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Register a new userThe perturbation of the Seiberg-Witten equations revisited
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We introduce a new class of perturbations of the Seiberg-Witten equations. Our perturbations offer flexibility in the way the Seiberg-Witten invariants are constructed and also shed a new light to LeBruns curvature inequalities.
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We define Seiberg-Witten equations on closed manifolds endowed with a Riemannian foliation of codimension 4. When the foliation is taut, we show compactness of the moduli space under some hypothesis satisfied for instance by closed K-contact manifolds. Furthermore, we prove some vanishing and non-vanishing results and we highlight that the invariants may be used to distinguish different foliations on diffeomorphic manifolds.
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.
Previously, Cristofaro-Gardiner, Hutchings and Ramos have proved that embedded contact homology (ECH) capacities can recover the volume of a contact 3-manifod in their paper the asymptotics of ECH capacities . There were two main steps to proving this theorem: The first step used an estimate for the energy of min-max Seiberg-Witten Floer generators. The second step used embedded balls in a certain symplectic four manifold. In this paper, stronger estimates on the energy of min-max Seiberg-Witten Floer generators are derived. This stronger estimate implies directly the ECH capacities recover volume theorem (without the help of embedded balls in a certain symplectic four manifold), and moreover, gives an estimate on its speed.
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 $t in mathbb{R} setminus { 0 }$.
In this paper we apply the idea of Higgs branch localization to 5d supersymmetric theories of vector multiplet and hypermultiplets, obtained as the rigid limit of $mathcal{N} = 1$ supergravity with all auxiliary fields. On supersymmetric K-contact/Sasakian background, the Higgs branch BPS equations can be interpreted as 5d generalizations of the Seiberg-Witten equations. We discuss the properties and local behavior of the solutions near closed Reeb orbits. For $U(1)$ gauge theories, we show the suppression of the deformed Coulomb branch, and the partition function is dominated by 5d Seiberg-Witten solutions at large $zeta$-limit. For squashed $S^5$ and $Y^{pq}$ manifolds, we show the matching between poles in the perturbative Coulomb branch matrix model, and the bound on local winding numbers of the BPS solutions.
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