Beijing Institute of Mathematical Sciences and Applications

Tetrahedron and 3D reflection equations are 3D generalizations of the Yang-Baxter and reflection equations in 2D quantum integrable systems. In this talk I explain how the quantum cluster algebras associated with quivers for the wiring diagrams of the Coxeter relations reproduce a number of existing solutions and lead to some new ones. (Joint work with <i>Rei INOUE</i> and <i>Yuji TERASHIMA</i>.)

I start with review of construction of integrable systems on the Poisson cluster varieties. A key point there is the Goncharov-Kenyon duality, and the self-dual subclass of these systems after deautonomization gives rise to $q$-difference Painlevé equations. However, to complete the known list of $q$-Painlevé systems, one has to extend this construction by cluster Hamiltonian reduction. I will also discuss briefly the relation of this construction with supersymmetric gauge theories.

Discrete Painlevé equations often appear in various guises in the theory of orthogonal polynomials and related probabilistic models. There is a classification of discrete Painlevé equations through a class of rational algebraic surfaces associated with affine root systems, so it is natural to ask how this matches with different weights on the orthogonal polynomials side.

This talk encloses the presentation of a linear system which explicitly involves the Planck constant $\hbar$. The compatibility condition of that linear system simultaneously yields quantum analogue of classical Painlevé-II equation and a quantum commutation relation between field variable $u(z)$ and variable $z$ as $zu-uz=-\frac{1}{2}\hbar u$, that commutation relation makes our system different from other matrix versions of Painlevé-II equation. This has been shown that under the classical limit as $\hbar\rightarrow0$ quantum linear system coincides to the Flaschka-Newell Lax Pair with its classical Painlevé-II equation.

By imposing Virasoro constraints to Drinfeld-Sokolov hierarchies, we obtain their solutions of Witten-Kontsevich and of Brezin-Gross-Witten types, and those characterized by certain ordinary differential equations of Painlevé type. We also show the existence of affine Weyl group actions on solutions of such Painlevé-type equations, which generalizes the theory of Noumi and Yamada on affine Weyl group symmetries of the Painlevé-type equations. This work is joint with <i>Si-Qi LIU</i> and <i>Youjin ZHANG</i>.

The problem of understanding the structure of the center of the small quantum group $u_q(g)$ associated to the Lie algebra $g$ and a root of unity $q$ is important in quantum and modular representation theory. The dimension of the center was known only in case $g=sl_2$ until 2016, when a remarkable link between the small quantum group for $g=sl_n$ and the space of the diagonal coinvariants of the symmetric group $S_n$ was discovered. We will consider the recent developments in this direction, relating the structure of the center of $u_q(g)$ with the geometry of the group $G$ and leading to a lower bound for the dimension of the center, which is conjecturally an equality in case $g=sl_n$. This is a joint work with <i>Qi YOU</i>, <i>Nicolas HEMELSOET</i> and <i>Oscar KIVINEN</i>.

I will discuss recent analyses of quantum group representations at arbitrary roots of $1$. In particular, I will explain how representations for the big quantum group are recovered as equivariant objects in the category of small quantum group representations, under an ``adjoint action" or the corresponding classical group. In this way one translates freely between the big and small settings, and also witnesses high levels of symmetry on the category of small quantum group representations. I will explain how our study is motivated by findings in mathematical physics, and in particular by analyses of topological twists for certain supersymmetric field theories.

The Muttalib--Borodin ensemble is a determinantal point process defined by the joint probability density function on $[0, +\infty)$ $$ \prod_{1 \leq i < j \leq n} (x_i - x_j)(x^{\theta}_i - x^{\theta}_j) \prod^n_{i = 1} x^{\alpha}_i e^{-nV(x_i)}. $$ It is proposed as a toy model for the quantum transport theory of disordered wires, and is also related to random matrix theory. It is a typical biorthogonal ensemble, and its limiting local statistics can be analyzed by biorthogonal polynomials. We note that its limit local distribution at the hard edge $0$ depends on $\theta$. In this talk, we show that the limit local distribution is universal for a general class of potential functions for rational $\theta$. Our method is the vector-valued Riemann-Hilbert problem. This is joint work with <i>Lun ZHANG</i>.

The interaction between fractional diffraction and parity-time ($\mathcal{PT}$) symmetry introduces unique properties to specific physical systems. In this study, we explore the impact of fractional diffraction on the formation of Bloch gap structures for the $\mathcal{PT}$ symmetric Hamiltonian. By combining linear periodic potentials with competing nonlinearities, we investigate a range of solitons, including fundamental gap, dipole, and vortical solitons. The stability of all soliton families is determined through linear stability analysis and direct simulations.

