Singular Monopoles from Cheshire Bows

Singular monopoles are nonabelian monopoles with prescribed Dirac-type singularities. All of them are delivered by the Nahm's construction. In practice, however, the effectiveness of the latter is limited to the cases of one or two singularities. We present an alternative construction of singular monopoles formulated in terms of Cheshire bows. To illustrate the advantages of our bow construction we obtain an explicit expression for one U(2) gauge group monopole with any given number of singularities of Dirac type.Comment: LaTeX, 34 pages, 8 figure

Phases of Five-dimensional Theories, Monopole Walls, and Melting Crystals

Moduli spaces of doubly periodic monopoles, also called monopole walls or monowalls, are hyperk\"ahler; thus, when four-dimensional, they are self-dual gravitational instantons. We find all monowalls with lowest number of moduli. Their moduli spaces can be identified, on the one hand, with Coulomb branches of five-dimensional supersymmetric quantum field theories on $\mathbb{R}^3\times T^2$ and, on the other hand, with moduli spaces of local Calabi-Yau metrics on the canonical bundle of a del Pezzo surface. We explore the asymptotic metric of these moduli spaces and compare our results with Seiberg's low energy description of the five-dimensional quantum theories. We also give a natural description of the phase structure of general monowall moduli spaces in terms of triangulations of Newton polygons, secondary polyhedra, and associahedral projections of secondary fans.Comment: 45 pages, 11 figure

The topology of asymptotically locally flat gravitational instantons

In this letter we demonstrate that the intersection form of the Hausel--Hunsicker--Mazzeo compactification of a four dimensional ALF gravitational instanton is definite and diagonalizable over the integers if one of the Kahler forms of the hyper-Kahler gravitational instanton metric is exact. This leads to the topological classification of these spaces. The proof exploits the relationship between L^2 cohomology and U(1) anti-instantons over gravitational instantons recognized by Hitchin. We then interprete these as reducible points in a singular SU(2) anti-instanton moduli space over the compactification leading to the identification of its intersection form. This observation on the intersection form might be a useful tool in the full geometric classification of various asymptotically locally flat gravitational instantons.Comment: 9 pages, LaTeX, no figures; Some typos corrected, slightly differs from the published versio

Singular Monopoles and Gravitational Instantons

We model A_k and D_k asymptotically locally flat gravitational instantons on the moduli spaces of solutions of U(2) Bogomolny equations with prescribed singularities. We study these moduli spaces using Ward correspondence and find their twistor description. This enables us to write down the K\"ahler potential for A_k and D_k gravitational instantons in a relatively explicit form.Comment: 22 pages, LaTe

One Monopole with k Singularities

We present all charge one monopole solutions of the Bogomolny equation with k prescribed Dirac singularities for the gauge groups U(2), SO(3), or SU(2). We analyze these solutions comparing them to the previously known expressions for the cases of one or two singularities.Comment: 12 pages, LaTe

On 6d N=(2,0) theory compactified on a Riemann surface with finite area

We study 6d N=(2,0) theory of type SU(N) compactified on Riemann surfaces with finite area, including spheres with fewer than three punctures. The Higgs branch, whose metric is inversely proportional to the total area of the Riemann surface, is discussed in detail. We show that the zero-area limit, which gives us a genuine 4d theory, can involve a Wigner-Inonu contraction of global symmetries of the six-dimensional theory. We show how this explains why subgroups of SU(N) can appear as the gauge group in the 4d limit. As a by-product we suggest that half-BPS codimension-two defects in the six-dimensional (2,0) theory have an operator product expansion whose operator product coefficients are four-dimensional field theories.Comment: 22 pages, 4 figure