seiberg-Witten geometry, $u$-plane integral, and blowup formula.

Results match known generating functions for Segre invariants on algebraic surfaces.

Abstract

We consider correlation functions of topologically twisted, $\mathcal{N}=2$ supersymmetric Yang-Mills theory with gauge group ${\rm SU}(2)$ and $N_f\leq 3$ massive hypermultiplets in the fundamental representation. For a smooth, compact, oriented four-manifold $X$ with $b_2^+>1$, the correlation functions are expressed in terms of a finite set of universal functions. The mass dependence of these functions encodes intersection numbers of the moduli space of instantons. We determine closed expressions for the universal functions by combining techniques of the Seiberg-Witten geometry, $u$-plane integral and the blowup formula. If $X$ is specialised to a complex algebraic surface $S$, the correlation functions can be identified with generating functions of Segre invariants for moduli spaces of sheaves on $S$. We verify that our results agree with the results by G\"ottsche and Kool for these generating functions.