Consider the exterior algebra of the first fundamental representation of the symplectic group $Sp(2n,\mathbb{C})$ and raise to the $k$-th tensor power. This space admits also an action of the symplectic group $Sp(2k,\mathbb{C})$, moreover the actions of these two symplectic groups commute. Therefore the space is decomposed multiplicity-free into the sum of tensor products of irreducible $Sp(2n,\mathbb{C})$- and $Sp(2k,\mathbb{C})$-representations. This is called skew Howe duality. Such a decomposition can be used to introduce a probability measure on Young diagrams that parameterize irreducible representations as the ratio of the dimension of the irreducible component to the dimension of the whole space. In this talk we will discuss the limiting behavior of this measure as $n,k\to\infty$ with a fixed ratio $k/n$. Although the limit shape of the diagram is related to the limit shape of Young diagrams for the skew $GL(2n),GL(2k)$-duality, the fluctuations of the first column are different.

First, I will recall the definitions of symmetric and non-symmetric Macdonald polynomials as orthogonal polynomials depending on two parameters $q,t$. Second, I substitute $t=0$ or $t=\infty$ and explain the relationship with representations of current Lie algebras and Iwahori subalgebras. Third, I will show how homological properties of these categories imply the analogs of the Peter-Weyl theorem for Iwahori subgroup and the new ``reciprocal" Macdonald-type identities. The talk is based on the joint paper with <a href='https://arxiv.org/abs/2307.02124' target='_blank'>arXiv:2307.02124</a> with <i>E.~FEIGIN</i>, <i>e.~MAKEDONSKYI</i> and <i>D.~ORR</i>.

This talk gives an introduction to the Stokes phenomenon of an irregular Knizhnik--Zamolodchikov at a second order pole, associated to a representation $L(\lambda)$ of $gl_n$. It then shows that the Stokes matrices of the equation define representation of $U_q(gl_n)$ on $L(\lambda)$, while the WKB approximation of the Stokes matrices give rise to $gl_n$-crystal structures on $L(\lambda)$.

Demazure modules are an important family of representations for the Borel group of a simple Lie group $B$. Here, we will focus on $B$ being the invertible lower triangular matrices. Demazure modules can be considered as partial irreducible representations of $GL_n$, and they have many constructions. Their characters are also known as key polynomials, with many connections and generalizations such as those given by colored lattice models. In this talk, we will be looking at an analogous set of polynomials called the shifted key polynomials that were recently introduced and motivated by Schubert calculus and the orbits of the orthogonal and symplectic groups in the flag variety. We will provide a few different descriptions of these polynomials, what we know about them, and a number of conjectures/open problems. In particular, we will give an interpretation of the shifted key polynomials as characters of certain subcrystals of highest weight crystals of the queer Lie superalgebra. This is based on joint work with <i>Eric MARBERG</i>; <a href='https://arxiv.org/abs/2302.04226' target='_blank'>arXiv:2302.04226</a>, <a href='https://arxiv.org/abs/2306.00336' target='_blank'>arxiv: 2306.00336</a>.

In 1978 Lepowsky and Wilson provided a construction of the affine Lie algebra $A^{(1)}_{1}$. They introduced an "exponential generating function'' (known as vertex operator) to span the algebra. Later, their vertex operator was extended by Date, Kashiwara and Miwa to generate tau function of the classical soliton solutions of the KdV equation. Apart from the classical solitons that are composed by usual exponential type plane wave factors, there exist "elliptic solitons'' composed by the Lamé-type function as the plane wave factors. Recently, we found vertex operators to generate tau functions for such elliptic solitons. In this talk, I will introduce the related tau functions, vertex operators, bilinear identities and reductions. However, the Lie algebras related to such vertex operators are not yet known, which is left open. The talk is mainly based on the paper <a href='https://arxiv.org/abs/2204.01240' target='_blank'>arxiv: 2204.01240</a>, collaborated with <i>Xing LI</i>.

In this talk the notion of boundary consistency which defines integrable boundary conditions for quad graph systems will be provided. We show how to solve such consistency which leads to classification of discrete integrable boundary conditions. Connections to discrete holomorphic functions, to initial boundary value problems on quad graph and to set theoretical reflection equations will also be discussed.

We propose a modification of the classical Schlesinger system where some of the independent variables become functions of the other variables. This construction is motivated by considering a special variation of a hyperelliptic curve related to generalized Chebyshev polynomials. We also construct an algebro-geometric solution to the constrained Schlesinger system in terms of such a family of hyperelliptic curves. This is a joint work with <i>Vladimir DRAGOVIC</i> (UTD).

