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Register a new userOn the singular sets of solutions to the Kapustin-Witten equations and the Vafa-Witten ones on compact Kahler surfaces
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Yuuji Tanaka
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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.
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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.
Arising from a topological twist of $mathcal{N} = 4$ super Yang-Mills theory are the Kapustin-Witten equations, a family of gauge-theoretic equations on a four-manifold parametrized by $tinmathbb{P}^1$. The parameter corresponds to a linear combination of two super charges in the twist. When $t=0$ and the four-manifold is a compact Kahler surface, the equations become the Simpson equations, which was originally studied by Hitchin on a compact Riemann surface, as demonstrated independently in works of Nakajima and the third-named author. At the same time, there is a notion of $lambda$-connection in the nonabelian Hodge theory of Donaldson-Corlette-Hitchin-Simpson in which $lambda$ is also valued in $mathbb{P}^1$. Varying $lambda$ interpolates between the moduli space of semistable Higgs sheaves with vanishing Chern classes on a smooth projective variety (at $lambda=0$) and the moduli space of semisimple local systems on the same variety (at $lambda=1$) in the twistor space. In this article, we utilise the correspondence furnished by nonabelian Hodge theory to describe a relation between the moduli spaces of solutions to the equations by Kapustin and Witten at $t=0$ and $t in mathbb{R} setminus { 0 }$ on a smooth, compact Kahler surface. We then provide supporting evidence for a more general form of this relation on a smooth, closed four-manifold by computing its expected dimension of the moduli space for each of $t=0$ and $t in mathbb{R} setminus { 0 }$.
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.
In this article, we consider a gauge-theoretic equation on compact symplectic 6-manifolds, which forms an elliptic system after gauge fixing. This can be thought of as a higher-dimensional analogue of the Seiberg-Witten equation. By using the virtual neighbourhood method by Ruan, we define an integer-valued invariant, a 6-dimensional Seiberg-Witten invariant, from the moduli space of solutions to the equations, assuming that the moduli space is compact; and it has no reducible solutions. We prove that the moduli spaces are compact if the underlying manifold is a compact Kahler threefold. We then compute the integers in some cases.
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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