This workshop will explore various beautiful mathematical structures making their appearance in the study of supersymmetric quantum field theory. There will be introductory lectures on Hitchin moduli spaces, knot homology and quantum integrable systems as well as research talks.

The workshop will be held in the Newhaven lecture theatre at the International Centre of Mathematical Sciences (ICMS) and is made possible by the Royal Society.

Contact Lotte Hollands for more information.


Monday Tuesday Wednesday Thursday
8:30 - 9:15 Registration
9:15 - 10:30 Gukov

The sound of knots and 3-manifolds

Barclay Prime is a restaurant in Philadelphia that became famous for its exquisite $100 cheesesteak. Served with a small bottle of Veuve Clicquot champagne, Barclay Prime’s cheesesteak is made of sliced Kobe beef, melted Taleggio cheese, shaved truffles, sauteed foie gras, caramelized onions and heirloom shaved tomatoes on a homemade brioche roll brushed with truffle butter and squirted with homemade mustard. Yet, in essence, it still is just a sandwich. One of the recent developments at the interface of physics and mathematics involves a number of very sophisticated ingredients from both fields. Yet, at its basic, it is just a familiar quantum mechanics that I will present to you on February 27.


Geometric construction of bethe eigenfunctions

A fundamental discovery of Nekrasov and Shatashvili equates quantum K-theory of a Nakajima quiver variety (as a commutative ring) with Bethe equations for a certain quantum affine Lie algebra. I will explain how to go make the next step and find the corresponding Bethe eigenfunctions and, more generally, solutions to qKZ and dynamical difference equations. This is a joint work with Mina Aganagic.


Introduction to moduli spaces of flat connections

We give an overview of the moduli spaces parametrizing flat connections over complex algebraic varieties: the character variety, the Hitchin moduli space of Higgs bundles, and the moduli space of algebraic bundles with integrable connection. We discuss the correspondences between them, the hyperkahler structure, the Hitchin map, the role of variations of Hodge structure. Further topics will include cohomology jump loci, quasiprojective varieties and parabolic structures, and current directions on the study of compactifications.


A skeletal introduction to homological mirror symmetry

Guided by structural features of the categories of coherent sheaves and exact lagrangians, I'll sketch a proof of homological mirror symmetry at the large complex structure limit.

10:30 - 11:00 Tea break Tea break Tea break Tea break
11:00 - 12:15 Nawata

Representations of DAHA from Hitchin moduli space

I will talk about physics approach to understand representation theory of double affine Hecke algebra (DAHA). DAHA can be realized as an algebra of line operators in 4d N=2* theory and therefore it appears as quantization of coordinate ring of Hitchin moduli space over once-punctured torus. Using 2d A-model on the Hitchin moduli space, I will explain relationship between representation category of DAHA and Fukaya category of the Hitchin moduli space.


Quiver gauge theories and quantum integrable systems


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(Sigma), G)/G of representations of the fundamental group of the base curve Sigma 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.


Surface defects in 4d supersymmetric field theories via integrable lattice model

In this talk, we consider surface defects in a certain class of 4d N=1 theories in relation with the integrable lattice model. The supersymmetric index of an N=1 theory realized by a brane tiling coincides with the partition function of an integrable 2d lattice model. We argue that a class of half-BPS surface defects in the 4d theory are represented in the lattice model as transfer matrices constructed from L-operators. For a surface defect labeled by the fundamental representation of SU(2) in the 4d theory with SU(2) gauge groups, we identify the relevant L-operator as that discovered by Sklyanin in the context of the eight-vertex model. We perform nontrivial checks against the residue computation in class S and class S_k theories. The corresponding transfer matrix unifies 2k difference operators obtained for class S_k theories into a one-parameter family of difference operators.

This talk is based on the collaboration with Junya Yagi, and on arXiv:1606.01041.

12:15 - 13:45 Lunch Lunch Lunch Lunch
13:45 - 15:00 Rasmussen

Knot homologies

I'll give an overview of knot homologies from a mathematical perspective. Exactly which topics I focus on will depend on what seems most interesting to the organizer and other participants.


Integrable lattice models from gauge theory

In a celebrated paper in 1989, E. Witten discovered a beautiful connection between knot invariants (such as the Jones polynomial) and three-dimensional Chern-Simons theory. Since there are similarities between knot theory and integrable models, it is natural to ask if there is also a gauge theory explanation for integrable models. The answer to this question was recently given by K. Costello in 2013. In this talk I will describe my ongoing work with K. Costello and E. Witten, which explains many results in integrable models from standard quantum field theory analysis of Costello's theory.


Abelianization in Chern-Simons theory

I will describe the notion of "abelianization" of flat connections over 2- and 3-manifolds, and an application of this notion to the topological field theory of classical Chern-Simons invariants. The notion of abelianization arose in joint work with Davide Gaiotto and Greg Moore, and the work on Chern-Simons is joint work in progress with Dan Freed.


The two-dimensional Landau-Ginzburg approach to link homology

I will give a very basic introduction to the two-dimensional Landau-Ginzburg approach to link homology through the web-based formalism. Some peculiarities and relation to Khovanov homology will be discussed.

