# Hitchin Systems in Mathematics and Physics

Conference Date:
Monday, February 13, 2017 (All day) to Friday, February 17, 2017 (All day)
Scientific Areas:
Mathematical Physics

Hitchin systems play an important role in Supersymmetric Gauge Theory, Algebra, and Geometry.  The purpose of the workshop is to bring together experts in these fields, which are interested in the subject.

Sponsorship for this workshop has been provided by:

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• Philip Boalch, Université Paris-Sud
• Sergio Cecotti, SISSA
• Sergey Cherkis, University of Arizona
• Ben Davison, University of Glasgow
• Emanuel Diaconescu, Rutgers University
• Tudor Dimofte, University of Cambridge
• Victor Ginzburg, University of Chicago
• Marco Gualtieri, University of Toronto
• Tamas Hausel, Institute of Science and Technology Austria
• Nigel Hitchin, University of Oxford
• Kazuki Hiroe, Josai University
• Mikhail Kapranov, Kavli Institute for Theoretical Physics
• Andrew Neitzke, University of Texas at Austin
• Nikita Nekrasov, Stony Brook University
• Francesco Sala, Kavli IPMU, University of Tokyo
• Alexander Soibelman, University of Southern California
• Szilard Szabo, Budapest University of Technology and Economics
• Rina Anno, Kansas State University
• Chris Beasley, Northwestern University
• Lakshya Bhardwaj, Perimeter Institute
• Philip Boalch, Université Paris-Sud
• Alexander Braverman, University of Toronto
• Dylan, Butson, Perimeter Institute
• Sergio Cecotti, SISSA
• Benoit Charbonneau, University of Waterloo
• Sergey Cherkis, University of Arizona
• Kevin Costello, Perimeter Institute
• Xinle Dai, Perimeter Institute
• Ben Davison, University of Glasgow
• Emanuel Diaconescu, Rutgers University
• Tudor Dimofte, University of Cambridge
• Shubham Dwivedi, University of Waterloo
• Davide Gaiotto, Perimeter Institute
• Panagiotis Gianniotis, Univeristy of Waterloo
• Victor Ginzburg, University of Chicago
• Marco Gualtieri, University of Toronto
• Tamas Hausel, Institute of Science and Technology Austria
• Justin Hilburn, University of Pennsylvania
• Nigel Hitchin, University of Oxford
• Kazuki Hiroe, Josai University
• Shengda Hu, Wilfrid Laurier University
• Theo Johnson-Freyd, Perimeter Institute
• Joel Kamnitzer, University of Toronto
• Mikhail Kapranov, Kavli Institute for Theoretical Physics
• Spiro Karigiannis, University of Waterloo
• Peter Koroteev, University of Minnesota
• Ian Le, Perimeter Institute
• Raeez Lorgat, Massachusetts Institute of Technology
• Anthony McCormick, University of Waterloo
• Sayed Faroogh Moosavian, Perimeter Institute
• Ruxandra Moraru, University of Waterloo
• Akos Nagy, University of Waterloo
• Andrew Neitzke, University of Texas at Austin
• Nikita Nekrasov, Stony Brook University
• Percy Paul, Perimeter Institute
• Surya Raghavendran, Perimeter Institute
• Miroslav Rapcak, Perimeter Institute
• Marcelo Rubio, National University of Cordoba
• Francesco Sala, Kavli IPMU, University of Tokyo
• Alexander Shapiro, University of Toronto
• Andrei Shieber, Perimeter Institute
• Alexander Soibelman, University of Southern California
• Yan Soibelman, Kansas State University
• David Svoboda, Perimeter Institute
• Szilard Szabo, Budapest University of Technology and Economics
• Jie Zhou, Perimeter Institute
• Yehao Zhou, Perimeter Institute

Monday, February 13, 2017

 Time Event Location 9:00 – 9:30am Registration Reception 9:30 - 9:35am Davide Gaiotto, Perimeter InstituteWelcome and Opening Remarks 9:35 – 10:30am Nigel Hitchin, University of OxfordCritical points and spectral curves Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00-12:00pm Sergey Cherkis, University of ArizonaGeneralizing Quivers: Bows, Slings, Monowalls Sky Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 3:00pm Marco Gualtieri, University of TorontoHolomorphic symplectic Morita equivalence and the generalized Kahler potential Sky Room 3:00 - 4:00pm Szilard Szabo, Budapest University of Technology & EconomicsNahm transformation for parabolic harmonic bundles on the projective line with regular residues Sky Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor 4:30 – 5:30pm Discussion Sky Room

