Opening comments

• Abhiram Kidambi (Kavli IPMU, U. Tokyo)

Khovanov Homology from Mirror Symmetry

• Mina Aganagic (U.C. Berkeley)

Khovanov showed, more than 20 years ago, that there is a deeper theory underlying the Jones polynomial. The knot categorification problem is to find a uniform description of this theory, for all gauge groups, which originates from physics. I found two solutions to this problem, related by a version of two dimensional homological mirror symmetry. They are based on two descriptions of the theory that lives on defects of the six dimensional (0,2) CFT, which are supported on a link times time. The theory turns out to be solvable explicitly. It is also more efficient, often exponentially so, than Khovanov's original approach.

Chiral algebras with exceptional finite symmetry groups

• Brandon Rayhaun (Stanford U.)

The classification of finite simple groups is a remarkable theorem of modern mathematics which says that every such group either a) belongs to one of three infinite families, or b) is one of 26 exceptional cases, which are called the sporadic groups. Of these 26 outliers, 20 of them appear as subquotients inside of the largest, which is called the monster. It is natural to ask what objects these groups act on by symmetries. In the case of the monster, it is a cherished result of mathematical physics that it arises as the automorphism group of a certain meromorphic conformal field theory of central charge 24: the moonshine module. We show that the method of coset conformal field theory can be effectively used to obtain chiral algebras which furnish several of the other sporadic groups as their symmetries. Moreover, these chiral algebras embed into one another in the same way as do their automorphism groups; that is to say, we have discovered a functorial assignment of subalgebras of the moonshine module to certain privileged subgroups of the monster.

3D rank-0 N=4 SCFTs and non-unitary TQFTs

• Dongmin Gang (Seoul Natl. U & APCTP)

I will talk about a recently proposed correspondence between 3D rank-0 N=4 SCFTs and 3D non-unitary TQFTs. Using the basic dictionaries of correspondence, we derive the lower bound on F (3-sphere free energy), F >= -1/2 log((5-\sqrt{5})/10)=0.6429. The talk is based on arXiv:2103.09283.

BPS (shifted) quiver Yangians and representations from colored crystals

• Wei Li (ITP, Chinese Academy of Sciences)

I will first explain how to construct BPS algebras for string theory on general toric Calabi-Yau threefolds, based on the 3D colored crystals that describe BPS states of the system. The resulting algebras are shifted quiver Yangians Y(Q,W) that are associated with the quiver and the superpotential of the theory. Then I will show how to construct representations of a shifted quiver Yangian from general subcrystals of the canonical crystal, and how the shape of the subcrystal determines the framing of the quiver.

On Supersymmetric Interface Defects, Brane Parallel Transport and Higgs-Coulomb Duality

• Dimitrii Galakhov (Kavli IPMU, U. Tokyo)

We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d N=(2,2) theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated Abelian categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors. We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects. Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold. Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space. The talk is based on arXiv:2105.07602

Graviton scattering and differential equations in automorphic forms

• Kim Klinger-Logan (Rutgers)

Green, Russo, and Vanhove have shown that the scattering amplitude for gravitons (hypothetical particles of gravity represented by massless string states) is closely related to automorphic forms through differential equations. Green, Miller, Russo, Vanhove, Pioline, and K-L have used a variety of methods to solve eigenvalue problems for the invariant Laplacian on different moduli spaces to compute the coefficients of the scattering amplitude of four gravitons. We will examine two methods for solving the most complicated of these differential equations on $SL_2(\mathbb{Z})\backslash\mathfrak{H}$. Time permitting, we will discuss recent work with S. Miller to improve upon his original method for solving this equation.

Discussion session Hosted on Gather Town

A Whipple formula revisited

• Ling Long (Louisiana State U.)

A well-known formula of Whipple relates certain hypergeometric values $_7F_6(1)$ and $_4F_3(1)$. In this paper we revisit this relation from the viewpoint of the underlying hypergeometric data $HD$, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple's formula when the hypergeometric data $HD$ are primitive and defined over rationals. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values $_7F_6(1)$ in Whipple's formula to the periods of modular forms.

Analytic Langlands correspondence for complex curves

• Edward Frenkel (U.C. Berkeley)

The Langlands correspondence for complex curves has been traditionally formulated in terms of sheaves rather than functions. Together with Pavel Etingof and David Kazhdan (arXiv:1908.09677, arXiv:2103.01509), we have formulated an analytic (or function-theoretic) version as a spectral problem for an algebra of commuting operators acting on half-densities on the moduli space Bun_G of G-bundles over a complex algebraic curve. This algebra is generated by the global differential operators on Bun_G (holomorphic and anti-holomorphic quantum Hitchin Hamiltonians) as well as integral operators, which are analytic analogues of the Hecke operators of the classical Langlands correspondence. We conjecture that the joint spectrum of this algebra (properly understood) can be identified with the set of opers for the Langlands dual group of G whose monodromy is in the split real form (up to conjugation). Furthermore, we give an explicit formula relating the eigenvalues of the Hecke operators and the global differential operators.

