Invited talks

Tomoyuki Arakawa (RIMS, Japan)

TITLE: 4D/2D duality and Moore-Tachikawa symplectic varieties

ABSTRACT: Recently, Beem et al. have discovered a 4d/2d duality which associates a VOA to any 4d $\mathcal N=2$ SCFT. We give a functorial construction of the chiral algebras of class $\mathcal S$, that is, the VOAs corresponding to the 4d $\mathcal N=2$ SCFTs called the theory of class $\mathcal S$.

Moreover, we show that those VOAs recover the Moore-Tachikawa symplectic varieties that have been constructed by Braverman, Finkelberg and Nakajima using the geometry of the affine Grassmannian.


Ivan Cherednik (ETH-ITS, Switzerland & UNC Chapel Hill, USA)

TITLE: Refined Rogers-Ramanujan sums and DAHA invariants of Hopf links

ABSTRACT: The product formulas $P_aP_b=\sum_c C_{ab}^cP_c$ for "remarkable" families of orthogonal polynomials $\{P_a\}$ have many applications. The $3j$-symbols, $2d$ TQFT, instanton sums, and counting holomorphic maps from bordered Riemann surfaces to "reasonable" CY $3$-folds with boundary in $3$ specific Lagrangian submanifolds (open GW invariants) are well-known examples. We will discuss such formulas for the family $\{P_b\theta^{\ell}\}$ for the Macdonald polynomials and the corresponding $\theta$-functions. The matrix entries of the (operators of) multiplication by $\theta^{\ell}$ in the basis $\{P_b\}$ are the key. They are sums of the DAHA-Jones polynomials of chains of Hopf $2$-links with some Rogers-Ramanujan-type weights (Ch, Danilenko). For $\mathfrak{gl}_n$, they are $a$-stable, i.e. depend on $q,t,a$, where the substitution $a=-t^n$ is the passage to $\mathfrak{gl}_n$, which gives sums in terms of the DAHA superpolynomials. When $t\to 0$, we arrive at Rogers-Ramanujan-type presentations for certain string functions of level $\ell$ (Ch, B.Feigin). Importantly, the level-rank duality for $\widehat{\mathfrak{gl}}_n$ can be seen directly from these sums. The theory is based on DAHA, but actually we will not need these algebras to state/explain the main results.


Alexander Givental (UC Berkeley, USA)

TITLE: GW-invariants: from ordinary to extra-ordinary.

ABSTRACT: I will discuss the question whether (or in what sense, or to what extent) Gromov-Witten invariants can be generalized from cohomological to cobordism-valued.


Sabir Gusein-Zade (Moscow State University, Russia)

TITLE: On a version of the Berglund-Hübsch-Henningson duality with non-abelian symmetry groups

ABSTRACT: P.Berglund, T.Hübsch and M.Henningson found a method to construct mirror symmetric Calabi-Yau manifolds using so-called invertible polynomials: see details below. They considered pairs $(f,G)$ consisting of an invertible polynomial f and a (finite abelian) group $G$ of diagonal symmetries of $f$. To a pair $(f,G)$ one associates the Berglund-Hübsch-Henningson (BHH) dual pair $(f^\vee,G^\vee)$. There were found some symmetries between dual invertible polynomials and dual pairs not related directly with the mirror symmetry. (E.g., some of them hold when the corresponding manifolds are not Calabi-Yau. One of them was the so-called equivariant Saito duality as a duality between Burnside rings. A.Takahashi suggested a conjectural method to find symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. The equivariant Saito duality was generalized to a case of non-abelian groups. It turns out that the corresponding symmetry holds only under a special condition on the action of the subgroup of the permutation group called here PC ("parity condition''). An inspection of data on Calabi-Yau threefolds obtained from quotients by non-abelian groups (taken from tables computed by X.Yu) shows that the pairs found on the basis of the method of Takahashi have symmetric pairs of Hodge numbers (and thus are hopefully mirror symmetric) if and only if they satisfy PC.

The talk is based on a joint work with W.Ebeling.


Askold Khovanskii (University of Toronto, Canada)

TITLE: GOOD COMPACTIFICATION THEOREM AND THE RING OF CONDITIONS OF $(\mathbb C^*)^n$

ABSTRACT: The ring of conditions of $(\mathbb C^*)^n$ is an intersection theory for algebraic cycles in $(\mathbb C^*)^n$. It was introduced by De Concini and Procesi. The famous Bernshtein--Koushnirenko theorem fits nicely into this theory. The ring of conditions can be reduced to cohomology rings of smooth toric varieties via the good compactification theorem.

According to the good compactification theorem for any algebraic variety $X\subset (\mathbb C^*)^n$ there is a complete toric variety $M\supset (\mathbb C^*)^n$ such that the closure of $X$ in $M$ does not intersect orbits in $M$ of codimension bigger than $\dim_\mathbb C X$. All proofs of this theorem I met in literature are rather involved.

