Schedule for: 24w5501 - Geometry and Physics of Quantum Toroidal Algebra

Breakfast is served daily between 7 and 9am in the Xianghu Lake National Tourist Resort

A brief introduction with important logistical information, technology instruction, and opportunity for participants to ask questions.

I will start by surveying several known facts concerning the XXZ type models related to quantum toroidal algebras of type gl(n). The Hamiltonians of these models are given by transfer matrices and can be studied by Bethe ansatz. Then I will discuss how these knowledge can be used to understand the quantum KdV flows - the integrable system whose Hamiltonians are given by integrals of some local Virasoro currents. In particular, I will describe some results which may be a step to the proof of the BLZ conjecture that the spectrum of quantum KdV flows is parameterized by Schroedinger equations with monster potential and apparent singularities.

In the former part, I will present a personal summary of the history of the development/applications of these algebras to Physics, such as the quantum Hall effect, Seiberg-Witten theory, and string theory. This part is based on a review article, arXiv:2309.07596 (with Nawata, Noshita, and Zhu). In the latter part, I will discuss a new application to the nonabelian Hall effect based on a generalization of the spin Calogero system. This part is based on arXiv:2401.03087 (with Jean-Emile Bourgine).

Lunch is served daily between 11:30am and 1:30pm in the Xianghu Lake National Tourist Resort

I will describe recent work with Vivek Shende and Peng Zhou, which relates the Fukaya category of a multiplicative abelian Coulomb branch (also known as a multiplicative hypertoric variety) to the Fukaya category of its associated toroidal hyperplane arrangement. I will say a little about how this relates to mirror symmetry for multiplicative Coulomb branches, and how this picture fits into wider conjectures on Fukaya categories of spaces with group actions.

I will first explain the BPS algebras for 4D N=2 theories in terms of cohomological Hall algebras. We conjecture that for a theory whose BPS spectrum admits a quiver description that is 2-acyclic with infinitely-mutable potential, the BPS algebra reduces to spherical shuffle algebra. I will then explain how to study the 1/2-BPS line defects in the theory as the bimodules of its BPS algebra.

It is by now a classic result that the Yangian/quantum toroidal gl_1 acts on the cohomology/K-theory of moduli spaces of sheaves on surfaces. We will survey the natural generalization of this construction to elliptic cohomology, with some interesting new features.

Symplectic singularities introduced by Beauville appear in various aspects of representation theory. On the other hand, symplectic singularities also arise in the context of quantum field theory in physics, particularly in the Higgs and Coulomb branches of three-dimensional theories, as well as in the Higgs branches of four-dimensional theories. Additionally, in vertex algebra theory, certain Poisson varieties called associated varieties are defined as geometric invariants, and they often turn out to be symplectic singularities. In such cases, vertex algebras can be regarded as chiral quantization of symplectic singularities. In particular, the 4D/2D duality proposed by Beem et al. in theoretical physics determines vertex algebras as invariants for superconformal four-dimensional theories. It is claimed that the Higgs branch of four-dimensional theories can be reconstructed as the associated variety of vertex algebras. Therefore, all vertex algebras arising from four-dimensional theories are supposed to be chiral quantization of symplectic singularities.

The Deligne-Simpson problem asks for a criterion for the existence of a connection on a punctured P^1 with prescribed singularities. I will explain joint work with Zhiwei Yun on a generalization of this problem for G-connections on P^1 with two singularities, only one of which is a regular singularity, where G is any complex reductive group. We give a geometric and a representation-theoretic existence criterion, and use them to solve this Deligne-Simpson problem in many cases. Perhaps surprisingly, the second criterion is given in terms of representations of the rational Cherednik algebra.

We elucidate the relation of the $q$-deformed N=2 superconformal algebra (SCA) to the deformation of $Y$-algebra (a.k.a, the corner vertex operator algebra), which is deduced from the quantum toroidal algebra of type $\hat{\mathfrak{gl}}_1$ as a deformation of the affine Yangian. In particular, we show how a free field representation of the $q$-deformed N=2 SCA is obtained by “twisting” the Wakimoto representation of the quantum affine algebra $U_q(\hat{\mathfrak{sl}}_2)$.

