Symplectic duality and Gauge theory

COVID-19 information for PI Residents and Visitors

Conference Date:

Thursday, April 7, 2016 (All day) to Sunday, April 10, 2016 (All day)

The purpose of this workshop is to explore the connections between symplectic duality, three-dimensional gauge theory and other subjects such a geometric Langlands duality and homological knot invariants.

Sponsorship for this workshop has been provided by:

Chris Beem, Institute for Advanced Study
Gwyn Bellamy, University of Glasgow
Roman Bezrukavnikov, Massachusetts Institute of Technology
Alexander Braverman, Perimeter Institute & University of Toronto
Tudor Dimofte, Perimeter Institute & University of California, Davis
Davide Gaioto, Perimeter Institute
Iordan Ganev, University of Texas
Joel Kamnitzer, University of Toronto
Yanki Lekili, King's College London
Ivan Loseu, Northeastern University
Jacob Matherne, Louisiana State University
Kevin McGerty, University of Oxford
David Nadler, University of California, Berkeley
Thomas Nevins, University of Illinois Urbana-Champaign
Lev Rozansky, University of North Carolina
Ben Webster, University of Virginia
Edward Witten, Institute for Advanced Study

Chris Beem, Institute for Advanced Study
Gwyn Bellamy, University of Glasgow
Roman Bezrukavnikov, Massachusetts Institute of Technology
Lakshya Bhardwaj, Perimeter Institute
Alexander Braverman, Perimeter Institute & University of Toronto
Mathew Bullimore, Institute for Advanced Study
Dylan Butson, Perimeter Institute
Kevin Costello, Perimeter Institute
Michele del Zotto, Harvard University
Tudor Dimofte, Perimeter Institute & University of California, Davis
Chris Dodd, Perimeter Institute
Anne Dranovski, University of Toronto
Laurent Freidel, Perimeter Institute
Chris Elliott, Northwestern University
Davide Gaiotto, Perimeter Institute
Iordan Ganev, University of Texas
Justin Hilburn, Perimeter Institute
Joel Kamnitzer, University of Toronto
Hee-Cheol Kim, Perimeter Institute
Heeyeon Kim, Perimeter Institute
Peter Koroteev, University of Minnesota
Yanki Lekili, King's College London
Ivan Loseu, Northeastern University
Jacob Matherne, Louisiana State University
Dalimil Mazac, Perimeter Institute
Kevin McGerty, University of Oxford
Seyed Faroogh Moosavian, Perimeter Institute
Dinakar Muthiah, University of Alberta
David Nadler, University of California, Berkeley
Thomas Nevins, University of Illinois Urbana-Champaign
James Pascaleff, University of Illinois Urbana-Champaign
Miroslav Rapcak, Perimeter Institute
Lev Rozansky, University of North Carolina
Ibrahim Shehzad, Perimeter Institute
Ben Webster, University of Virginia
Alexander Weeks, University of Toronto
Edward Witten, Institute for Advanced Study
Oded Yacobi, University of Sydney
Philsang Yoo, Northwestern University
Guojun Zhang, Perimeter Institute
Jie Zhou, Perimeter Institute

Thursday, April 7, 2016

Time	Event	Location
10:30-11:00am	Registration	Reception
11:00-12:00pm	Gwyn Bellamy, University of Glasgow Intro to Quantization	Sky Room
12:00 – 1:30pm	Lunch	Bistro – 1^st Floor
1:30 – 2:30pm	Iordan Ganev, University of Texas The wonderful compactification for quantum groups	Sky Room
2:30 – 3:30pm	Coffee Break	Bistro – 1^st Floor
3:30 – 4:30pm	Jacob Matherne, Louisiana State University Derived geometric Satake equivalence, Springer correspondence, and small representations	Sky Room
4:30 – 5:30pm	Kevin McGerty, University of Oxford Springer theory and symplectic resolutions	Sky Room

Friday, April 8, 2016

Time	Event	Location
9:00 – 10:00am	Thomas Nevins, University of Illinois at Urbana-Champaign Moduli of vacua and representations of Cherednik algebras	Sky Room
10:00 – 11:00am	Coffee Break	Bistro – 1st Floor
11:00 – 12:00pm	Chris Beem, Institute for Advanced Study Quantization of hyperkahler cones and superconformal symmetry in three dimensions	Sky Room
12:00 – 1:30pm	Lunch	Bistro – 2^nd Floor
1:30 – 2:30pm	Yanki Lekili, King's College London Koszul duality patterns in Floer theory	Sky Room
2:30 – 3:30pm	Coffee Break	Bistro – 1st Floor
3:30 – 4:30pm	Ben Webster, University of Virginia Symplectic duality for hyperkähler quotients	Sky Room
4:30 – 5:30pm	Joel Kamnitzer, University of Toronto From MV polytopes to the BFN construction of the Coulomb branch	Sky Room
6:00pm onwards	Banquet	Bistro – 2^nd Floor