I will present a new version of the good compactification theorem which is stronger than the usual one. Its proof is very transparent. It uses Newton polyhedra and Puiseux power series in several variables.

If time permitted I will discuss two geometric descriptions of the ring of conditions. Tropical geometry provides the first description. The second one can be formulated in terms of the volume function on the cone of convex polyhedra in $\mathbb R^n$.


Igor Krichever (Columbia University, USA and Skoltech, Russia)

TITLE: The Bethe ansatz equations and integrable system of particles.

ABSTRACT: In the talk based on a joint work with A.Varchenko a new approach for the construction of solutions of the Bethe ansatz equations of the $\widehat{\frak{sl}}_N$ XXX quantum integrable model, associated with the trivial representation of $\widehat{\frak{sl}}_N$ will be presented.

It is based on interplay with the theory of coherent rational Ruijesenaars-Schneider systems. For that we develop in full generality the spectral transform for the rational Ruijesenaars-Schneider system.


Maxim Nazarov (York University, UK)

TITLE: Representations of Yangians via Howe duality

ABSTRACT: The aim of this talk is to explain how a discovery made by Tarasov and Varchenko resulted in solving a long standing problem in Invariant Theory. Back in 2002 they established a correspondence between the "extremal cocycle" on the Weyl group of $\mathfrak{gl}_m$ introduced by Zhelobenko, and the intertwining operators of tensor products of fundamental modules over the Yangian $\operatorname{Y}(\mathfrak{gl}_n)$. This correspondence is based on the duality between the two general linear Lie algebras $\mathfrak{gl}_m$ and $\mathfrak{gl}_n\,$. Khoroshkin and I extended this correspondence to other classical dual pairs. Instead of $\mathfrak{gl}_m$ we took the symplectic Lie algebra $\mathfrak{sp}_{2m}$ and the orthogonal Lie algebra $\mathfrak{so}_{2m}\,$. Instead of $\operatorname{Y}(\mathfrak{gl}_n)$ we employed the twisted Yangians of $\mathfrak{so}_n$ and $\mathfrak{sp}_n\,$. These twisted Yangians are coideal subalgebras in the Hopf algebra $\operatorname{Y}(\mathfrak{gl}_n)\,$. We used our correspondence to give explicit realisations of irreducible finite dimensional modules over the twisted Yangians, as images of intertwining operators. For $\operatorname{Y}(\mathfrak{gl}_n)$ such realisations were given in 1997 by Akasaka and Kashiwara proving a conjecture of Cherednik of 1987.

Working with the twisted Yangians instead of $\operatorname{Y}(\mathfrak{gl}_n)$ led us to a new general result. Together with Vinberg, for any complex reductive Lie algebra $\mathfrak{g}$ and any locally finite $\mathfrak{g}\,$-module $V$, we extended to the tensor product $\operatorname{U}(\mathfrak{g})\otimes V$ the Harish-Chandra description of $\mathfrak{g}$-invariants in the universal enveloping algebra $\operatorname{U}(\mathfrak{g})$.Instead of the shifted action of the Weyl group of $\mathfrak{g}$ on $\operatorname{U}(\mathfrak{h})$, we used the Zhelobenko operators on the space of coinvariants of $\operatorname{U}(\mathfrak{g})\otimes V$ relative to the left action of $\mathfrak{n}$and the right action of $\mathfrak{n}'$. Here $\mathfrak{n}\oplus\mathfrak{h}\oplus\mathfrak{n}'$ is a triangular decomposition of $\mathfrak{g}\,$. We also extended to $\operatorname{S}(\mathfrak{g})\otimes V$ the Chevalley restriction theorem which describes $\mathfrak{g}$-invariants in the symmetric algebra $\operatorname{S}(\mathfrak{g})$. When $\mathfrak{g}=\mathfrak{sp}_{2m}$ or $\mathfrak{g}=\mathfrak{so}_{2m}$ our general result implies the irreduciblity of images of intertwining operators for the twisted Yangians. When $\mathfrak{g}=\mathfrak{gl}_m$ it gives a new proof of the Cherednik conjecture for the Yangian $\operatorname{Y}(\mathfrak{gl}_n)$. When $\mathfrak{g}$ is arbitrary, it solves a problem open since the work of Chevalley of 1955.


Andrei Okounkov (Columbia University, USA)

TITLE: Monodromy and derived equivalences.

ABSTRACT: This is a report on a joint work with R. Bezrukavnikov linking monodromy of quantum differential equations with the quantizations of symplectic resolutions in large prime characteristic.


Eric Rains (Caltech, USA)

TITLE: Noncommutative geometry and special functions

ABSTRACT: Many interesting classes of special functions either satisfy a nice (linear) differential/difference equation (e.g., hypergeometric functions), or parameterize a family of such equations (Painlevé, Garnier, etc.). Thus in general, one would like to understand the family of equations with specified singularities, either to know when it is rigid, or to understand natural "isomonodromy" maps between two such moduli spaces. I'll describe how to translate these questions to questions about sheaves on noncommutative surfaces, and explain some of the consequences for special functions.


