## Arithmetic Geometry and Quantum Field Theory, Past Events

22 April, 2020

David Ben-Zvi (UT Austin)

Title: Relative Langlands Duality

Background talks at MSRI

29 April, 2020

Yiannis Sakellaridis (JHU)

Title: Background on periods and L-functions

Abstract: Following up on David Ben-Zvi's talk from last week, I will give an overview of the problems and conjectures on automorphic periods, L-functions, and spherical varieties that were around before our joint work with David and Akshay, and will compare them with the geometric versions that David presented. I will assume that people have watched David's MSRI talks (at least the first one), in order to go into more detail.

Notes

6 May, 2020

Arnav Tripathy (Harvard)

Title: Nonalgebraic attractor points on higher-dimensional Calabi-Yau manifolds

Abstract: I'll discuss an exploration in transcendental number theory motivated by physical considerations. Supergravity attractor flow is an important mechanism that conjecturally provides a countable, equidistributed set of points in Calabi-Yau moduli spaces analogous to the theory of special points of Shimura varieties, and the Attractor Conjecture of Greg Moore postulates that these points are algebraic, i.e. defined over number fields. I'll discuss negative results in this direction for higher-dimensional Calabi-Yau manifolds based on Pila-Wilkie counting arguments. This is joint work with Josh Lam.

13 May, 2020

Matthew Emerton (Chicago)

Title: An introduction to the Langlands correspondence

Abstract: The Langlands program began as a letter from Robert Langlands to Andre Weil in the late 60's, explaining certain constructions and conjectures in the theory of automorphic forms. Langlands was particularly interested in defining L-functions for automorphic forms in generality, *and* in having the definition be in the form of an Euler product, as was the case with Artin L-functions in algebraic number theory. His success in this led him to a series of conjectures, the key such being his functoriality and reciprocity conjectures. Reciprocity is the core of what is often called the Langlands correspondence'', relating L-functions arising in arithmetic (Artin L-functions, Hasse--Weil L-functions) to Langlands L-functions of automorphic forms. The L-functions are the shadows of more structured objects on each side --- Galois representations (and/or motives) on the arithmetic side, and the conjectural Langlands group on the automorphic side. Arthur's conjectures are a variation on Langlands' conjectures (not a revision or correction, just a reformulation and extension) in the context of L^2-automorphic forms, which incorporate the celebrated Arthur SL_2''. This SL_2 can be thought of as a Lefschetz SL_2 on the motivic side of the correspondence, and mirror symmetry --- interchanging the Lefschetz SL_2 with a monodromy SL_2 --- has an interpretation on the automorphic side of the Langlands correspondence as switching large'' and small'' composition factors of induced representations. I will try to explain these ideas in as elementary and direct way as possible, focusing on some key examples and concepts, rather than the precise (and often somewhat elaborate) definitions and technically correct formulations. I'm also very to take audience input as to what they would like to hear about, either during the talk or in the discussion afterwards!

20 May, 2020

Matthew Emerton (Chicago)

An Introduction to the Langlands Correspondence, Part II

Zoom recording Password: 9f$8&57R 27 May, 2020 Georgios Pappas (Michigan State) Title: Symplectic constructions for l-adic local systems and their deformations Abstract: I will discuss various constructions for l-adic local systems over algebraic curves which can be viewed as arithmetic analogues of more familiar topological constructions for representations of the fundamental groups of 3-manifolds that fiber over a circle. Slides 3 June, 2020 Jeff Harvey (Chicago) Title: Conformal Field Theories with Sporadic Group Symmetry Abstract: This seminar will be based on joint work with Jin-Beom Bae, Kimyeong Lee and Brandon Rayhaun. The Monster VOA/CFT has a c=24 stress tensor, but it also possesses many stress tensors or conformal vectors of lower central charge. For example, it contains 48 commuting c=1/2 conformal vectors. This allows one to decompose the Monster VOA into subVOAs that possess sporadic automorphism groups for a number of the sporadic groups that appear as subquotients of the Monster. A number of techniques are used to compute conjectured characters of these VOAs including Hecke operators for rational conformal field theories, modular linear differential equations and Rademacher sums. Many of the examples we find are connected to McKay's$\hat E_8$correspondence for the Monster VOA. Zoom recording Password: 6V!$V6#E

