QFT for Mathematicians 2022

20 Jun 2022, 08:30 → 30 Jun 2022, 17:00 America/Toronto

Ben Webster (Perimeter Institute & University of Waterloo), Davide Gaiotto (Perimeter Institute), Kevin Costello (Perimeter Institute), Philsang Yoo (Tsinghua University), Tudor Dimofte (University of California, Davis)

In recent decades, interaction between mathematics and quantum field theory has proven to be fertile for both fields. However, differences in language, training, and perspective have significantly hampered effective communication between the two parties. The aim of this workshop series is to tear down this wall, and help mathematicians to learn the way physicists think about QFT and strings and eventually ``speak'' QFT and strings fluently. This year we will focus on algebraic, geometric, and homological structures in supersymmetric gauge theories. We will use 3-dimensional N=4 gauge and N=2 theories --- which have seen an explosion of interest in both physical and mathematical communities in recent years --- as a key source of new examples, exploring their connections with integrability, boundary vertex algebras and elliptic cohomology, knot homology, 2d mirror symmetry, and the geometric Langlands program.  

The first week of this event will consist of an in person school and the second week will consist of a virtual conference.

Perimeter Institute will make every effort to host the school as an in-person event.  However, we reserve the right to change to an online program to align with changes in regulations due to the COVID-19 pandemic

PIRSA:  https://pirsa.org/C22011

 

Territorial Land Acknowledgement

Perimeter Institute acknowledges that it is situated on the traditional territory of the Anishinaabe, Haudenosaunee, and Neutral peoples.

Perimeter Institute is located on the Haldimand Tract. After the American Revolution, the tract was granted by the British to the Six Nations of the Grand River and the Mississaugas of the Credit First Nation as compensation for their role in the war and for the loss of their traditional lands in upstate New York. Of the 950,000 acres granted to the Haudenosaunee, less than 5 percent remains Six Nations land. Only 6,100 acres remain Mississaugas of the Credit land.

We thank the Anishinaabe, Haudenosaunee, and Neutral peoples for hosting us on their land. 

problems (1).pdf

ProblemSession_4d.pdf

Monday, 20 June

08:30 Registration

1 Welcome and Opening Remarks

Speaker: Kevin Costello (Perimeter Institute)

2 3d Theories and Twists I

Speaker: Kevin Costello (Perimeter Institute)

10:30 Coffee Break

3 3d Gauge Theory and Elliptic Stable Envelopes I

Speaker: Andrei Okounkov (Columbia University)

12:30 Lunch

Problem Session

15:00 Coffee Break

4 Classical BV Formalism and Topological Quantum Field Theory

Speaker: Philsang Yoo (Seoul National University)

Tuesday, 21 June

5 A and B models in 3d and 4d I

Speaker: Justin Hilburn (Perimeter Institute)

10:30 Coffee Break

6 3d B Models and Knot Homology I

Speaker: Lev Rozansky (University of North Carolina, Chapel Hill)

12:30 Lunch

Problem Session

15:00 Coffee Break

7 On Boundary VOA's

Speaker: Davide Gaiotto (Perimeter Institute)

17:00 BBQ

Wednesday, 22 June

8 3d Gauge Theory and Elliptic Stable Envelopes II

Speaker: Mykola Dedushenko (Stony Brook University)

10:30 Coffee Break

9 3d Theories and Twists II

Speaker: Tudor Dimofte (University of Edinburgh)

12:30 Lunch

Problem Session

15:00 Coffee Break

10 Open Discussion

Thursday, 23 June

11 3d Gauge Theory and Elliptic Stable Envelopes III

Speaker: Andrei Okounkov (Columbia University)

10:30 Coffee Break

12 Geometrical structures in 4d N=2 class S theories

I will give an introduction to 4d N=2 class-S theories. I will describe the construction of such theories, the roles played by extended defects such as line defects and surface defects, as well as connections to Hitchin systems.

Speaker: Fei Yan (Rutgers University)

12:30 Lunch

Problem Session

15:00 Coffee Break

13 3d Gauge Theory and Elliptic Stable Envelopes IV

Speaker: Mykola Dedushenko (Stony Brook University)

Friday, 24 June

14 3d B Models and Knot Homology II

Speaker: Lev Rozansky (University of North Carolina, Chapel Hill)

10:30 Coffee Break

15 A and B Models in 3d and 4d II

Speaker: Justin Hilburn (Perimeter Institute)

12:30 Lunch

Problem Session

15:00 Coffee Break

16 Open Discussion

Monday, 27 June

17 Welcome and Opening Remarks

Speaker: Ben Webster (Perimeter Institute & University of Waterloo)

18 Generalised Langlands, VOAs, and (generalised) tau-functions

In the first part of my talk I'll briefly review some aspects of the relations between N=4, d=4 SYM and vertex operator algebras (VOAs) discussed in recent work of Gaiotto and collaborators. The resulting picture predicts conjectural generalisations of the geometric Langlands correspondence. We will focus on a class of examples figuring prominently in recent work of Creutzig-Dimofte-Garner-Geer, labelled by parameters n (rank) and k. For the case k=1,n=2 we will point out that the conformal blocks of the relevant VOA, twisted by local systems, represent sections of natural holomorphic line bundles over the moduli spaces of local systems closely related to the isomonodromic tau functions. Observing the crucial role of (quantised) cluster algebras in the definition of the holomorphic line bundles suggests natural generalisations of this story to higher values of the parameters k and n.