Nikolai Reshetikhin (UC Berkeley, USA)

TITLE: Superintegrable systems of Calogero-Moser type on moduli spaces of flat connections

ABSTRACT: We present a new class of superintegrable systems on moduli spaces of flat connections on principal $G$-bundles over surfaces. Poisson commuting Hamiltonians for such systems are $G$-invariant functions of holonomies along any systems of non-intersecting, non-selfintersecting curves o a surface. Equations of motion for such systems can be solved by a projection method, similarly to Calogero-Moser systems and theory generalizations. Relativistic spin Calogero-Moser-Sutherland systems and corresponding spin Ruijenaars-Schneider systems appear when the surface is a torus with a puncture. At the end we will discuss the universal counterpart of such systems on the space of chord diagrams. This is a joint work with S. Artamonov.


Richard Rimanyi (UNC at Chapel Hill, USA)

TITLE: Schubert calculus in equivariant elliptic cohomology

ABSTRACT: Assigning characteristic classes to singular varieties is an effective way of studying the enumerative properties of the singularities. Initially one wants to consider the so-called fundamental class in H, K, or Ell, but it turns out that in Ell such class is not well defined. However, a deformation of the notion of fundamental class (under the name of Chern-Schwartz-MacPherson class in H, motivic Chern class in K) extends to Ell, due to Borisov-Libgober. To make sense of the Borisov-Libgober class for a wider class of singularities we introduce a version of it, which now necessarily depends on new (`dynamical’) variables. We obtain that this elliptic class of Schubert varieties satisfies two different recursions (Bott-Samelson, and R-matrix recursions). The second one relates elliptic Schubert calculus with Felder-Tarasov-Varchenko weight functions, and Aganagic-Okounkov stable envelopes. The duality between the two recursions is an incarnation of 3d mirror symmetry (symplectic duality). Joint work with A. Weber.


Vadim Schechtman (Paul Sabatier University, France)

TITLE: Stochastic matrices, polyhedra, and configuration spaces

ABSTRACT: I will discuss some combinatorial constructions and conjectures having their origin in the classification of perverse sheaves over configuration spaces. Joint work in progress with Mikhail Kapranov.


Vera Serganova (UC Berkeley, USA)

TITLE: The Jacobson-Morozov theorem for Lie superalgebras and semisimplification functors.

ABSTRACT: Famous Jacobson-Morozov theorem claims that every nilpotent element of a semisimple Lie algebra g can be embedded into $\mathfrak{sl}(2)$-triple inside $\mathfrak g$. Let $\mathfrak g$ be a Lie superalgebra with reductive even part and $x$ be an odd element of $\mathfrak g$ with non-zero nilpotent $[x,x]$. We give necessary and sufficient condition when $x$ can be embedded in $\mathfrak{osp}(1|2)$ inside g. The proof follows the approach of Etingof and Ostrik and involves semisimplification functor for tensor categories. Furthermore, for every odd $x$ in $\mathfrak g$ such that $[x,x]$ is either semisimple or nilpotent we construct a symmetric monoidal functor between categories of representations of certain superalgebras. We discuss some properties of these functors and applications of them to representation theory of superalgebras with reductive even part.

(Joint work in progress with Inna Entova-Aizenbud).


Vitaly Tarasov (IUPUI, USA)

TITLE: Norms of Bethe vectors via the Wronski map

ABSTRACT: One of the central statements of the Bethe ansatz in integrable models is the Gaudin-Korepin formula for the norm of properly normalized Bethe vectors as the determinant of the Jacobi matrix of the standard Bethe ansatz equations. This determinant is also the (discrete) Hessian of the corresponding master function. Another form of the Bethe ansatz equations is given via the (discrete) Wronski map. I will present a formula for the norm of the Bethe vectors through the differential of the (discrete) Wronski map. Together with the Gaudin-Korepin formula, it provides a relation between the Hessian of the master function and the differential of the Wronski map.


Alexander Varchenko (UNC at Chapel Hill, USA)

TITLE: Equivariant quantum differential equation, Stokes bases, and K-theory of a projective space

ABSTRACT: I will discuss the equivariant quantum differential equation of the projective space $\mathbb P^{n-1}$. The differential equation has two singular points: a regular singular point at $0$ and an irregular singular point at $\infty$. I will describe the monodromy properties of the differential equation in terms of the equivariant K-theory algebra of $\mathbb P^{n-1}$, in particular I will relate the Stokes bases and matrices at infinity with suitable exceptional collections in the K-theory algebra.