15:00 - 15:30 Tea break Tea break Tea break Tea break
15:30 - 16:45 Marino

The spectral theory of quantum mirror curves

I will present a correspondence between open and closed topological strings on toric Calabi-Yau manifolds, on one hand, and the spectral theory of a new family of trace class operators on the real line, on the other hand. These operators are obtained by quantizing the mirror curves to the toric CYs, and they are exactly solvable by using open and closed BPS invariants of the CY. Conversely, the spectral problem makes it possible to define a non-perturbative completion of the topological string. If time permits, I will explain the relation of this framework to resurgent trans-series.


From Dimers to Quantum K-theory

Considering dimer configurations on the honeycomb lattice we define two different solutions of the quantum Yang-Baxter equation. Using techniques from quantum integrable systems, such as Baxter’s Q-operator and the Bethe ansatz, we define a generalised quantum Schubert calculus which describes several known cases in the literature, for example the equivariant K-theory ring of Grassmannians. We also explicitly construct a hitherto unknown ring, which we conjecture to be the quantum equivariant K-theory of Grassmannians. (This has recently been confirmed - using different methods - by Buch, Chaput, Mihalcea and Perrin who consider all cominuscule varieties.) Our description via a quantum integrable system has the advantage of revealing an underlying “quantum group structure”. This structure is akin to the Yangian structures appearing in the work by Maulik and Okounkov on general Nakajima varieties.

This is joint work with Vassily Gorbounov, Aberdeen.


Relating the geometric Langlands correspondence to AGT


Twisted Hilbert Space of 3d N = 2 Gauge Theories

I will talk about the description of 3d N = 2 gauge theories on R x C with a topological twist on a Riemann surface C as a supersymmetric quantum mechanics on R. I will focus on a mathematical description of the Hilbert space of supersymmetric ground states in U(1) gauge theories, demonstrating invariance under three-dimensional mirror symmetry and reproducing the twisted index on S^1 x C.

16:45 - 18:00 Wine reception

Venue and Travel

The workshop takes place at the ICMS in town. Some useful tips for traveling to Edinburgh can be found here. In particular, if you travel to Edinburgh Airport take the Airport Shuttle to town. Accomodation for speakers is closeby at the Jurys Inn.


Aghaei, Nezhla University of Bern
Alim, Murad University of Hamburg
Allegretti, Dylan University of Sheffield
Aravanis, Christos University of Sheffield
Atiyah, Sir Michael University of Edinburgh
Beck, Florian Mathematical Institute Hamburg
Belliard, Raphael IPhT CEA Saclay
Benetti Genolini, Pietro University of Oxford
Boalch, Philip CNRS Paris-Sud
Booth, Matt University of Edinburgh
Borot, Gaetan Max Planck Institute for Mathematics
Braden, Harry University of Edinburgh
Bullimore, Mathew University of Oxford
Bunk, Severin Heriot-Watt University
Cirafici, Michele Instituto Superior Tecnico
Cooke, Juliet University of Edinburgh
Feigin, Misha University of Glasgow
Figiel, Troy DESY
Fluder, Martin Caltech
Fringuelli, Roberto University of Edinburgh
Galakhov, Dimitrii University of California, Berkeley
Gukov, Sergei California Institute of Technology
Hollands, Lotte Heriot-Watt University
Inglima, Sergio Heriot-Watt University
Jardim, Marcos IMECC - UNICAMP
Jordan, David University of Edinburgh
Kameyama, Masaya Nagoya University
Kidwai, Omar University of Oxford
Kim, Heeyeon Perimeter Institute for Theoretical Physics
Korff, Christian University of Glasgow
Korpas, Georgios Trinity College Dublin
Lanini, Martina University of Edinburgh
Lekili, Yanki Kings College London
Mariño, Marcos University of Geneva
Maruyoshi, Kazunobu Seikei University
McAteer, Dermot Heriot-Watt University
Mirza Amraji, Moein King's College London
Mkrtchyan, Anna University of Edinburgh
Müller, Lukas Heriot-Watt University
Nawata, Satoshi Fudan University
Neitzke, Andrew University of Texas at Austin
Nguyen, Trang University of Edinburgh
Nieri, Fabrizio Uppsala University
Okounkov, Andrei Columbia University
Okounkova, Anna Columbia University
Palazzo, David University of Glasgow
Panfil, Miłosz University of Warsaw
Pawelkiewicz, Michal IPhT, CEA
Pestun, Vasily Institut des Hautes Études Scientifiques
Rasmussen, Jacob University of Cambridge
Reynolds, Ruth University of Edinburgh
Romao, Nuno University of Augsburg
Ross, Calum Heriot-Watt University
Rueter, Philipp Heriot-Watt University
Saberi, Ingmar Universität Heidelberg
Saemann, Christian Heriot-Watt University
Samuelson, Peter University of Edinburgh
Shende, Vivek University of California, Berkeley
Simpson, Carlos CNRS, Universite de Nice
Strachan, Ian University of Glasgow
Tabler, Alexander Ludwig-Maximilians University of Munich
Teschner, Jörg University of Hamburg
Vasko, Petr University of Warsaw
Vogrin, Martin University of Hamburg
Weelinck, Tim University of Edinburgh
Weston, Robert Heriot-Watt University
Yagi, Junya University of Warsaw
Yamazaki, Masahito University of Tokyo
Zapata, Carlos University of Edinburgh