Title and Abstract

■ Murray Batchelor (ANU): Free parafermions I & II

Abstract: In 1989 Rodney Baxter discovered a simple Z(N) quantum spin chain with eigenvalue spectrum given by a direct product of N x N diagonal matrices. In 2014 Paul Fendley showed that this structure corresponds to previously elusive free parafermions. For N = 2 this Z(N) model reduces to the familiar quantum Ising chain in a transverse field described by free fermions. For N > 2 the model is non-Hermitian, which opens up some interesting questions. Why? Because non-Hermitian systems are expected to behave differently to Hermitian systems due to the fact that non-Hermitian hamiltonians describe the dynamics of physical systems that are not conservative. Specifically, Hermiticity guarantees that the energy spectrum is real and that time evolution is probability preserving. There are many examples of integrable (exactly solved) Hermitian hamiltonians, but integrable non-Hermitian models are relatively rare. [An important exception is the class of non-Hermitian models whose hamiltonians are PT symmetric, ensuring a real eigenspectrum.] In these talks I will discuss the background to the free parafermion Z(N) spin chain and describe some recent progress towards calculating physical properties.

■ Ryo Fujita (Kyoto): Geometric realization of Dynkin quiver type quantum affine Schur-Weyl duality

Abstract: Associated with a Dynkin quiver, Kang-Kashiwara-Kim constructed a bimodule over the quantum affine algebra and the quiver Hecke (KLR) algebra of the corresponding ADE type as a generalized version of the quantum affine Schur-Weyl duality. They proved that the functor induced from the bimodule gives an isomorphism between Grothendieck rings of certain monoidal categories of finite-dimensional modules. In this talk, we realize this bimodule via the equivariant K-theory of Nakajima's graded quiver varieties. This is a Dynkin quiver analogue of Ginzburg-Reshetikhin-Vasserot's geometric realization of the usual quantum affine Schur-Weyl duality. As a result, we can prove that Kang-Kashiwara-Kim's functor actually gives an equivalence of monoidal categories.

■ Koichi Harada (U of Tokyo): Plane partition realization of (web of) W-algebra minimal models

Abstract: Recently, Gaiotto and Rapcak (GR) proposed a new family of the vertex operator algebra (VOA) as the symmetry appearing at an intersection of five-branes to which they refer as Y algebra. Prochazka and Rapcak, then proposed to interpret Y algebra as a truncation of affine Yangian whose module is directly connected to plane partitions (PP). They also developed GR's idea to generate a new VOA by connecting plane partitions through an infinite leg shared by them and referred to it as the web of W-algebra (WoW). We demonstrate that double truncation of PP gives the minimal models of such VOAs. For a single PP, it generates all the minimal model irreducible representations of W-algebra. We find that the rule connecting two PPs is more involved than those in the literature when the U(1) charge connecting two PPs is negative. For the simplest nontrivial WoW, N=2 superconformal algebra, we demonstrate that the improved rule precisely reproduces the known character of the minimal models.

■ Shinsuke Iwao (Tokai): K-Peterson isomorphism and relativistic Toda lattice

Abstract: The Peterson isomorphism is a ring isomorphism between the (equivariant) homology of the affine Grassmanninan and the quantum cohomology of the flag variety. This isomorphism has been discovered by Peterson and proved by Lam-Shimozono. The main topic of this talk is its K-theoretic analog, where the K-homology ring of the affine Grassmannian and the quantum K-theory of the flag variety are related with each other. In this talk, I will introduce some explicit construction of the `K-theoretic Peterson isomorphism' by using techniques in the field of classical integrable systems. In fact, the construction relies on Kostant's method for calculating algebraic solutions for the Ruijsenaars’ relativistic Toda lattice. This is a joint work with Takeshi Ikeda and Toshiaki Maeno.

■ Hitoshi Murakami (Tohoku): Volume conjecture for knots

Abstract: By using the R-matrix associated with the N-dimensional irreducible representation of sl_2, one can define the N-dimensional colored Jones polynomial for knots. The volume conjecture states that a certain limit of the colored Jones polynomial a knot would give the volume of its complement. In the talk I will introduce the conjecture and its generalizations.

■ Zengo Tsuboi (OCAMI): Solutions of the reflection equation and Baxter Q-operators for open spin chains

Abstract: The augmented $q$-Onsager algebra is a coideal subalgebra of the quantum affine algebra $U_{q}(\hat{sl}_2)$. We consider intertwining relations of the augmented $q$-Onsager algebra, and obtain generic (diagonal) boundary $K$-operators in terms of the Cartan element of $U_{q}(sl_2)$. These $K$-operators solve the reflection equation. Taking appropriate limits of these $K$-operators in Verma modules, we derive $K$-operators for Baxter Q-operators for open spin chains. We also discuss the generalization of these $K$-operators to the higher rank algebra $U_{q}(\hat{gl}_N)$ case.