Alexander Veselov (Loughborough University, UK and Moscow State University, Russia)

TITLE: On integrability, geometrization and knots

ABSTRACT: After Arnold the classical integrability is usually understood in the Liouville sense as the existence of sufficiently many Poisson commuting integrals. About 20 years ago it was discovered that this does not exclude the chaotic behaviour of the system, which may even have positive topological entropy.

I will review the situation with Liouville integrability in relation with Thurston’s geometrization programme.

Using as the main example the geodesic flows on the 3-folds with $\mathrm{SL}(2,\mathbb R)$-geometry, I will show that the corresponding phase space contains two open regions with integrable and chaotic behaviour respectively.

A particular case of such 3-folds the modular quotient $\mathrm{SL}(2,\mathbb R)/\mathrm{SL}(2,\mathbb Z)$, which is known, after Quillen, to be equivalent to the complement in 3-sphere of the trefoil knot.

I will explain that the remarkable results of Ghys about modular and Lorenz knots can be naturally extended to the integrable region, where these knots are replaced by the cable knots of trefoil.

The talk is partly based on a recent joint work with Alexey Bolsinov and Yiru Ye.

Contributed talks

Ana Agore (Vrije Universiteit, Belgium & Institute of Mathematics of the Romanian Academy, Romania)

TITLE: Equivalences of (co)module algebra structures over Hopf algebras

ABSTRACT: Module and comodule algebras over Hopf algebras appear in many areas of mathematics and physics. An important class of examples arises from (affine) algebraic geometry: If $G$ is an affine algebraic group $G$ acting morphically on an affine algebraic variety \(X\), then the algebra $A$ of regular functions on $X$ is an $H$-comodule algebra where $H$ is the algebra of regular functions on $G$. At the same time $A$ is an $\mathrm U(\mathfrak g)$-module algebra where $\mathrm U(\mathfrak g)$ is the universal enveloping algebra of the Lie algebra $\mathfrak g$ of the algebraic group $G$. Taking this into account, one may view a (not necessarily commutative) (co)module algebra as an action of a quantum group on a non-commutative space.

The above point of view was also considered by Y.I. Manin, who proved the existence of a universal coacting Hopf algebra $\mathrm{aut}(A)$ on an algebra $A$, which plays the role of a symmetry group in non-commutative geometry. In order to classify all (co)module algebra structures on a given algebra $A$, one therefore should understand the Manin-Hopf algebra $\mathrm{aut}(A)$, as well as its quotients. However, finding an explicit description of $\mathrm{aut}(A)$ seems to be a very wild problem. Therefore, we propose a refinement of Manin's construction by studying comodule algebras up to support equivalence, which generalizes in a natural way (weak) equivalence of gradings. We show that for each equivalence class of comodule algebra structures on a given algebra $A$, there exists a unique universal Hopf algebra $H$ together with an $H$-comodule structure on $A$, which factors any other equivalent comodule algebra structure on $A$. We study support equivalence and these universal Hopf algebras for group-gradings, Hopf-Galois extensions, algebraic groups and cocommutative Hopf algebras. We argue how the notion of support equivalence can be used to reduce the classification problem of Hopf algebra (co)actions.

Based on a joint work with Alexey Gordienko and Joost Vercruysse (arXiv:1812.04563).


Guilherme Almeida (SISSA, Italy)

TITLE: Differential Geometry of Orbit space of Extended Jacobi Group $A_n$

ABSTRACT: We define certain extensions of Jacobi groups of $A_n$, prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces.


Sigiswald Barbier (Ghent University, Belgium)

TITLE: The blocks of the periplectic Brauer algebra.

ABSTRACT: Certain algebras can be visualised using diagrams. Examples include the symmetric group algebra, the Temperley-Lieb algebra and the Brauer algebra. In this talk I will introduce the periplectic Brauer algebra $A_n$ using diagrams and determine its blocks in arbitrary characteristic (different from two). We find the surprising result that there is only one block if $n$ is not too small compared to the positive characteristic.

The motivation to study this periplectic Brauer algebra is representation theory of the periplectic Lie superalgebra, which is related to it via a Schur-Weyl type duality.


Luan Bezerra (IUPUI, US)

TITLE: Quantum toroidal superalgebras associated with $\mathfrak{gl}_{m|n}$.

ABSTRACT: We introduce the quantum toroidal superalgebras $E_{m|n}(q_1,q_2,q_3)$, which are affinizations of quantum affine algebras $U_q \widehat{\mathfrak{gl}}_{m|n}$. We give an evaluation map from $E_{m|n}(q_1,q_2,q_3)$ to the quantum affine algebra $U_q \widehat{\mathfrak{gl}}_{m|n}$ with $q^2=q_2$ at level $c=q_3^{(m-n)/2}$, and a bosonic realization of level one $E_{m|n}(q_1,q_2,q_3)$-modules.


Rekha Biswal (Max Planck Insititute, Germany)

TITLE: Demazure multiplicities of excellent filtrations in higher rank and cone theta functions.

