Marc Herzlich
We prove that the round metric on the sphere has the largest first eigenvalue of the Dirac operator among all metrics that are larger than it. As a corollary, this gives an alternative proof of an extremality result for scalar curvature due to M. Llarull.
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In this paper we consider the Vafa-Witten equations on closed, oriented and smooth 4-manifolds, and construct a set of perturbation terms to establish the transversality of the perturbed Vafa-Witten equations at the general part of the solutions. Then we show that for a generic choice of the perturbation terms, this part of the moduli space for the structure group $SU(2)$ or $SO(3)$ on a closed 4-manifold is a smooth manifold of dimension zero.
Previously we finish the establishment of the transversality of the general part of the Vafa-Witten moduli spaces, in this paper, we deal with the rest, i.e., the reduced part. We consider Vafa-Witten equation on closed, oriented and smooth Riemann 4-manifolds with $C\equiv0$, and construct perturbation to establish the transversality of the perturbed equation. Then we show that for a generic choice of the perturbation terms, the moduli space of solutions to the perturbed reduced Vafa-Witten equation for the structure group $SU(2)$ or $SO(3)$ on a closed 4-manifold is a smooth manifold of dime…
In this paper we consider a variation of the Vafa-Witten equations on compact, oriented and smooth 4-manifolds, and construct a set of perturbation terms to establish the transversality of that equations. The new perturbed equations provide us a priori estimates of the solutions, while the original reduced Vafa-Witten equations does not. By applying the a priori estimates we show that the singular of the solutions can be removed, and then construct the Ulhenbeck closure of the moduli spaces.
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 para…
We prove that the largest first eigenvalue of the Dirac operator among all hermitian metrics on the complex projective space of odd dimension $m$, larger than the Fubini-Study metric is bounded by $(2m(m+1))^{1/2}$.
We give an optimal upper bound for the first eigenvalue of the untwisted Dirac operator on a compact symmetric space G/H with rk G-rk H\le 1 with respect to arbitrary Riemannian metrics. We also prove a rigidity statement.
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…
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 Hitchin's 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 c…
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 $(\mathcal{E}, φ)$, where $\mathcal{E}$ is a locally-free sheaf over a surface $X$ and $φ$ 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 …
A simple perturbation of Vafa-Witten equations and a transversality result
该研究针对Vafa-Witten方程引入简单扰动，证明在结构群为SU(2)或SO(3)、扰动参数具一般性的情况下，扰动后模空间的满秩部分满足横截性条件，为相关模空间研究提供了横截性结果支撑。
This article provides a summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of nested Hilbert schemes of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.
On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with compact fixed locus. Applying virtual localisation we define invariants constant under deformations. When the vanishing theorem of Vafa-Witten holds, the result is the (signed) Euler characteristic of the moduli space of instantons. In general there are other, rational, contributions. Calculations of these on surfaces with positive canonical bundle recover th…
This paper is the the third part of a series of paper whose aim is to use of the framework of \emph{twisted spectral triples} to study conformal geometry from a noncommutive geometric viewpoint. In this paper we reformulate the inequality of Vafa-Witten \cite{VW:CMP84} in the setting of twisted spectral triples. This involves a notion of Poincaré duality for twisted spectral triples. Our main results have various consequences. In particular, we obtain a version in conformal geometry of the original inequality of Vafa-Witten, in the sense of an explicit control of the Vafa-Witten bound under co…
