Multi-Instantons, Supersymmetry and Topological Field Theories

TL;DR

This work addresses nonperturbative contributions in $N=2$ Super Yang-Mills by showing an exact equivalence between the constrained instanton method and the topologically twisted approach (Topological Yang-Mills). It develops an extended BRST framework that remains nilpotent in the presence of a nonzero scalar vev and rewrites instanton actions as BRST variations, so Seiberg--Witten prepotential coefficients arise from integrals of total derivatives over the instanton moduli space ${\mathscr{M}}^{+}$. Through a detailed ADHM realization of the BRST algebra on ${\mathscr{M}}^{+}$, the authors derive the instanton measure as a determinant of a moduli-dependent matrix $K$, providing a geometrical route to nonperturbative amplitudes and reproducing known results for $k=1$ and $k=2$. The framework emphasizes boundary contributions from ${\partial}{\mathscr{M}}^{+}$ and suggests a natural, potentially simplifying dilute-gas interpretation for small instantons in calculating the Seiberg--Witten data. Overall, the paper offers a unified, geometrically transparent method for nonperturbative effects in SUSY gauge theories with practical ties to the SW prepotential and low-energy dynamics.

Abstract

In this letter we argue that instanton-dominated Green's functions in N=2 Super Yang-Mills theories can be equivalently computed either using the so-called constrained instanton method or making reference to the topological twisted version of the theory. Defining an appropriate BRST operator (as a supersymmetry plus a gauge variation), we also show that the expansion coefficients of the Seiberg-Witten effective action for the low-energy degrees of freedom can be written as integrals of total derivatives over the moduli space of self-dual gauge connections.