$N=2$ Topological Yang-Mills Theories and Donaldson's Polynomials

TL;DR

This paper reexamines $N=2$ topological Yang-Mills theories on Kähler surfaces, clarifies their symmetry via Dolbeault models, and computes Donaldson's polynomial invariants, connecting physical theories with mathematical invariants.

Contribution

It provides a new interpretation of $N=2$ topological Yang-Mills theories using Dolbeault models and explicitly computes Donaldson's polynomials on specific cohomology groups.

Findings

Clarified $N=2$ symmetry in terms of Dolbeault models

Realized both algebraic and non-algebraic parts of Donaldson invariants

Calculated Donaldson's polynomials on $H^{2,0}(S,\mathbb{Z}) \oplus H^{0,2}(S,\mathbb{Z})$

Abstract

The $N=2$ topological Yang-Mills and holomorphic Yang-Mills theories on simply connected compact K\"{a}hler surfaces with $p_g\geq 1$ are reexamined. The $N=2$ symmetry is clarified in terms of a Dolbeault model of the equivariant cohomology. We realize the non-algebraic part of Donaldson's polynomial invariants as well as the algebraic part. We calculate Donaldson's polynomials on $H^{2,0}(S,\BZ)\oplus H^{0,2}(S,\BZ)$.