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 $K_S\le0$ 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 Vafa-Witten equations on an oriented Riemannian 4- manifold are first order, non-linear equations for a pair of connection on a principle SO(3) bundle over the 4-manifold and a self-dual 2-form with values in the associated Lie algebra bundle. The main theorem in this paper characterizes in part the behavior of sequences of solutions to the Vafa-Witten equations which have no convergent subsequence. The paper proves that a renormalization of a subsequence of the self-dual 2-form components converges on the complement of a closed set with Hausdorff dimension at most 2, with the limit being …
Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification
该研究在完整Vafa-Witten方程的模空间设定下重新研究Vafa-Witten理论，从物理推导新型不变量、Floer同调及几何Langlands对应等结果，揭示了数学多分支与物理拓扑场论的联系。
On Vafa-Witten equations over Kaehler manifolds
本文研究紧凯勒流形上Vafa-Witten方程解的解析性质，证明模空间在C*不变轨迹上的紧性类似Hermitian-Yang-Mills联络，并通过重整化Higgs场和谱覆盖收敛给出Taubes秩二解结果的新证明。
Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification (String Math 2022 Proceedings)
本文是Vafa-Witten理论相关研究论文的精简版，将刊载于美国数学学会出版的《弦数学2022会议论文集》，原论文对应arXiv:2203.17115。
Non-convergent sequences of solutions to the massive Vafa-Witten equations with 'interesting' $\mathbb{Z}/2\mathbb{Z}$ self-dual harmonic 2-form limits
本文在2维环面与亏格大于1的Riemann曲面乘积上构造了Vafa-Witten方程的发散解序列，经重整化后收敛于一个包含余维2子流形及实线丛的“有趣”$\mathbb{Z}/2\mathbb{Z}$自对偶调和2-形式数据集。
Recovering complex structure from $(n,0)$-form with implications for Vafa-Witten equation
本文研究从类似标量 $(n,0)$-形式的丛值形式恢复 $\mathbb{Z}_2$-（几乎）复结构，在 $4$-流形和四元数-Kähler 流形上可全局恢复，并应用于 Vafa-Witten 方程，表明无相容 $\mathbb{Z}_2$-复结构时解必平凡。
Refined $\mathrm{SU}(3)$ Vafa-Witten invariants and modularity
本文针对满足特定条件的光滑曲面提出精细化 SU(3) Vafa-Witten 不变量猜想公式，修正既有方案并证明其满足精细化 S-对偶模性，通过模空间虚拟亏格计算与嵌套 Hilbert 概型结果提供证据。
Virtual refinements of the Vafa-Witten formula
本文提出 Vafa-Witten 公式的虚拟精细化猜想，将秩 2 层模空间的虚拟 $χ_y$-亏格用 descendent Donaldson 不变量和 Seiberg-Witten 不变量表示，并通过局部化计算验证该猜想。
Spectral flow calculations for reducible solutions to the massive Vafa-Witten equations
本文针对有向紧致黎曼4-流形上的Vafa-Witten方程，研究其可约解序列定义的椭圆算子谱流的上下界，发现谱流绝对值在部分情形有界、部分情形发散，并引入了具有独立价值的局部化与切除计算技术。
Vafa-Witten Equations and Conformal Geometry
本文通过共形不变性和Bochner型估计，建立了闭4-流形上Vafa-Witten方程非平凡解存在的几何与分析约束，包括Yamabe常数与自对偶Weyl张量的不等式，并应用于正爱因斯坦流形和降维对应。
$\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
本文基于S-对偶性和虚拟退化轨迹理论，提出SU(r) Vafa-Witten配分函数的结构公式，并利用theta函数、Seiberg-Witten不变量及Ramanujan连分数给出r=4,5的新表达式，同时预测稳定层模空间的虚拟Euler特征。
Refined Vafa-Witten invariants for toric surfaces from supersymmetric localization in 5D gauge theory
研究五维\(\mathcal{N}=1\) \(U(N)\)超对称杨 - 米尔斯理论在复Kähler曲面相关空间上的配分函数，扩展前人工作，通过路径积分定域及处理极点，发现非偶数陈类贡献与精细化Vafa-Witten不变量一致，偶数陈类情况待明确。
Higher rank flag sheaves on Surfaces and Vafa-Witten invariants
本文研究了曲面上全纯三元组及链的模空间，在Schmitt稳定性下建立完美形变-阻碍理论，并通过wallcrossing将其与Higgs pairs及局部DT不变量相联系，从而计算高秩旗层对Vafa-Witten配分函数的贡献。
In this article, we consider the Kapustin-Witten equations on a closed $4$-manifold. We study certain analytic properties of solutions to the equations on a closed manifold. The main result is that there exists an $L^{2}$-lower bound on the extra fields over a closed four-manifold satisfying certain conditions if the connections are not ASD connections. Furthermore, we also obtain a similar result about the Vafa-Witten equations.
We offer a new construction of Lagrangian submanifolds for the Gopakumar-Vafa conjecture relating the Chern-Simons theory on the 3-sphere and the Gromov-Witten theory on the resolved conifold. Given a knot in the 3-sphere its conormal bundle is perturbed to disconnect it from the zero section and then pulled through the conifold transition. The construction produces totally real submanifolds of the resolved conifold that are Lagrangian in a perturbed symplectic structure and correspond to knots in a natural and explicit way. We prove that both the resolved conifold and the knot Lagrangians in …
In this article, we study the Kapustin-Witten equations on a closed, simply-connected, four-dimensional manifold which were introduced by Kapustin and Witten. We use the Taubes' compactness theorem in arXiv:1307.6447v4 to prove that if $(A,ϕ)$ is a smooth solution of Kapustin-Witten equations and the connection $A$ is closed to a $generic$ ASD connection $A_{\infty}$, then $(A,ϕ)$ must be a trivial solution. We also prove that the moduli space of the solutions of Kapustin-Witten equations is non-connected if the connections on the compactification of moduli space of ASD connections are all $ge…
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