Polyakov Path Integral as Modular Parametrization (?): CM Elliptic Curves as Target Spaces

• Taizan Watari (Kavli IPMU, U. Tokyo)

An L-function is defined for an algebraic variety with defining equations written only with algebraic numbers as coefficients. For some classes of such varieties, a correspondence between those L-functions and modular forms has been observed as pure mathematics. It is a natural question whether the modular transformation in g=1 string-theory amplitudes has something to do with this phenomenon. We argue for the class of varieties called elliptic curves of Shimura type (implying complex multiplication) that the modular transformation can indeed be regarded as that of the g=1 world sheet, and that certain class of g=1 correlation functions yield functions on the complex upper half plane, just like the theory of modular parametrization does. Roles played by choice of the target space metric, arithmetic model dependence, and the needs for average within the ideal class group will also be discussed along the way. This presentation is based on two joint works with Satoshi Kondo: https://arxiv.org/abs/1912.13294 and https://arxiv.org/abs/1801.07464

Matrix integrals & the two-sphere partition function

• Beatrix Mühlmann (U. Amsterdam)

We explore the two-sphere partition function in two-dimensional quantum gravity coupled to conformal matter and the relation to matrix integrals. We discuss how such two-dimensional models may shed light on Euclidean gravity with a positive cosmological constant.

Narain to Narnia

• Ida Zadeh (ICTP Trieste)

Recently, a new holographic correspondence was discovered between an ensemble average of toroidal conformal field theories in two dimensions and an abelian Chern-Simons theory in three dimensions coupled to topological gravity. I will discuss a generalisation of this duality for three families of conformal field theories and show that the correspondence works for toroidal orbifolds but not for K3/Calabi-Yau sigma-models and not always for the minimal models. For toroidal orbifolds, the holographic correspondence is extended to correlation functions of twist operators by using topological properties of rational tangles in the three-dimensional ball.

Quantization by Branes And Geometric Langlands

• Edward Witten (IAS, Princeton)

Discussion Hosted via Gather Town

2d Categorical Wall-Crossing With Twisted Masses, And An Application To Knot Invariants

• Gregory Moore (Rutgers)

We review how supersymmetric quantum mechanics naturally leads to several standard constructions in homological algebra. We apply these ideas to 2d Landau-Ginzburg models with (2,2) supersymmetry to discuss wall-crossing. Some aspects of the web formalism are reviewed and applied to the categorification of the Cecotti-Vafa wall-crossing formula for BPS invariants. We then sketch the generalization to include twisted masses. In the final part of the talk we sketch how some of these ideas give a natural framework for understanding a recent conjecture of Garoufalidis, Gu, and Marino and lead to potentially new knot invariants. The talk is based on work done with Ahsan Khan and recent discussions with Ahsan Khan, Davide Gaiotto, and Fei Yan.

One-Level density for cubic characters over the Eisenstein field

• Chantal David (Concordia U.)

Katz and Sarnak conjectured that statistics on zeroes of a family of L-functions on the critical line should match statistics on eigenvalues of characteristic polynomials of a group of random matrices, where the group is chosen according to the properties of the family. For example, the family of L-functions attached to quadratic Dirichlet characters corresponds to symplectic matrices, and evidence for the conjecture of Katz and Sarnak was obtained by proving that the one-level density of zeroes of quadratic Dirichlet L-functions matches the one-level density for eigenvalues of characteristic polynomials of symplectic matrices, for special test functions (with limited support of the Fourier transform) by Ozluk and Snyder in 1999. Since the support of the Fourier transform of the test function is large enough, they can deduce that more than $93.75 \%$ of the L-functions attached to quadratic Dirichlet characters are such that $L(1/2, \chi) \neq 0$, giving evidence for a well-known conjecture of Chowla. The full conjecture of Katz-Sarnak (without any restrictions on the support of the Fourier transform) implies that $100 \%$ of the L-functions attached to quadratic Dirichlet characters are such that $L(1/2, \chi) \neq 0$. We will review those results and consider the case of L-functions attached to cubic Dirichlet characters. We prove the first result towards the Katz and Sarnak conjecture for test functions with support of the Fourier transform large enough to obtain a positive proportion of L-functions attached to cubic Dirichlet characters such that $L(1/2, \chi) \neq 0$. Joint work with Ahmet M. Guloglu.