Speaker: Joerg Teschner (University of Hamburg & DESY)

10:30 Break

19 Generalised Langlands, VOAs, and (generalised) tau-functions cont.

Speaker: Joerg Teschner (University of Hamburg & DESY)

11:30 Break

20 Defects, quantum trace map, and exact WKB

I will continue the discussions on line defects and surface defects in class S theories, making connections to the construction of the quantum trace map, as well as to the exact WKB method for higher order ODEs.

Speaker: Fei Yan (Rutgers University)

13:00 Break

21 Discussion on knot homology

Speakers: Ben Webster (Perimeter Institute & University of Waterloo), Lev Rozansky (University of North Carolina, Chapel Hill), Tudor Dimofte (University of California, Davis)

Tuesday, 28 June

22 Holomorphic BF theory, 4d SYM and the analytic Geometric Langlands program

I will review the embedding of the analytic Geometric Langlands program in four-dimensional gauge theory.

Speaker: Davide Gaiotto (Perimeter Institute)

10:30 Break

23 Holomorphic BF theory, 4d SYM and the analytic Geometric Langlands program cont.

Speaker: Davide Gaiotto (Perimeter Institute)

11:30 Break

24 Abelian 3d mirror symmetry and boundary conditions

3d mirror symmetry predicts an equivalence between A- and B-twists of a pair of dual 3d N=4 theories. Essentially the strongest invariants one can produce of the resulting 3-dimensional topological field theories are their 2-categories of boundary conditions. The B-side 2-category was first described by Kapustin-Rozansky-Saulinas, but the 2-categorical structure on A-side boundary conditions has not previously been understood. For abelian gauge theories with matter, we propose a model for the 2-category of A-type boundary conditions using Kapranov-Schechtman's "perverse schobers," and we prove a 3d mirror equivalence between dual 2-categories. By reducing to lower-dimensions, we can recover both the BFN construction and the BLPW Koszul duality for hypertoric categories O. This is joint work with Justin Hilburn and Aaron Mazel-Gee.

Speaker: Benjamin Gammage (Harvard University)

13:00 Break

25 Discussion on 4d N=2

Speakers: Ahsan Khan (Institute for Advanced Study), Benjamin Gammage (Harvard University), Justin Hilburn (Perimeter Institute), Tudor Dimofte (University of California, Davis)

Wednesday, 29 June

26 Field theory in the 1/2 Omega-background

In this lecture I'll discuss various aspects of 4d N=2 and 5d N=1 supersymmetric QFT's in the 1/2 Omega-background (and along the way try to emphasize some relations to the 3d N=2 theories discussed in this workshop). Central to this story is the Nekrasov instanton partition function (or topological string partition function) in this background, which we will obtain through abelianization as an integral of a ratio of Wronskians of certain special solutions to the relevant Schrodinger equation. We will argue that a slight generalization of the above partition function solves an associated Riemann-Hilbert problem and defines a section of a distinguished line bundle over the moduli space of flat connections.

Speaker: Lotte Hollands (Heriot-Watt University)

10:30 Break

27 Field theory in the 1/2 Omega-background cont.

Speaker: Lotte Hollands (Heriot-Watt University)

11:30 Break

28 Fun with a holomorphic theory on twistor space

I will summarize recent work with K. Costello, in which a local holomorphic theory on twistor space furnishes an isomorphism between 1. correlators of a 2d chiral algebra and 2. form factors (scattering amplitudes in the presence of a local operator insertion) of a 4d non-unitary CFT, with physics applications.

Speaker: Natalie Paquette (University of Washington)

13:00 Break

29 Discussion on 3d A-model

Speakers: Davide Gaiotto (Perimeter Institute), Fei Yan (Rutgers University), Joerg Teschner (University of Hamburg & DESY), Lotte Hollands (Heriot-Watt University)

Thursday, 30 June

30 The relative Langlands program via gauge theory

I will discuss a close parallel between Gaiotto and Witten's S-duality for supersymmetric boundary conditions in 4d N=4 SYM and the relative Langlands program, an enhancement of the Langlands program that was developed to provide a framework for the theory of integral representations of L-functions. A special and conjecturally self-dual class of boundary conditions is provided by quantizations of "small" or "multiplicity-free" hamiltonian spaces called hyperspherical varieties. I'll explain how a hyperspherical variety produces objects of interest in all the different settings of the Langlands program (local / global, geometric / arithmetic) and a collection of conjectures providing S-dual descriptions of these objects. The talk is based on forthcoming joint work with Yiannis Sakellaridis and Akshay Venkatesh.

Speaker: David Ben-Zvi (University of Texas, Austin)

10:30 Break

31 The relative Langlands program via gauge theory cont.

Speaker: David Ben-Zvi (University of Texas, Austin)

11:30 Break

32 Lie superalgebras and S-duality

We present a series of (partly proven) conjectures describing geometric realizations of categories of (finite-dimensional) representations of quantum super-groups U_q(g) corresponding to Lie super-algebras g with reductive even part and a non-degenerate invariant form.
We shall also discuss the meaning of these conjectures from the point of view of local geometric Langlands correspondence as well as a connection to the work of Ben-Zvi, Sakellaridis and Venkatesh.
Based on joint works with M.Finkelberg, V.Ginzburg and R.Travkin as well as the work of R.Travkin and R.Yang.

Speaker: Alexander Braverman (PI & U of Toronto)

13:00 Break

33 Discussion on Langlands

Speakers: Alexander Braverman (PI & U of Toronto), David Ben-Zvi (University of Texas, Austin), Davide Gaiotto (Perimeter Institute), Philsang Yoo (Seoul National University)