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This talk will be on joint work with Robert Laugwitz and Milen Yakimov (arXiv:2307.14764) that is motivated by obtaining solutions to the quantum reflection equation (qRE). To start, given a braided monoidal category C and C-module category M, we introduce a version of the Drinfeld center Z(C) of C adapted for M. We refer to this category as the "reflective center" E_C(M) of M. Just like Z(C) is a canonical braided monoidal category attached to C, we show that E_C(M) is a canonical braided module category attached to M. We will also discuss the case when C is the category of modules over a quasitriangular Hopf algebra H, and show how quantum K-matrices arise in this setting (thus yielding solutions to the qRE).
Vladimir Kovalchuk constructed a two parameter family of VOA that have an affine sp(2) as subalgebra and in each even positive conformal weight a singlet and at each odd one an sp(2)-triplet. This structure is the third universal family of W-algebras after the W-infinity algebra and its even spin analogue and quotients of one-parameter subfamilies of Vladimir algebras are often realized by cosets of certain W-algebras of orthosymplectic type. Very much like in the W-infinity case and in the even spin case it is expected that there are quotients that are rational and lisse. I will explain that this expectation is indeed true and it is given by a novel family of rational and lisse cosets. This is joint work with Vlad Kovalchuk and Andy Linshaw.
The Feigin-Tipunin algorithm is a combinatorial and iterative algorithm introduced by the speaker recently to construct and study log VA(-module)s that have the q-series valued quantum invariant of 3-mfds called homological block introduced by S. Gukov et al. as their characters.
The key point is that at each step of the iteration, a geometric representation theory can be used as in the usual Feigin-Tipunin construction, so that the representation theory of the log-VA can be studied without having to examine its complicated algebraic structure. In this talk, after describing the speaker's previous work, it will be explained that if the FT algorithm can be applied to a certain lattice VOA-module, we can construct a log VA-module with the homological block of the (N+2)-Seifert 3-mfd as its character.
In physics, the webs of W-algebras are introduced as the vertex algebras associated with the (p, q)-webs of interfaces in the topologically twisted N = 4 super Yang-Mills theory. This class of vertex algebras are generalizations and extensions of the so-called vertex algebras at the corner, or equivalently the affine cosets of hook-type W-algebras in type A. Although not proven yet in general, it has recently been noticed that the webs of W-algebras actually recover the general W-algebras in type A through the ``reduction by stages". In this talk, I will present the first non-trivial examples and some interesting phenomena found in the course both in rational and irrational cases. The talk is based on a joint work with T. Creutzig, J. Fasquel, and A. Linshaw.
The study and classification of algebra objects in modular tensor categories has a strong motivation from the conformal field theoretical point of view, these objects being related to e.g. full theories [Fuchs-Runkel-Schweigert] and extensions of vertex operator algebras [Huang-Kirillov-Lepowsky, Creutzig-Kanade-McRae for superalgebras]. In this talk, I will present a classification of rigid, Frobenius algebras in the so-called Dijkgraaf-Witten categories, which we achieved using Frobenius monoidal functors. Joint work with Robert Laugwitz and Sam Hannah, based on SIGMA 19 (2023), 075, 42 pages.
Due to a celebrated result by Deligne, symmetric tensor categories of moderate growth over (algebraically closed) fields of characteristic zero correspond to categories of representations of affine algebraic supergroups. Once we move to positive characteristic, we need to take into account the Verlinde category Ver_p: Coulembier-Etingof-Ostrik proved recently that every such symmetric tensor category is the one of representations of an algebraic group in Ver_p, under some restrictions. Thus, we wonder how to describe algebraic groups in Ver_p, which in turn correspond to pairs of usual algebraic groups and Lie algebras in Ver_p, as described by Venkatesh.
This leads to the question of how to obtain Lie algebras in Ver_p. This talk is based on joint works with J. Plavnik and G. Sanmarco where we look for examples of these Lie algebras. We prove the existence of contragredient Lie algebras in symmetric tensor algebras generalizing Kac-Moody construction of Lie (super) algebras, which at the same time give a description of some examples obtained previously by 'semi simplifying' usual Lie algebras and provide new Lie algebras in Ver_p.
