Welcome and Opening Remarks

• Kevin Costello (Perimeter Institute)

3d Theories and Twists I

• Kevin Costello (Perimeter Institute)

3d Gauge Theory and Elliptic Stable Envelopes I

• Andrei Okounkov (Columbia University)

Classical BV Formalism and Topological Quantum Field Theory

• Philsang Yoo (Seoul National University)

A and B models in 3d and 4d I

• Justin Hilburn (Perimeter Institute)

3d B Models and Knot Homology I

• Lev Rozansky (University of North Carolina, Chapel Hill)

On Boundary VOA's

• Davide Gaiotto (Perimeter Institute)

3d Gauge Theory and Elliptic Stable Envelopes II

• Mykola Dedushenko (Stony Brook University)

3d Theories and Twists II

• Tudor Dimofte (University of Edinburgh)

Open Discussion

3d Gauge Theory and Elliptic Stable Envelopes III

• Andrei Okounkov (Columbia University)

Geometrical structures in 4d N=2 class S theories

• Fei Yan (Rutgers University)

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.

3d Gauge Theory and Elliptic Stable Envelopes IV

• Mykola Dedushenko (Stony Brook University)

3d B Models and Knot Homology II

• Lev Rozansky (University of North Carolina, Chapel Hill)

A and B Models in 3d and 4d II

• Justin Hilburn (Perimeter Institute)

Open Discussion

Welcome and Opening Remarks

• Ben Webster (Perimeter Institute & University of Waterloo)

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

• Joerg Teschner (University of Hamburg & DESY)

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.

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

• Joerg Teschner (University of Hamburg & DESY)

Defects, quantum trace map, and exact WKB

• Fei Yan (Rutgers University)

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.

Discussion on knot homology

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

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

• Davide Gaiotto (Perimeter Institute)

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

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

• Davide Gaiotto (Perimeter Institute)

Abelian 3d mirror symmetry and boundary conditions

• Benjamin Gammage (Harvard University)

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.

Discussion on 4d N=2

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

Field theory in the 1/2 Omega-background

• Lotte Hollands (Heriot-Watt University)

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.

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

• Lotte Hollands (Heriot-Watt University)

Fun with a holomorphic theory on twistor space

• Natalie Paquette (University of Washington)

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.

Discussion on 3d A-model

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

The relative Langlands program via gauge theory

• David Ben-Zvi (University of Texas, Austin)

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.

The relative Langlands program via gauge theory cont.

• David Ben-Zvi (University of Texas, Austin)

Lie superalgebras and S-duality

• Alexander Braverman (PI & U of Toronto)

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.

Discussion on Langlands

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