Tuesday, February 14, 2017

 Time Event Location 9:30 – 10:30am Tudor Dimofte, University of CambridgeA mathematical definition of 3d indices Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00-12:00pm Tamas Hausel, Institute of Science and Technology AustriaPerverse Hirzebruch y-genus of Higgs moduli spaces Sky Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 3:00pm Alexander Soibleman, University of Southern CaliforniaMotivic Classes for Moduli of Connections Sky Room 3:00 - 4:00pm Ben Davison, University of GlasgowBPS algebras and twisted character varieties Sky Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor 4:30 – 5:30pm Discussion Sky Room

Wednesday, February 15, 2016

 Time Event Location 9:30 – 10:30am Kazuki Hiroe, Josai UniversityOn index of rigidity Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00-12:00pm Victor Ginzburg, University of ChicagoSymplectic geometry related to G/U and Sicilian theories' Sky Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 3:30pm ColloquiumNigel Hitchin, University of OxfordThe Hitchin system, past and present Theater 3:30 – 4:00pm Coffee Break Bistro – 1st Floor 4:00 – 5:00pm Discussion Sky Room 6:00pm Banquet Bistro – 2nd Floor

Thursday, February 16, 2016

 Time Event Location 9:30 – 10:30am Mikhail Kapranov, Kavli InstituteGeometric interpretation of Witten's d-bar equation Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00-12:00pm Nikita Nekrasov, Stony Brook UniversityHow I learned to stop worrying and to love both instantons and anti-instantons Sky Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 3:00pm Francesco Sala, Kavli IPMU, University of TokyoHiggs sheaves on a curve and Cohomological Hall algebras Sky Room 3:00 - 4:00pm Philip Boalch, Université Paris-SudWild character varieties, meromorphic Hitchin systems and Dynkin diagrams Sky Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor 4:30 – 5:30pm Discussion Sky Room

Friday, February 17, 2017

 Time Event Location 9:30 – 10:30am Emanuel Diaconescu, Rutgers UniversityTBA Sky Room 10:30 – 11:00am Coffee Break Bistro – 1st Floor 11:00-12:00pm Andrew Neitzke, University of Texas at AustinAbelianization in complex Chern-Simons theory and a hyperholomorphic line bundle Sky Room 12:00 – 2:00pm Lunch Bistro – 2nd Floor 2:00 – 3:00pm Sergio Cecotti, SISSAFQHE and Hitchin Systems on Modular Curves Sky Room 4:00 – 4:30pm Coffee Break Bistro – 1st Floor 4:30 – 5:30pm Wrap Up Discussion Sky Room

Philip Boalch, Université Paris-Sud

Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams

In 1987 Hitchin discovered a new family of algebraic integrable systems, solvable by spectral curve methods. One novelty was that the base curve was of arbitrary genus. Later on it was understood how to extend Hitchin's viewpoint, allowing poles in the Higgs fields, and thus incorporating many of the known classical integrable systems, which occur as meromorphic Hitchin systems when the base curve has genus zero. However, in a different 1987 paper, Hitchin also proved that the total space of his integrable system admits a hyperkahler metric and (combined with work of Donaldson, Corlette and Simpson) this shows that the differentiable manifold underlying the total space of the integrable system has a simple description as a character variety Hom(pi_1(Ʃ), G)/G of representations of the fundamental group of the base curve Ʃ into the structure group G. This misses the main cases of interest classically, but it turns out there is an extension. In work with Biquard from 2004 Hitchin's hyperkahler story was extended to the meromorphic case, upgrading the speakers holomorphic symplectic quotient approach from 1999. Using the irregular Riemann--Hilbert correspondence the total space of such integrable systems then has a simple explicit description in terms of monodromy and Stokes data, generalising the character varieties. The construction of such wild character varieties'', as algebraic symplectic varieties, was recently completed in work with D. Yamakawa, generalizing the author's construction in the untwisted case (2002-2014).  For example, by hyperkahler rotation, the wild character varieties all thus admit special Lagrangian fibrations.  The main aim of this talk is to describe some simple examples of wild character varieties including some cases of complex dimension 2, familiar in the theory of Painleve equations, although their structure as new examples of complete hyperkahler manifolds (gravitational instantons) is perhaps less well-known. The language of quasi-Hamiltonian geometry will be used and we will see how this leads to relations to quivers, Catalan numbers and triangulations, and in particular how simple examples of gluing wild boundary conditions for Stokes data leads to duplicial algebras in the sense of Loday.  The new results to be discussed are joint work with R. Paluba and/or D. Yamakawa.