17 June, 2020

Tudor Dimofte (UC Davis)

Title: Algebraic structure of boundary conditions in (T)QFT

Abstract: I will give a broad review some of the more algebraic aspects of boundary conditions in (topological twists of) quantum field theories in various dimensions. I will aim focus on gauge theories (as relevant, e.g., for geometric Langlands), and try to explain some of the mathematical data in terns of which boundary conditions are characterized. I will also try to touch on the action of dualities on boundary conditions (such as electric-magnetic duality in 4d Yang-Mills), optimistically connecting with David Ben-Zvi's and Yiannis Sakellaridis's talks from previous weeks.

Notes

Video

24 June, 2020

Tudor Dimofte (UC Davis)

Title: Algebraic structure of boundary conditions in (T)QFT, Part II

8 July, 2020

Magnus Carlson (Hebrew University)

Title: Arithmetic field theories with finite coefficients

Abstract: In this talk I will discuss arithmetic field theories with finite coefficients. Arithmetic field theories are analogous to field theories for 3-manifolds in the same sense number fields are analogous to 3-manifolds. I will start by describing the general idea behind what an arithmetic field theory is, first focusing on arithmetic Dijkgraaf-Witten theory. I will then proceed by defining arithmetic BF-theory and explain how this is a field theory that is non-trivial even in the non-orientable situation. I will give examples showing how path integrals can be calculated for these field theories and relate these path integrals to classical arithmetic invariants. I will also explain how one can define arithmetic field theories on number fields with a non-trivial boundary. This talk is based on joint work with Minhyong Kim.

Video

23 September, 2020

Johannes Walcher (Heidelberg)

Title: On the rationality of MUMs and 2-functions

Abstract: Points of maximal unipotent monodromy in Calabi-Yau moduli space play a central role in mirror symmetry, and also harbor some interesting arithmetic. In the classic examples, suitable expansion coefficients of the (all-genus) prepotential (in polylogarithms) under the mirror map are integers with an enumerative interpretation on the mirror manifold. This correspondence should be expected to extend to periods relative to algebraic cycles capturing the enumerative geometry relative to Lagrangian submanifolds. This expectation is challenged, however, when the mixed degeneration is not defined over Q. After musing about compatibility with mirror symmetry, I will discuss two recent results that sharpen these questions further: The first is a theorem proven by Felipe Mueller which states that the coefficients of rational 2-functions are necessarily contained in an abelian number field. (As defined in the talk, 2-functions are formal power series whose coefficients satisfy a natural Hodge theoretic supercongruence.) The second are examples worked out in collaboration with Boenisch, Klemm, and van Straten, of MUMs that are themselves not defined over Q.

Slides
Video

7 October, 2020

David Jordan (Edinburgh)

Title: Defects, boundaries and monads in Betti quantum geometric Langlands

Abstract: The Betti geometric Langlands TFT introduced by Ben-Zvi and Nadler has a quantum analog, which we introduced with Ben-Zvi and Brochier. This is a fully extended 3-dimensional TFT which captures the (0,1,2,3)-dimensional structures in the 4-dimensional Kapustin-Witten twist of N=4 d=4 SYM gauge theory (to what extent it is well-defined also in dimension 4 is an interesting open question). In this talk I'll focus on the algebraic formalism for doing computations in dimension 2, using boundary and defect structures, the excision property, and the machinery of monadic reconstruction. I'll sketch the construction (due to Haugseng and Johnson-Freyd--Scheimbauer) of the 4-category which houses all these constructions, and I'll outline how to do computations. In particular, I'll outline how the the Alekseev-Grosse-Schomerus algebras and the Fock-Goncharov quantum cluster algebras can be extracted from the formalism. [Note: I expect non-trivial overlap from a talk I gave in WHCGP; my talk in AGQFT won't assume anyone attending has seen that talk, but I'll nevertheless strive to give an independent and more detailed algebraic perspective to the one given there.]