Quantum Mechanics of bipartite ribbon graphs and Kronecker coefficients

• Sanjaye Ramgoolam (QMUL & U. Witwatersrand)

I describe a family of algebras $K(n)$, one for every positive integer n, related to the group algebra of the symmetric group $S_n$. These algebras have a basis labelled by bi-partite ribbon graphs with $n$ edges. They also have a decomposition into matrix blocks labelled by triples of Young diagrams with $n$ boxes, with matrix block size equal to the Kronecker coefficient $C$ for the triple. This leads to algorithms for the determination of sub-lattices in the lattice of ribbon graphs, of dimensions $C^2$ and $C$, equipped with bases constructed from null vectors of integer matrices. Some of the algorithms are realised in quantum mechanical systems where the quantum states are bipartite ribbon graphs. Using the known connections between bipartite ribbon graphs and Belyi maps, these quantum systems have an interpretation as models of quantum membranes.

QFT's for Non-Semisimple TQFT's

• Tudor Dimofte (U.C. Davis & Edinburg U.)

Thirty years ago, work of Witten and Reshetikhin-Turaev activated the study of quantum invariants of links and three-manifolds. A cornerstone of subsequent developments, leading up to our current knot-homology conference, was a three-pronged approach involving 1) quantum field theory (Chern-Simons); 2) rational VOA's (WZW); and 3) semisimple representation theory of quantum groups. The second and third perspectives have since been extended, to logarithmic VOA's and related non-semisimple quantum-group categories. I will propose a family of 3d quantum field theories that similarly extend the first perspective to a non-semisimple (and more so, derived) regime. They support boundary VOA's whose module categories equivalent to modules for small quantum groups at even roots of unity. This is joint work with T. Creutzig, N. Garner, and N. Geer, and also related to recent work of Gukov-Hsin-Nakajima-Park-Pei-Sopenko.

Three Avatars of Mock Modularity

• Atish Dabholkar (ICTP Trieste)

Mock theta functions were introduced by Ramanujan in his famous last letter to Hardy in 1920 but were properly understood only recently with the work of Zwegers in 2002. I will describe three manifestations of this apparently exotic mathematics in three important physical contexts of holography, topology and duality where mock modularity has come to play in important role. In particular, I will derive a holomorphic anomaly equation for the indexed partition function of a two-dimensional CFT2 dual to AdS3 that counts the black hole degeneracies, and for Vafa-Witten partition function for twisted four dimensional N=4 super Yang-Mills theory on CP2 for the gauge group SO(3) that counts instantons. The holomorphic kernel of this equation is not modular but `mock modular’ and one obtains correct modular properties only after including certain `anomalous’ nonholomorphic boundary contributions. This phenomenon can be related to the holomorphic anomaly of the elliptic genus of a two-dimensional noncompact supersymmetric sigma model, and in a simpler context of quantum mechanics to the Atiyah-Patodi-Singer eta invariant. Mock modularity is thus essential to exhibit modular symmetries expected from the AdS3/CFT2 holographic equivalence in quantum gravity and the S-duality symmetry of four-dimensional quantum gauge theories.

Discussion session Hosted via Gather Town

Quantum invariants, q-series, DAHA

• Kazuhiro Hikami (Kyushu U.)

We will review some properties of colored Jones polynomial and WRT invariants from the viewpoint of the volume conjecture based on Habiro's expansion.

Modularity, Integrability, and Logarithmic Invariants

• Sergei Gukov (Caltech)

The goal of the talk is to explain new ways in which exotic types of modularity, associated with log-CFT's, appear in 3d supersymmetric theories, 3-manifold topology, and lattice integrable models.

Chern-Simons Invariants from Ensemble Averages

• Matthew Dodelson (Kavli IPMU, U. Tokyo)

I will discuss the holographic duality between a free boson CFT associated with an integral lattice and Chern-Simons theory. The boundary CFT has a moduli space, and averaging over the moduli space reproduces the partition function of Chern-Simons theory in the bulk, as a result of a theorem derived by Siegel. For odd lattices the bulk theory is given by a spin Chern-Simons theory.

Mathieu Moonshine: Quarter BPS states at the Kummer point and nearby

• Anne Taormina (Durham U.)

The construction of $\mathbb{Z}_2$ orbifolds of toroidal conformal field theories (CFTs) is induced by the Kummer construction of a K3 surface. These theories provide a vantage point from which to study the quarter BPS states of K3 theories which are at the heart of the Mathieu Moonshine phenomenon. In this talk, we argue that a non-trivial SU(2) action on a subspace of quarter BPS states in these orbifold CFTs governs the pairing of states that lift from the BPS bound upon a given type of deformation.

Towards a generic space of BPS states for K3

• Katrin Wendland (Freiburg U.)

This talk will focus on the notion of a "generic space of states" of K3 theories, which can serve as an ingredient of the symmetry surfing idea in Mathieu Moonshine.