Sergio Cecotti, SISSA

FQHE and Hitchin Systems on Modular Curves

Sergey Cherkis, University of Arizona

Generalizing Quivers: Bows, Slings, Monowalls

Quivers emerge naturally in the study of instantons on flat four-space (ADHM), its orbifolds and their deformations, called ALE space (Kronheimer-Nakajima).  Pursuing this direction, we study instantons on other hyperkaehler spaces, such as ALF, ALG, and ALH spaces.  Each of these cases produces instanton data that organize, respectively, into a bow (involving the Nahm equations), a sling (involving the Hitchin equations), and a monopole wall (Bogomolny equation).

Ben Davison, University of Glasgow

BPS algebras and twisted character varieties

In this talk I will explain how a perverse filtration on the Kontsevich-Soibelman cohomological Hall algebra enables us to define the Lie algebra of BPS states associated to a smooth algebra with potential.  I will then explain what this means for character varieties, and in particular, how to build the "genus g Kac-Moody Lie algebra" out of the cohomology of representations of the fundamental group of a surface.

Tudor Dimofte, University of Cambridge

A mathematical definition of 3d indices

3d field theories with N=2 supersymmetry play a special role in the evolving web of connections between geometry and physics originating in the 6d (2,0) theory. Specifically, these 3d theories are associated to 3-manifolds M, and their vacuum structure captures the geometry of local systems on M. (Sometimes M arises as a cobordism between two surfaces C, C', in which case the 3d theories encode some functorial relation between the geometry of Hitchin systems on C and C'.) I would like to explain some of the mathematics of 3d N=2 theories. In particular, I would like to explain how Hilbert spaces in these theories arise as Dolbeault cohomology of certain moduli spaces of bundles. One application is a homological interpretation of the "pentagon relation" relating flips of triangulation on a surface.

Victor Ginzburg, University of Chicago

Symplectic geometry related to G/U and Sicilian theories'

We construct an action of the Weyl group on the affine closure of the cotangent bundle on G/U. The construction involves Hamiltonian reduction with respect to the universal centralizer' and an interesting Lagrangian variety, the Miura variety. A closely related construction produces symplectic manifolds which play a role in Sicilian theories' and whose existence was conjectured by Moore and Tachikawa. Some of  these constructions may be reinterpreted, via the Geometric Satake,  in terms of the affine grassmannian.

Marco Gualtieri, University of Toronto

Holomorphic symplectic Morita equivalence and the generalized Kahler potential

Since the introduction of generalized Kahler geometry in 1984 by Gates, Hull, and Rocek in the context of two-dimensional supersymmetric sigma models, we have lacked a compelling picture of the degrees of freedom inherent in the geometry.  In particular, the description of a usual Kahler structure in terms of a complex manifold together with a Kahler potential function is not available for generalized Kahler structures, despite many positive indications in the literature over the last decade.  I will explain recent work showing that a generalized Kahler structure may be viewed in terms of a Morita equivalence between holomorphic Poisson manifolds; this allows us to solve the problem of existence of a generalized Kahler potential.

Tamas Hausel, Institute of Science and Technology Austria

Perverse Hirzebruch y-genus of Higgs moduli spaces

I will discuss in the framework of the P=W conjecture, how one can conjecture formulas for the perverse Hirzebruch y-genus of Higgs moduli spaces. The form of the conjecture raises the possibility that they can be obtained as the partition function of a 2D TQFT.

