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The relative Langlands program via gauge theory cont.

David Ben-Zvi The University of Texas at Austin
June 30, 2022

PIRSA:22060092
Mathematical Physics
The relative Langlands program via gauge theory

David Ben-Zvi The University of Texas at Austin
June 30, 2022

PIRSA:22060091
Mathematical Physics
Discussion on 4d N=2
- Davide Gaiotto Perimeter Institute for Theoretical Physics
- Lotte Hollands Heriot-Watt University
- Joerg Teschner Deutsches Elektronen-Synchrotron DESY
- Fei Yan Rutgers University
June 29, 2022

PIRSA:22060090
Mathematical Physics
Field theory in the 1/2 Omega-background cont.

Lotte Hollands Heriot-Watt University
June 29, 2022

PIRSA:22060088
Mathematical Physics
Field theory in the 1/2 Omega-background

Lotte Hollands Heriot-Watt University
June 29, 2022

PIRSA:22060087
Mathematical Physics
Discussion on 3d A-model
- Tudor Dimofte University of Edinburgh
- Benjamin Gammage Harvard University
- Justin Hilburn
- Ahsan Khan Institute for Advanced Study (IAS)
June 28, 2022

PIRSA:22060086
Mathematical Physics
Abelian 3d mirror symmetry and boundary conditions

Benjamin Gammage Harvard University
June 28, 2022

PIRSA:22060085
Mathematical Physics
Holomorphic BF theory, 4d SYM and the analytic Geometric Langlands program

Davide Gaiotto Perimeter Institute for Theoretical Physics
June 28, 2022

PIRSA:22060083
Mathematical Physics
Discussion on knot homology
- Tudor Dimofte University of Edinburgh
- Lev Rozansky University of North Carolina at Chapel Hill
- Ben Webster University of Waterloo
June 27, 2022

PIRSA:22060082
Mathematical Physics
Defects, quantum trace map, and exact WKB

Fei Yan Rutgers University
June 27, 2022

PIRSA:22060081
Mathematical Physics
Generalised Langlands, VOAs, and (generalised) tau-functions cont.

Joerg Teschner Deutsches Elektronen-Synchrotron DESY
June 27, 2022

PIRSA:22060080
Mathematical Physics
Generalised Langlands, VOAs, and (generalised) tau-functions

Joerg Teschner Deutsches Elektronen-Synchrotron DESY
June 27, 2022

PIRSA:22060079
Mathematical Physics

The relative Langlands program via gauge theory cont.

David Ben-Zvi The University of Texas at Austin
June 30, 2022

PIRSA:22060092
Mathematical Physics
The relative Langlands program via gauge theory

David Ben-Zvi The University of Texas at Austin
June 30, 2022

PIRSA:22060091
Mathematical Physics
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.
Discussion on 4d N=2
- Davide Gaiotto Perimeter Institute for Theoretical Physics
- Lotte Hollands Heriot-Watt University
- Joerg Teschner Deutsches Elektronen-Synchrotron DESY
- Fei Yan Rutgers University
June 29, 2022

PIRSA:22060090
Mathematical Physics
Field theory in the 1/2 Omega-background cont.

Lotte Hollands Heriot-Watt University
June 29, 2022

PIRSA:22060088
Mathematical Physics
Field theory in the 1/2 Omega-background

Lotte Hollands Heriot-Watt University
June 29, 2022

PIRSA:22060087
Mathematical Physics
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.
Discussion on 3d A-model
- Tudor Dimofte University of Edinburgh
- Benjamin Gammage Harvard University
- Justin Hilburn
- Ahsan Khan Institute for Advanced Study (IAS)
June 28, 2022

PIRSA:22060086
Mathematical Physics
Abelian 3d mirror symmetry and boundary conditions

Benjamin Gammage Harvard University
June 28, 2022

PIRSA:22060085
Mathematical Physics
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.
Holomorphic BF theory, 4d SYM and the analytic Geometric Langlands program

Davide Gaiotto Perimeter Institute for Theoretical Physics
June 28, 2022

PIRSA:22060083
Mathematical Physics
I will review the embedding of the analytic Geometric Langlands program in four-dimensional gauge theory.
Discussion on knot homology
- Tudor Dimofte University of Edinburgh
- Lev Rozansky University of North Carolina at Chapel Hill
- Ben Webster University of Waterloo
June 27, 2022

PIRSA:22060082
Mathematical Physics
Defects, quantum trace map, and exact WKB

Fei Yan Rutgers University
June 27, 2022

PIRSA:22060081
Mathematical Physics
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.
Generalised Langlands, VOAs, and (generalised) tau-functions cont.

Joerg Teschner Deutsches Elektronen-Synchrotron DESY
June 27, 2022

PIRSA:22060080
Mathematical Physics
Generalised Langlands, VOAs, and (generalised) tau-functions

Joerg Teschner Deutsches Elektronen-Synchrotron DESY
June 27, 2022

PIRSA:22060079
Mathematical Physics
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.

Title	Date	Collection	Type	Info
The relative Langlands program via gauge theory cont.	2022‑06‑30	QFT for Mathematicians 2022	Conference	View details
The relative Langlands program via gauge theory	2022‑06‑30	QFT for Mathematicians 2022	Conference	View details
Discussion on 4d N=2	2022‑06‑29	QFT for Mathematicians 2022	Conference	View details
Field theory in the 1/2 Omega-background cont.	2022‑06‑29	QFT for Mathematicians 2022	Conference	View details
Field theory in the 1/2 Omega-background	2022‑06‑29	QFT for Mathematicians 2022	Conference	View details
Discussion on 3d A-model	2022‑06‑28	QFT for Mathematicians 2022	Conference	View details
Abelian 3d mirror symmetry and boundary conditions	2022‑06‑28	QFT for Mathematicians 2022	Conference	View details
Holomorphic BF theory, 4d SYM and the analytic Geometric Langlands program	2022‑06‑28	QFT for Mathematicians 2022	Conference	View details
Discussion on knot homology	2022‑06‑27	QFT for Mathematicians 2022	Conference	View details
Defects, quantum trace map, and exact WKB	2022‑06‑27	QFT for Mathematicians 2022	Conference	View details
Generalised Langlands, VOAs, and (generalised) tau-functions cont.	2022‑06‑27	QFT for Mathematicians 2022	Conference	View details
Generalised Langlands, VOAs, and (generalised) tau-functions	2022‑06‑27	QFT for Mathematicians 2022	Conference	View details

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The relative Langlands program via gauge theory cont.

The relative Langlands program via gauge theory

Discussion on 4d N=2

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

Field theory in the 1/2 Omega-background

Discussion on 3d A-model

Abelian 3d mirror symmetry and boundary conditions

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

Discussion on knot homology

Defects, quantum trace map, and exact WKB

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

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

The relative Langlands program via gauge theory cont.

The relative Langlands program via gauge theory

Discussion on 4d N=2

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

Field theory in the 1/2 Omega-background

Discussion on 3d A-model

Abelian 3d mirror symmetry and boundary conditions

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

Discussion on knot homology

Defects, quantum trace map, and exact WKB

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

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