The Donaldson-Witten function for gauge groups of rank larger than one

TL;DR

This paper explores the computation of topological correlation functions in supersymmetric Yang-Mills theory with higher-rank gauge groups, linking them to Seiberg-Witten invariants and vacua structures, with applications to superconformal theories and large N expansions.

Contribution

It provides explicit formulas for correlation functions in higher-rank gauge groups, extending previous results and connecting topological invariants with supersymmetric vacua.

Findings

Correlation functions are topologically invariant despite noncompact field space.

Expressed correlators in terms of Seiberg-Witten invariants and classical cohomology.

Explicit formulas for SU(N) gauge groups on manifolds of simple type.

Abstract

We study correlation functions in topologically twisted $\mathcal{N}=2, d=4$ supersymmetric Yang-Mills theory for gauge groups of rank larger than one on compact four-manifolds $X$. We find that the topological invariance of the generator of correlation functions of BRST invariant observables is not spoiled by noncompactness of field space. We show how to express the correlators on simply connected manifolds of $b_{2,+}(X)>0$ in terms of Seiberg-Witten invariants and the classical cohomology ring of $X$. For manifolds $X$ of simple type and gauge group SU(N) we give explicit expressions of the correlators as a sum over $\mathcal{N}=1$ vacua. We describe two applications of our expressions, one to superconformal field theory and one to large $N$ expansions of SU(N) $\mathcal{N}=2, d=4$ supersymmetric Yang-Mills theory.