Mathematical Physics Seminar, Mondays

The seminar on mathematical physics will be held on Mondays from 10:00 – 11:00am ET on Zoom. Please email the seminar organizers to learn how toattend. This year’s Seminar will be organized by Yoosik Kim (yoosik@cmsa.fas.harvard.edu), Tsung-Ju Lee (tjlee@cmsa.fas.harvard.edu), and Yang Zhou (yangzhou@cmsa.fas.harvard.edu).

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The list of speakers for the upcoming academic year will be posted below and updated as details are confirmed. Titles and abstracts for the talks will be added as they are received. 

Spring 2021:

DateSpeakerTitle/Abstract
2/1/2021Choa Dongwook
(KIAS)

Video
Title: Fukaya category of Landau-Ginzburg orbifolds.

Abstract: Landau-Ginzburg orbifold is just another name for a holomorphic function W with its abelian symmetry G. Its Fukaya category can be viewed as a categorification of a homology group of its Milnor fiber. In this introductory talk, we will start with some classical results on the topology of isolated singularities and its Fukaya-Seidel category. Then I will explain a new construction for such category to deal with a non-trivial symmetry group G. The main ingredients are classical variation map and the Reeb dynamics at the contact boundary. If time permits, I will show its application to mirror symmetry of LG orbifolds and its Milnor fiber. This is a joint work with C.-H. Cho and W. Jeong
2/8/2021Jérémy Guéré (Fourier Institute)Title: Congruences on K-theoretic Gromov-Witten invariants

Abstract: K-theoretic Gromov-Witten invariants of smooth projective varieties have been introduced by YP Lee, using the Euler characteristic of a virtual structure sheaf. In particular, they are integers. In this talk, I look at these invariants for the quintic threefold and I will explain how to compute them modulo 41, using the virtual localization formula under a finite group action, up to genus 19 and degree 40.
2/15/2021Zhiwei Zheng (Max Planck Institute)

Video
Title: Some new results on automorphisms of hypersurfaces

Abstract: It is natural to study automorphisms of hypersurfaces in projective spaces. In this talk, I will discuss a new approach to determine all possible orders of automorphisms of smooth hypersurfaces with fixed degree and dimension. Then we consider the specific case of cubic fourfolds, and discuss the relation with Hodge theory.
2/22/2021Yu-Shen Lin (Boston University)

Video
Title: Full SYZ Conjecture for del Pezzo Surfaces and Rational Elliptic Surfaces

Abstract: Strominger–Yau–Zaslow conjecture predicts the existence of special Lagrangian fibrations on Calabi–Yau manifolds. The conjecture inspires the development of mirror symmetry while the original conjecture has little progress. In this talk, I will confirm the conjecture for the complement of a smooth anti-canonical divisor in del Pezzo surfaces. Moreover, I will also construct the dual torus fibration on its mirror. As a consequence, the special Lagrangian fibrations detect a non-standard semi-flat metric and some Ricci-flat metrics that don’t obviously appear in the literature. This is based on a joint work with T. Collins and A. Jacob.
3/1/2021Carlos S. Shahbazi (Hamburg University)

Video
TitleMathematical supergravity and its applications to differential geometry.

Abstract: I will discuss the recent developments in the mathematical theory of supergravity that lay the mathematical foundations of the universal bosonic sector of four-dimensional ungauged supergravity and its Killing spinor equations in a differential-geometric framework.  I will provide the necessary context and background. explaining the results pedagogically from scratch and highlighting several open mathematical problems which arise in the mathematical theory of supergravity, as well as some of its potential mathematical applications. Work in collaboration with Vicente Cortés and Calin Lazaroiu.
3/8/2021Miguel Moreira (ETH)

Video
Title: Virasoro constraints for stable pairs.