Taking the Limit: Quantum Modular Forms in Moonshine, Physics and Topology

• Miranda Cheng (U. Amsterdam)

Often it is useful to study the "boundary condition", namely the behaviour near the cusps, of functions on the upper-half plane. This leads naturally to the concept of quantum modular forms, which are functions on rational numbers that have rather mysterious weak modular properties generalizing modular forms and mock modular forms. In this talk I will discuss how considerations of boundary conditions lead to interesting application of quantum modular forms in the inter-connected subjects of moonshine, physics and topology.

Two new avatars of Thompson Moonshine

• Jeffrey Harvey (U. Chicago)

Among other things I will discuss the relationship between Thompson moonshine at weight 1/2 and Generalized Monstrous Moonshine at weight zero.

Discussion session

Supersymmetric Flux Compactifications and Calabi-Yau Modularity

• Richard Nally (Stanford U.)

Many familiar constructions in string theory are rooted in the complex geometry of the compact dimensions. On the other hand, many recent advances in mathematics come from arithmetic geometry, where we consider the properties of varieties over smaller fields such as Q. In this talk, following recent work (arXiv:2001.06022, arXiv:2010.07285) with S. Kachru and W. Yang, I will explain how string theory can be related to arithmetic. In particular, I will argue that supersymmetric flux vacua admit arithmetic structures closely related to those of elliptic curves, and moreover that these arithmetic structures are related to the geometry of the F-theory description of the flux compactification.

From Little Strings to M5-branes and Number Theory via a Quasi-Topological Sigma Model on Loop Group

• Meng-Chwan Tan (NUS Singapore)

We unravel the ground states and left-excited states of the A_{k-1} N=(2,0) little string theory. Via a theorem by Atiyah, these sectors can be captured by a supersymmetric quasi-topological sigma model on CP^1 with target space the based loop group of SU(k). The ground states, described by L^2-cohomology classes, form modules over an affine Lie algebra, while the left-excited states, described by chiral differential operators, form modules over a toroidal Lie algebra. We also apply our results to unravel the 1/2 and 1/4 BPS sectors of the M5-brane worldvolume theory, which spectrum we find to be captured by cousins of modular and automorphic forms, respectively, that reveal an intrinsic S- and T-duality of the worldvolume theory.

Arborescent and non-arborescent knots & links

• Ramadevi Pichai (IIT Mumbai)

Finding colored HOMFLY-PT invariants for knots carrying arbitrary colors still needs new ideas. I will briefly discuss the computational methods of obtaining colored HOMFLY-PT invariants of arborescent knots/links and non-arborescent knots/links and their limitations. Some of our recent works on mutant knots and hybrid weaving knots will also be presented.

Interpolation, integrals, and indices

• Andrei Okounkov (Columbia U.)

There is an interesting topology behind such classical questions as interpolation and solving linear q-difference equations by integrals. It has to do with counting algebraic curves in some very specific geometries, which can be also phrased as computing indices in certain (2+1) dimensional supersymmetric QFTs. The talk will be an introduction to this circle of ideas.

On the arithmetic of Calabi-Yau manifolds: periods, zeta functions and attractor varieties

• Xenia de la Ossa (Oxford U.)

In this seminar I will discuss the arithmetic of Calabi-Yau 3-folds. The main goal is to explore whether there are questions of common interest in this context to physicists, number theorists and geometers. The main quantities of interest in the arithmetic context are the numbers of points of the manifold considered as a variety over a finite field. We are interested in the computation of these numbers and their dependence on the moduli of the variety. The surprise for a physicist is that the numbers of points over a finite field are also given by expression that involve the periods of a manifold. The number of points are encoded in the local zeta function, about which much is known in virtue of the Weil conjectures. I will discuss interesting topics related to the zeta function and the appearance of modularity for one parameter families of Calabi-Yau manifolds. A topic I will stress is that for these families there are values of the parameter for which the manifold becomes singular and for these values the zeta function degenerates and exhibits modular behaviour. I will report (on joint work with Philip Candelas, Mohamed Elmi and Duco van Straten) on an example for which the quartic numerator of the zeta function factorises into two quadrics at special values of the parameter which satisfy an algebraic equation with coefficients in Q (so independent of any particular prime), and for which the underlying manifold is smooth. We note that these factorisations are due to a splitting of the Hodge structure and that these special values of the parameter are rank two attractor points in the sense of type IIB supergravity. Modular groups and modular forms arise in relation to these attractor points. To our knowledge, the rank two attractor points that were found by the application of these number theoretic techniques, provide the first explicit examples of such points for Calabi-Yau manifolds of full SU(3) holonomy.

Borcherds Algebras and 2d String Theory

• Sarah Harrison (McGill U.)

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 talk about the construction of new BKMs superalgebras arising from self-dual vertex operator algebras of central charge 12, and comment on their relation to physical string theories in 2 dimensions. Based on work with N. Paquette, D. Persson, and R. Volpato.

Discussion session Hosted via Gather Town