I will present results of joint work with T. Arakawa, on representation theory of simple affine W-algebras. For so-called exceptional W-algebras, the category of representations acquires the structure of a modular tensor category, and in this talk I will describe the modular data and fusion rules for some cases. In many cases the modular data matches that of quantum groups at roots of unity, but in other cases, the results are quite mysterious.
We develop a general approach to reduction along strong Dirac maps, which are a broad generalization of Poisson moment maps. We recover a number of familiar constructions and we give several new reduction procedures, including a multiplicative analogue of Whittaker reduction.
The principal W-algebras Wk(π°π©n), as well as the cosets Com(Vk(π€π©n), Vk(π°π©n+1)) for n β₯ 2, are all quotients of a universal 2-parameter vertex operator algebra (VOA) which is freely generated of type W(2,3,4, ...). This VOA admits many other interesting 1-parameter quotients which can be identified (up to an extra Heisenberg field) with the Gaiotto-Rapcak Y-algebras. We consider a similar construction in type C, namely, the cosets Com(Vk(π°π2n), Vk(π°π2n+2)), for n β₯ 2. This gives rise to a 2-parameter VOA which is freely generated of type W(13, 2, 33, 4,...), which we expect to be the universal VOA of this type. The universal algebra admits 8 infinite families of 1-parameter quotients, which are analogues of the Gaiotto-Rapcak Y-algebras. Assuming that the universal algebra has exactly two parameters, which is ongoing work to prove, we present some applications including new rationality results for W-(super)algebras.
To any double Poisson algebra we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers.
I'll discuss how we can understand finite W-algebras of type A as Coulomb branches of quiver gauge theories, and the insights this gives us on their representation theory and geometry. If I have time, I may also engage in some irresponsible speculation about the BCD case.
Let X be a Poisson variety with a Hamiltonian G-action and H be a normal subgroup of G. Then X//G is obtained by a (Hamiltonian) reduction of X//H by the induced G/H-action under suitable assumptions, called reduction by stages. We apply for the Slodowy slices and show that the Slodowy slice associated to (π€,πͺ) is obtained by a reduction of the Slodowy slice associated to (π€,πͺβ) for a simple Lie algebra π€ and nilpotent orbits πͺ, πͺβ such that πͺ > πͺβ with some conditions. The quantum cases imply that the finite/affine W-algebras associated to (π€,πͺ) are obtained by W-algebras associated to (π€,πͺβ), which proves a conjecture of Morgan in finite cases and gives a conjectural generalization of results of Madsen and Ragoucy in affine cases.
For a given pair of a simple finite basic Lie superalgebra and its odd nilpotent element, one can construct the corresponding N=1 SUSY vertex algebra called SUSY W-algebra. The structure of any SUSY W-algebra is quite complicated but SUSY W-algebra associated with π°π©(n+1|n) and its odd principal nilpotent fpr is relatively simple. In particular, π°π©(n+1|n) is the only simple basic Lie superalgebra which admits principal π°π©(2|1)-embedding and it gives rise to an N=2 primary superconformal structure of the classical SUSY W-algebra for π°π©(n+1|n) and fpr. In the first part of this talk, I will introduce the notions of quantum and classical SUSY W-algebra and their basic properties. In the second part, I will explain the recent result on N=2 primary superconformal superconformal structure of the classical SUSY W-algebra associated with π°π©(n+1|n) and f
For each complex reflection group Ξ one can attach a canonical symplectic singularity β³Ξ. Motivated by the 4D/2D duality discovered by Beem et al., Bonetti, Menegheli and Rastelli conjectured the existence of a supersymmetric vertex operator algebra WΞ whose associated variety is isomorphic to β³Ξ. We prove this conjecture when the complex reflection group Ξ is the symmetric group SN, by constructing a sheaf of β-adic vertex algebras on the Hilbert schemes of N points in the plane. In physical terms, the vertex operator algebra WSN corresponds, by the 4D/2D duality, to the 4-dimensional N=4 super Yang-Mills theory with gauge group SLN.
Let V be an affine vertex algebra of some simple Lie algebra π€ and some level. Let KL be the category of V-modules whose conformal weight spaces are integrable π€-modules. A famous result of Kazhdan and Lusztig tells us that for almost all levels KL is a braided tensor category and as such equivalent to a category of weight modules of the quantum group Uq(π€) of π€ for suitable q.
It is desired to have similar results for suitable categories of W-algebras and superalgebras. In particular one wants to understand tensor structure and equivalences to quantum supergroups.
I will outline how to prove such statements and illustrate this in some examples.
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