■ Oleksandr Tsymbaliuk (Yale): Shifted quantum affine algebras and applications to integrable systems

Abstract: In the recent series of papers by Braverman-Finkelberg-Nakajima a mathematical construction of the Coulomb branches of 3d N=4 quiver gauge theories was proposed (they are symplectic dual to the corresponding well-understood Higgs branches). They can be also realized as slices in the affine Grassmannian and therefore admit a multiplication.

In these two talks, I will discuss the quantizations of these Coulomb branches and their K-theoretic analogues, and the (conjectural) down-to-earth realization of these quantizations via shifted Yangians and shifted quantum affine algebras. Those admit a coproduct quantizing the aforementioned multiplication of slices. In type A, they also act on equivariant cohomology/K-theory of parabolic Laumon spaces. In the simplest case of sl(2) they are also related to the Q-systems.

Both of the aforementioned shifted algebras are deeply related to the integrable systems. In particular, the study of the Bethe subalgebras associated to the antidominantly shifted Yangians of 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 sl(2) give rise to a new family of 3^{n-2} q-Toda systems of sl(n), generalizing the well-known one due to Etingof and Sevostyanov. I will also explain how one can generalize the latter construction to produce exactly 3^{rk(g)-1} modified q-Toda systems for any semisimple Lie algebra g.

These talks are based on the joint works with M. Finkelberg, R. Gonin, A. Weekes, and a joint project with R. Frassek and V. Pestun.

■ Weiqiang Wang (Virginia): Quantum symmetric pairs

Abstract: As a quantization of symmetric pairs, a quantum symmetric pair consists of a quantum group U and its coideal subalgebra Ui (called an i-quantum group). Note that U itself is an example of an i-quantum group associated to the symmetric pair of diagonal type. In recent years, a number of fundamental constructions for quantum groups, e.g., R-matrix, canonical bases, geometric realizations and Hall algebras, have found generalizations in the setting of i-quantum groups. The goal of our lectures is to present some selected topics in this direction.

In Lecture 1, we will concentrate on a distinguished class of i-quantum groups Ui (which should be viewed as the i-analogue of type A quantum groups). We establish a Schur type duality between Ui and Hecke algebra of type B and develop new (i-version of) canonical bases for Ui-modules; as an application we formulate and establish a Kazhdan-Lusztig theory for the ortho-symplectic Lie superalgebras.

In Lecture 2, we will describe various combinatorics arising from new algebraic structures of the i-quantum groups, such as i-divided powers and Serre presentations.

■ Hideya Watanabe (Tokyo Inst of Tech): Highest weight theory for i-quantum groups of type A

Abstract: Quantum symmetric pairs $(U,Ui)$ have played important roles in many branches of mathematics and physics such as integrable systems, low-dimensional topology, and representation theory. However, the classification of finite-dimensional modules over an i-quantum group is still unknown except a few cases. In this talk, I explain recent progress on the classification problem by means of highest weight theory (when $U$ is of type A and $Ui$ is general).

■ Akihito Yoneyama (U of Tokyo): Matrix product solution to the reflection equation associated with a coideal subalgebra of $U_q(A_{n-1}^{(1)})$

Abstract: It is known that the intertwiner of the representations of Drinfeld-Jimbo affine quantum algebras and their coideal subalgebras gives infinitely many solutions of Yang-Baxter and the reflection equation, although latter story seems less familiar. However, they are only characterization: they do not necessarily mean some handy explicit formula. In this talk, we will report our recent result on new explicit solutions to the reflection equation, which is characterized in terms of the representations on the generalized q-Onsager coideal subalgebras of $U_q(A_{n-1}^{(1)})$. The elements of K-matrices are given in the form of the matrix product of some q-boson valued operator and they are related to three dimensional integrability.

■ Yutaka Yoshida (IPMU): Equivariant U(n) Verlinde algebra from Bethe/Gauge correspondence

Abstract: Various 4d/2d dualities are obtained by compactification of six dimensional (2,0) theory on Riemann surface. When we focus on the Coulomb branch limit of the lens space superconformal index in 4d side, the relevant two dimensional topological field theory on Riemann surface is believed to be topologically twisted Chern-Simons-matter theory with an adjoint chiral multiplet with R-charge r=2 on the direct product of circle and Riemann surface. We compute topologically twisted indices of U(n) and SU(n) Chern-Simons-matter theories in terms of supersymmetric localization and quantum integrability. By using supersymmetric localization technique, the path integral of Chern-Simons-matter theory localizes to the Bethe equation of q-boson model which allows us to represent an algebra of Wilson loops as one parameter deformation of Verlinde algebra (equivariant Verlinde algebra) by C. Korff. We confirm the proposed duality for SU(2) and SU(3) with lower levels. This talk is based on a joint work with Hiroaki Kanno and Katsuyuki Sugiyama.