ABSTRACT: In this talk, I will present recent ongoing joint work with Vyjayanthi Chari, Peri Shereen and Jeffrey wand concerning the multiplicities of level two Demazure modules in the excellent filtration of local Weyl modules.

As an application of that we will express the character of level two Demazure modules as alternating sum of local Weyl modules, in particular as an alternating sum of non-symmetric Macdonald polynomials specialized at $t=0$. Moreover, the generating functions of Demazure multiplicities will be connected to cone theta functions.


Ilija Buric (DESY, Germany)

TITLE: Superconformal blocks and Calogero-Moser systems

ABSTRACT: Conformal blocks are an important ingredient of a conformal field theory which allow for a decomposition of correlation functions. For theories in more than two dimensions, the blocks have been the subject of many investigations, starting with the work of Dolan and Osborn, who determined the four-point blocks for scalar operators in an even dimensional space. Less progress has been made in theories with supersymmetry. In this talk I will present a harmonic analysis approach to conformal partial waves, which casts them in the form of eigenfunctions of a Schroedinger problem of Calogero-Moser type. In particular, the blocks for a large class of superconformal theories will be constructed as eigenfunctions of a spin Calogero-Moser Hamiltonian perturbed by a nilpotent potential term. The talk is based on the joint work with Volker Schomerus and Evgeny Sobko.


Marijana Butorac (University of Rijeka, Croatia)

TITLE: Quasi-particle bases of principal subspaces of representations of affine Lie algebras

ABSTRACT: Quasi-particle bases of the principal subspaces of standard modules of affine Lie algebras, introduced by B. Feigin and A. Stoyanovsky, provide an interpretation of the sum side of various Rogers-Ramanujan type identities. In this talk we describe the construction of combinatorial bases of principal subspaces of standard module of highest weight $k\Lambda_0$ and of the universal vacuum principal vertex algebra $N(k\Lambda_0)$, for all positive integral levels $k$ in the case of affine Lie algebras of type $D$, $E$ and $F$. From combinatorial bases, we obtain characters of principal subspaces and some new combinatorial identities.

This talk is based on a joint work with Slaven Kožić.


Tanmay Deshpande (TIFR, Mumbai, India)

TITLE : A Verlinde formula for twisted conformal blocks.

ABSTRACT: Let $G$ be a finite group. I will begin by describing a categorical Verlinde formula for $G$-crossed modular categories, which computes the fusion coefficients in such a category in terms of certain unitary matrices known as crossed $S$-matrices. I will then apply it to the setting of a twisted affine Lie algebra to compute the dimensions of the associated twisted conformal blocks in terms of crossed $S$-matrices, generalizing the classical Verlinde formula to the twisted setting.

This is ongoing joint work with S. Mukhopadhyay.


Laszlo Feher (University of Szeged, Hungary)

TITLE: Reduction of a bi-Hamiltonian hierarchy on $T^*U(n)$ to spin Ruijsenaars--Sutherland models

ABSTRACT: We first exhibit two compatible Poisson structures on the cotangent bundle of the unitary group $U(n)$ in such a way that the invariant functions of the $\mathfrak u(n)^*$-valued momenta generate a bi-Hamiltonian hierarchy. One of the Poisson structures is the canonical one and the other one arises from embedding the Heisenberg double of the Poisson--Lie group $U(n)$ into $T^*U(n)$, and subsequently extending the embedded Poisson structure to the full cotangent bundle. We then apply Poisson reduction to the bi-Hamiltonian hierarchy on $T^*U(n)$ using the conjugation action of $T^*U(n)$, for which the ring of invariant functions is closed under both Poisson brackets. We demonstrate that the reduced hierarchy belongs to the overlap of well-known trigonometric spin Sutherland and spin Ruijsenaars--Schneider type integrable many-body models, which receive a bi-Hamiltonian interpretation via our treatment.


Nikolay Grantcharov (U Chicago, US)

TITLE: Extension quiver for Lie superalgebra $\mathfrak q(3)$

ABSTRACT: In this talk we will discuss the queer Lie superalgebra $\mathfrak q(n)$ and its representations. We will give a description of the blocks of the category of finite-dimensional $\mathfrak q(3)$ (super)modules by providing their Ext-quivers.


Chenliang Huang (IUPUI, US)

TITLE: The solutions of $\mathfrak{gl}(m|n)$ Bethe ansatz equation and rational pseudodifferential operators

ABSTRACT: We consider the $\mathfrak{gl}(m|n)$ Gaudin Bethe ansatz equation associated to a weight in a tensor product of polynomial modules. To a solution of $\mathfrak{gl}(m|n)$ Gaudin Bethe ansatz equation, we associate a rational pseudodifferential operator. The rational pseudodifferential operator is invariant under the reproduction procedure. We expect that the coefficients of the expansion of the rational pseudodifferential operator are eigenvalues of the higher Gaudin Hamiltonians acting on the corresponding Bethe vector.


