This online colloquium features geometers and physicists presenting current research on a wide range of topics in the interface of the two fields. The talks are aimed at a broad audience. They will take place via Zoom on selected Mondays at 3pm Eastern, noon Pacific, 5pm BRT. Each session features a 60 minute talk, followed by 15 minutes for questions and discussion. You may join the meeting 15 minutes in advance. Questions and comments may be submitted to the moderator via the chat interface during the talk, or presented in person during the Q&A session. These colloquia will be recorded and will be available (linked from this page) asap after the event.
As an alternative to Zoom, you may watch a live stream of the lecture at our: YouTube streaming site.
To receive announcements about the colloquia (including Zoom links to individual meetings), sign up for the WHCGP mailing list here.
Organizing committee: Tudor Dimofte, Ron Donagi, Dan Freed, Sheldon Katz, Dave Morrison, Andy Neitzke.
(Indexed at researchseminars.org.)
Go to Past Talks (2020). Past Talks (2021) Past Talks (2022) Past Talks (2023) Past Talks (2024)| May 6, 2024 |
Jacques Distler (UT Austin)TBAAbstract: |
| February 12, 2024 |
Tom Bridgeland (Sheffield)Geometry from Donaldson-Thomas invariantsAbstract: For a few years now I've been working on a programme which aims to use Donaldson-Thomas invariants to define geometric structures on spaces of stability conditions. This turns out to be closely related to recent work by physicists on non-perturbative completions of topological string partition functions. In this overview talk I will try to explain the mathematical background to this story and survey some of the known results. Video of lecture OR Video of lecture (alternate version, same content as the other) |
March 11, 2024 |
Miranda Cheng (Univ. of Amsterdam/Academica Sinica Taiwan)Revisiting 3d ModularityAbstract: A few years ago, new topological invariants of closed three-manifolds, based on 3d SCFT obtained by compactifying on these manifolds, have been proposed. Subsequently, we have observed an interesting relation between quantum modular forms and vertex operator algebras and these so-called \hat Z -invariants. In this talk I will revisit a few key aspects of this relation between \hat Z -invariants and modular-type objects, including the role of the SL_2(Z) representation, the mock modular invariants, and the connection to vertex operator algebras. This talk is based on joint work with Ioana Coman, Davide Passaro, Piotr Kucharski and Gabriele Sgroi. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| April 15, 2024 |
Brian Williams (Boston University)A survey of holomorphic quantum field theoryAbstract: A holomorphic quantum field theory is one which is sensitive to the complex structure of spacetime; in complex dimension one this is simply a (chiral) conformal field theory. I will give examples of holomorphic QFTs which exist in higher dimensions, some of which appear within the context of supersymmetry. I will then survey recent approaches to a rigorous axiomatization of holomorphic quantum field theory which build off of well-combed methods in CFT, such as the theory of vertex algebras. Finally, I will highlight some open problems in holomorphic QFT as well as applications to the understanding of the ubiquitous six-dimensional superconformal field theory. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| January 30, 2023 |
Xin Sun (University of Pennsylvania)Random surfaces, planar lattice models, and conformal field theoryAbstract: Liouville quantum gravity (LQG) is a theory of random surfaces that originated from string theory. Schramm Loewner evolution (SLE) is a family of random planar curves describing scaling limits of many 2D lattice models at their criticality. Before the rigorous study via LQG and SLE in probability, random surfaces and scaling limits of lattice models have been studied via another approach in theoretical physics called conformal field theory (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics on random surfaces and planar lattice models, including the law of the random modulus of the scaling limit of uniform triangulation of the annular topology, and the crossing formula for critical planar percolation on an annulus. I will then present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT. Based on joint works with Ang, Holden, Remy, Xu, and Zhuang. Video of lecture |
| February 13, 2023 |
Thomas Grimm (Utrecht)The Tameness of Quantum PhysicsAbstract: While physicists have learned to accept the many wild phenomena of quantum theories, one might hope that the mathematical structure underlying these theories is more tame and inherently geometric in nature. The aim of this colloquium is to introduce a general tameness principle, using o-minimal structures originating in mathematical logic, and argue that it is common to many well-defined quantum theories. We will discuss quantum field theories and show the tameness of perturbative scattering amplitudes. At the non-perturbative level, tameness depends on the high-energy definition of the physical theory and might be seen as a condition that arises from consistency with quantum gravity. In fact, all well-understood effective theories derived from string theory appear to be tame. As one direct application we will see how this property is key in the mathematical proof of an almost 20-year-old finiteness conjecture for string theory vacua. We will close by formulating our expectations on the tameness of conformal field theories and their observables. |
| February 27, 2023 |
Marcos Marino (Geneva)Resurgence and non-perturbative topological stringsAbstract: The theory of resurgence provides a precise and unified mathematical formulation of the non-perturbative sectors of a physical theory, based solely on its perturbative expansion. In recent years it has been applied to topological quantum field theories and topological strings, leading to many insights and results. For example, it has been found that, in many cases, the Stokes constants appearing in resurgent analysis are related to BPS invariants. In this talk I will first review the resurgence program in topological theories, and then I will consider recent progress on its implementation in topological string theories. To do this, I introduce the analogue of instanton calculus in Kodaira-Spencer theory, based on trans-series solutions to the holomorphic anomaly equations. This leads to exact, (spacetime) multi-instanton amplitudes of the topological string on arbitrary Calabi-Yau manifolds, and gives non-perturbative corrections in the string coupling constant. These results also suggest that, in the full quantum theory, the moduli of the Calabi-Yau are quantized in units of the string coupling constant. As an application of these results, I present the all-orders, large genus asymptotics for the topological string free energies on the famous quintic Calabi-Yau. Video of lecture |
| March 13, 2023 |
Clark Barwick (Edinburgh)Condensed/pyknotic structures: a gentle introductionAbstract: Condensed (alias pyknotic) structures provide a formalism in which algebraic structures can interact gracefully with topological structures. I will introduce the ideas via some simple examples. If time permits, I will explain how these structures arise when one contemplates field theories in p-adic contexts. |
| Marh 27, 2023 |
Sakura Schafer-Nameki (Oxford)Theta-defects and Non-Invertible SymmetriesAbstract: I will give an overview of recent works constructing global generalized symmetries, in particular non-invertible symmetries, using so-called theta defects (generalizing the notion of theta angle). These provide a unified framework to characterize symmetry categories of QFTs in various dimensions. This is work in collaboration with Lakshya Bhardwaj, Lea Bottini and Apoorv Tiwari over the last year. |
| April 10, 2023 |