Video

14 October, 2020

Jonathan Heckman (Penn)

Title: Physical Discretization and Arithmetic Geometry

Abstract: We present a speculative proposal for formulating physically discretized theories using characteristic p geometries. The resulting path integral formalism for physics in characteristic p retains more symmetries than standard lattice formulations. By way of example we illustrate how this works for some bosonic, fermionic, and supersymmetric fields theories. Time permitting, we discuss some potential applications such as defining a physical notion of a Weil cohomology theory, geometric engineering for characteristic p and arithmetic varieties, as well as an information theoretic interpretation of our results. Based on a working paper available at jjheckman.com/research

Slides

21 October, 2020

Sergei Gukov (Caltech)

Title: Arithmetic Topology and Chiral Algebras

Abstract: Arithmetic topology is a program (somewhat similar in spirit to the Langlands program) that aims to bridge number theory and 3-manifold topology. Suggested by David Mumford and Yuri I. Manin, and developed further by Barry Mazur, Alexander Reznikov, and Mikhail Kapranov, among others, this correspondence relates number fields to closed orientable 3-manifolds, prime ideals to knots, etc. On the other hand, when we learn rational chiral CFT we can't miss many intriguing connections to number theory. For example, a simple consequence of the fusion algebra is that generalized quantum dimensions are elements of an algebraic number field attached to a rational chiral CFT. So, if 3-manifolds and chiral CFTs correspond to number fields, could there be a direct correspondence between 3-manifolds and chiral algebras?

Slides
Video

4 November, 2020

Clark Barwick (Edinburgh)

Toward Geometric Foundations for Arithmetic Field Theories

Abstract: Following Gukov's lead from two weeks ago, I want to describe a program to develop some homotopical "shadows" of arithmetic topology that are sufficient to define arithmetic field theories in the sense of Kim. More precisely, I want to describe the following challenge: Construct the stratified homotopy type of the Ran space of a compactification of Spec O_K for a number field K. I will try to explain how the story of "exodromy" offers some interesting insights already at Step 1.

Slides
Video

18 November, 2020, 3 PM GMT (CHANGE OF TIME!)

Michael Harris (Columbia)

Title: Categorification of the Langlands correspondence and Iwasawa theory

Abstract: The aim is to imagine an analogue for number fields of the categorical Langlands correspondence for the function fields of curves. In the framework developed in the recent paper of Arinkin, Gaitsgory, Kazhdan et al., the Langlands parametrization of V. Lafforgue is recovered by applying a trace construction to the action of Frobenius on the (still conjectural) categorical correspondence for a curve C over the algebraic closure of a finite field k. In our construction, the function field k(C) is replaced by a number field K, the constant field extension of k(C) is replaced by the cyclotomic Zp-extension $K_\infty$ of K, and Frobenius is replaced by a generator $\gamma$ of the Galois group $Gal(K_\infty/K)$. Assuming a (purely hypothetical) categorical correspondence in this setting, taking the trace of $\gamma$ yields a Langlands parametrization of cuspidal cohomological automorphic representations, together with an action of derived deformation rings on the cohomology of locally symmetric spaces, as in the work of Galatius-Venkatesh. Although there seems to be no prospect of constructing such a categorical correspondence in the near future, the project sheds new light on work of Hida and Burungale-Clozel on deformations of representations of the absolute Galois group of $K_\infty$, and raises novel questions about the cohomology of locally symmetric spaces attached to towers of number fields.

Video

25 November, 2020

Pavel Mnev (Notre Dame)

Title: Chern-Simons theory on cylinders and generalized Hamilton-Jacobi actions

Abstract: We study the perturbative path integral of Chern-Simons theory on a cylinder [0,1]x Sigma with a holomorphic polarization on the boundaries, in the context of Batalin-Vilkovisky quantization (or rather its variant compatible with cutting-gluing, BV-BFV ). We find that, in the case of non-abelian 3D Chern-Simons, the fiber BV integral for the system produces the gauged WZW model on Sigma. Classically, the result corresponds to computing generalized Hamilton-Jacobi for Chern-Simons theory on cylinder a a generating function (in an appropriate sense) for the evolution relation induced on the boundary conditions by the equations of motion. A similar setup applied to 7D abelian Chern-Simons on a cylinder [0,1] x Sigma, with Sigma a Calabi-Yau of (real) dimension 6, with a linear polarization on one side and a nonlinear (Hitchin) polarization on the other side, is related to the Kodaira-Spencer (a.k.a. BCOV) theory. In the talk, I will introduce the concept of generalized Hamilton-Jacobi functions in the example of classical mechanics with constraints described by an equivariant moment map and proceed to discuss the examples above. This is a report on a joint work with Alberto S. Cattaneo and Konstantin Wernli.

