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Register a new userStable sheaves with twisted sections and the Vafa-Witten equations on smooth projective surfaces
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Authors
Yuuji Tanaka
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This article describes a Hitchin-Kobayashi style correspondence for the Vafa-Witten equations on smooth projective surfaces. This is an equivalence between a suitable notion of stability for a pair $(mathcal{E}, varphi)$, where $mathcal{E}$ is a locally-free sheaf over a surface $X$ and $varphi$ is a section of $text{End} (mathcal{E}) otimes K_{X}$; and the existence of a solution to certain gauge-theoretic equations, the Vafa-Witten equations, for a Hermitian metric on $mathcal{E}$. It turns out to be a special case of results obtained by Alvarez-Consul and Garcia-Prada. In this article, we give an alternative proof which uses a Mehta-Ramanathan style argument originally developed by Donaldson for the Hermitian-Einstein problem, as it relates the subject with the Hitchin equations on Riemann surfaces, and surely indicates a similar proof of the existence of a solution under the assumption of stability for the Donaldson-Thomas instanton equations described in arXiv:0805.2192 on smooth projective threefolds; and more broadly that for the quiver vortex equation on higher dimensional smooth projective varieties.
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This article finds a structure of singular sets on compact Kahler surfaces, which Taubes introduced in the studies of the asymptotic analysis of solutions to the Kapustin-Witten equations and the Vafa-Witten ones originally on smooth four-manifolds. These equations can be seen as real four-dimensional analogues of the Hitchin equations on Riemann surfaces, and one of common obstacles to be overcome is a certain unboundedness of solutions to these equations, especially of the Higgs fields. The singular sets by Taubes describe part of the limiting behaviour of a sequence of solutions with this unboundedness property, and Taubes proved that the real two-dimensional Haussdorff measures of these singular sets are finite. In this article, we look into the singular sets, when the underlying manifold is a compact Kahler surface, and find out that they have the structure of an analytic subvariety in this case.
We propose a definition of Vafa-Witten invariants counting semistable Higgs pairs on a polarised surface. We use virtual localisation applied to Mochizuki/Joyce-Song pairs. For $K_Sle0$ we expect our definition coincides with an alternative definition using weighted Euler characteristics. We prove this for deg $K_S<0$ here, and it is proved for $S$ a K3 surface in cite{MT}. For K3 surfaces we calculate the invariants in terms of modular forms which generalise and prove conjectures of Vafa and Witten.
The $mathrm{SU}(r)$ Vafa-Witten partition function, which virtually counts Higgs pairs on a projective surface $S$, was mathematically defined by Tanaka-Thomas. On the Langlands dual side, the first-named author recently introduced virtual counts of Higgs pairs on $mu_r$-gerbes. In this paper, we instead use Yoshiokas moduli spaces of twisted sheaves. Using Chern character twisted by rational $B$-field, we give a new mathematical definition of the $mathrm{SU}(r) / mathbb{Z}_r$ Vafa-Witten partition function when $r$ is prime. Our definition uses the period-index theorem of de Jong. $S$-duality, a concept from physics, predicts that the $mathrm{SU}(r)$ and $mathrm{SU}(r) / mathbb{Z}_r$ partitions functions are related by a modular transformation. We turn this into a mathematical conjecture, which we prove for all $K3$ surfaces and prime numbers $r$.
We consider a set of gauge-theoretic equations on closed oriented four-manifolds, which was introduced by Vafa and Witten. The equations involve a triple consisting of a connection and extra fields associated to a principal bundle over a closed oriented four-manifold. They are similar to Hitchins equations over compact Riemann surfaces, and as part of the resemblance, there is no $L^2$-bound on the curvature without an $L^2$-bound on the extra fields. In this article, however, we observe that under the particular circumstance where the curvature does not become concentrated and the limiting connection is not locally reducible, one obtains an $L^2$-bound on the extra fields.
We prove a Freed-Uhlenbeck style generic smoothness theorem for the moduli space of solutions to the Vafa--Witten equations on a closed symplectic four-manifold by using a method developed by Feehan for the study of the $PU(2)$-monopole equations on smooth closed four-manifolds. We introduce a set of perturbation terms to the Vafa--Witten equations, and prove that the moduli space of solutions to the perturbed Vafa-Witten equations on a closed symplectic four-manifold for the structure group $SU(2)$ or $SO(3)$ is a smooth manifold of dimension zero for a generic choice of the perturbation parameters.
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