Guangbo Xu (Texas A&M)Integer-valued Gromov-Witten invariantsAbstract: Gromov-Witten invariants are generally rational numbers because of the contribution of holomorphic curves with nontrivial symmetries. Recently joint with Shaoyun Bai, we developed an idea of Fukaya-Ono which allows us to obtain refined integer-valued Gromov-Witten invariants for all symplectic manifolds. These integers are roughly the number of holomorphic curves with no symmetries. In this talk I will sketch the mathematical construction of our integral invariants and our speculation about their relations with Gopakumar-Vafa invariants for Calabi-Yau threefolds. |
| April 24, 2023 |
Pavel Safronov (Edinburgh)Holomorphic Morse theory and moduli spaces of flat connections on 3-manifoldsAbstract: For a real function (more generally, a closed one-form) the Hilbert space of supersymmetric quantum mechanics has a description in terms of the twisted de Rham complex or the Morse complex. In the holomorphic case there is also the cohomology of the sheaf of vanishing cycles. I will describe various relations between these objects as well as their connection to deformation (Batalin—Vilkovisky) quantization. I will apply these results to the complex Chern—Simons functional of a 3-manifold and outline a relationship between the corresponding cohomology and more topological invariants such as skein modules. This is based on works (partially in progress) joint with B. Williams, F. Naef and S. Gunningham. |
| May 8, 2023 |
Sam Raskin (UT Austin)Title Recent progress in the geometric Langlands programAbstract: The geometric Langlands conjecture of Beilinson-Drinfeld describes the category of D-modules on the moduli stack of G-bundles on a smooth projective curve in terms of spectral data involving the dual group. It can be thought of as an analogue of Langlands's conjectures in the unramified function field case, but where all of the objects have purely algebro-geometric nature. In this talk, I will describe on-going work with Dennis Gaitsgory settling this conjecture. In this talk, I will try to give an introduction to the subject and to outline some of the starting points of our work. |
| September 11, 2023 |
Nathan Seiberg (IAS)Emanant SymmetriesAbstract: Based on joint work with Meng Cheng (arXiv:2211.12543), with Shu-Heng Shao (arXiv: 2307.02534), and with Shu- Heng Shao and Sahand Seifnashri (to appear), we will discuss some aspects of global symmetries and their ‘t Hooft anomalies. We will define a notion of an emanant global symmetry. It is not a symmetry of the UV theory, but unlike emergent (accidental) symmetries, it is not violated by any relevant or irrelevant operators in the IR theory. It is an exact symmetry of the low-energy theory. We will demonstrate this notion in several well-known examples. We will discuss in detail the Majorana chain, the transverse field Ising model, a continuum system with a chemical potential, and the Heisenberg chain. In all these models, we will find emanant symmetries. In one case, it is a non-invertible emanant symmetry. |
| October 16, 2023 |
John Pardon (SCGP)Universally counting curves in Calabi--Yau threefoldsAbstract: Enumerating curves in algebraic varieties traditionally involves choosing a compactification of the space of smooth embedded curves in the variety. There are many such compactifications, hence many different enumerative invariants. I will propose a "universal" (very tautological) enumerative invariant which takes values in a certain Grothendieck group of 1-cycles. It is often the case with such "universal" constructions that the resulting Grothendieck group is essentially uncomputable. But in this case, the cluster formalism of Ionel and Parker shows that, in the case of threefolds with nef anticanonical bundle, this Grothendieck group is freely generated by local curves. This reduces the MNOP conjecture (in the case of nef anticanonical bundle and primary insertions) to the case of local curves, where it is already known due to work of Bryan--Pandharipande and Okounkov--Pandharipande. |
| November 13, 2023 |
Davide Gaiotto (Perimeter Institute)Twisted holography and the 't Hooft expansion of vertex operator algebrasAbstract: Twisted holography relates the large N 't Hooft expansion of OPEs and correlation functions of certain chiral algebras/VOAs and the B-model topological string theory on 3d Calabi-Yau manifolds with an SL(2) global symmetry. Given a 2d Calabi-Yau cone X, possibly non-geometric, a chiral algebra can be produced as the world-volume theory of N branes in C x X and the dual 3d CY is expected to arise from the back-reaction of the same collection of branes. I will discuss how the large N combinatorics in the chiral algebras reconstruct algebraically the C x X geometry and its back-reaction. Video of lecture OR Video of lecture (alternate version, same content as the other) |
December 11, 2023 |
Sabrina Pasterski (Perimeter Institute)Celestial Holography from Bottom-up to Top-downAbstract: The Celestial Holography program encompasses recent efforts to understand the flat space hologram in terms of a CFT living on the celestial sphere. A key development instigating these efforts came from understanding how soft limits of scattering encode infinite dimensional symmetry enhancements corresponding to the asymptotic symmetry group of the bulk spacetime. Historically, the construction of the bulk-boundary dual pair has followed bottom up approach matching symmetries on both sides. Recently, however, there has been exciting progress in formulating top down descriptions using insights from twisted holography. In this talk we will cover salient aspects of the celestial construction, the status of the dictionary, and active research directions. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| January 24, 2022 |
Hirosi Ooguri (Cal Tech/Kavli IPMU)Symmetry in QFT and GravityAbstract: I will review aspects of symmetry in quantum field theory and combine them with the AdS/CFT correspondence to derive constraints on symmetry in quantum gravity. The quantum gravity constraints to be discussed include the no-go theorem on global symmetry, the completeness of gauge charges, and the decomposition of high energy states into gauge group representations. OR Video of lecture (alternate version, same content as the other) |
| February 7, 2022 |
Emily Cliff (Université de Sherbrooke)Moduli spaces of principal 2-group bundles and a categorification of the Freed--Quinn line bundleAbstract: A 2-group is a higher categorical analogue of a group, while a smooth 2-group is a higher categorical analogue of a Lie group. An important example is the string 2-group in the sense of Schommer-Pries. We study the notion of principal bundles for smooth 2-groups, and investigate the moduli "space" of such objects. In particular in the case of flat principal bundles for a finite 2-group over a Riemann surface, we prove that the moduli space gives a categorification of the Freed--Quinn line bundle. This line bundle has as its global sections the state space of Chern--Simons theory for the underlying finite group. We can also use our results to better understand the notion of geometric string structures (as previously studied by Waldorf and Stolz--Teichner). This is based on joint work with Dan Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.
Video of lecture OR Video of lecture (alternate version, same content as the other) |
| February 28, 2022 |
Eric Zaslow (Northwestern)Framing DualityAbstract: I will describe joint work in progress with Linhui Shen and Gus Schrader, in which we study moduli spaces of Fukaya objects and conjecture about their relationship to BPS and open Gromov-Witten invariants. This unabashedly synthesize previous works of many other groups, whom I will credit in the talk. I will try to highlight some new aspects: 1) a definition of phases and framings and their combinatorial origins; 2) conjectures on open Gromov-Witten invariants for Lagrangians bounding certain Legendrian surfaces; 3) a “framing duality” relating Dondaldson-Thomas and open Gromov-Witten invariants.
Video of lecture OR Video of lecture (alternate version, same content as the other) |
March 14, 2022 |
Joerg Teschner (DESY)The complex geometry of topological string partition functionsAbstract: The goal of this talk will be to review some aspects of a program inspired by work of Tom Bridgeland aiming at a non-perturbative characterisation of topological string partition functions. The program is based on two main ingredients: The complex geometry of the underlying moduli spaces on the one hand, and cluster algebra structures defined by BPS- or DT-invariants on the other hand. The general picture is nicely illustrated by the Borel summation of the conifold partition functions recently studied with M. Alim, A. Saha and I. Tulli, with Stokes jumps of the partition functions getting related to wall-crossing phenomena in the theory of DT-invariants. Based on this and other examples we will propose a conjectural characterisation of the partition functions for local Calabi-Yau manifolds, generalising earlier proposals by Marino and collaborators, and related to earlier proposals by Alexandrov, Pioline and collaborators based on the geometry of hypermultiplet moduli spaces.
Video of lecture OR Video of lecture (alternate version, same content as the other) |
| March 28, 2022 |
Washington Taylor (MIT)Middle intersection forms on singular elliptic Calabi-Yau fourfolds, and applications to the standard model and mirror symmetryAbstract: Recent work with Jefferson and Turner indicates that the intersection form on the vertical part of middle cohomology of singular elliptic Calabi-Yau fourfolds is independent of resolution. This suggests that the intersection structure should have a natural definition even in these singular geometries. The resulting intersection form has a simple block-diagonal structure in terms of Kodaira singularities and the geometry of the base of the elliptic fibration; the talk will describe applications of this intersection form to analysis of chiral matter and standard model constructions in F-theory as well as new insights into mirror symmetry, in particular providing in some cases a complete description of the intersection form on H_4 (X, Z) for a smooth Calabi-Yau fourfold including both vertical and horizontal parts. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| April 11, 2022 |
Daniel Halpern-Leistner (Cornell)Infinite dimensional geometric invariant theory and gauged Gromov-Witten theoryAbstract: Harder-Narasimhan (HN) theory gives a structure theorem for holomorphic vector bundles on a Riemann surface. A bundle is either semistable, or it admits a canonical filtration whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary moduli problems in algebraic geometry I will discuss work in progress with Andres Fernandez Herrero and Eduardo Gonzalez to apply this general machinery to the moduli problem of gauged maps from a curve C to a G-variety X, where G is a reductive group. Our main immediate application is to use HN theory for gauged maps to compute generating functions for K-theoretic gauged Gromov-Witten invariants. This problem is interesting more broadly because it can be formulated as an example of an infinite dimensional analog of the usual set up of geometric invariant theory, which has applications to other moduli problems.
Video of lecture OR Video of lecture (alternate version, same content as the other) |
| April 25, 2022 |
Constantin Teleman (U.C. Berkeley)Coulomb branches and Drinfeld centersAbstract: We discuss a construction of Coulomb branches of a compact Lie group G from the Toda integrable systems and speculate on their origins as Drinfeld centers of modifications of (a variant of) the tensor category of topological representations of G. Video of Lecture OR Video of lecture (alternate version, same content as the other) |
| May 9, 2022 |
Heeyeon Kim (Rutgers University)Path integral derivations of K-theoretic Donaldson invariantsAbstract: We discuss path integral derivations of topologically twisted partition functions of 5d SU(2) supersymmetric Yang-Mills theory on M4 x S1, where M4 is a smooth closed four-manifold. Mathematically, they can be identified with the K-theoretic version of the Donaldson invariants. In particular, we provide two different path integral derivations of their wall-crossing formula for b_2^+(M4)=1, first in the so-called U-plane integral approach, and in the perspective of instanton counting. We briefly discuss the generalization to b_2^+(M4)>1.
Video of lecture OR Video of lecture (alternate version, same content as the other) |
| September 12, 2022 |
Edward Witten (Institute for Advanced Study)An Algebra of Observables for de Sitter SpaceAbstract: De Sitter space is the maximally symmetric solution of Einstein's equations with positive cosmological constant. It is also perhaps the simplest example of a spacetime with a cosmological horizon, and this, as explained long ago by Gibbons and Hawking, leads to perplexing questions about quantum field theory and gravity in a de Sitter spacetime. I will describe a von Neumann algebra of Type II_1 that describes the observations made by an observer in de Sitter space. This provides an abstract answer to some of the questions.
Video of lecture OR Video of lecture (alternate version, same content as the other) |
| October 3, 2022 |
Ciprian Manolescu (Stanford University)What is a Floer homotopy type?Abstract: Floer homology is an important tool in both symplectic geometry and low-dimensional topology. I will briefly review the different versions of Floer homology (Hamiltonian, Lagrangian, Yang-Mills, Seiberg-Witten, Khovanov), and then describe the program to refining them by producing (stable) homotopy types whose homology is Floer homology. Among the applications of the resulting Floer homotopy are a proof of the existence of non-triangulable manifolds in high dimensions (due to the presenter) and the recent proof of the Arnold conjecture with Z/p coefficients (due to Abouzaid and Blumberg). Recommended reference: "Floer homotopy theory, revisited" by Ralph Cohen, https://arxiv.org/pdf/1901.08694.pdf
Video of lecture OR Video of lecture (alternate version, same content as the other) |
| October 17, 2022 |
Mike Douglas (Harvard CMSA)How will we do mathematics in 2030 ?Abstract: We make the case that over the coming decade, computer assisted reasoning will become far more widely used in the mathematical sciences. This includes interactive and automatic theorem verification, symbolic algebra, and emerging technologies such as formal knowledge repositories, semantic search and intelligent textbooks. After a short review of the state of the art, we survey directions where we expect progress, such as mathematical search and formal abstracts, developments in computational mathematics, integration of computation into textbooks, and organizing and verifying large calculations and proofs. For each we try to identify the barriers and potential solutions.
Video of lecture OR Video of lecture (alternate version, same content as the other) |
| October 31, 2022 |
Herman Verlinde (Princeton)S-duality and Mirror Symmetry in TTbar deformed CFTAbstract: After a brief review of TTbar conformal field theory, I will study the grand-canonical partition sum of TTbar deformed symmetric product CFT. We will find that it admits a natural extension that exhibits an S-duality symmetry under an PSL(2,Z)-duality group that exchanges strong and weak TTbar coupling, and an analog of mirror symmetry that exchanges to modular shape of the torus with the complexified TTbar coupling. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| November 14, 2022 |
Sara Pasquetti (University of Milan, Bicocca)The local dualisation algorithm at workAbstract: I will present an algorithm to construct mirror and more general SL(2,Z) duals of 3d N=4 quiver theories and of their 4d uplifts. The algorithm uses a set of basic duality moves and the properties of the duality-walls providing a generalisation of the Kapustin-Strassler local dualisation to the non-abelian case. All the basic duality moves can be derived by iterative applications of Seiberg-like dualities, hence our algorithm implies that mirror and SL(2,Z) dualities can be derived assuming only Seiberg duality. I will also discuss the case of bad theories, where the dualisation algorithm allows us to extract non-trivial information on the quantum moduli space. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| November 28, 2022 |
Dalimil Mazac (IAS)Automorphic Spectra and the Conformal BootstrapAbstract: I will describe a close analogy between the spectral geometry of hyperbolic manifolds and conformal field theory. A hyperbolic d-manifold gives rise to a Hilbert space which is a unitary representation of the conformal group in d-1 dimensions. Elements of this Hilbert space can be thought of as local operators living in a (d-1)-dimensional spacetime. Their scaling dimensions are related to the Laplacian eigenvalues on the manifold. The operators satisfy an operator product expansion. One can define correlation functions and derive conformal bootstrap equations constraining the spectrum. As an application, I will use conformal bootstrap techniques to obtain new rigorous bounds on the first positive Laplacian eigenvalue of hyperbolic orbifolds. In two dimensions, these bounds allow us to determine the set of first eigenvalues attained by all hyperbolic orbifolds. Based on joint work with Petr Kravchuk and Sridip Pal and ongoing work with James Bonifacio. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| December 12, 2022 |
Natalie Paquette (University of Washington)Koszul duality & twisted holography for asymptotically flat spacetimesAbstract: Koszul duality has been understood in recent years to characterize order-type defects in twists of supersymmetric field theories. This notion has been generalized, from a physical point of view, by studying couplings between D-branes and closed string theories in the topological string. Computing the D-brane backreaction, and studying the resulting open/closed string duality, is the purview of the twisted holography program. Twisted holography seeks to study supersymmetric sectors of the AdS/CFT correspondence using these methods, and leverage the appropriate generalization of Koszul duality to elucidate the bulk/boundary map. When applying these methods to a topological string configuration on twistor space, one can construct an instance of twisted holography in which a 2d chiral algebra, supported on the ``celestial sphere'', is dual to a 4d theory in an asymptotically flat spacetime. This is the first such top-down example of holography in an asymptotically flat spacetime. This talk describes joint work done, variously, with Kevin Costello, Brian Williams, and Atul Sharma. |
| January 11, 2021 |
Mike Freedman (Microsoft Research/UCSB)Quantum Codes and Systolic freedomAbstract: In work with Hastings we find a two-way street between quantum error correcting codes and Riemannian manifolds. A recent advance in coding theory allows us to produce the first example of a manifolds with Z_2-power law-systolic freedom. Specifically we find, for any e>0, a sequence of appropriately scaled 11D Riemannian manifolds M_i, so that for any dual 4 and 7 dimensional Z_2-cycles, X_i and Y_i, resp. $ Vol_4(X_i)*Vol_7(Y_i) > (Vol_11(M_i))^(5/4-e) $. Video of lecture
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| January 25, 2021 |
Albrecht Klemm (Bonn)Calabi-Yau modularity and Feynman GraphsAbstract: Using the GKZ system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana integrals with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel \hat b-class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the Γb-class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius κ-constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We find the exact differential equation for the graph integrals with arbitrary value for the dimensional regularisation (d-\epsilon) parameter and extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential ideal for arbitrary masses. Using a recent p-adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins. Video of lecture |
| February 8, 2021 |
Nikita Nekrasov (SCGP, Stony Brook)Beauty of the defectsAbstract: Surface and point-like defects in supersymmetric gauge theories in four dimensions are studied with applications to quantum/classical correspondence. In particular, the GIL formula for the tau-function of Painleve VI is explained using the blow-up method in the context of the BPS/CFT correspondence, while the spin chain generalisation of Kharchev-Lebedev wavefunction of periodic Toda chain is obtained via wallcrossing. Based on several works, in particular on papers in collaboration with Saebyeok Jeong; Norton Lee; Oleksandr Tsymbaliuk; as well as S. Jeong and N. Lee Video of lecture |
| February 22, 2021 |
Tamas Hausel (IST Austria)Mirror symmetry for Langlands dual Higgs bundles at the tip of the nilpotent coneAbstract: I will explain what we can prove and what we conjecture about the mirror of Hecke transformed Hitchin section motivated by symmetry ideas of Kapustin-Witten. The talk is based on arXiv:2101.08583 joint with Hitchin. |
| March 8, 2021 |
Thomas Dumitrescu (UCLA)2-Group Global Symmetry in Quantum Field TheoryAbstract: Higher-form generalizations of global symmetries play an important role in Quantum Field Theory (QFT). In general, symmetries of different form degrees need not be independent; instead, they can form a higher group. In this talk I will illustrate this phenomenon by explaining why many simple Lagrangian QFTs in four and six dimensions enjoy 2-group global symmetries. I will then apply this understanding to deduce new general results about (typically non-Lagrangian) SCFTs in six dimensions. |
| March 22, 2021 |
David Ben-Zvi (U. Texas, Austin)Electric-Magnetic Duality between Periods and L-functionsAbstract: I will describe joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which ideas originating in quantum field theory are applied to a problem in number theory. A fundamental tool in number theory, the relative Langlands program, is centered on the representation of L-functions of Galois representations as integrals of automorphic forms. However, the data that naturally index these period integrals (spherical varieties for a reductive group G) and the L-functions (representations of the Langlands dual group G^) don't seem to line up, making the search for integral representations somewhat of an art. We present an approach to this problem via the Kapustin-Witten interpretation of the [geometric] Langlands correspondence as electric-magnetic duality for 4-dimensional supersymmetric gauge theory. Namely, we rewrite the relative Langlands program as duality in the presence of boundary conditions. As a result the partial correspondence between periods and L-functions is embedded in a natural duality between Hamiltonian actions of the dual groups. |
| April 5, 2021 |
Arnav Tripathy (Harvard)K3 metricsIt has long been an open problem to explicitly produce a Ricci-flat metric on a (non-toroidal) compact manifold. I'll discuss two approaches to this problem, related by a version of 3d mirror symmetry, for K3 manifolds. This is joint work with M. Zimet. |
| April 19, 2021 |
Cody Long and Cumrun Vafa (Harvard)Non-Holomorphic Cycles and Non-BPS Black BranesAbstract: We discuss extremal non-BPS black holes and strings arising in M-theory compactifications on Calabi-Yau threefolds, obtained by wrapping M2 branes on non-holomorphic 2-cycles and M5 branes on non-holomorphic 4-cycles. Using the attractor mechanism we compute the black hole mass and black string tension, leading to a conjectural formula for the asymptotic volumes of connected, locally volume-minimizing representatives of non-holomorphic, even-dimensional homology classes in the threefold, without knowledge of an explicit metric. In the case of divisors we find examples where the volume of the representative corresponding to the black string is less than the volume of the minimal piecewise-holomorphic representative, predicting recombination for those homology classes and leading to stable, non-BPS strings. We also show how to compute the central charges of non-BPS strings in F-theory via a near-horizon AdS3 limit in 6d which, upon compactification on a circle, account for the asymptotic entropy of extremal nonsupersymmetric 5d black holes (i.e., the asymptotic count of non-holomorphic minimal 2-cycles). Video of lecture |
| May 3, 2021 |
Fei Yan (Rutgers)Two tales of networks and quantizationAbstract: I will describe two quantization scenarios. The first scenario involves the construction of a quantum trace map computing a new link "invariant" (with possible wall-crossing behavior) for links L in a 3-manifold M, where M is a Riemann surface C times a real line. This construction computes familiar link invariants in a new way, moreover it unifies that computation with the computation of protected spin characters counting ground states with spin for line defects in 4d N=2 theories of class-S. Certain networks on C play an important role in the construction. The second scenario concerns the study of Schroedinger equations and their higher order analogues, which could arise in the quantization of Seiberg-Witten curves in 4d N=2 theories. Here similarly certain networks play an important part in the exact WKB analysis for these Schroedinger-like equations. At the end of my talk I will also try to sketch a possibility to bridge these two quantization scenarios. The first part of the talk is based on joint work with A. Neitzke; the final sketch is based on discussions with D. Gaiotto, G. Moore and A. Neitzke. |
| May 17, 2021 |
Richard Thomas (Imperial College)Nonabelian DT theory from abelian DT theoryAbstract: Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. Joyce's generalised DT invariants count semistable sheaves on X. I will describe ongoing work with Soheyla Feyzbakhsh with the eventual aim of writing the generalised DT invariants in any rank r in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X. Along the way we express rank r DT invariants in terms of rank 0 invariants counting D4-D2-D0 branes. These invariants are predicted by S-duality to be governed by (vector-valued mock) modular forms. Based partly on arXiv:2007.03037 and arXiv:2103.02915 . |
| June 7, 2021 |
Zohar Komargodski (SCGP, Stony Brook)Higher central charges and gapped boundariesAbstract: The chiral central charge is a famous diagnostic of edge modes on the boundary of 2+1 dimensional topological phases. We show that many theories with a vanishing chiral central charge nevertheless cannot admit a gapped boundary. We define higher chiral central charges and investigate their properties. |
| June 21, 2021 |
no meeting, due to Strings 2021
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| July 12, 2021 |
Sarah Harrison (McGill)New BPS algebras from superstring compactificationsAbstract: Borcherds Kac-Moody (BKM) algebras are a generalization of familiar Kac-Moody algebras with imaginary simple roots. On the one hand, they were invented by Borcherds in his proof of the monstrous moonshine conjectures and have many interesting connections to new moonshines, number theory and the theory of automorphic forms. On the other hand, there is an old conjecture of Harvey and Moore that BPS states in string theory form an algebra that is in some cases a BKM algebra and which is based on certain signatures of BKMs observed in 4d threshold corrections and black hole physics. I will briefly review the construction of new BKMs superalgebras arising from self-dual vertex operator algebras of central charge 12, and then discuss recent work showing how they arise as algebras of BPS states in physical string theories in 2 dimensions, as well as their connection with automorphic forms. Based on work with N. Paquette, D. Persson, and R. Volpato. This can be seen as a follow-up to a talk given by N. Paquette at this series this past October. |
| July 26, 2021 |
Theo Johnson-Freyd (Perimeter/Dalhousie)Semisimple higher categoriesAbstract: Semisimple higher categories are a quantum version of topological spaces (behaving sometimes like homotopy types and sometimes like manifolds) in which cells are attached along superpositions of other cells. Many operations from topology make sense for semisimple higher categories: they have homotopy sets (not groups), loop spaces, etc. For example, the extended operators in a topological sigma model form a semisimple higher category that can be thought of as a type of "cotangent bundle" of the target space. The "symplectic pairing" on this "cotangent bundle" is measured an S-matrix pairing aka Whitehead bracket defined on the homotopy sets of any (pointed connected) semisimple higher category, and the nondegeneracy of this pairing is a type of Poincare or Atiyah duality. This is joint work in progress with David Reutter.
Video of lecture OR Video of lecture (alternate version, same content as the other) |
| September 13, 2021 |
Pavel Etingof (MIT)Hecke operators over local fields and an analytic approach to the geometric Langlands correspondenceAbstract: I will review an analytic approach to the geometric Langlands correspondence, following my work with E. Frenkel and D. Kazhdan, arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. This approach was developed by us in the last couple of years and involves ideas from previous and ongoing works of a number of mathematicians and mathematical physicists, Kontsevich, Langlands, Teschner, and Gaiotto-Witten. One of the goals of this approach is to understand single-valued real analytic eigenfunctions of the quantum Hitchin integrable system. The main method of studying these functions is realizing them as the eigenbasis for certain compact normal commuting integral operators the Hilbert space of L2 half-densities on the (complex points of) the moduli space Bun_G of principal G-bundles on a smooth projective curve X, possibly with parabolic points. These operators actually make sense over any local field, and over non-archimedian fields are a replacement for the quantum Hitchin system. We conjecture them to be compact and prove this conjecture in the genus zero case (with parabolic points) for G=PGL(2). I will first discuss the simplest non-trivial example of Hecke operators over local fields, namely G=PGL(2) and genus 0 curve with 4 parabolic points. In this case the moduli space of semistable bundles Bun_G^{ss} is P^1, and the situation is relatively well understood; over C it is the theory of single-valued eigenfunctions of the Lame operator with coupling parameter -1/2 (previously studied by Beukers and later in a more functional-analytic sense in our work with Frenkel and Kazhdan). I will consider the corresponding spectral theory and then explain its generalization to N>4 points and conjecturally to higher genus curves.
Video of Lecture OR Video of lecture (alternate version, same content as the other) Slides of Lecture (first part); link to Slides of Lecture (second part) |
September 27, 2021 |
Nathan Seiberg (IAS)Comments on Lattice vs. Continuum Quantum Field TheoryAbstract: There is an interesting interplay between continuum quantum field theory (QFT) and lattice systems. First, as in condensed-matter physics, we start at short distances (UV) with a lattice model and our goal is to find its long distance (IR) behavior. The lore is that this behavior is captured by a continuum QFT. Conversely, as is more common in high-energy physics and mathematical physics, the lattice theory is a first step toward a rigorous definition of the continuum theory. Despite enormous progress over the past decades, these two directions of the interplay between the lattice and the continuum face interesting challenges. Here, motivated by recently discovered theoretical phases of matter (including the XY-plaquette model and models of fractons), we will address two aspects of the relation between the lattice in the UV and the continuum in the IR. We will present lattice models exhibiting topological properties of continuum theories, like winding symmetries, ‘t Hooft anomalies, and duality. We will use this approach to clarify the subsystem global symmetries of some of the recently discovered exotic models. We will also discuss some more dynamical aspects of these systems and in particular their enigmatic UV/IR mixing; i.e., some long-distance properties are sensitive to short-distance details. Video of Lecture OR Video of lecture (alternate version, same content as the other) |
| October 11, 2021 |
Ryan Thorngren (Harvard University)A Tour of Categorical SymmetryAbstract: A categorical symmetry is a category acting as a symmetry of a QFT. These symmetries correspond to the topological operators in the QFT. I'll try to motivate this definition with some simple examples. Then I'll discuss a bulk-boundary correspondence which in finite situations allows us to classify gapped phases with categorical symmetry and define things like anomalous symmetries and gauging. I'll conclude with some more examples of topological operators in c = 1 CFTs and describe a Noether theorem for continuous categorical symmetries. Video of lecture OR Video of lecture (alternate version, same content as the other) |
October 25, 2021 |
Ibrahima Bah (Johns Hopkins University)Non-supersymmetric smooth solitonic solutions in Einstein-Maxwell type theoriesAbstract: In this talk I will present a recent framework to construct and study smooth horizon-less non-supersymmetric solutions in gravity with interesting topologies. These live in backgrounds that are 4d Minkowski with tori of various dimensions. I will discuss the physical mechanism that allows for their existence and comment on their classical and thermodynamic stability. I will describe a family of these constructions that resolve certain curvature singularities. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| November 8, 2021 |
Tony Pantev (University of Pennsylania)Geometry and topology of wild character varietiesAbstract: Wild character varieties parametrize monodromy representations of flat meromorphic connections on compact Riemann surfaces. They are classical objects with remarkable geometric and topological properties. I will recall how intrinsic geometric structures resolve singularities of wild character varieties and will show that their algebraic symplectic structures extend naturally to the resolutions. This is based on a new universal method for producing symplectic structures which is a joint work with Arinkin and Toen. I will also describe recent joint works with Chuang, Diaconescu, Donagi, and Nawata in which we use string dualities to extract cohomological invariants of twisted wild character varieties from BPS counts on Calabi-Yau threefolds and refined Chern-Simons invariants of torus knots. Video of lecture OR Video of lecture (alternate version, same content as the other) |
| November 22, 2021 |
Lotte Hollands (Heriot-Watt University)Partition functions, BPS states and abelianizationAbstract: In this talk I will re-express the Nekrasov-Shatashvili partition function for a four-dimensional N=2 gauge theory as an integral of a ratio of Wronskians of solutions to the relevant oper equation, with the AD2 theory and the pure SU(2) theory as two main examples. This motivates the definition of a generalized Nekrasov-Shatashvili partition function for any four-dimensional N=2 theory of class S, and makes a connection with abelianization and exact WKB analysis. We will end with some remarks regarding the five-dimensional generalization and the relation to similar mathematical structures underlying the topological string partition function. This talk is based on 2109.14699 and work in progress. Video of Lecture OR Video of lecture (alternate version, same content as the other) |
| December 6, 2021 |
Clay Córdova (University of Chicago)Non-Invertible Duality DefectsAbstract: For any quantum system invariant under gauging a higher-form global symmetry, we construct a non-invertible topological defect by gauging in only half of spacetime. This generalizes the Kramers-Wannier duality line in 1+1 dimensions to higher space- time dimensions. We focus on the case of a one-form symmetry in 3+1 dimensions, and determine the fusion rule. From a direct analysis of one-form symmetry protected topological phases, we show that the existence of certain kinds of duality defects is intrinsically incompatible with a trivially gapped phase. We give an explicit realization of this duality defect in the free Maxwell theory where it is realized by a Chern-Simons coupling between the gauge fields from the two sides. Video of Lecture OR Video of lecture (alternate version, same content as the other) |
| January 24, 2022 |
Hirosi Ooguri (Cal Tech/Kavli IPMU)Symmetry in QFT and GravityAbstract: I will review aspects of symmetry in quantum field theory and combine them with the AdS/CFT correspondence to derive constraints on symmetry in quantum gravity. The quantum gravity constraints to be discussed include the no-go theorem on global symmetry, the completeness of gauge charges, and the decomposition of high energy states into gauge group representations.
Video of lecture OR Video of lecture (alternate version, same content as the other)
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| April 13, 2020 |
Edward Witten (IAS)Volumes and Random MatricesAbstract: I will describe recent results relating two-dimensional gravity and supergravity; volumes of moduli spaces of Riemann surfaces and super Riemann surfaces; and random matrix ensembles. See https://arxiv.org/abs/1903.11115 by Saad, Shenker, and Stanford; https://arxiv.org/abs/1907.03363 by Stanford and me. |
| April 27, 2020 |
Kevin Costello (Perimeter Institute)Topological strings, twistors, and SkyrmionsAbstract: It has long been known that holomorphic field theories on twistor space lead to "physical" field theories on Minkowski space. In this talk I will discuss a type I (unoriented) version of the topological B model on twistor space. The corresponding theory on Minkowski space is a sigma-model with target the group SO(8). This is a variant of the Skyrme model that appears as the low-energy effective theory of mesons in QCD. (The group SO(8) appears because of the Green-Schwarz mechanism in the topological string). The origin of this model in the topological string implies many remarkable properties. For one thing, the model is, in a certain sense, integrable. Further, although the Lagrangian is power-counting non-renormalizable, counter-terms at all loops can be uniquely fixed. |
| May 11, 2020 |
Mark Gross (Cambridge)Intrinsic Mirror SymmetryAbstract: I will talk about joint work with Bernd Siebert, proposing a general mirror construction for log Calabi-Yau pairs, i.e., a pair (X,D) with D a "maximally degenerate" boundary divisor and K_X+D=0, and for maximally unipotent degenerations of Calabi-Yau manifolds. We accomplish this by constructing the coordinate ring or homogeneous coordinate ring respectively in the two cases, using certain kinds of Gromov-Witten invariants we call "punctured invariants", developed jointly with Abramovich and Chen. |
| May 18, 2020 |
Miranda Cheng (Univ. of Amsterdam/National Taiwan University)Quantum Modularity from 3-ManifoldsAbstract: Quantum modular forms are functions 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. Inspired by the 3d-3d correspondence in string theory, 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. |
| June 1, 2020 |
Davide Gaiotto (Perimeter Institute)Integrable Kondo problems and affine Geometric LanglandsAbstract: I will present some work on integrable line defects in WZW models and their relation to 4d CS theory, the IM/ODe correspondence and affine generalizations of Geometric Langlands constructions. |
| June 15, 2020 |
Maxim Kontsevich (IHES)Space-time analyticity in QFTAbstact: I will talk on a joint work with Graeme Segal. We propose a new axiomatics for unitary quantum field theory which includes both Lorentzian and Euclidean signatures for curved space-time manifolds. The key to the definition is certain open domain in the space of complex-valued symmetric bilinear forms on a real vector space. The justification comes from holomorphic convexity (lower bound) and from higher gauge theories (upper bound). |
| June 22, 2020 |
Anton Kapustin (Cal Tech)From gapped phases of matter to Topological Quantum Field Theory and back againAbstract: I will review the connection between gapped phases of matter and Topological Quantum Field Theory (TQFT). Conjecturally, this connection becomes 1-1 correspondence if one restricts to a special class of phases and TQFTs (namely, invertible ones). A related conjecture is that the space of all lattice Hamiltonians describing Short-Range Entangled phases of matter is an infinite loop space. These conjectures predict that the space of lattice Hamiltonians has non-trivial cohomology in particular dimensions. We test this by constructing closed differential forms on the space of gapped lattice Hamiltonians following a suggestion by Kitaev. These differential forms can be regarded as a higher-categorical generalization of the curvature of the Berry connection and correspond to Wess-Zumino-Witten forms in field theory. |
| June 29, 2020 |
No meeting due to Strings 2020
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| July 6, 2020 |
Nima Arkani-Hamed (IAS)Spacetime, Quantum Mechanics and Clusterhedra at InfinityAbstract: Elementary particle scattering is perhaps the most basic physical process in Nature. The data specifying the scattering process defines a "kinematic space", associated with the on-shell propagation of particles out to infinity. By contrast the usual approach to computing scattering amplitudes, involving path integrals and Feynman diagrams, invokes auxilliary structures beyond this kinematic space--local interactions in the interior of spacetime, and unitary evolution in Hilbert space. This description makes space-time locality and quantum-mechanical unitarity manifest, but hides the extraordinary simplicity and infinite hidden symmetries of the amplitude that have been uncovered over the past thirty years. The past decade has seen the emergence of a new picture, where scattering amplitudes are seen as the answer to an entirely different sort of mathematical question involving "positive geometries" directly in the kinematic space, making surprising connections to total positivity, combinatorics and geometry of the grassmannian, and cluster algebras. The hidden symmetries of amplitudes are made manifest in this way, while locality and unitarity are seen as derivative notions, arising from the "factorizing" boundary structure of the positive geometries. This was first see in the story of "amplituhedra" and scattering amplitudes in planar N=4 SYM theory. In the past few years, a similar structure has been seen for non-superysmmetric "bi-adjoint" scalar theories with cubic interactions, in any number of dimensions. The positive geometries through to one-loop order are given by "cluster polytopes"--generalized associahedra for finite-type cluster algebras--with a simple description involving "dynamical evolution" in the kinematic space. Extending these ideas involves understanding cluster algebras associated with triangulations of general Riemann surfaces. These cluster algebras are infinite, reflecting the infinite action of mapping class group. One of the manifestations of this infinity is that the "g-vector fan" of the cluster algebra is not space-filling, making it impossible to define cluster polytopes, and obstructing the connection with positive geometries and scattering amplitudes. Remarkably, incorporating non-cluster variables, associated with closed loops in the Riemann surfaces, suggests a natural way of modding out by the mapping class group, canonically compactifying the cluster complex, and associating it with "clusterhedron" polytopes. Clusterhedra are conjectured to exist for all surfaces, providing the positive geometry in kinematic space for scattering amplitudes in the bi-adjoint scalar theory to all loop orders and all orders in the 1/N expansion. In this talk I will give a simple, self-contained overview of this set of ideas, assuming no prior knowledge of scattering amplitudes or cluster algebras.
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| July 13, 2020 |
Mina Aganagic (UC Berkeley)Knot categorification from mirror symmetry, via string theoryAbstract: I will describe two approaches to categorifying quantum link invariants which work uniformly for all simple Lie algebras, and originate from geometry and string theory. A key aspect of both approaches is that it is manifest that decategorification gives the quantum link invariants one set out to categorify. Many ingredients that go into the story have been found by mathematicians earlier, but string theory spells out how they should be put together for a uniform framework for knot categorification. The first approach is based on derived categories of coherent sheaves on resolutions of slices in affine Grassmannians. Some elements of it have been discovered by mathematicians earlier and others are new. The second approach is perhaps more surprising. It uses symplectic geometry and is related to the first by two dimensional (equivariant) mirror symmetry. Unlike previous symplectic geometry based approaches, it produces a bi-graded homology theory. In both cases, mirror symmetry, and techniques developed to understand it play a crucial role. I will explain the string theory origin of the two approaches, and the relation to another string theory based approach, due to Witten. |
| July 20, 2020 |
Greg Moore (Rutgers)Breaking News About, Topologically Twisted Rank One N=2* Supersymmetric Yang-Mills Theory On Four-Manifolds, Without SpinAbstract: I will report on work in progress with Jan Manschot. We generalize previous results concerning a topological theory in four dimension that generalizes both the Donaldson invariants and the Vafa-Witten invariants. In contrast with previous studies we include an arbitrary background spin-c structure with connection. The Coulomb branch measure involves non-holomorphic topological couplings to the background spin-c connection. (This violates some folklore). Using some novel identities for the $N=2*$ prepotential, the Coulomb branch integral can be evaluated explicitly using the theory of mock modular and Jacobi forms. For $b_2^+>1$ the path integral can be written explicitly in terms of Seiberg-Witten invariants and modular functions of the ultraviolet coupling. We discuss the orbit of partition functions of the three rank one $N=2*$ theories under the action of S-duality. Video of Lecture |
| July 27, 2020 |
No meeting due to String-Math 2020
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| August 3, 2020 |
Sakura Schafer-Nameki (Oxford)5d SCFTs: Symmetries and Moduli SpacesAbstract: I will report on recent developments in 5d SCFTs, studying their global symmetries, 0- and higher-form, M-theory on a canonical singularity. We provide a geometric characterization of the Coulomb and Higgs branch moduli spaces and connect this to recent work on magnetic quivers in 3d. |
| August 10, 2020 |
