Supersymmetric Yang-Mills Theory On A Four-Manifold

TL;DR

This work shows how Donaldson invariants on Kahler four-manifolds can be computed by embedding topological Donaldson theory into a twisted, mass-gap like setting derived from N=1 supersymmetric Yang-Mills theory. By twisting N=2 to expose BRST structure, introducing a holomorphic mass term via a Kahler form, and exploiting a mass gap and vacuum structure, the author derives generating functions for Donaldson invariants that agree with known mathematical results in key cases (e.g., K3, simple-type manifolds). The approach clarifies how canonical divisor geometry and cosmic-string-like two-dimensional sectors control the topological correlators through intersection data, yielding a Gaussian-plus-exponentials structure constrained by symmetry and global data. This provides a physical framework that reproduces and explains intricate mathematical invariants, and extends to higher-rank groups and general Kahler geometries, with implications for understanding four-manifold topology through supersymmetric field theory.

Abstract

By exploiting standard facts about $N=1$ and $N=2$ supersymmetric Yang-Mills theory, the Donaldson invariants of four-manifolds that admit a Kahler metric can be computed. The results are in agreement with available mathematical computations, and provide a powerful check on the standard claims about supersymmetric Yang-Mills theory.