We elucidate the relation of the $q$-deformed N=2 superconformal algebra (SCA) to the deformation of $Y$-algebra (a.k.a, the corner vertex operator algebra), which is deduced from the quantum toroidal algebra of type $\hat{\mathfrak{gl}}_1$ as a deformation of the affine Yangian. In particular, we show how a free field representation of the $q$-deformed N=2 SCA is obtained by “twisting” the Wakimoto representation of the quantum affine algebra $U_q(\hat{\mathfrak{sl}}_2)$.

Associated to any root datum, there is an elliptic affine Hecke algebra defined by Ginzburg, Kapranov, and Vasserot. In this talk, we present a Fourier-Mukai functor from the representation category of the elliptic affine Hecke algebra to the corresponding category associated with the Langlands dual root datum. To achieve this connection, we employ the elliptic Hecke algebra with dynamical parameters as an intermediary. As an application, we obtain a bijection between irreducible flat representations of the elliptic affine Hecke algebra and nilpotent Higgs bundles on the elliptic curve. This is based on joint work with Changlong Zhong.

We start from a review of the elliptic quantum group U_{q,p}(\widehat{sl}}_N) and its correspondences to the elliptic cohomology of the cotangent bundle to the partial flag variety and to the deformed W-algebras. We emphasize the different roles of the two vertex operators defined by the two different co-algebra structures associated with the standard comultiplication ¥Delta and the Drinfeld comultiplication ¥Delta^D. Then we discuss the elliptic quantum toroidal algebra U_{t_1,t_2,p}(gl_{1,tor}) and construct the two vertex operators associated with ¥Delta^D and ¥Delta. By using them, we show the same correspondence of U_{t_1,t_2,p}(gl_{1,tor}) to the Jordan quiver W-algebras (an operator version of the qq-character) and to the elliptic cohomology of the instanton moduli spaces M(n,r). The former further yields the instanton calculus for the 5d and 6d lifts of the 4d N=2^* SUSY gauge theory, whereas the latter yields the shuffle product formula for the elliptic stable envelopes, the K-theoretic vertex functions and L-operators satisfying the RLL=LLR^* relation with R and R^* being the elliptic dynamical instanton R-matrices. If time allows, we also discuss their higher rank extensions, the elliptic quantum toroidal algebra U_{t_1,t_2,p}(gl_{N,tor}) and its connections to the A^{(1)}_{N-1} and A_¥infty quiver varieties. This talk is based on the works done with Kazuyuki Oshima and Andrey Smirnov.

We'll first review some basic facts about COHAs attached to quivers and algebraic surfaces. Then, we'll provide an explicit description of the latter in terms of analogues of W-algebras, and we'll discuss some consequences.

Gauge origami is a generalized supersymmetric quiver gauge theory where intersecting D- branes appear. The simplest setup is the gauge origami system in C^4, where D2/D4/D6/D8 branes are present. We demonstrate that the contour integral formulas have free field interpretations, leading to the operator formalism of qq-characters associated with each D-brane. The qq-characters of D2 and D4-branes correspond to screening charges and generators of the affine quiver W-algebra, respectively. On the other hand, the qq-characters of D6 and D8 branes give new types of qq-characters, where the monomial terms are characterized by plane partitions and solid partitions. The D4, D6, D8 qq-characters automatically reproduce the partition function of the spiked instanton, tetrahedron instanton, and the magnificent four which eventually establish the BPS/CFT correspondence. We also discuss the relation with quantum toroidal algebras and generalizations. This talk is based on arXiv: 2310.08545, 2404.17061.

We present results on the structure and representation theory of quantum toroidal algebras Uq(g_tor) in untwisted types. In particular, we'll first construct an action of the extended double affine braid group, with a surprising finite presentation for Uq(g_tor) acting as a crucial ingredient in the proof. This will be used to exhibit horizontal-vertical symmetries of Uq(g_tor), in the form of anti-involutions and Miki automorphisms which exchange its horizontal and vertical subalgebras. We'll finish by discussing representation theoretic applications (work in progress), and deducing congruence group actions of the central extension of SL(2,Z) on Uq(g_tor).