Aleksei Ilin (Higher School of Economics, Russia)

TITLE: Bethe subalgebras in Yangians

ABSTRACT: We define the family of commutative Bethe subalgebras of the Yangian of any simple Lie algebra parameterized by the corresponding adjoint group $G$. We also extend the parameter space of Bethe subalgebras to the wonderful compactification of $G$. Then we study the properties of these subalgebras.

Based on arXiv:1810.07308.


Hitoshi Konno (Tokyo University of Marine Sciences and Technology, Japan)

TITLE: Elliptic Quantum Groups and Deformed W-algebras

ABSTRACT: It has been known from the very beginning that the Drinfeld realization $U_{q,p}(\widehat{\mathfrak g})$ of the (dynamical) elliptic quantum group has a deep relationship to deformation of the W-algebras of coset type. In this talk we discuss a relationship of the vertex operators of $U_{q,p}(\widehat{\mathfrak g})$ associated with the Drinfeld coproduct to the generating function of the deformed W-algebras including the elliptic quantum toroidal algebra cases.


Jian-Rong Li (University of Graz, Austria)

TITLE: Quantum affine algebras and Grassmannians

ABSTRACT: I will talk about our recent work (Joint with Wen Chang, Bing Duan, and Chris Fraser) on quantum affine algebras of type A and Grassmannian cluster algebras.

Let $\mathfrak g=\mathfrak{sl}_n$ and $U_q(\widehat{\mathfrak g})$ the corresponding quantum affine algebra. Hernandez and Leclerc proved that there is an isomorphism $\Phi$ from the Grothendieck ring $R_l\widehat{\mathfrak g}$ of a certain subcategory $C_l\widehat{\mathfrak g}$ of finite dimensional $U_q(\widehat{\mathfrak g})$-modules to a certain quotient $S_{n,n+l+1}$ of a Grassmannian cluster algebra (certain frozen variables are sent to 1). We proved that this isomorphism induced an isomorphism between the monoid of dominant monomials and the monoid of rectangular semi-standard Young tableaux. Using the isomorphism, we defined $\mathrm{ch}(T)$ in $S_{n,n+l+1}$ for every rectangular tableau $T$.

Using the isomorphism and the results of Kashiwara, Kim, Oh, and Park and the results of Qin, we proved that every cluster monomial (resp. cluster variable) in a Grassmannian cluster algebra is of the form $\mathrm{ch}(T)$ for some real (resp. prime real) rectangular semi-standard Young tableau $T$.

We translated a formula of Arakawa-Suzuki and Lapid-Minguez to the setting of $q$-characters and obtained an explicit $q$-character formula for a finite dimensional $U_q(\widehat{\mathfrak g})$-module. These formulas are useful in studying real modules, prime modules, and compatibility of two cluster variables.

We described the mutations of a Grassmannian cluster algebra using semi-standard Young tableaux and described the mutations of modules.


Andras Lorincz (MPI, Germany)

TITLE: Equivariant $\mathcal D$-modules on varieties with finitely many orbits

ABSTRACT: Let an algebraic group $G$ act on a variety $X$ with finitely many orbits. In this talk I will discuss some results on $G$-equivariant coherent $\mathcal D$-modules. The category of equivariant $\mathcal D$-modules is equivalent to the category of finite-dimensional representations of a quiver, which we describe in some special cases, when the quivers turn out to be of finite or tame representation type. We apply these results to local cohomology modules supported in orbit closures by describing their explicit $\mathcal D$-module and $G$-module structures.


Kang Lu (IUPUI, US)

TITLE: On the $\mathfrak{gl}(1|1)$ supersymmetric XXX spin chains

ABSTRACT: We study the $\mathfrak{gl}(1|1)$ supersymmetric XXX spin chains. We show that there exists a bijective correspondence between common eigenvectors (up to proportionality) of Bethe subalgebra in any cyclic tensor product of polynomial evaluation modules of the $\mathfrak{gl}(1|1)$ Yangian and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace has dimension 1. We also show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz and express the transfer matrices associated to symmetrizers and anti-symmetrizers in terms of the first transfer matrix and the center of the Yangian.


Zhipeng Lu (University of Gottingen, Germany)

TITLE: Flatness of the commutator map on $\mathrm{SL}_n$​

ABSTRACT: In this short talk, I will briefly introduce the flatness property of the commutator map on special linear groups, which roughly states that $[a,b]=aba^{-1}b^{-1}$ on $\mathrm{SL}_n(\mathbb C)$ has almost all its fibers having the same dimension $n^2-1$ for any $n\geq 2$, except for some scalar matrices. By a commonly used technique in algebraic geometry or model theory, it is equivalent to an analogous result on finite groups $\mathrm{SL}_n(\mathbb F_q)$. Then the approach mainly counts on J.A.Green's irreducible character formula of finite general linear groups and a character estimate by Liebeck et al. Time permits, I will talk about the meaning of the result in a topological setting.

