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A simple perturbation of Vafa-Witten equations and a transversality result

Bo Dai, Ren Guan

TL;DR

The paper studies a simple perturbation of the Vafa-Witten equations on closed 4-manifolds with $SU(2)$ or $SO(3)$ bundles. It shows that for generic perturbations $\tau$, the full-rank part of the perturbed moduli space is a smooth, oriented, zero-dimensional manifold by proving transversality of the parameterized map $F$. The key analytic step is establishing the surjectivity of $dF_{(\tau,A,B,C)}$ at zeros with $B$ of rank $3$ via an $L^2$-adjoint analysis and unique continuation, which implies transversality and enables Sard-Smale genericity. This provides a solid analytic foundation for rank-3 Vafa-Witten moduli spaces and supports potential invariants defined from the perturbed theory.

Abstract

We consider a simple perturbation of the Vafa-Witten equations, and prove that for generic perturbation parameter, the full rank part of the perturbed Vafa-Witten moduli space satisfies transversality condition, when the structure group is $SU(2)$ or $SO(3)$.

A simple perturbation of Vafa-Witten equations and a transversality result

TL;DR

The paper studies a simple perturbation of the Vafa-Witten equations on closed 4-manifolds with or bundles. It shows that for generic perturbations , the full-rank part of the perturbed moduli space is a smooth, oriented, zero-dimensional manifold by proving transversality of the parameterized map . The key analytic step is establishing the surjectivity of at zeros with of rank via an -adjoint analysis and unique continuation, which implies transversality and enables Sard-Smale genericity. This provides a solid analytic foundation for rank-3 Vafa-Witten moduli spaces and supports potential invariants defined from the perturbed theory.

Abstract

We consider a simple perturbation of the Vafa-Witten equations, and prove that for generic perturbation parameter, the full rank part of the perturbed Vafa-Witten moduli space satisfies transversality condition, when the structure group is or .

Paper Structure

This paper contains 2 sections, 4 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Prove of the main theorem

Key Result

Theorem 1.1

Let $X$ be a closed, oriented, smooth, Riemannian 4-manifold, and $P\to X$ be a principal $SU(2)$- or $SO(3)$-bundle. Then for generic perturbation $\tau\in C^r(X,\mathfrak{gl}(\Lambda^{2,+}))$, where $r>k\geq 2$ are integers, the full rank part of the moduli space, $\mathcal{M}_{VW,\tau}^{(3)}(P)$,

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof