Related papers: The $\mathbb{R}$ invariant solutions to the Kapust…
A non-negative integer labeled set of model solutions to the $\mathbb{R}$-invariant Kapustin-Witten equations on $(0,\infty)\times \mathbb{R}^2\times \mathbb{R}$ plays a central role in Edward Witten's program to interpret the colored Jones…
In this note, we classify all solutions to the $\mathrm{SU(n)}$ Kapustin-Witten equations on $S^1\times\Sigma \times \mathbb{R}^+$, where $\Sigma$ is a compact Riemann surface, with Nahm pole singularity at $S^1\times\Sigma \times \{0\}$.…
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…
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…
In this paper we prove existence for model knot solutions to the dimensionally reduced twisted Kapustin-Witten equations on $\mathbb{R}^3_+$ for any twisting parameter $t\in(0,\infty)$. We start with the explicit solutions for $t = 1$…
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…
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…
This is a written version of lectures that I would have given myself about aspects of the differential operator that is obtained from the linearized Kapustin-Witten equations on the product of the half-line with a compact, oriented,…
It is conjectured that the coefficients of the Jones polynomial can be computed by counting solutions of the KW equations on a four-dimensional half-space, with certain boundary conditions that depend on a knot. The boundary conditions are…
In this paper, we study the dimensionally reduced twisted Kapustin-Witten equations on the product of a compact Riemann surface $\Sigma$ with $\mathbb{R}^+$. The main result is a Kobayashi-Hitchin type correspondence between the space of…
Solutions to both the diffeomorphism and the hamiltonian constraint of quantum gravity have been found in the loop representation, which is based on Ashtekar's new variables. While the diffeomorphism constraint is easily solved by…
We elucidate further properties of the novel family of polynomial time knot polynomials devised by Bar-Natan and van der Veen based on the Gaussian calculus of generating series for noncommutative algebras. These polynomials determine all…
We provide a definitive treatment, including sharp decay and the precise late-time asymptotic profile, for generic solutions of linear wave equations with a (singular) inverse-square potential in (3+1)-dimensional Minkowski spacetime. Such…
In this work we study existence, asymptotic behaviour and stability properties of $O(m) times O(n)$ invariant solutions of the Allen-Cahn equation $Delta u+u(1-u^2)=0$ in $R^m times R^n$ with $m,n ge 2$, $m+n ge 8$. We exhibit four families…
The expectation value of Wilson loop operators in three-dimensional SO(N) Chern-Simons gauge theory gives a known knot invariant: the Kauffman polynomial. Here this result is derived, at the first order, via a simple variational method.…
We propose a gauge model of quantum electrodynamics (QED) and its nonabelian generalization from which we derive knot invariants such as the Jones polynomial. Our approach is inspired by the work of Witten who derived knot invariants from…
This paper deals with Schr\"{o}dinger equations with potentials which are time-dependent non-smooth and at most quadratic growth. In the case where potentials are smooth with respect to spatial variables, fundamental solutions have explicit…
Chern-Simons gauge theory for compact semisimple groups is analyzed from a perturbation theory point of view. The general form of the perturbative series expansion of a Wilson line is presented in terms of the Casimir operators of the gauge…
In this paper we introduce invariants of semi-free Hamiltonian actions of $S\sp 1$ on compact symplectic manifolds (which satisfy some technical conditions related to positivity) using the space of solutions to certain gauge theoretical…
We obtain new solutions of topologically massive gravity. We find the general Kundt solutions, which in three dimensions are spacetimes admitting an expansion-free null geodesic congruence. The solutions are generically of algebraic type…