Saturday, April 9, 2016

Time	Event	Location
9:00 – 10:00am	Edward Witten, Institute for Advanced Study More On Gauge Theory And Geometric Langlands	Sky Room
10:00 – 11:00am	Coffee Break	Bistro – 1st Floor
11:00 – 12:00pm	Roman Bezrukavnikov, Massachusetts Institute of Technology Coherent sheaves on symplectic resolutions and highest weight categories	Sky Room
12:00 – 1:30pm	Lunch	Bistro – 1^st Floor
1:30 – 2:30pm	David Nadler, University of California, Berkeley Mirror symmetry for pairs of pants	Sky Room
2:30 – 3:30pm	Break
3:30 – 4:30pm	Ivan Loseu, Northeastern University Derived equivalences for Rational Cherednik algebras	Sky Room
4:30 – 5:30pm	Lev Rozansky, University of North Carolina HOMFLY-PT link homology and the Gukov-Witten tame ramification defect in the Kapustin-Witten TQFT	Sky Room
7:00pm onwards	Dinner	Proof

Sunday, April 10, 2016

Time	Event	Location
9:00 – 10:00am	Alexander Braverman, Perimeter Institute & University of Toronto Cyclotomic Cherednik algebras and quantized Coulomb branches	Sky Room
10:00 – 11:00am	Tudor Dimofte, Perimeter Institute & University of California, Davis Davide Gaiotto, Perimeter Institute TBA	Sky Room

Chris Beem, Institute for Advanced Study

Quantization of hyperkahler cones and superconformal symmetry in three dimensions

I will describe a connection between the algebra of local operators in three-dimensional N=4 superconformal field theories and the problem of deformation quantization for hyperkahler cones. I will point out some apparently surprising mathematical consequences of this correspondence.

Roman Bezrukavnikov, Massachusetts Institute of Technology

Coherent sheaves on symplectic resolutions and highest weight categories

I will discuss some features of the derived categories of coherent sheaves on a symplectic resolution of singularities that become manifest once the variety is reduced to positive characteristic and quantized. I will emphasize the features that are relevant to (either classical or 3D) mirror symmetry.

Alexander Braverman, Perimeter Institute & University of Toronto

Cyclotomic Cherednik algebras and quantized Coulomb branches

This talk will be devoted to the discussion of quantized Coulomb branches of 3d N=4 guage theories(as was defined mathematically in the work of Finkelberg, Nakajima and the presenter) in the case of quiver gauge theories attached to the Jordan quiver. I will show that the resulting non-commutative algebra is the spherical subalgebra in the cyclotomic Cherednik algebra.

I will also discuss a possible extension of this result to the case of 4d theories (in this case new algebras appear) and possible applications to representation theory.

Based on work in progress with Etingof, Finkelberg, Kodera and Nakajima.

Iordan Ganev, University of Texas

The wonderful compactification for quantum groups

The wonderful compactification of a group links the geometry of the group to the geometry of its partial flag varieties, encodes the asymptotics of matrix coefficients for the group, and captures the rational degenerations of the group. It plays a crucial role in several areas of geometric representation theory and related fields. In this talk, we review several constructions of the wonderful compactification and its relevant properties. We then introduce quantum group versions of the wonderful compactification and its stratification by G x G orbits. A key player in our approach is the Vinberg semigroup, which arises as a Rees construction and forms a multi-cone over the wonderful compactification.

Joel Kamnitzer, University of Toronto

From MV polytopes to the BFN construction of the Coulomb branch

Components of cores of quiver varieties and MV cycles both provide geometric bases for representations of semisimple Lie algebras. One of the goals of symplectic duality is to relate these two geometric constructions. These constructions can be related on a combinatorial level using crystals and MV polytopes. Based on this combinatorics, about 6 year ago, Allen Knutson and I conjectured a way to produce coordinate rings of MV cycles using quiver varieties. The Braverman-Finkelberg-Nakajima construction of the Coulomb branch provides us a framework for understanding this conjecture.

Yanki Lekili, King's College London

Koszul duality patterns in Floer theory

We study symplectic invariants of the open symplectic manifolds X_Γ obtained by plumbing cotangent bundles of 2-spheres according to a plumbing tree Γ. For any tree Γ, we calculate (DG-)algebra models of the Fukaya category F(X_Γ) of closed exact Lagrangians in X_Γ and the wrapped Fukaya category W(X_Γ). When Γ is a Dynkin tree of type An or Dn (and conjecturally also for E6 , E7, E8 ), we prove that these models for the Fukaya category F(X_Γ) and W(X_Γ) are related by (derived) Koszul duality. As an application, we give explicit computations of symplectic cohomology of X_Γ for Γ = An, Dn , based on the Legendrian surgery formula.