It is well known that tau functions of KP hierarchy are parameterized by points in Sato's Universal Grassmannian manifold (UGM). These tau functions have Schur expansions with coefficients satisfying Pl\"ucker relations. In this talk we will show that all tau functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. These tau functions are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). They have natural Schur-Q expansions with coefficients satisfying Cartan-Pl\"ucker relations. As a byproduct this parameterization gives a constructive description for complex pure spinors du E. Cartan. As an application, we reprove a theorem due to A.~Alexandrov which states that tau functions of KdV satisfy BKP up to rescaling of the time parameters by 2. We prove this by showing that the KdV hierarchy can be viewed as 4-reduction of the BKP hierarchy. This interpretation gives complete characterization for the KdV orbits inside the BKP hierarchy. This talk is based on preprint <a href='https://arxiv.org/abs/2210.03307' target='_blank'>arXiv:2210.03307</a>.

In this talk, we investigate the relations between the matrix-valued Cauchy bi-orthogonal polynomials and the integrable systems. We give a formal definition for matrix-valued Cauchy bi-orthogonal polynomials and their quasi-determinant expressions. Then we derive a four-term recurrence relation for the matrix-valued Cauchy bi-orthogonal polynomials and show the coefficients in the recurrence relation could also be written in terms of quasi-determinants. Moreover, we introduce proper time flows and hence derive a novel noncommutative integrable lattice. We provide a Lax pair to this novel noncommutative integrable lattice and verify it through a direct method by quasi-determinant identities.

Hurwitz numbers are essentially just the numbers of transitive factorizations of a given element of the symmetric group into a product of a given number of transpositions. Such factorizations correspond to isomorphism classes of complex covers of a Riemann sphere of a given degree and genus, so no wonder that Hurwitz numbers have to be related to the geometry of moduli spaces of complex curves.

In my talk I will discuss an integrable hierarchy of nonlinear differential difference equations which was suggested in the work of A.~Zabrodin and I.~Krichever \\ \href{https://arxiv.org/abs/2210.12534}{arxiv: 2210.12534}. It is a sub-hierarchy of the 2D Toda lattice defined by imposing a constraint to the Lax operators of the latter. This constraint is similar to the one imposed on KP hierarchy to determine BKP sub-hierarchy. Authors named it a Toda lattice with constraint of type B. There are several known solutions for this hierarchy: algebra-geometric (<a href='https://arxiv.org/abs/2210.12534' target='_blank'>arxiv: 2210.12534</a>), (general) elliptic (<a href='https://arxiv.org/abs/2302.12085' target='_blank'>arxiv: 2302.12085</a>) and soliton ones (<a href='https://arxiv.org/abs/2303.17467' target='_blank'>arxiv: 2303.17467</a>). I will focus on the tau function of this hierarchy and bilinear equations for it (<a href='https://arxiv.org/abs/2303.17467' target='_blank'>arxiv: 2303.17467</a>). There are two types of equations which encode the whole hierarchy: the first one is Hirota type integral equation and the second one is a Hirota-Miwa type discrete equation, which is equivalent to the fully discrete BKP equation first appeared in the work of Miwa (<a href='https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-58/issue-1/On-Hirotas-difference-equations/10.3792/pjaa.58.9.full target='_blank'>10.3792/pjaa.58.9</a>).

We consider integrable models with $o_{2n+1}$ symmetry. Within the framework of Algebraic Bethe Ansatz, we construct their Bethe vectors and compute their scalar products. The calculation is done using also the current presentation of Yangians.

Enumeration of quarter-plane lattice walks is a classical combinatorial problem. In algebraic combinatorics, an efficient approach is the kernel method and the solution is written in a form with $[x^>]$ operators. It is straight forward to obtain the number of configurations from this solution form. Another approach is through Carleman type Riemann boundary value problem. The solution is in an integral form with a conformal gluing function. In this talk, I will show the relation between these two approaches and propose a combinatorial insight for the conformal gluing function in RBVP approach.

First I will recall Verlinde-Verlinde's work on pure gravity that the main contribution to the amplitude comes from contact interactions. Moveover, they show that the corresponding contact term algebra is isomorphic to the (half of) Virasoro algebra. We will consider such a contact algebra in the theory of chiral deformations of CFT on elliptic curves, and derive the relations between iterated surface integrals and iterated $A$-cycle intergrals, which recover Dijkgraaf's master equation in a rigorously mathematical way. Inside we use various integrability conditions: Dijkgraaf's integrable condition, $S_n$-symmetry of ($n$-point) conformal blocks, Fubini's Theorem of regularized integrals. This talk is based on the joint work with <i>Zhengping GUI</i> and <i>Si LI</i>.

Quantum integrable systems enjoy the rich physical meanings and beautiful mathematical structures, thus have important applications in condensed matter, theoretical and mathematical physics. In this talk, I will introduce the quantum integrable systems without $U(1)$-symmetry and the off-diagonal Bethe Ansatz method. I will focus on the eigenvalues, eigenstates, exact physical quantities in the thermodynamic limit and $t-W$ scheme.