Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Special Issue on Geometry and Physics of Hitchin Systems

The Guest Editors for this special issue are

Lara B. Anderson (Virginia Tech, USA)
Laura P. Schaposnik (University of Illinois at Chicago, USA)

This volume is based on the minicourses given during the graduate workshops organized by L. Schaposnik as part of the “Geometry and Physics of Higgs Bundles” series which began in 2015 (and on several occasions co-organized by L. Anderson). The latest of these meetings was held at the Simons Center for Geometry and Physics at Stony Brook University as part of a semester-long program on the “Geometry and Physics of Hitchin Systems” in Spring 2019, co-organized by L. Anderson and L. Schaposnik. The goal of these workshop series has been to bring together researchers at all career stages to address open problems and to introduce young researchers to a rich and complex field that spans diverse areas of mathematics and physics. In addition, it has been a goal of the workshops to highlight the important work being done by female and minority researchers in the field and the workshops were organized in cooperation with the Association for Women in Mathematics.

Higgs bundles were first introduced by Nigel Hitchin in 1987 as solutions of the dimensionally reduced Yang–Mills self-duality equations on a Riemann surface and generalized by Carlos Simpson. Algebraically, a Higgs bundle on a compact Riemann surface $\Sigma$ of genus $g\geq 2$ is a pair $(E,\Phi)$ where $E$ is a holomorphic vector bundle on $\Sigma$, and the Higgs field $\Phi$ is a holomorphic 1-form in $H^{0}(\Sigma, \operatorname{End}(E)\otimes K_{\Sigma})$; here $K_\Sigma$ is the cotangent bundle of $\Sigma$ and $\operatorname{End}(E)$ the endomorphisms of $E$. Non-abelian Hodge theory gives a correspondence between representations of the fundamental group of the surface into a classical complex Lie group $G_\mathbb{C}$ and the moduli space of Higgs bundles with a $G_\mathbb{C}$ structure satisfying a stability condition.

A natural structure on the moduli spaces $\mathcal{M}_{G_\mathbb{C}}$ of $G_\mathbb{C}$-Higgs bundles is the Hitchin fibration which takes values in a certain collection of holomorphic differentials on $\Sigma$. The generic fibre of this map is an abelian variety, and the coefficients of the characteristic polynomial of $\Phi$ define the map to the base and determine the spectral curve $S$ associated to $\Phi$. The generic fibre over $S$ for classical Higgs bundles is the Jacobian variety of $S$. Given a Higgs bundle, its associated spectral data is given by the curve $S$ and the corresponding line bundle in $\operatorname{Jac}(S)$. Spectral data can also be defined for Higgs bundles whose structure group is a real form $G$ of the complex group $G_\mathbb{C}$.

Higgs bundles have found many remarkable applications in quite different areas of mathematics and physics, including gauge theory, Kähler and hyper-Kähler geometry, surface group representations, integrable systems, and non-abelian Hodge theory. In particular, the hyper-Kähler metric and the integrable system immediately yield many examples of special Lagrangian fibrations, a key ingredient in the SYZ (Strominger–Yau–Zaslow) approach to mirror symmetry.

Many fruitful applications of Higgs bundles include relationships to Langlands duality for Lie groups by Hausel–Thaddeus and Donagi–Pantev and realizations of geometric Langlands in string theory by Kapustin and Witten. Moreover, Higgs bundles and the Hitchin fibration were a key ingredient in the proof by Ngô of the fundamental lemma, a crucial component of the number-theoretic Langlands. In addition, recent developments in string theory have linked singular limits of Hitchin systems (including so-called wild/irregular Hitchin systems) with degenerate limits of Calabi–Yau integrable systems. This includes deep connections to so-called “T-brane” solutions in F-theory.

The papers in this volume span a wide range of these topics and provide hopefully useful introductions to both current areas of investigation and foundational aspects of the subject. An introduction to representation theory of Hitchin systems, including harmonic maps and character varieties, appears through the reviews of Li and Collier. In addition, the review of Alessandrini touch upon the developing topic of geometric structures and surface group representations as they are linked to Higgs bundles.

The structure of the Higgs bundle moduli space itself is studied in the articles of Rayan and Hoskins. This includes links to quiver varieties and hyperpolygon spaces as well as the topology and detailed geometry for the moduli space itself. Finally, the reviews of Beck, Logares, and Frederickson touch upon singular and degenerating limits of Higgs bundles (Logares) and themes of particular interest in string theory including links between Hitchin and Calabi–Yau integrable systems (Beck), and asymptotic limits of the Higgs bundle moduli space related to physical conjectures by Gaiotto, Moore and Neitzke (Frederickson).

The eight articles appearing in this volume (and the workshops which hosted them) would not have been possible without the generous support of a number of institutions and organizations. First, we would like to acknowledge the support of NSF DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network) and NSF RTG grant DMS-1246844 and NSF CAREER Award DMS 1749013. In addition, we would like to thank the University of Illinois, Chicago, the American Institute of Mathematics (AIM), the University of Oxford, and the Simons Center for Geometry and Physics for hosting these events. Finally, we are grateful to the lecturers and the diverse participants of these workshops for making this collection possible.

Lara B. Anderson and Laura Schaposnik         

Papers in this Issue:

Higgs Bundles and Geometric Structures on Manifolds
Daniele Alessandrini
SIGMA 15 (2019), 039, 32 pages [ abs   pdf ]
An Introduction to Higgs Bundles via Harmonic Maps
Qiongling Li
SIGMA 15 (2019), 035, 30 pages [ abs   pdf ]
On Higgs Bundles on Nodal Curves
Marina Logares
SIGMA 15 (2019), 024, 16 pages [ abs   pdf ]
Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space
Laura Fredrickson
SIGMA 15 (2019), 018, 20 pages [ abs   pdf ]
Studying Deformations of Fuchsian Representations with Higgs Bundles
Brian Collier
SIGMA 15 (2019), 010, 32 pages [ abs   pdf ]
Aspects of Calabi-Yau Integrable and Hitchin Systems
Florian Beck
SIGMA 15 (2019), 001, 25 pages [ abs   pdf ]
Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces
Steven Rayan
SIGMA 14 (2018), 129, 18 pages [ abs   pdf ]
Parallels between Moduli of Quiver Representations and Vector Bundles over Curves
Victoria Hoskins
SIGMA 14 (2018), 127, 46 pages [ abs   pdf ]

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