Video

2 December, 2020

David Treuman (Boston College)

Title: Symplectic, or mirrorical, look at the Fargues-Fontaine curve

Abstract: Homological mirror symmetry describes Lagrangian Floer theory on a torus in terms of vector bundles on the Tate elliptic curve. A version of Lekili and Perutz's works "over Z[[t]]", where t is the Novikov parameter. I will review this story and describe a modified form of it, which is joint work with Lekili, where the Floer theory is altered by a locally constant sheaf of rings on the torus. When the fiber of this sheaf of rings is perfectoid of characteristic p, and the holonomy around one of the circles in the torus is the pth power map, it is possible to specialize to t = 1, and the resulting theory there is described in terms of vector bundles on the equal-characteristic-version of the Fargues-Fontaine curve.

Video

9 December, 2020, 11 AM GMT (CHANGE OF TIME!)

Miranda Cheng (Amsterdam)

Title: Quantum modular forms from three-manifolds

Abstract: Quantum modular forms are functions defined on rational numbers that have rather mysterious weak modular properties. Mock modular forms and false theta functions are examples of holomorphic functions on the upper-half plane which lead to quantum modular forms. Generalising the Witten-Reshetikhin-Turaev invariants, a new topological invariants named homological blocks for (in particular plumbed) three-manifolds have been proposed a few years ago. My talk aims to explain the recent observations on the quantum modular properties of the homological blocks, as well as the relation to logarithmic vertex algebras. The talk will be based on a series of work in collaboration with Sungbong Chun, Boris Feigin, Francesca Ferrari, Sergei Gukov, Sarah Harrison, and Gabriele Sgroi.

Video

### 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

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.

Video

Slides

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.

Video

Slides

### Schedule, Spring, 2021

2 June, 2021, 20:00 UK time

Title: Conformal blocks and factorisable sheaves

Abstract: Conformal blocks ("one half" of correlation functions) are solutions of certain remarkable partial differential equations discovered by physicists in the mid-1980-ies. Soon afterwards it was found that they are closely related to quantum groups. On the other hand these differential equations may be expressed as a Gauss-Manin connection. This allowed one to interpret, using the Lefschetz vanishing cycles, representations of quantum groups as (complexes of) sheaves over configuration spaces. I will briefly review these results which have found later on some unexpected applications.

Slides

Video

9 June, 2021, 20:00 UK time

Roman Bezrukavnikov (MIT)

Title: Local geometric Langlands and roots of unity

Abstract: I will review relations between quantum groups at a root of unity and categories appearing in local geometric Langlands. The talk will be based on old papers with Arkhipov, Ginzburg, Braverman, Gaitsgory and Mirkovic, and work in progress with Boixeda Alvarez, McBreen and Yun.

16 June, 2021, 17:00 UK time

(NOTE CHANGE OF TIME!)

Mikhail Finkelberg (HSE University)

Title: Factorizable sheaves and local systems of conformal blocks

Abstract: This is a continuation of the talk of Vadim Schechtman on June 2nd. I will explain how the theory of factorizable sheaves implies the motivic property of local systems of conformal blocks of WZW models.

Video

Slides

23 June, 2021, 20:00 UK time

Dennis Gaitsgory (Harvard)

Title: The Bezrukavnikov-Finkelberg-Schechtman theory from the point of view of Geometric Langlands

Abstract: We will try to recast the construction from [BFS] (including the semi-infinite modular functor and its connection with the WZW theory) from the point of view of (quantum) Geometric Langlands Theory.

Video

Notes