Kazuki Hiroe, Josai University

On index of rigidity

The index of rigidity was introduced by Katz as the Euler characteristic of de Rham cohomology of End-connection of a meromorphic connection on curve. As its name suggests, the index valuates the rigidity of the connection on curve. Especially, in P^1 case, this index makes a significant contribution together with middle convolution. Namely Katz showed that regular singular connection on P^1 can be reduced to a rank 1 connection by middle convolution if and only if the index of rigidity is 2. After that, the work of Crawley-Boevey gave an interpretation of the index of rigidity and the Katz' algorithm from the theory of root system. Namely, he gave a realization of moduli spaces of regular singular connections on a trivial bundle as quiver varieties. In this setting the index of rigidity can be naturally computed by the Euler form of quiver, and the Katz algorithm can be understood as a special example of the theory of Weyl group orbits of positive roots of the quiver. I will give an overview of this story with a generalization to the case of irregular singular connections. Moreover, I will introduce an algebraic curve associated to a linear differential equation on Riemann surface as an analogy of the spectral curve of Higgs bundle. And compare some indices of singularities of differential equation and its associated curve,  Milnor numbers and Komatsu-Malgrange irregularities. Finally as a corollary of this comparison of local indices, I will give a comparison between cohomology of the curve and de Rham cohomology of the differential equation and show the coincidence of the index of rigidity and the Euler characteristic of the associated curve.

Nigel Hitchin, University of Oxford

Critical points and spectral curves

Critical values of the integrable system correspond to singular spectral curves. In this talk we shall discuss critical points, points in the moduli space where one of the Hamiltonian vector fields vanishes. These involve torsion-free sheaves on the spectral curve instead of line bundles and a lifting to a 3-manifold which fibres over the cotangent bundle. The case of rank 2 will be described in more detail.

Colloquium:  The Hitchin system, past and present

The talk will be a survey of Higgs bundles, their moduli spaces and the associated fibration structure from a historical, and personal,  point of view.

Mikhail Kapranov, Kavli Institute

Geometric interpretation of Witten's d-bar equation

The Witten d-bar equation is a generalization of the parametrized holomorphic curve equation associated to a holomorphic function (superpotential) on a Kahler manifold X. It plays a central role in the work of Gaiotto-Moore-Witten on the "algebra of the infrared".

The talk will explain an "intrinsic" point of view on the equation as a condition on a real surface S embedded into X (i.e., not involving any parametrization of S). This is possible if S is not a holomorphic curve in the usual sense.

Andrew Neitzke, University of Texas at Austin

Abelianization in complex Chern-Simons theory and a hyperholomorphic line bundle

I will describe an approach to classical complex Chern-Simons theory via "abelianization", relating flat SL(N)-connections over a manifold of dimension d <= 3 to flat GL(1)-connections over a branched N-fold cover.  This is joint work with Dan Freed. When applied in dimension d=2 this construction leads to an alternative description of a hyperholomorphic line bundle over the moduli space of Higgs bundles, studied e.g. by Haydys, Hitchin, Alexandrov-Persson-Pioline.

Nikita Nekrasov, Stony Brook University

How I learned to stop worrying and to love both instantons and anti-instantons

Francesco Sala, Kavli IPMU, University of Tokyo

Higgs sheaves on a curve and Cohomological Hall algebras

Cohomological Hall algebras associated with preprojective algebras of quivers play a preeminent role in geometric representation theory and mathematical physics. In the present talk, I will introduce and describe CoHAs associated with the stack of Higgs sheaves on a smooth projective curve. Moreover, I will address the connections with representation theory and gauge theory. (This is a joint work with Olivier Schiffmann.)

Alexander Soibleman, University of Southern California

Motivic Classes for Moduli of Connections

In their paper, "On the motivic class of the stack of bundles",  Behrend and Dhillon were able to derive a formula for the class of a stack of vector bundles on a curve in a completion of the K-ring of varieties.  Later, Mozgovoy and Schiffmann performed a similar computation in order to obtain the number of points over a finite field in the moduli space of twisted Higgs bundles.  We will briefly introduce motivic classes.  Then, following Mozgovoy and Schiffmann's argument, we will outline an approach for computing motivic classes for the moduli stack of vector bundles with connections on a curve. This is a work in progress with Roman Fedorov and Yan Soibelman.