This is a joint work with Professor Michael Larsen.


Akane Nakamura (Josai University, Japan)

TITLE: Recovering linear from nonlinear

ABSTRACT: This talk is based on joint work with Eric Rains. One of the important aspects of the integrable systems is that these nonlinear systems possess linear problems. However, it is not easy to find a linear problem (Lax equation) just by looking at the nonlinear equations.

In this talk, we will explain a way to recover a linear problem from the nonlinear autonomous 4-dimensional Painlevé-type systems (the Hitchin systems). Our way is to compare generic degenerations of the families of curves arising from the nonlinear problem (i.e., the boundary divisors adjoined in the compactification of the Liouville tori) and curves appearing in the linear side (the spectral curves). We have proved that the Jacobian of generic curve of these systems has unique principal polarization, so that we can recover curves.


Nikita Nikolaev (Univesity of Geneva, Switzerland)

TITLE: Abelianisation of $\mathrm{SL}(2)$-Connections and the Hitchin System

ABSTRACT: Recently, I began developing the mathematical theory of abelianisation for meromorphic connections (arXiv:1902.03384) analogous to abelianisation of Higgs bundle (aka spectral correspondence): it is correspondence between connections on vector bundles over a curve $X$ and connections on line bundles over a spectral cover $S\to X$. My current work in preparation is developing the same abelianisation procedure for lambda-connections (these are families of connections which degenerate to a Higgs field for $\lambda = 0$), where I show in particular that lambda-abelianisation recovers the spectral corresponds at $\lambda = 0$. I will speculate about what I think this means for the spaces of Stokes data and the Hitchin integrable system.


Veronica Pedic (University of Zagreb)

TITLE: On fusion rules for the Weyl vertex algebra modules

ABSTRACT: We describe fusion rules in the category of weight modules for the Weyl vertex algebra. This way we confirm the conjecture on fusion rules based on the Verlinde formula. We explicitly construct the intertwining operators appearing in the formula for fusion rules. We present a result which relates irreducible weight modules for the Weyl vertex algebra to the irreducible modules for the affine Lie superalgebra $\widehat{\mathfrak{gl}(1 \vert 1)}$.

This is joint work with Drazen Adamovic.


Shifra Reif (Bar-Ilan University, Israel)

TITLE: Grothendieck rings for periplectic Lie superalgebras

ABSTRACT: The Grothendieck ring of the category of finite dimensional representations over a simple Lie algebra can be described via the character map, as a ring of functions invariant under the action of the Weyl group. This result was generalized to basic Lie superalgebras by A. N. Sergeev and A. P. Veselov with additional invariance conditions.

In this talk, we will generalize the theorem of Sergeev and Veselov to periplectic Lie superalgebras and describe their Grothendieck rings. In particular, we show that for the periplectic Lie supergroup $P(n)$, the ring of supercharacters of finite dimensional representations is isomorphic to to the ring of symmetric Laurent polynomials in $x_1,\dots,x_n$ for which the evaluation $x_1=x_2^{-1}=t$ is independent of $t$.


Claudia Rella (University of Oxford, UK)

TITLE: Motivic Scattering Amplitudes

ABSTRACT: Recently developed approaches to scattering amplitudes in quantum field theory highlight underlying geometrical structures which allow to interpret Feynman amplitudes as periods of motives. Techniques in algebraic geometry are applied to the motivic version of Feynman integrals to investigate their geometric properties and to give information about their numerical value. I will present the main results of the application of motivic Galois theory in the Tannakian formalism to primitive log-divergent Feynman diagrams in $\phi^4$ theory.


Gabriele Rembado (ETH Zurich, Switzerland)

TITLE : Quantisation of isomonodromy connections

ABSTRACT : The Knizhnik--Zamolodchikov equations are the differential equations satisfied by the correlators in the 2-dimensional Wess--Zumino--Novikov--Witten model for Conformal Field Theory. Mathematically, they constitute a flat connection in a vector bundle, encoded by a nonautonomus integrable quantum system. Reshetikhin and Harnad have shown that the KZ Hamiltonians can be obtained from the quantisation of the isomonodromic deformations of meromorphic connections with simple poles on the Riemann sphere. In this talk we will discuss some extension of this procedure for connections with irregular singularities. This yields new integrable quantum systems including the Casimir connection of De Concini--Millson--Toledano Laredo and the dynamical connection of Felder--Markov--Tarasov--Varchenko.


Natasha Rozhkovskaya (Kansas State University, US).

