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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 ParisSud
 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 ParisSud
 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 JohnsonFreyd, 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 Institute 

9:35 – 10:30am 
Nigel Hitchin, University of Oxford 
Sky Room 
10:30 – 11:00am 
Coffee Break 
Bistro – 1^{st} Floor 
11:0012:00pm 
Sergey Cherkis, University of Arizona 
Sky Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Marco Gualtieri, University of Toronto 
Sky Room 
3:00  4:00pm 
Szilard Szabo, Budapest University of Technology & Economics 
Sky Room 
4:00 – 4:30pm 
Coffee Break 
Bistro – 1^{st} Floor 
4:30 – 5:30pm 
Discussion 
Sky Room 
Tuesday, February 14, 2017
Time 
Event 
Location 
9:30 – 10:30am 
Tudor Dimofte, University of Cambridge 
Sky Room 
10:30 – 11:00am 
Coffee Break 
Bistro – 1^{st} Floor 
11:0012:00pm 
Tamas Hausel, Institute of Science and Technology Austria 
Sky Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Alexander Soibleman, University of Southern California 
Sky Room 
3:00  4:00pm 
Ben Davison, University of Glasgow 
Sky Room 
4:00 – 4:30pm 
Coffee Break 
Bistro – 1^{st} Floor 
4:30 – 5:30pm 
Discussion 
Sky Room 
Wednesday, February 15, 2016
Time 
Event 
Location 
9:30 – 10:30am 
Kazuki Hiroe, Josai University 
Sky Room 
10:30 – 11:00am 
Coffee Break 
Bistro – 1^{st} Floor 
11:0012:00pm 
Victor Ginzburg, University of Chicago 
Sky Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 3:30pm 
Colloquium 
Theater 
3:30 – 4:00pm 
Coffee Break 
Bistro – 1^{st} Floor 
4:00 – 5:00pm 
Discussion 
Sky Room 
6:00pm 
Banquet 
Bistro – 2^{nd} Floor 
Thursday, February 16, 2016
Time 
Event 
Location 
9:30 – 10:30am 
Mikhail Kapranov, Kavli Institute 
Sky Room 
10:30 – 11:00am 
Coffee Break 
Bistro – 1^{st} Floor 
11:0012:00pm 
Nikita Nekrasov, Stony Brook University 
Sky Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Francesco Sala, Kavli IPMU, University of Tokyo 
Sky Room 
3:00  4:00pm 
Philip Boalch, Université ParisSud 
Sky Room 
4:00 – 4:30pm 
Coffee Break 
Bistro – 1^{st} Floor 
4:30 – 5:30pm 
Discussion 
Sky Room 
Friday, February 17, 2017
Time 
Event 
Location 
9:30 – 10:30am 
Emanuel Diaconescu, Rutgers University 
Sky Room 
10:30 – 11:00am 
Coffee Break 
Bistro – 1^{st} Floor 
11:0012:00pm 
Andrew Neitzke, University of Texas at Austin 
Sky Room 
12:00 – 2:00pm 
Lunch 
Bistro – 2^{nd} Floor 
2:00 – 3:00pm 
Sergio Cecotti, SISSA 
Sky Room 
4:00 – 4:30pm 
Coffee Break 
Bistro – 1^{st} Floor 
4:30 – 5:30pm 
Wrap Up Discussion 
Sky Room 
Philip Boalch, Université ParisSud
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 RiemannHilbert 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 (20022014). 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 wellknown. The language of quasiHamiltonian 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 fourspace (ADHM), its orbifolds and their deformations, called ALE space (KronheimerNakajima). 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 KontsevichSoibelman 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 KacMoody 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 3manifolds 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 twodimensional 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 ygenus of Higgs moduli spaces
I will discuss in the framework of the P=W conjecture, how one can conjecture formulas for the perverse Hirzebruch ygenus 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 Endconnection 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 CrawleyBoevey 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 KomatsuMalgrange 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 torsionfree sheaves on the spectral curve instead of line bundles and a lifting to a 3manifold 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 dbar equation
The Witten dbar 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 GaiottoMooreWitten on the "algebra of the infrared".
Andrew Neitzke, University of Texas at Austin
Abelianization in complex ChernSimons theory and a hyperholomorphic line bundle
I will describe an approach to classical complex ChernSimons theory via "abelianization", relating flat SL(N)connections over a manifold of dimension d <= 3 to flat GL(1)connections over a branched Nfold 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, AlexandrovPerssonPioline.
Nikita Nekrasov, Stony Brook University
How I learned to stop worrying and to love both instantons and antiinstantons
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 Kring 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 Dmodules 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 L2cohomology, 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 semisimple case we show that the transformation is a hyperKaehler isometry.
FQHE and Hitchin Systems on Modular Curves
Abelianization in complex ChernSimons theory and a hyperholomorphic line bundle
I will describe an approach to classical complex ChernSimons theory via "abelianization", relating flat SL(N)connections over a manifold of dimension d
TBA
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.
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.)
How I learned to stop worrying and to love both instantons and antiinstantons
Geometric interpretation of Witten's dbar equation
The Witten dbar 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 GaiottoMooreWitten 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.
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.
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.
On index of rigidity
The index of rigidity was introduced by Katz as the Euler characteristic of de Rham cohomology of Endconnection 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.
Pages
Scientific Organizers:
 Kevin Costello, Perimeter Institute
 Davide Gaiotto, Perimeter Institute
 Yan Soibelman, Kansas State University