Szilard Szabo, Budapest University of Technology & Economics

Nahm transformation for parabolic harmonic bundles on the projective line with regular residues

I will define a generalization of the classical Laplace transform for D-modules on the projective line to parabolic harmonic bundles with finitely many logarithmic singularities with regular residues and one irregular singularity, and show some of its properties. The construction involves on the analytic side L2-cohomology, and it has algebraic de Rham and Dolbeault interpretations using certain elementary modifications of complexes.  We establish stationary phase formulas, in patricular a transformation rule for the parabolic weights. In the regular  semi-simple case we show that the transformation is a hyper-Kaehler isometry.

## FQHE and Hitchin Systems on Modular Curves

Friday Feb 17, 2017
Speaker(s):
Scientific Areas:

## Abelianization in complex Chern-Simons theory and a hyperholomorphic line bundle

Friday Feb 17, 2017
Speaker(s):

I will describe an approach to classical complex Chern-Simons theory via "abelianization", relating flat SL(N)-connections over a manifold of dimension d

Scientific Areas:

## TBA

Friday Feb 17, 2017
Speaker(s):
Scientific Areas:

## Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams

Thursday Feb 16, 2017
Speaker(s):

In 1987 Hitchin discovered a new family of algebraic integrable systems, solvable by spectral curve methods. One novelty was that the base curve was of arbitrary genus. Later on it was understood how to extend Hitchin's viewpoint, allowing poles in the Higgs fields, and thus incorporating many of the known classical integrable systems, which occur as meromorphic Hitchin systems when the base curve has genus zero.

Scientific Areas:

## Higgs sheaves on a curve and Cohomological Hall algebras

Thursday Feb 16, 2017
Speaker(s):

Cohomological Hall algebras associated with preprojective algebras of quivers play a preeminent role in geometric representation theory and mathematical physics. In the present talk, I will introduce and describe CoHAs associated with the stack of Higgs sheaves on a smooth projective curve. Moreover, I will address the connections with representation theory and gauge theory. (This is a joint work with Olivier Schiffmann.)

Scientific Areas:

## How I learned to stop worrying and to love both instantons and anti-instantons

Thursday Feb 16, 2017
Speaker(s):
Scientific Areas:

## Geometric interpretation of Witten's d-bar equation

Thursday Feb 16, 2017
Speaker(s):

The Witten d-bar equation is a generalization of the parametrized holomorphic curve equation associated to a holomorphic function (superpotential) on a Kahler manifold X. It plays a central role in the work of Gaiotto-Moore-Witten on the "algebra of the infrared".

The talk will explain an "intrinsic" point of view on the equation as a condition on a real surface S embedded into X (i.e., not involving any parametrization of S). This is possible if S is not a holomorphic curve in the usual sense.

Scientific Areas:

## The Hitchin system, past and present

Wednesday Feb 15, 2017
Speaker(s):

The talk will be a survey of Higgs bundles, their moduli spaces and the associated fibration structure from a historical, and personal, point of view.

Scientific Areas:

## Symplectic geometry related to G/U and Sicilian theories'

Wednesday Feb 15, 2017
Speaker(s):

We construct an action of the Weyl group on the affine closure of the cotangent bundle on G/U. The construction involves Hamiltonian reduction with respect to the universal centralizer' and an interesting Lagrangian variety, the Miura variety. A closely related construction produces symplectic manifolds which play a role in `Sicilian theories' and whose existence was conjectured by Moore and Tachikawa. Some of these constructions may be reinterpreted, via the Geometric Satake, in terms of the affine grassmannian.

Scientific Areas:

## On index of rigidity

Wednesday Feb 15, 2017
Speaker(s):

The index of rigidity was introduced by Katz as the Euler characteristic of de Rham cohomology of End-connection of a meromorphic connection on curve. As its name suggests, the index valuates the rigidity of the connection on curve. Especially, in P^1 case, this index makes a significant contribution together with middle convolution. Namely Katz showed that regular singular connection on P^1 can be reduced to a rank 1 connection by middle convolution if and only if the index of rigidity is 2.

Scientific Areas:

## Pages

Scientific Organizers:

• Kevin Costello, Perimeter Institute
• Davide Gaiotto, Perimeter Institute
• Yan Soibelman, Kansas State University