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.

Past Events


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.

Notes

Video



20 January, 2021, 20:00 GMT

Nick Rozenblyum (Chicago)

Title: Spectral decomposition in geometric Langlands

Abstract: 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.

Video



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.

Video

Notes



3 February, 2021, 20:00 GMT

Edward Witten (IAS, Princeton)

Title: Branes, Quantization, and Geometric Langlands

Video

Slides



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.



Video

Slides



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)

Video

Slides






24 February, 2021

Katrin Wendland (Freiburg)

Title: Counting half and quarter BPS states - and their geometric counterparts

Abstract: 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.

Video

Slides




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.

Video

Notes




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

Abstract: 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.