Arithmetic Quantum Field Theory Program
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Dates: Feb. 5–Mar. 29, 2024
Location: Harvard CMSA, 20 Garden Street, Cambridge MA 02138
Organizers:
 David BenZvi (University of Texas Austin)
 Solomon Friedberg (Boston College)
 Natalie Paquette (University of Washington Seattle)
 Brian Williams (Boston University)
This program will feature a weekly seminar series, workshops, and a conference.
The object of the program is to develop and disseminate exciting new connections emerging between quantum field theory and algebraic number theory, and in particular between the fundamental invariants of each: partition functions and Lfunctions.
On one hand, there has been tremendous progress in the past decade in our understanding of the algebraic structures underlying quantum field theory as expressed in terms of the geometry and topology of lowdimensional manifolds, both on the level of states (via the AtiyahSegal / BaezDolan / Lurie formalism of extended, functorial field theory) and on the level of observables (via the Beilinson–Drinfeld / Costello–Gwilliam formalism of factorization algebras). On the other hand, Weil’s Rosetta Stone and the Mazur–Morishita–Kapranov–Reznikov arithmetic topology (the “knots and primes” dictionary) provide a sturdy bridge between the topology of 2 and 3manifolds and the arithmetic of number fields. Thus, one can now port over quantum field theoretic ideas to number theory, as first proposed by Minhyong Kim with his arithmetic counterpart of ChernSimons theory. Most recently, the work of BenZvi–Sakellaridis–Venkatesh applies an understanding of the Langlands program as an arithmetic avatar of electricmagnetic duality in fourdimensional gauge theory to reveal a hidden quantum mechanical nature of the theory of $L$functions.
The program will bring together a wide range of mathematicians and physicists working on adjacent areas to explore the emerging notion of arithmetic quantum field theory as a tool to bring quantum physics to bear on questions of interest for the theory of automorphic forms, harmonic analysis and Lfunctions. Conversely, we will explore potential geometric and physical consequences of arithmetic ideas, for example, the Langlands correspondence theory of Lfunctions for 3manifolds.
Schedule
The first week of the program will feature several lecture series aimed at a broad local community of mathematicians and physicists, aiming to introduce the main ideas underlying our program and help establish a common reference point.
The program will host a weekly seminar series on Fridays.
The speakers will be selected with the aim of covering a wide panorama of the subjects over the course of the program.
The program will conclude with a weeklong Conference on Arithmetic Quantum Field Theory March 25–29, 2024.
All lectures take place in Room G10, Harvard CMSA, 20 Garden Street Cambridge.
Week 1: Feb. 5–9, 2024
Abstract: In this lecture series we will introduce some of the themes underlying the CMSA program on Arithmetic Quantum Field Theory taking place this winter and the upcoming conference March 2529, 2024.
Some of the themes we plan to discuss include:
Structures in QFT (like factorization for observables and functorial QFT for states and their relation to geometric / deformation quantization) that are sufficiently algebraic and formal to allow for arithmetic analogs.
The setup of arithmetic topology as a bridge between the background of QFT to that of arithmetic (both “global” and “local”), including the “middle realm” of positive characteristic function fields.
Questions and structures in arithmetic that have been / might be amenable to inspiration from QFT, in particular the theory of Lfunctions and the Langlands program.
Monday, Feb. 5, 2024  
11:00 am – 12:00 pm  Minhyong Kim  Arithmetic topology and field theory (Slides part 1 pdf) 
1:30 – 2:30 pm  Brian Williams  Algebraic quantum field theory (Lecture Notes) 
2:30 – 3:30 pm  David BenZvi  The Langlands program via arithmetic QFT 
Wednesday, Feb. 7, 2024  
11:00 am – 12:00 pm  Minhyong Kim  Arithmetic topology and field theory (Slides part 2 pdf) 
2:30 – 3:30 pm  Brian Williams  Algebraic quantum field theory (Lecture Notes) 
Thursday, Feb.8, 2024  
2:30 – 3:30 pm  Minhyong Kim  Arithmetic topology and field theory (Slides part 3 pdf) 
4:00 – 5:00 pm  David BenZvi  The Langlands program via arithmetic QFT 
Friday, Feb. 9, 2024  
1:00 – 2:00 pm  Brian Williams  Algebraic quantum field theory (Lecture Notes) 
2:00 – 3:00 pm  David BenZvi  The Langlands program via arithmetic QFT 
3:30 – 4:30 pm  David BenZvi  The Langlands program via arithmetic QFT 
Monday, Feb. 26, 2024  
1:00 – 2:00 pm  Omer Offen (Brandeis)  Period integrals of automorphic forms and the residue method 
Tuesday, Feb. 27, 2024  
2:00 – 3:00 pm  Wei Zhang (MIT)  Shtuka special cycles and their generating series 
Friday, March 1, 2024  
11:00 am – 12:00 pm  Chen Wan (Rutgers Newark)  Some examples of the relative Langlands duality 
2:00 pm – 3:00 pm  Peng Shan (Tsinghua)  Skein algebras and quantized Coulomb branches 
Program Visitors
 Mina Aganagic, University of California, Berkeley
 AnneMarie Aubert, Institut de Mathématiques de JussieuParis Rive Gauche, March 1529
 Clark Barwick, University of Edinburgh, February 19March 15
 Alexander Braverman, Perimeter Institute
 Alejandra Castro, Cambridge University, March 2529
 YoungJu Choie, Pohang University of Science and Technology, February 1216; March 2228
 John Francis, Northwestern University, March 114
 David Gaiotto, Perimeter Institute, March 2529
 Jayce Getz, Duke University, March 1822
 Ezra Getzler, Northwestern University, March 1122
 Sam Gunningham, Montana State University, February 912
 Sarah Harrison, Northeastern University
 Dihua Jiang, University of Minnesota, February 29March 9
 Tasho Kaletha, University of Michigan, March 1220
 Minhyong Kim, University of Edinburgh, February 129
 Axel Kleinschmidt, Max Planck Institute for Gravitational Physics, Potsdam, March 1828
 Kim KlingerLogan, Kansas State University, March 2529
 Kobi Kremnitzer, Oxford University, March 2529

Baiying Liu, Purdue University, March 1316
 Steven Miller, Rutgers University
 Greg Moore, Rutgers University, February 59
 David Nadler, University of California, Berkeley, March 1730
 Bảo Châu Ngô, University of Chicago, March 2529
 George Pappas, Michigan State University, March 2529
 Daniel Persson, Chalmers Institute of Technology, March 2529
 Sam Raskin, Yale University, March 2629
 Yiannis Sakellaridis, Johns Hopkins University, March 1822
 Peng Shan, Tsinghua University, February 12April 14
 Akshay Venkatesh, Institute for Advanced Study
 Roberto Volpato, University of Padova, February 410
 Chen Wan, Rutgers University, February 29March 9
 Fei Yan, Brookhaven National Laboratory, March 1829
 Xinwen Zhu, Stanford University