Abstract: The theory of stable pairs (PT) with descendents, defined on a 3-fold X, is a sheaf theoretical curve counting theory. Conjecturally, it is equivalent to the Gromov-Witten (GW) theory of X via a universal (but intricate) transformation, so we can expect that the Virasoro conjecture on the GW side should have a parallel in the PT world. In joint work with A. Oblomkov, A. Okounkov, and R. Pandharipande, we formulated such a conjecture and proved it for toric 3-folds in the stationary case. The Hilbert scheme of points on a surface S might be regarded as a component of the moduli space of stable pairs on S x P1, and the Virasoro conjecture predicts a new set of relations satisfied by tautological classes on S[n] which can be proven by reduction to the toric case.
3/15/2021Spring break
3/22/2021Ying Xie (Shanghai Center for Mathematical Sciences)Title: Derived categories for Grassmannian flips

Abstract: Flip is a fundamental surgery operation for constructing minimal models in higher-dimensional birational geometry. In this talk, I will introduce a series of flips from Lie theory and investigate their derived categories. This is a joint program with Conan Leung.
3/29/2021Emanuel Scheidegger (Peking University)TitleOn the quantum K-theory of the quintic.

Abstract: Quantum cohomology is a deformation of the cohomology of a projective variety governed by counts of stable maps from a curve into this variety. Quantum K-theory is in a similar way a deformation of K-theory but also of quantum cohomology, It has recently attracted attention in physics since a realization in a physical theory has been found. Currently, both the structure and examples in quantum K-theory are far less understood than in quantum cohomology.
We will explain the properties of quantum K-theory in comparison with quantum cohomology, and we will discuss the examples of projective space and the quintic hypersurface in P^4.
4/5/2021Gaëtan Borot (HU Berlin)

Video
TitleTopological recursion in 4d N = 2 supersymmetric gauge theories

Abstract: According to the Alday-Gaiotto-Tachikawa conjecture (proved in this case by Schiffman and Vasserot), the instanton partition function in 4d N = 2 SU(r) supersymmetric gauge theory on P^2 with equivariant parameters \epsilon_1,\epsilon_2 is the norm of a Whittaker vector for W(gl_r) algebra. I will explain how these Whittaker vectors can be computed (at least perturbatively in the energy scale) by topological recursion for \epsilon_1 +\epsilon_2 = 0, and by a non-commutation version of the topological recursion in the Nekrasov-Shatashvili regime where \epsilon_1/\epsilon_2 is fixed. This is a joint work to appear with Bouchard, Chidambaram and Creutzig.
4/12/2021Fei Yan (Rutgers)Title: Networks and quantization

Abstract: I will describe two quantization scenarios. The first scenario involves the construction of a quantum trace map computing a 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 unifies the computation of familiar link invariant with the refined counting of framed BPS states 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 scenarios.
4/19/2021Hazel Mak (Brown University)

TitleBranching Rules and Young Tableaux Methods: 10D & 11D Supergravity

Abstract: In this talk, I will review 4D, N = 1 off-shell supergravity. Then I present explorations to construct 10D and 11D supergravity theories in two steps. The first step is to decompose scalar superfield into Lorentz group representations which involves branching rules and related methods. Interpretations of component fields by Young tableaux methods will be presented. The second step is to implement an analogue of Breitenlohner’s approach for 4D supergravity to 10D and 11D theories.
4/26/2021Owen Gwilliam (UMass. Amherst)

Video
TitleTopological-holomorphic field theories and their BV quantizations

Abstract: Topological field theories and holomorphic field theories have each had a substantial impact in both physics and mathematics, so it is natural to consider theories that are hybrids of the two, which we call topological-holomorphic and denote as THFTs. Examples include Kapustin’s twist of N=2, D=4 supersymmetric Yang-Mills theory and Costello’s 4-dimensional Chern-Simons theory. In this talk about joint work with Rabinovich and Williams, I will define THFTs, describe several examples, and then explain how to quantize them rigorously and explicitly, by building on techniques of Si Li.  Time permitting, I will indicate how these results offer a novel perspective on the Gaudin model via 3-dimensional field theories.

