Nahm Transform for Periodic Monopoles¶and ?=2 Super Yang–Mills Theory

Abstract

We study Bogomolny equations on R 2 × S 1 . Although they do not admit nontrivial finite-energy solutions, we show that there are interesting infinite-energy solutions with Higgs field growing logarithmically at infinity. We call these solutions periodic monopoles. Using the Nahm transform, we show that periodic monopoles are in one-to-one correspondence with solutions of Hitchin equations on a cylinder with Higgs field growing exponentially at infinity. The moduli spaces of periodic monopoles belong to a novel class of hyperkähler manifolds and have applications to quantum gauge theory and string theory. For example, we show that the moduli space of k periodic monopoles provides the exact solution of N = 2 super Yang-Mills theory with gauge group SU (k) compactified on a circle of arbitrary radius.

Periodic Monopoles with Singularities and 𝒩=2 Super-QCD

“…For example, Nahm transform takes monopoles on R 3 with finite energy to solutions of the so-called Nahm equations, which are the reduction of the self-duality equation to one dimension. Periodic monopoles without singularities are mapped to solutions of the Hitchin equations on a cylinder [5]. We will see that Nahm transform establishes a one-to-one correspondence between periodic monopoles with singularities and solutions of Hitchin equations on a cylinder with singularities.…”

“…Their moduli spaces are asymptotically locally flat hyperkähler manifolds which can be used to solve N = 4 d = 3 gauge theories with matter. The present work can be viewed as an extension of both [6] and [5].…”

“…For example, certain finite N = 2 gauge theories (the socalled quiver theories) are solved in terms of instantons on R 2 × T 2 [20,19]. N = 2 super-Yang-Mills with gauge group SU(k) and no hypermultiplets are solved in terms of monopoles on R 2 × S 1 [5]. We will see below that…”

“…The operator D is Fredholm for s ∈ (R ×Ŝ 1 )\K because D † D has a mass gap. The Weitzenbock formula [5] implies that the operator D † D is positivedefinite on the space of functions of rapid decrease. It is also easy to see that all elements of the L 2 kernel of D must be decreasing rapidly at infinity for s / ∈ K, and therefore the L 2 kernel of D is empty for s / ∈ K. It follows that the L 2 kernel of D † is a vector bundle on (R ×Ŝ 1 )\K of rank −IndD.…”

Monopole wall

“…This D-brane connection is partly responsible for a particular interest in periodic assemblages of monopoles. For example, there have been studies of monopole chains [3], where the underlying theory (such as the Nahm transform) is well-developed. However, for monopole sheets or walls, where the fields are doubly-periodic, much less is known.…”