Hitchin Moduli Space Research Articles

Non-relativistic intersecting branes, Newton-Cartan geometry and AdS/CFT

We discuss non-relativistic variants of four-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 super-Yang-Mills theory obtained from generalised Newton-Cartan geometric limits of D3-branes in ten-dimensional spacetime. We argue that the natural interpretation of these limits is that they correspond to non-relativistic D1-branes or D3-branes intersecting the original D3-branes. The resulting gauge theories have dynamics that reduce to quantum mechanics on monopole moduli space or two-dimensional sigma-models on Hitchin moduli space respectively. We show that these theories possess interesting infinite-dimensional symmetries and we discuss the dual AdS geometries.

The Haydys monopole equation

We study complexified Bogomolny monopoles using the complex linear extension of the Hodge star operator; these monopoles can be interpreted as solutions to the Bogomolny equation with a complex gauge group. Alternatively, these equations can be obtained from dimensional reduction of the Haydys instanton equations to three dimensions, thus we call them Haydys monopoles. We find that (under mild hypotheses) the smooth locus of the moduli space of finite energy Haydys monopoles on $\mathbb{R}^3$ is a K\"ahler manifold containing the ordinary Bogomolny moduli space as a minimal Lagrangian submanifold -- an $A$-brane. Moreover, using a gluing construction we construct an open neighborhood of this submanifold modeled on a neighborhood of the zero section in the tangent bundle to the Bogomolny moduli space. This is analogous to the case of Higgs bundles over a Riemann surface, where the (co)tangent bundle of holomorphic bundles canonically embeds into the Hitchin moduli space. These results contrast immensely with the case of finite energy Kapustin--Witten monopoles for which we have shown a vanishing theorem in [12].

Asymptotic Geometry of the Hitchin Metric

We study the asymptotics of the natural $L^2$ metric on the Hitchin moduli space with group $G = \mathrm{SU}(2)$. Our main result, which addresses a detailed conjectural picture made by Gaiotto, Neitzke and Moore \cite{gmn13}, is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric from \cite{gmn13}. We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. Very recent work by Dumas and Neitzke indicates that the convergence rate for the metric is exponential, at least in certain directions.

Geometric quantization of the Hitchin system

This paper is about geometric quantization of the Hitchin system. We quantize a Kahler form on the Hitchin moduli space (which is half the first Kahler form defined by Hitchin) by considering the Quillen bundle as the prequantum line bundle and modifying the Quillen metric using the Higgs field so that the curvature is proportional to the Kahler form. We show that this Kahler form is integral and the Quillen bundle descends as a prequantum line bundle on the moduli space. It is holomorphic and hence one can take holomorphic square integrable sections as the Hilbert space of quantization of the Hitchin moduli space.

Surface operators and separation of variables

Alday, Gaiotto, and Tachikawa conjectured relations between certain 4d N = 2 supersymmetric field theories and 2d Liouville conformal field theory. We study generalizations of these relations to 4d theories with surface operators. For one type of surface operators the corresponding 2d theory is the WZW model, and for another type — the Liouville theory with insertions of extra degenerate fields. We show that these two 4d theories with surface operators exhibit an IR duality, which reflects the known relation (the so-called separation of variables) between the conformal blocks of the WZW model and the Liouville theory. Furthermore, we trace this IR duality to a brane creation construction relating systems of M5 and M2 branes in M-theory. Finally, we show that this duality may be expressed as an explicit relation between the generating functions for the changes of variables between natural sets of Darboux coordinates on the Hitchin moduli space.

Chern-Simons theory and S-duality

We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example, the space of flat SL(2, C ) connections on M is identified with the space of supersymmetric vacua in a dual 3d gauge theory. The hidden symmetry h → -4π^2/h of SL(2) Chern-Simons theory can be identified as the S-duality transformation of N=4 super-Yang-Mills theory (obtained by compactifying the five-brane theory on a torus); whereas the mapping class group action in Chern-Simons theory on a three-manifold M with boundary C is realized as S-duality in 4d N=2 super-Yang-Mills theory associated with the Riemann surface C. We illustrate these symmetries by considering simple examples of 3-manifolds that include knot complements and punctured torus bundles, on the one hand, and mapping cylinders associated with mapping class group transformations, on the other. A generalization of mapping class group actions further allows us to study the transformations between several distinguished coordinate systems on the phase space of Chern-Simons theory, the SL(2) Hitchin moduli space.

Compactification of moduli of Higgs bundles

In this paper we consider a canonical compactification of Hitchin's moduli space of stable Higgs bundles with fixed determinant of odd degree over a Riemann surface, producing a projective variety by gluing in a divisor at infinity. We give a detailed study of the compactified space, the divisor at infinity and the moduli space itself. In doing so we reprove some assertions of Laumon and Thaddeus on the nilpotent cone.

Vanishing of intersection numbers on the moduli space of Higgs bundles

In this paper we consider the topological side of a problem which is the analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish, i.e. there are no topological L^2 harmonic forms on Hitchin's space. This result generalizes the well known vanishing of the Euler characteristic of the moduli space of rank 2 stable bundles of fixed determinant of odd degree over the given Riemann surface. Our proof shows that the vanishing of all intersection numbers in the compactly supported cohomology of Hitchin's space is given by relations analogous to Mumford's relations in the cohomology ring of the moduli space of stable bundles.