Seminar on Arithmetic Geometry and Quantum Field Theory
Organisers: Jeff Harvey and Minhyong Kim
Wednesdays, 20:00 London time, approximately 30 weeks per year
If you are interested in attending this online seminar, please send an email to Minhyong Kim.
Schedule, Winter, 2021
Online Mini-Conference on the Geometric Langlands Correspondence
13 January to 17 February, 2021
This virtual conference will extend over 6 weeks with one talk per week.
We will start out with a three-week
mini-course by Sam Raskin, Nick Rosenblyum, and Dennis Gaitsgory.
This will be followed by lectures by Edward Witten, Edward Frenkel, and David Kazhdan. If you are not on the regular mailing list for the seminar on arithmetic geometry and quantum field theory but would like to attend this conference, write to Minhyong Kim.
13 January, 2021, 20:00 GMT
Sam Raskin (UT Austin)
Title: Geometric Langlands for l-adic sheaves
Abstract: In celebrated work, Beilinson-Drinfeld formulated a categorical analogue of the Langlands program for unramified automorphic forms. Their conjecture has appeared specialized to the setting of algebraic D-modules: non-holonomic D-modules play a prominent role in known constructions.
In this talk, we will discuss a categorical conjecture suitable in other geometric settings, including l-adic sheaves. One of the main constructions is a suitable moduli space of local systems. Subsequent talks of Rozenblyum and Gaitsgory will discuss applications to unramified automorphic forms for function fields. This is joint work with Arinkin, Gaitsgory, Kazhdan, Rozenblyum, and Varshavsky.
20 January, 2021, 20:00 GMT
Nick Rozenblyum (Chicago)
Title: Spectral decomposition in geometric Langlands
We will describe a version of spectral decomposition in the setting of geometric Langlands. Specifically, we will
explain how a version of higher categorical trace applied to the category of representations of the Langlands dual
group gives an action on the automorphic category of the category of quasi-coherent sheaves on the moduli space of
local systems. We will introduce the trace conjecture (to be discussed in Gaitsgory's talk) which
gives, upon additionally taking the categorical trace of Frobenius, V. Lafforgue's spectral decomposition of the
space of automorphic forms as well as expected structures on the cohomologies of shtukas, which give rise to a
localization of the space of automorphic forms on the moduli space of arithmetic local systems.
This is joint work with Arinkin, Gaitsgory, Kazhdan, Raskin, and Varshavsky.
27 January, 2021, 20:00 GMT
Dennis Gaitsgory (Harvard)
Title: Automorphic forms as categorical trace
Abstract: In this talk we will tie together the material of the previous two talks. We will explain
how to obtain the space of automorphic functions by the procedure of categorical trace.
Furthermore, assuming the "restricted" form of the geometric Langlands conjecture, we
will obtain an explicit expression for the space of unramified automorphic functions in
terms of spectral data on the Langlands dual side. This is joint work with Arinkin, Kazhdan,
Raskin, Rozenblyum and Varshavsky.
3 February, 2021, 20:00 GMT
Edward Witten (IAS, Princeton)
Title: Branes, Quantization, and Geometric
10 February, 2021, 20:00 GMT
Edward Frenkel (Berkeley)
Title: An analytic version of the Langlands correspondence for complex curves
Abstract: The Langlands correspondence for complex curves was traditionally formulated in terms of sheaves rather than functions. In 2018, Robert Langlands asked whether it is possible to construct a function-theoretic version. Together with Pavel Etingof and David Kazhdan, we have formulated a function-theoretic version as a spectral problem for an algebra of commuting operators acting on (a dense subspace of) the Hilbert space of half-densities on the moduli space of G-bundles over a complex algebraic curve. These operators include (self-adjoint extensions of) differential operators (both holomorphic and anti-holomorphic) as well as integral operators, which are analytic analogues of the Hecke operators. I will start with a brief introduction to both the sheaf-theoretic and function-theoretic versions and explain in what sense they complement each other. I will then present some of the results and conjectures from my joint work with Etingof and Kazhdan.
The talk will be independent from the talks given by the speaker in the past two weeks.
17 February, 2021, 18:00 GMT (CHANGE OF TIME!)
David Kazhdan (Hebrew)
Title: A proposal of a categorical construction of the algebraic version of L2(BunG)
24 February, 2021
Katrin Wendland (Freiburg)
Counting half and quarter BPS states - and their geometric counterparts
The BPS spectrum of quantum field theories with extended supersymmetry is key to constructing invariants that are reincarnations of geometric or topological invariants. In this talk, we will focus on the complex elliptic genus and its refinements on K3 surfaces and on the non-compact singular spaces that model the singularities which can occur on such K3 surfaces.
The results presented here have mostly been obtained in collaboration with Anne Taormina or with Yuhang Hou.
3 March, 2021
Theo Johnson-Freyd (Dalhousie/Perimeter)
Title: Higher Galois closures
Abstract: I will describe a mostly-conjectural picture of the higher-categorical separable closure of \RR. In particular, I will speculate about unitary topological field theory, higher analogues of spin-statistics, homotopy groups of spheres, and the j-homomorphism.
10 March, 2021
Erik Panzer (Oxford)
Title :Multiple Zeta Values in Deformation Quantization
Abstract: In 1997, Kontsevich constructed a universal quantization of every
Poisson manifold as a formal power series. Its coefficients are given
as integrals over moduli spaces of marked holomorphic discs. In joint
work with Peter Banks and Brent Pym, we show that these integrals
always evaluate to multiple zeta values, which are interesting
transcendental numbers that appear in several other contexts. I will
motivate and define deformation quantization, illustrate Kontsevich's
formula and explain our result and discuss some ideas of the proof.
17 March, 2021
Shamit Kachru (Stanford)
Title: Modularity of (rational) flux vacua
Compactifications of type IIb string theory on Calabi-Yau threefolds admit choices of background three-form fluxes, specified by selecting pairs of integral 3-forms on the manifold. In the presence of such fluxes, special points in the moduli space of complex structures on the Calabi-Yau emerge as (energetically) preferred by the physics.
In this talk, we describe a happy coincidence:
the manifolds which are preferred, in the case the Calabi-Yau is rational,
are precisely those whose associated point counts yield weight-2 cusp forms in accord with certain modularity conjectures.
These weight-2 forms naturally hint at the presence of a hidden elliptic curve in the physics and geometry,
and we show that a physics construction (known as the "F-theory lift" of the model)
makes the presence of the hidden elliptic curve manifest.