The Seiberg-Witten prepotential and the Euler class of the reduced moduli space of instantons

TL;DR

This paper links the n-instanton contribution to the Seiberg-Witten prepotential in N=2 supersymmetric Yang-Mills theory with the Euler class of the instanton moduli space, using equivariant cohomology and bundle theory.

Contribution

It provides a novel geometric interpretation of the instanton contributions as integrals of Euler classes over moduli spaces, connecting physical quantities with topological invariants.

Findings

Representation of instanton contributions as integrals of equivariantly exact forms.

Reformulation of the integral as a product of closed two-forms related to Euler classes.

Insight into the geometric structure of the instanton moduli space in supersymmetric gauge theories.

Abstract

The n-instanton contribution to the Seiberg-Witten prepotential of 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.