Fall 2020:

DateSpeakerTitle/Abstract
9/14/2020Lino Amorim (Kansas State University)Title: Non-commutative Gromov-Witten invariants

Abstract:  I will describe an analogue of Saito’s theory of primitive forms for Calabi-Yau A-infinity categories. Under some conditions on the Hochschild cohomology of the category, this construction recovers the (genus zero) Gromov-Witten invariants of a symplectic manifold from its Fukaya category. This includes many compact toric manifolds, in particular projective spaces.
9/21/2020Yuhan Sun (Rutgers)Title: Displacement energy of Lagrangian 3-spheres

Abstract:  We study local and global Hamiltonian dynamical behaviors of some Lagrangian submanifolds near a Lagrangian sphere S in a symplectic manifold X. When dim S = 2, we show that there is a one-parameter family of Lagrangian tori near S, which are nondisplaceable in X. When dim S = 3, we obtain a new estimate of the displacement energy of S, by estimating the displacement energy of a one-parameter family of Lagrangian tori near S.
9/28/2020Shota Komatsu (CERN)Title: Wilson loops as matrix product states

Abstract:  In this talk, I will discuss a reformulation of the Wilson loop in large N gauge theories in terms of matrix product states. The construction is motivated by the analysis of supersymmetric Wilson loops in the maximally super Yang–Mills theory in four dimensions, but can be applied to any other large N gauge theories and matrix models, although less effective. For the maximally super Yang–Mills theory, one can further perform the computation exactly as a function of ‘t Hooft coupling by combining our formulation with the relation to integrable spin chains. 
10/5/2020Ming Zhang (UBC)Title: Verlinde/Grassmannian correspondence and applications.

Abstract: In the 90s’, Witten gave a physical derivation of an isomorphism between the Verlinde algebra of $GL(n)$ of level $l$ and the quantum cohomology ring of the Grassmannian $\text{Gr}(n,n+l)$. In the joint work arXiv:1811.01377 with Yongbin Ruan, we proposed a K-theoretic generalization of Witten’s work by relating the $\text{GL}_{n}$ Verlinde numbers to the level $l$ quantum K-invariants of the Grassmannian $\text{Gr}(n,n+l)$, and refer to it as the Verlinde/Grassmannian correspondence.

The correspondence was formulated precisely in the aforementioned paper, and we proved the rank 2 case (n=2) there. In this talk, I will discuss the proof for arbitrary rank. A new technical ingredient is the virtual nonabelian localization formula developed by Daniel Halpern-Leistner.  At the end of the talk, I will describe some applications of this correspondence. 
10/12/2020Cancelled -Columbus Day
10/19/2020Ben Gammage (Harvard)Title: 3d mirror symmetry for abelian gauge groups

Abstract: 3d mirror symmetry is a proposed duality relating a pair of 3-dimensional supersymmetric gauge theories. Various consequences of this duality have been heavily explored by representation theorists in recent years, under the name of “symplectic duality”. In joint work in progress with Justin Hilburn, for the case of abelian gauge groups, we provide a fully mathematical explanation of this duality in the form of an equivalence of 2-categories of boundary conditions for topological twists of these theories. We will also discuss some applications to homological mirror symmetry and geometric Langlands duality. 
10/26/2020Cancelled
11/2/2020Haoyu Sun (Berkeley)Title: Double-Janus linear sigma models and generalized quadratic reciprocity 

Abstract: We study the supersymmetric partition function of a 2d linear sigma-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kähler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d N=4 U(1) Super-Yang-Mills theory compactified on a mapping torus (T^2 fibered over S^1) times a circle with an SL(2,Z) duality wall inserted on S^1, but our setup has minimal supersymmetry. The partition function depends on two independent elements of SL(2,Z), one describing the duality twist, and the other describing the geometry of the mapping torus. It is topological and can be written as a multivariate quadratic Gauss sum. By calculating the partition function in two different ways, we obtain identities relating different quadratic Gauss sums, generalizing the Landsberg-Schaar relation. These identities are a subset of a collection of identities discovered by F. Deloup. Each identity contains a phase which is an eighth root of unity, and we show how it arises as a Berry phase in the supersymmetric Janus-like configuration. Supersymmetry requires the complex structure to vary along a semicircle in the upper half-plane, as shown by Gaiotto and Witten in a related context, and that semicircle plays an important role in reproducing the correct Berry phase.
11/9/2020An Huang (Brandeis)Titlep-adic strings, Einstein equations, Green’s functions, and Tate’s thesis

Abstract: I shall discuss a recent work on how p-adic strings can produce perturbative quantum gravity, and an adelic physics interpretation of Tate’s thesis. 
11/16/2020
10:00am ET
Matt Kerr (WUSTL)TitleDifferential equations and mixed Hodge structures

Abstract: We report on a new development in asymptotic Hodge theory, arising from work of Golyshev–Zagier and Bloch–Vlasenko, and connected to the Gamma Conjectures in Fano/LG-model mirror symmetry.  The talk will focus exclusively on the Hodge/period-theoretic aspects through two main examples. 
Given a variation of Hodge structure M on a Zariski open in P^1, the periods of the limiting mixed Hodge structures at the punctures are interesting invariants of M.  More generally, one can try to compute these asymptotic invariants for iterated extensions of M by “Tate objects”, which may arise for example from normal functions associated to algebraic cycles. The main point of the talk will be that (with suitable assumptions on M) these invariants are encoded in an entire function called the motivic Gamma function, which is determined by the Picard-Fuchs operator L underlying M. In particular, when L is hypergeometric, this is easy to compute and we get a closed-form answer (and a limiting motive).  In the non-hypergeometric setting, it yields predictions for special values of normal functions; this part of the story is joint with V. Golyshev and T. Sasaki.
11/23/2020

11:30am ET
Kyoung-Seog Lee (U of Miami)Title: Derived categories and motives of moduli spaces of vector bundles on curves

Abstract: Derived categories and motives are important invariants of algebraic varieties invented by Grothendieck and his collaborators around 1960s. In 2005, Orlov conjectured that they will be closely related and now there are several evidences supporting his conjecture. On the other hand, moduli spaces of vector bundles on curves provide attractive and important examples of algebraic varieties and there have been intensive works studying them. In this talk, I will discuss derived categories and motives of moduli spaces of vector bundles on curves. This talk is based on joint works with I. Biswas and T. Gomez.
11/30/2020Zijun Zhou (IPMU)Title: 3d N=2 toric mirror symmetry and quantum K-theory

Abstract: In this talk, I will introduce a new construction for the K-theoretic mirror symmetry of toric varieties/stacks, based on the 3d N=2 mirror symmetry introduced by Dorey-Tong. Given the toric datum, i.e. a  short exact sequence 0 -> Z^k -> Z^n -> Z^{n-k} -> 0, we consider the toric Artin stack of the form [C^n / (C^*)^k]. Its mirror is constructed by taking the Gale dual of the defining short exact sequence. As an analogue of the 3d N=4 case, we consider the K-theoretic I-function, with a suitable level structure, defined by counting parameterized quasimaps from P^1. Under mirror symmetry, the I-functions of a mirror pair are related to each other under the mirror map, which exchanges the K\”ahler and equivariant parameters, and maps q to q^{-1}. This is joint work with Yongbin Ruan and Yaoxiong Wen.
12/7/2020Thomas Grimm (Utrecht)Title: Moduli Space Holography and the Finiteness of Flux Vacua

Abstract: In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using asymptotic Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The bulk reconstruction from the boundary data is provided by the sl(2)-orbit theorem of Schmid and Cattani, Kaplan, Schmid, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us, in work with C. Schnell, to prove that the number of self-dual flux vacua is finite

For a listing of previous Mathematical Physics Seminars, please click here.

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