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
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.
Zoom recording Password: 9G^zo?%4
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
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.
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.
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
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
23 September, 2020
Johannes Walcher (Heidelberg)
Title: On the rationality of MUMs and 2-functions
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.
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.]
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
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?
4 November, 2020
Clark Barwick (Edinburgh)
Toward Geometric Foundations for Arithmetic Field Theories
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.
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.
25 November, 2020
Pavel Mnev (Notre Dame)
Chern-Simons theory on cylinders and generalized Hamilton-Jacobi actions
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.
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.
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.
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.
Schedule, Spring, 2021
2 June, 2021, 20:00 UK time
Vadim Schechtman (Toulouse)
Title: Conformal blocks and factorisable sheaves
Conformal blocks ("one half" of correlation functions) are solutions of certain remarkable partial
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
I will briefly review these results which have found later on some unexpected applications.
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.
23 June, 2021, 20:00 UK time
Dennis Gaitsgory (Harvard)
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.