arXiv research

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 …

Paper finds instantons for Kapustin-Witten equations on a specific manifold.

The Kapustin-Witten equations relate to nonabelian Hodge theory on Kähler surfaces.

Lectures on linearized Kapustin-Witten equations on half-line.

In the present paper, we find a system of non-linear ODEs that gives rotationally invariant solutions to the Kapustin-Witten equations in 4-dimensional Euclidean space. We explicitly solve these ODEs in some special cases and find decaying rational solutions, which provide solutions to the Kapustin-Witten equations. Th…

This expository article introduces the Kapustin-Witten equations to mathematicians. We discuss the connections between the Complex Yang-Mills equations and the Kapustin-Witten equations. In addition, we show the relation between the Kapustin-Witten equations, the moment map condition and the gradient Chern-Simons flow.…

The paper establishes a Lagrangian correspondence linking different geometric structures on complex varieties.

We consider solutions of Kapustin-Witten equation with Nahm pole boundary on $S^3\times \mathbb{R}^+$. These solutions are usually called Nahm pole solutions. In this paper, we will prove that there exists a constant $C>0$ such that $\|F_A\|_{L^2}\leq C$ for any Nahm pole solution $(A,φ)$.

In the present paper, we establish a gluing construction for the Nahm pole solutions to the Kapustin-Witten equations over manifolds with boundaries and cylindrical ends. Given two Nahm pole solutions with some convergence assumptions on the cylindrical ends, we prove that there exists an obstruction class for gluing t…

This paper supplies a new characterization of the Kapustin Witten equation solutions on $(0,\infty) \times \mathbb{R}^2 \times \mathbb{R}$ that play a key role in Edward Witten's program to obtain the Jones polynomial knot invariants using solutions to the same equation on $(0,\infty) \times S^3$.

We study solutions to the Kapustin--Witten equations on ALE and ALF gravitational instantons. On any such space and for any compact structure group, we prove asymptotic estimates for the Higgs field. We then use it to prove a vanishing theorem in the case when the underlying manifold is $\mathrm{R}^4$ or $\mathrm{R}^3 …

In this note, we classify all solutions to the $\mathrm{SU(n)}$ Kapustin-Witten equations on $S^1\timesΣ\times \mathbb{R}^+$, where $Σ$ is a compact Riemann surface, with Nahm pole singularity at $S^1\timesΣ\times \{0\}$. We provide a similar classification of solutions with generalized Nahm pole singularities along a …

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 c…

For a 3-manifold $X$ and compact simple Lie group $G$, we study the expansions of polyhomogeneous Nahm pole solutions to the Kapustin-Witten equations over $X\times (0,+\infty)$. Let $y$ be the coordinate of $(0,+\infty)$, we prove that the sub-leading terms of a polyhomogeneous Nahm pole solution is smooth to the boun…

The Higgs field growth is studied on special geometric spaces, confirming a conjecture.

Paper proves existence of knot solutions for specific equations.

The Kapustin-Witten equations on R^4 are equations for a pair of connection on the product principle SU(2) bundle and 1-form with values in the product Lie algebra bundle. The 1-form is the Higgs field. A dichotomy is proved to the effect that either the averaged norm of the Higgs field on large radius spheres grows fa…

This paper describes the behavior of sequences of solutions to the Kapustin-Witten equations with Nahm pole asymptotics on the product of the half-line with a compact, oriented, Riemannian 3-manifold. These sequences have sub-sequences that either converge to another solution after acting term-wise by an automorphism o…

We explain how, starting with a stack of D4-branes ending on an NS5-brane in type IIA string theory, one can, via T-duality and the topological-holomorphic nature of the relevant worldvolume theories, relate (i) the lattice models realized by Costello's 4d Chern-Simons theory, (ii) links in 3d analytically-continued Ch…

The paper introduces a new family of instanton-invariants for four-manifolds and connects them to Khovanov homology.

Teichmüller TQFT is a unitary 3d topological theory whose Hilbert spaces are spanned by Liouville conformal blocks. It is related but not identical to PSL(2,R) Chern-Simons theory. To physicists, it is known in particular in the context of 3d-3d correspondence and also in the holographic description of Virasoro conform…

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 …

In this paper, we study the dimensionally reduced twisted Kapustin-Witten equations on the product of a compact Riemann surface $Σ$ with $\mathbb{R}^+$. The main result is a Kobayashi-Hitchin type correspondence between the space of tilted Nahm pole solutions and the moduli space of Beilinson-Drinfeld opers. This corro…

Explains a property of algebras related to quantum field theories.

In this article, we study the analytical properties of the solutions of the complex Yang-Mills equations on a closed Riemannian four-manifold $X$ with a Riemannian metric $g$. The main result is that if $g$ is $good$ and the connection is an approximate ASD connection, then the extra field has a positive lower {bound}.…

Constructs Lagrangian correspondences for Higgs bundles and holomorphic connections.

Growth of spinors in 4D and 3D generalized Seiberg-Witten equations.

Let $M$ be a holomorphic symplectic Kähler manifold equipped with a Lagrangian fibration $π$ with compact fibers. The base of this manifold is equipped with a special Kähler structure, that is, a Kähler structure $(I, g, ω)$ and a symplectic flat connection $\nabla$ such that the metric $g$ is locally the Hessian of a …

This paper explores adiabatic solutions of Haydys-Witten equations for knot homology.

We develop a Kobayashi-Hitchin correspondence for the extended Bogomolny equations, i.e., the dimensionally reduced Kapustin-Witten equations, on the product of a compact Riemann surface $Σ$ with ${\mathbb R}^+_y$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these…

Let $X$ be a compact manifold, $D$ a real elliptic operator on $X$, $G$ a Lie group, $P\to X$ a principal $G$-bundle, and ${\mathcal B}_P$ the infinite-dimensional moduli space of all connections $\nabla_P$ on $P$ modulo gauge, as a topological stack. For each $[\nabla_P]\in{\mathcal B}_P$, we can consider the twisted …

Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an orie…

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to th…

Introduces generalized hyperpolygons and their geometric and algebraic properties.

The paper connects function theory, dynamics, and ergodic theory via Thurston's theory.

Survey of Floer theories and their connections.

Lectures on topological field theories and differential cohomology.

The paper defines strong emergence in field theories and proves it exists between certain theories.

This is the first paper in a series introducing a generalized Fredholm theory in a new class of smooth spaces called polyfolds. The theory will be illustrated in upcoming papers by applications to Floer Theory, Gromov-Witten Theory and Symplectic Field Theory.

Unified Higgs bundle vacua from M-theory on Spin(7) spaces.

Researchers find new $G_2$-conifolds in $M$-theory with potential field theory duals.

We survey three different ways in which K-theory in all its forms enters quantum field theory. In Part 1 we give a general argument which relates topological field theory in codimension two with twisted K-theory, and we illustrate with some finite models. Part 2 is a review of pfaffians of Dirac operators, anomalies, a…

Distributivity in algebraic structures appeared in many contexts such as in quasigroup theory, semigroup theory and algebraic knot theory. In this paper we give a survey of distributivity in quasigroup theory and in quandle theory.

New theory captures framing anomaly in gauge theory.

Survey on algebraic K- and L-theory conjecture.

Study pin manifolds using Clifford linear Dirac operator and KO-theory.

Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes rema…

In this paper, we construct a new homology theory for semi-groups satisfying the self distributivity axiom or the idempotency axiom. Next, we consider the geometric realization corresponding to the homology theory. We continue with the comparison of this homology theory with one term and two term (rack) homology theori…