The local operators of a unitary 4d N=2 SCFT in twisted Schur cohomology form a vertex operator algebra (VOA). By "local operator" we mean one associated with a point in space-time. We show that to every 4d N=2 SCFT there is associated a vertex algebra containing the VOA of local twisted Schur operators as a proper subalgebra. The new vertex operators of this larger vertex algebra are associated with certain extended operators (line, surface, etc.) in twisted Schur cohomology. Though we can compute some partial results in simple SCFTs, the structure of these extended vertex algebras is still largely mysterious.
It is known by the works of Adamović and Perše that the affine simple vertex algebras associated with G2 and B3 at level -2 can be conformally embedded into L−2(D4).
In this talk, I will present a join work with Tomoyuki Arakawa, Xuanzhong Dai, Justine Fasquel, Bohan Li on the classification to the irreducible highest weight modules of these vertex algebras.
I will also describe their associated varieties: it turns out that the associated variety of that corresponding to G2 is the orbifold of the associated variety of that corresponding to D4 by the symmetric group of degree 3 which is the Dynkin diagram automorphism group of D4. This provides a new interesting example of associated variety satisfying a number of conjectures in the context of orbifold vertex algebras. It is interesting to notice that these vertex algebra also appear as the vertex operator algebras corresponding to rank one Argyres–Douglas theories in four dimension with flavour symmetry G2 and B3.
Quantum vertex algebras are deformations of vertex algebras introduced by Etingof and Kazhdan in 1998. They are families of vertex operators with relations deformed by a solution of the quantum Yang-Baxter equation. There is also a dual notion of quantum vertex coalgebras which deform vertex coalgebras. In this talk I will explain how to construct two distinct structures of a quantum vertex coalgebra on the Yangian associated to any simple finite-dimensional complex Lie algebra, and how to induce quantum vertex algebra structures on the dual Yangian. This talk is based on ongoing work with Alex Weekes and Curtis Wendlandt.
Quantum hamiltonian reduction refers to a collection of functors that map the module category of a given affine vertex algebra to those of its associated W-algebras. Some of these functors are reasonably well understood and then the representation theory of the W-algebra is accessible. But some are not. Inverse quantum hamiltonian reduction is a recent discovery that there (sometimes) exist functors in the opposite direction: from a given W-algebra module category to that of another W-algebra, which may be the affine vertex algebra itself. I will give an overview of the simplest example, which connects the module categories of the Virasoro and sl2 minimal model vertex operator algebras.
In this talk, I want to give a pedagogical overview of Zhu's algebras and then describe an alternative approach to understanding their algebraic structure that has resurfaced in 3D-2D correspondences. First, I will introduce the approach to Zhu's algebras in physics and the resulting calculational framework. Following that, I'll describe the link between Zhu's algebras and Yangians and briefly discuss how these links are realised in physics if time permits.
A star-product on a Poisson algebra A is an associative product * such that A := (A, *) is a (formal) quantization of A. Famous examples are the Moyal-Weyl and Gutt star product, which quantize the symmetric algebra of a symplectic vector space and of a Lie algebra, respectively. Suppose that we can realize A and A as the Zhu algebras of a Poisson vertex algebra V and of a vertex algebra V, respectively. A chiralization of * is a deformation of the Poisson vertex algebra structure on V, that becomes the star-product * after applying the Zhu-functor. In this talk, I will explain the general framework and some general results on the problem, and then I’ll show how to compute some explicit formulae for the chiralization of some classical star-products, like the Moyal-Weyl and Gutt star products. This talk is based on arXiv:2308.13412.
Creutzig-Diaconescu-Ma conjectured that there exists a homomorphism from the shifted affine Yangian to the universal enveloping algebra of an iterated W-algebra. They also conjectured that this homomorphism will induce a resolution of the generalized AGT conjecture. In this talk, I will explain how to construct a homomorphism from the affine Yangian of type A to the universal enveloping algebra of a W-algebra including the non-rectangular W-algebra. I expect that this homomorphism can be extended to the shifted affine Yangian.