Ben Webster (Waterloo)3d mirror symmetry and its discontentsAbstract: One of the central topics of the interaction between QFT and math is mirror symmetry for 2d theories. This theory has a more mysterious and exotic friend one dimension higher, sometimes called 3d mirror symmetry, which relates two 3-dimensional theories with N=4 supersymmetry. For roughly a decade, I struggled to understand this phenomenon without understanding what most of the words in the previous sentence meant. Eventually, I wised up and based on work of Braverman, Finkelberg, Nakajima, Dimofte, Gaiotto, Hilburn and others, I actually did learn a little bit, and will now try to explain to you what I learned. This knowledge has some interesting payoffs in the mathematics related to 3d theories, such as an understanding of Bezrukavnikov and Kaledin's noncommutative resolutions of the Coulomb branch, and explaining a lot of interesting Koszul dualities between category O's. |
| August 17, 2020 |
Marco Gualtieri (Univ. of Toronto)Branes in symplectic groupoidsAbstract: After reviewing coisotropic A-branes in symplectic manifolds and their role in mirror symmetry and geometric quantization, I will explain how the problem of holomorphic quantization of Poisson brackets may be recast, and in some cases solved, as a problem of computing morphisms between coisotropic branes in symplectic groupoids. This is joint work with Francis Bischoff and Joshua Lackman. |
| August 24, 2020 |
Laura Fredrickson (Stanford/U. Oregon)The asymptotic geometry of the Hitchin moduli spaceAbstract: Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkahler metric. An intricate conjectural description of its asymptotic structure appears in the work of physicists Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently. I will discuss some recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures. |
| September 14, 2020 |
David Jordan (Edinburgh)Quantum geometric Langlands as a fully extended TFTI will survey several recent works realizing Betti geometric Langlands and its quantization as fully extended TFT's. In physics terms this is most closely related to the Kapustin-Witten twist of N=4 d=4 SYM at generic values of \Psi. I will outline numerous applications to quantum topology, most notably to a conjecture of Witten on finite-dimensionality of skein modules, a conjecture of Bonahon-Wong concerning skein modules at root-of-unity parameters, a proposal of Ben-Zvi concerning cluster varieties and factorization homology, and an appearance of the type-A spherical DAHA from a once-marked torus. Finally, I will present a novel conjectural appearance of S-duality in the study of skein modules of 3-manifolds. |
| September 21, 2020 |
Lauren Williams (Harvard)How is the hypersimplex related to the amplituhedron?Abstract: In 1987, Gelfand-Goresky-MacPherson-Serganova made a beautiful connection between the geometry of the Grassmannian and convex polytopes, via the moment map; the moment map image of the Grassmannian Gr(k,n) is a polytope known as the hypersimplex Delta(k,n). In 2013, motivated by the desire to give a geometric basis for the computation of scattering amplitudes in N=4 SYM, Arkani-Hamed and Trnka introduced the amplituhedron A(n,k,m) as the image of the positive Grassmannian Gr+(k,n) under a linear map Z from R^n to R^{k+m} which is totally positive. While the case m=4 is most relevant to physics, the amplituhedron makes sense for any m. In my talk I will explain some strange parallels between the positroidal subdivisions of the hypersimplex Delta(k+1,n) and the m=2 amplituhedron A(n,k,2). One link is provided by the positive tropical Grassmannian. Attributions: based on joint works with Tomek Lukowski, Matteo Parisi, and David Speyer. Disclaimer: I'm neither a geometer nor a physicist. |
| October 5, 2020 |
Natalie Paquette (IAS)Borcherds-Kac-Moody algebras, 2d strings, & other curiositiesAbstract: In this talk we will extol the virtues of compactifying critical string theory down to few noncompact spacetime dimensions (particularly two). These string vacua possess rich groups of dualities. BPS-saturated quantities, which mathematically are described by automorphic forms, are invariant under such duality transformations. Further, such BPS states can furnish representations of interesting algebras, such as infinite-dimensional Lie algebras. In this talk, we explore some particularly nice, concrete examples, which employ holomorphic super vertex operator algebras in our `compactification' theories. The BPS states in these models organize into representations of algebras, which we prove are (new) examples of Borcherds-Kac-Moody superalgebras. Video of Lecture |
| October 19, 2020 |
Djordje Radicevic (Brandeis)The Lattice-Continuum Correspondence in Quantum MechanicsAbstract: It is very well known that long-distance correlation functions of many lattice systems can be calculated from continuum QFTs. Making this correspondence more precise --- identifying continuum operators that correspond to individual lattice operators, or exhibiting the lattice origins of subtler continuum phenomena like operator product expansions --- has proven quite daunting. In this talk, I will report on recent progress in this direction, using quantum mechanics (QFT in 0+1 dimensions) as an example. I will show how a finite but large quantum system can be systematically reduced to an Ersatz continuum theory, using both Hamiltonian and path integral formalisms. Along the way I will point out the lattice origins of several familiar continuum concepts, including contact terms, scale invariance, and the distinction between compact and noncompact theories. I will also stress the limitations imposed on the emergent continuum theory by its lattice progenitor --- for instance, any supersymmetric continuum theory emerging from a finite theory must have a vanishing Witten index. Video of Lecture |
| November 2, 2020 |
Christopher Beem (Oxford)Hall-Littlewood Chiral Rings and Derived Higgs BranchesAbstract: I will discuss a relatively novel algebraic structure arising in four-dimensional N=2 superconformal field theories: the Hall-Littlewood Chiral Ring (HLCR). The HLCR is an enhancement of the more familiar Higgs branch chiral ring (which encodes the Higgs branch of the moduli space of vacua as an algebraic variety). The HLCR in gauge theories is constructed as the cohomology of a kind of BRST complex, which allows it to be identified with the ring of functions on the derived Higgs branch (in the sense of derived algebraic geometry). I will describe the solution of the HLCR cohomology problem for a large class of Lagrangian theories (the class S theories of type A1), which illustrate some interesting phenomena. This talk is based on work in progress with Diego Berdeja Suárez. Video of Lecture |
| November 16, 2020 |
Sergei Gukov (Cal Tech)Rokhlin, quantum groups, and BPS statesAbstract: What do ADO polynomials, cobordism invariants, and affine Grassmannians have in common? We will discuss how these seemingly different objects can be put under one roof of a BPS q-series that, on the one hand, can be thought of as a 3d analogue of the Vafa-Witten partition function and, on the other hand, is associated to quantum groups at generic q where Verma modules with arbitrary complex weights play an important role. Video of Lecture |
| November 30, 2020 |
Sourav Chatterjee (Stanford)Strong mass gap implies quark confinementAbstract: I will show that if a lattice gauge theory has exponential decay of correlations under arbitrary boundary conditions (which I call strong mass gap), and the gauge group has a nontrivial center, then Wilson's area law holds. |
| December 14, 2020 |
Harold Williams (USC)Canonical Bases for Coulomb BranchesAbstract: Following work of Kapustin-Saulina and Gaiotto-Moore-Neitzke, one expects half-BPS line defects in a 4d N=2 field theory to form a monoidal category with a rich structure. In general, a mathematical definition of this category is not known. In this talk we discuss an algebro-geometric proposal in the case of gauge theories with polarizable matter. The proposed category is the heart of a nonstandard t-structure on the dg category of coherent sheaves on the derived Braverman-Finkelberg-Nakajima space of triples. We refer to its objects as Koszul-perverse coherent sheaves, as this t-structure interpolates between the perverse coherent t-structure and certain t-structures appearing in the theory of Koszul duality (specializing to these in the case of a pure gauge theory and an abelian gauge theory, respectively). As a byproduct, this defines a canonical basis in the associated quantized Coulomb branch by passing to classes of irreducible objects. This is joint work with Sabin Cautis. Video of Lecture |