arXiv research

Formula derived for special q-hypergeometric series at roots of unity.

Constructs units in cyclotomic fields from Bloch groups, proving Nahm's conjecture.

Nahm sums are $q$-series of a special hypergeometric type that appear in character formulas in Conformal Field Theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in Quantum Knot Theory, namely we prove the stability of th…

New boundary conditions for knot polynomials solved via elliptic KW equations.

The paper establishes a correspondence between solutions of extended Bogomolny equations and Higgs bundles.

Study on octonionic Nahm's equations and their moduli space properties.

Study of continuous symmetries in Nahm data and BPS monopoles.

Study explores Nahm-Schmid equations and their connection to hypersymplectic geometry.

This paper proves a bound on the energy of Nahm pole solutions on $S^3 imes\mathbb{R}^+$.

Given two hyperkähler manifolds $M$ and $N$ and a quaternionic instanton on their product, a hyperkähler Nahm transform can be defined, which maps quaternionic instantons on $M$ to quaternionic instantons on $N$. This construction includes the case of Nahm transform for periodic instantons on $\bR^4$, the Fourier-Mukai…

New non-commutative geometric Nahm transform for ASD connections.

Study of Nahm pole solutions over 3-manifolds, proving smoothness at boundary.

Study of $D_2$ ALF manifolds via hyperkahler quotients of affine spaces.

Nahm's equations studied in broader context, linking to sheaves and spectral curves.

The paper constructs solutions to Kapustin-Witten equations with Nahm poles over manifolds with boundaries.

Paper proves bijection of periodic instantons to singular monopoles.

The paper extends Nahm transforms to Spin(7) structures on tori, finding obstructions.

The paper classifies solutions to Kapustin-Witten equations with specific singularities.

We discuss the correspondence between Nahm's equations, the Basu-Harvey-Terashima equations, and Lie superalgebras.

A new transform links rotating calorons to solutions of a differential equation.

We review the construction known as the Nahm transform in a generalized context, which includes all the examples of this construction already described in the literature. The Nahm transform for translation invariant instantons on $\real^4$ is presented in an uniform manner. We also analyze two new examples, the first o…

We use Nahm data to give a description of a candidate for the universal hyperkahler implosion with respect to a compact Lie group

Transforms instantons to monopoles for torus products.

The paper studies sequences of solutions to Kapustin-Witten equations with Nahm pole asymptotics.

We prove that the Fourier--Laplace--Nahm transform for connections on the projective line is a hyper-Kähler isometry.

We present the Nahm transform of the doubly-periodic instantons introduced in math.DG/9909069, converting them into certain meromorphic solutions of Hitchin's equations over an elliptic curve.

We construct the Nahm transform for Higgs bundles over a Riemann surface of genus at least 2 as hyperholomorphic connections on the total space of the tangent bundle of its dual Jacobian.

This paper is a discussion of relations between some free-boundary problems and infinite dimensional Lie groups; particularly a version of Nahm's equations for the group of Hamiltonian diffeomorphisms in two dimensions.

Researchers create BPS monopoles with any desired symmetry breaking.

The paper connects Nahm's equations to rational maps between projective spaces.

The Nahm pole boundary condition for certain gauge theory equations in four and five dimensions is defined by requiring that a solution should have a specified singularity along the boundary. In the present paper, we show that this boundary condition is elliptic and has regularity properties analogous to more standard …

Develops twistorial approach to define monopoles on R^5.

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-b…

We study the asymptotic behaviour of doubly periodic instantons with square-integrable curvature. Then, we establish the equivalence given by the Nahm transform between the doubly periodic instantons with square integrable curvature and the wild harmonic bundles on the dual torus.

We study supersymmetric deformations of N = 4 quantum mechanics with a Kahler target space admitting a holomorphic isometry. We show that the twisted mass deformation generalises to a deformation constructed from matrix-valued functions of the moment map, which obey the Nahm equations. We also explain how N = 4 supersy…

Instantons on various spaces can be constructed via a generalization of the Fourier transform called the ADHM-Nahm transform. An explicit use of this construction, however, involves rather tedious calculations. Here we derive a simple formula for instantons on a space with one periodic direction. It simplifies the ADHM…

The study connects monopole chains to Higgs bundles and classifies symmetric chains.

In this paper, we complete the proof of an equivalence given by Nye and Singer of the equivalence between calorons (instantons on $S^1\times R^3$) and solutions to Nahm's equations over the circle, both satisfying appropriate boundary conditions. Many of the key ingredients are provided by a third way of encoding the s…

The moduli space of solutions to Nahm's equations of rank (k,k+j) on the circle, and hence, of SU(2) calorons of charge (k,j), is shown to be equivalent to the moduli of holomorphic rank 2 bundles on P^1xP^1 trivialized at infinity with c_2=k and equipped with a flag of degree j along P^1x{0}. An explicit matrix descri…

Study of twisted Kapustin-Witten equations on Riemann surfaces.

Develops correspondence for Bogomolny equations with specific boundary conditions.

We show that any cyclically symmetric monopole is gauge equivalent to Nahm data given by Sutcliffe's ansatz, and so obtained from the affine Toda equations. Further the direction (the Ercolani-Sinha vector) and base point of the linearising flow in the Jacobian of the spectral curve associated to the Nahm equations ari…

Paper characterizes solutions to Kapustin Witten equations on specific domain.

Paper localizes ADHM construction for ASD instantons over smooth domains.

Lectures given at the summer school on Algebraic Groups, Goettingen, June 27 - July 15 2005

In this work we are concerned with reduction of the ASD-equations to the Riemann sphere, that is integrable connections with a harmonic metric, or equivalently Higgs bundles with a Hermitian-Einstein metric. In the first chapter, we introduce the type of singularities we allow for the solutions to have, and announce th…

We define a Fourier-Mukai transform for a triple consisting of two holomorphic vector bundles over an elliptic curve and a homomorphism between them. We prove that in some cases the transform preserves the natural stability condition for a triple. We also define a Nahm transform for solutions to natural gauge-theoretic…

Monads link instantons and bow solutions in complex geometry.