The coefficients of the Seiberg-Witten prepotential as intersection numbers (?)

TL;DR

This paper expresses the n-instanton contribution to the Seiberg-Witten prepotential as intersection numbers on the instanton moduli space, linking supersymmetric gauge theory to topological invariants.

Contribution

It provides a novel geometric interpretation of the instanton contributions as intersection numbers, connecting physical calculations to topological and geometric structures.

Findings

Representation of instanton contributions as intersection numbers

Identification of the two-form as an Euler class representative

Commentary on the relation to Liouville theory and moduli spaces

Abstract

The $n$-instanton contribution to the Seiberg-Witten prepotential of ${\bf N}=2$ supersymmetric $d=4$ Yang Mills theory is represented as the integral of the exponential of an equivariantly exact form. Integrating out an overall scale and a U(1) angle the integral is rewritten as $(4n-3)$ fold product of a closed two form. This two form is, formally, a representative of the Euler class of the Instanton moduli space viewed as a principal U(1) bundle, because its pullback under bundel projection is the exterior derivative of an angular one-form. We comment on a recent speculation of Matone concerning an analogy linking the instanton problem and classical Liouville theory of punctured Riemann spheres.