Vafa-Witten Equations and Conformal Geometry

Abstract

In this article, we establish geometric and analytic constraints imposed by the existence of nontrivial solutions to the Vafa-Witten equations on closed 4-manifolds. Using conformal invariance and refined Bochner-type estimates, we first prove an inequality relating the Yamabe constant $Y(g)$ to the $L^2$-norm of the self-dual Weyl tensor: $Y(g)\le 2\sqrt{6}\Vert W_g^{+}{\Vert }_{L^2}$; when $Y(g)>0$, this yields a topological lower bound ${\int }_M|W_g^{+}{|}^2\ge \frac{4}{3}\ddot{I}{€}^2(2\ddot{I}‡(M)+3\ddot{I}ƒ(M))$. In the equality case, we show that the manifold must be Kähler with nonnegative scalar curvature and that the connection is reducible. As an application, for positive Einstein manifolds with $\mathrm{Ric}=3g$ admitting an irreducible Vafa-Witten solution, we obtain a sharp volume bound and prove the manifold cannot be Kähler. Through dimensional reduction $S^1\times N$, we establish a one-to-one correspondence between stable flat connections on a closed 3-manifold $N$ and $S^1$-invariant Vafa-Witten solutions, which yields a new estimate for the Yamabe constant $Y(g_{S^1\times N})\le 2\sqrt{6\ddot{I}€}({\int }_N|\mathrm{Ric}(g_N)-\frac{1}{3}{R}_{g_N}{g}_N{|}^2{)}^{1/2}$. Finally, under a regularity assumption that every anti-self-dual connection in the compactified moduli space is regular, we prove an energy gap: there exists $\epsilon (g,P)>0$ such that any Vafa-Witten solution satisfies either $F_A^{+}\equiv 0$ or $\Vert F_A^{+}{\Vert }_{L^2}\ge \epsilon$.

All comments are welcome. 24 pages