TITLE: Vertex operators presentation of generalized Hall-Littlewood polynomials

ABSTRACT: Generalization of the vertex operators presentation of Hall-Littlewood polynomials leads to a new family of symmetric functions that depend on two sets of parameters $a=(a_1,a_2,\dots) $ and $b=(b_1, b_2,\dots)$. We describe the properties of these functions along with the orthogonality relations in the extended ring of symmetric functions $\Lambda[a, b]$ that lead to solutions of KP bilinear identities.

This is joint work with H. Naruse.


Libor Snobl (Czech Technichal University in Prague, Czech Republic)

TITLE: Superintegrability in the presence of magnetic fields

ABSTRACT: In my talk I shall review our recent results concerning classical superintegrability with magnetic fields. In particular, I shall focus on two concrete examples in three spatial dimensions:

1) 3-parameter family of maximally superintegrable systems with constant magnetic field, which are quadratically minimally superintegrable and which for rational values of one of its parameters kappa=m/n (where m,n are incommensurable) possess an additional integral of the order $m+n-1$; and

2) 6-parameter family of minimally superintegrable systems with the magnetic field of the form

\[B= b_m /r^3 .(x,y,z) + b_n/r^3 .( xz, yz, r^2+ z^2 )+(0,0,b_z),\]

where $r= \sqrt{x^2+y^2+z^2}$ which seems to possess closed bounded trajectories (based on numerical experiments), thus hinting at a hypothetical maximal superintegrability.

The talk is based on articles published in SIGMA 14 (2018) 092 and J. Phys. A: Math. Theor. 52 (2019) 195201 in collaboration with Antonella Marchesiello and Sebastien Bertrand.


Oleksandr Tsymbaliuk (Yale University, US)

TITLE: Integrable systems via shifted quantum groups

ABSTRACT: In the recent papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of 3d $\mathcal N=4$ quiver gauge theories was proposed, whose quantization is conjecturally described via the so-called shifted Yangians and shifted quantum affine algebras.

The goal of this talk is to explain how both of these shifted algebras provide a new insight towards integrable systems via the RLL realization. In particular, the study of Bethe subalgebras associated to the antidominantly shifted Yangians of $\mathfrak{sl}(n)$ provides an interesting plethora of integrable systems generalizing the famous Toda and DST systems. As another interesting application, the shifted quantum affine algebras in the simplest case of $\mathfrak{sl}(2)$ give rise to a new family of $3^{n-2}$ $q$-Toda systems of $\mathfrak{sl}(n)$, generalizing the well-known one due to Etingof and Sevostyanov.

Time permitted, I will also explain how one can generalize the latter construction to produce exactly $3^{\mathrm{rk}(\mathfrak g)-1}$ modified $q$-Toda systems for any semisimple Lie algebra $\mathfrak g$.

These talk is based on the joint works with M. Finkelberg, R. Gonin and a current project with R. Frassek, V. Pestun.


Filipp Uvarov (IUPUI, US)

TITLE: Duality for Bethe algebras acting on polynomials in anticommuting variables.

ABSTRACT: We consider the actions of the current Lie algebras $\mathfrak{gl}(n)[t]$ and $\mathfrak{gl}(k)[t]$ on the space of polynomials in $kn$ anticommuting variables. We show that the images of the Bethe algebras under these actions coincide. To prove the statement we use the Bethe ansatz description of the eigenvectors and eigenvalues of the actions of the Bethe algebras.


Charles Young (University of Hertfordshire, UK)

TTILE: Affine Gaudin models and hypergeometric integrals of motion

ABSTRACT: To any Kac-Moody algebra, one can associate a quantum Gaudin model. For algebras of finite type, the study of the Bethe ansatz has yielded deep results including isomorphisms between the commutative algebra of Hamiltonians (called the Gaudin or Bethe algebra) and algebras of functions on spaces of opers for the Langlands dual algebra. One would like analogous results for affine-type Gaudin models, since they are closely related (in various ways) to integrable quantum field theories. I will discuss opers in affine type, and show that the functions on spaces of affine opers take the form of certain hypergeometric-type integrals. That leads to a conjecture that there exists a hierarchy of higher affine Gaudin Hamiltonians, also given by such integrals. I will describe this conjecture, and give some supporting evidence coming from GKO-type coset constructions of the Virasoro and W$_3$ algebras.

This talk is based on joint work with Sylvain Lacroix and Benoit Vicedo, in the papers arXiv:1804.01480, arxiv:1804.06751 and work in progress.


Pablo Zadunaisky (University of Buenos Aires and Jacons University, Argentina)

TITLE: Gelfand-Tsetlin presentations and Schubert Calculus

ABSTRACT : I will present a connection between Gelfand-Tsetlin representation theory and Schubert calculus, developed in joint work with V. Futorny, D. Grantcharov and L. E. Ramírez. This connection allowed us to give very explicit, Gelfand-Tsetlin like realisations of representations of a wide number of algebras related to $\mathfrak{gl}(n,\mathbb C)$. If time allows I will mention some details regarding an extension of this work to the representation theory of the plactic algebra.