We discuss joint work with Tom Gannon, showing that the algebra D(SL_n/U) of differential operators on the base affine space of SL_n is the quantized Coulomb branch of a certain 3d N = 4 quiver gauge theory. In the semiclassical limit this confirms a conjecture of Dancer-Hanany-Kirwan on the universal hyperkähler implosion of SL_n. We also formulate and prove a generalization interpreting an arbitrary unipotent reduction of the cotangent bundle of SL_n as a Coulomb branch. These results also provide a new interpretation of the Gelfand-Graev Weyl group symmetry of D(SL_n/U).

For the Yangian of sl_2, higher spin representations are tensor products of the evaluation pullback of the $\ell_i+1$-dimensional irreducible representations of sl_2, where $\ell_i$ are the highest weights. In my talk, I will give a geometric realization of the higher spin representations in terms of the critical cohomology of representations of the quiver with potential of Bykov and Zinn-Justin. I will also talk about the construction of R-matrices via the lattice model and the weight functions. This is based on my joint work with Paul Zinn-Justin.

I consider a particular combinations of intertwining operators of quantum toroidal algebra of type gl(1), which correspond to branes spirally wrapping a compactified dimension. Curiously these operators reproduce two interesting objects in other areas of mathematical physics: 1) the K-theoretica vertex function counting sheaves on C^3, 2) non-stationary elliptic Ruijsenaars-Schneider wavefunctions recently proposed by Shiraishi. I would also give some ideas on the possible interpretations (both physical and algebraic) of the spiraling branes.

The skein algebra of a surface is a noncommutative algebra that quantizes the SL2-character variety of the surface. In this talk, I will describe a relationship between skein algebras and the quantized K-theoretic Coulomb branches of Braverman, Finkelberg, and Nakajima. This is based on joint work with Peng Shan and Hyun Kyu Kim.

Classical q-shuffle algebras provide combinatorial models for the positive half U_q(n) of a finite quantum group. We define a loop version of this construction, yielding a combinatorial model for the positive half U_q(Ln)of a quantum loop group. In particular, we construct a PBW basis of U_q(Ln) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the U_q(n) case. We also connect this to Enriquez' degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding U_q(Ln) -> A in terms of pole and wheel conditions. In the second part of the talk, we shall discuss the shuffle interpretations of fused currents proposed by Ding-Khoroshkin. This is based on joint works with Andrei Negut.

Following the development of the AGT correspondence, new relations between free field representations of quantum groups and W- algebras were obtained. The simplest one is the homomorphism between the level $(N,0)$ horizontal representation of the quantum toroidal gl(1) algebra and (dressed) q-deformed $W_N$ algebras. In this talk, I will explain how to extend this type of relations to the Wakimoto representations of quantum affine sl(N) algebras using the 'surface defect' deformation of the quantum toroidal sl(N) algebra.

Kostka polynomials are ubiquitous in representation theory, and the theory of Kostka polynomials are particularly rich for symmetric groups. Kostka polynomials of complex reflection groups are defined and studied by Shoji around 2000, in which he conjectured that one particular case of his generalized Kostka polynomials of the wreath product of the symmetric group with a cyclic group (denoted by $G(\ell,1,n)$ by historical reason) behave like the case of symmetric groups. Some part of his conjectures were settled by the works of Finkelberg-Ionov, Hu, and Shoji around 2018 by a combination of geometric and combinatorial interpretations. Here we present a new approach this conjecture using representation theory, that resolves the above mentioned Shoji’s conjecture completely. This exhibits a unique family of ``Kostka polynomials of $G(\ell,1,n)$” that satisfies the properties transported from the case of symmetric groups in an ideal way. Here we remark that our family is different from (a specialization of) the wreath Macdonald polynomials introduced by Haiman, nor the ones employed by Lusztig in his study of unipotent characters of finite groups of Lie types (that exists for $\ell = 2$). This talk is based on arXiv:2404.0626