Ivan Loseu, Northeastern University

Derived equivalences for Rational Cherednik algebras

The representation theory of Hecke algebras of Weyl group is a classical topic in Lie representation theory. Sometimes, the categories of modules have infinite homological dimension (a.k.a. singular), for example, this happens when the parameters are roots of unity of small enough order. Ginzburg, Guay, Opdam and Rouquier constructed highest weight covers of the categories of modules over Hecke algebras (that should be thought as resolutions of singularities). These covers are categories O over Rational Cherednik algebras. In 2005, Rouquier has conjectured that all these covers are derived equivalent (a result usually anticipated for nice resolutions in Algebraic geometry). I will introduce the Rational Cherednik algebras, their categories O and the KZ functor. Then I will sketch the proof of Rouquier's conjecture. Time permitting I will explain an application to the computation of supports of the simple modules in categories O for Weyl groups of classical types.

Jacob Matherne, Lousiana State University

Derived geometric Satake equivalence, Springer correspondence, and small representations

Two major theorems in geometric representation theory are the geometric Satake equivalence and the Springer correspondence, which state:

1. For G a semisimple algebraic group, we can realize Rep(G) as intersection cohomology of the affine Grassmannian for the Langlands dual group.

2. For W a Weyl group, we can realize Rep(W) as intersection cohomology of the nilpotent cone.

In the late 90s, M. Reeder computed the Weyl group action on the zero weight space of the irreducible representations of G, thereby relating Rep(G) to Rep(W). More recently, P. Achar, A. Henderson, and S. Riche have established a functorial relationship between the two phenomena above. In my talk, I will discuss my thesis work which extends their functorial relationship to the setting of mixed, derived categories.

Kevin McGerty, University of Oxford

Springer theory and symplectic resolutions

We will describe how an analogue of Springer's theory of Weyl group representations can be defined for a symplectic resolution of singularities, and explain what aspects of the classical theory survive in this more general set-up. For finite type Nakajima quiver varieties we will show how one recovers the Weyl group action of Lusztig, Nakajima and Maffei. This is joint work with T. Nevins.

David Nadler, University of California, Berkeley

Mirror symmetry for pairs of pants

We will discuss homological mirror symmetry for n-dimensional pairs of pants and their mirror Landau-Ginzburg models. Our emphasis will be on methods that reduce the calculation of branes to simple combinatorial configurations.

Thomas Nevins, University of Illinois at Urbana-Champaign

Moduli of vacua and representations of Cherednik algebras

The moduli of vacua of N=2* gauge theory is known to be the Calogero-Moser (CM) integrable system. I will introduce this system, describe some features of its geometry, and explain its relation to representations of Cherednik algebras. I will discuss work in progress on geometric Langlands duality for the CM system. Finally, I hope to describe some aspects of the relevant geometry that seem worth understanding better.

Lev Rozansky, University of North Carolina

HOMFLY-PT link homology and the Gukov-Witten tame ramification defect in the Kapustin-Witten TQFT

This is a joint work with A. Oblomkov. Kapustin and Witten presented the geometric Langlands duality as a particular manifestation of an electric-magnetic duality of a special 4d Yang-Mills TQFT. Gukov and Witten described a special 2d defect within this TQFT which corresponds to a tame ramification appearing in Langlands duality. This GW defect is characterized by some `irrelevant' parameters which braid as one goes along the defect. We will present the evidence that the space of states of a 3d sphere containing a defect-unknot, whose irrelevant parameters are braided, is isomorphic to the HOMFLY-PT homology of the link which is the closure of the irrelevant braid. We will also relate this isomorphism to a conjectured relation between the HOMFLY-PT link homology and coherent sheaves on Hilbert schemes of C^2.

The calculation of the space of states of the defect-unknot is based on the Kapustin-Setter-Vyas description of the 2-category of S^1 within the Kapustin-Witten TQFT, the identification of the object corresponding to a disk with a single ramification point and the description of the braid group action on the category of endomorphisms of this object.

Ben Webster, University of Virginia

Symplectic duality for hyperkähler quotients

It has been conjectured by Braden, Licata, Proudfoot and the presenter that the Higgs and Coulomb branches of a 3d N=4 supersymmetric gauge theory for a reductive group G and representation T^*V will have associated category O's (formed by A-branes on the branches with specified boundary conditions) which are Koszul dual. I'll present preliminary work proving a version of this result. The relevant object on the Higgs branch is a convolution algebra in equivariant cohomology, which is the Ext algebra of a natural semi-simple perverse sheaf on V/G; on Coulomb side, we'll consider certain the endomorphism algebras of projective objects in the category of weight modules over the quantized Coulomb branch, as presented by Braverman, Finkelberg and Nakajima. There is a simple and explicit isomorphism between these algebras (after a suitable completion on the Higgs side) which gives a new and uniform proof of all known cases of symplectic duality.

Edward Witten, Institute for Advanced Study

More On Gauge Theory And Geometric Langlands

In this talk, I will explain some topics from the paper with the same title as the talk, including the relation between Hitchin's equations and the A-model and how to concretely verify that an A-brane dual to a B-brane supported at a point on Hitchin's moduli space is a eigenbrane for the 't Hooft operators.

Scientific Organizers:

Alexander Braverman, Perimeter Institute & University of Toronto
Kevin Costello, Perimeter Institute
Tudor Dimofte, Perimeter Institute & University of California, Davis
Davide Gaiotto, Perimeter Institute