arXiv research

Study of twisted Kapustin-Witten equations on Riemann surfaces.

Paper proves existence of knot solutions for specific equations.

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

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.…

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

Paper characterizes solutions to Kapustin Witten equations on specific domain.

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…

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

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 …

Study Kapustin--Witten equations on ALE and ALF spaces, proving asymptotic estimates and vanishing theorems.

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…

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…

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

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

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…

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 …

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…

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 $(A,φ)$.

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…

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 …

The study finds a positive lower bound for the extra field in Yang-Mills equations on closed 4-manifolds.

Constructs Lagrangian correspondences for Higgs bundles and holomorphic connections.

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…

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

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

Study of Haydys monopoles using complexified Bogomolny equations.

We prove a priori interior $C^{2,α}$ estimates for solutions of fully nonlinear elliptic equations of twisted type. For example, our estimates apply to equations of the type convex + concave. These results are particularly well suited to equations arising from elliptic regularization. As application, we obtain a new pr…

Holomorphic Lagrangian subvarieties are toric fibrations over their projections in special Kähler bases.

Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, interesting first and higher flows of twi…

The paper studies loxodromes on twisted surfaces in a specific 3D space.

Develops a new parabolic equation for surfaces, proving long-time existence and convergence.

Study characterizes 2-Killing vector fields on complex spacetimes.

We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special Kähler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also…

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 …

Study on G2 structures with torsion and existence of solutions.

Explains a property of algebras related to quantum field theories.

Solves supercritical dHYM on projective manifolds with specific conditions.

A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kaelher, quiver bundles admit natural gauge-theoretic equat…

Study on symmetries of differential equations using gauge transformations and coverings.

On a five dimensional simply connected Sasaki-Einstein manifold, one can construct Yang-Mills theories coupled to matter with at least two supersymmetries. The partition function of these theories localises on the contact instantons, however the contact instanton equations are not elliptic. It turns out that these equa…

Proves well-posedness for Einstein equations with specific boundary conditions.

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…

Simplified A-polynomial calculation for twisted knots.

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…

Study rolling dynamics with random slipping and twisting using large deviation principles.

Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.

We investigate a class of Leibniz algebroids which are invariant under diffeomorphisms and symmetries involving collections of closed forms. Under appropriate assumptions we arrive at a classification which in particular gives a construction starting from graded Lie algebras. In this case the Leibniz bracket is a deriv…