Tag - Finite W-algebras

Victor Kac: Unitary representations of minimal W-algebras

1st September, 2023

To each non-zero nilpotent orbit of a simple finite-dimensional Lie superalgebra 𝔀 with a non-degenerate invariant bilinear form one associates a simple vertex algebra, called a quantum affine W-algebra. In the simplest case 𝔀=𝔰𝔩2 one gets the Virasoro vertex algebra.

For the smallest simple Lie superalgebras 𝔀 one gets by this construction all N = 1,2,3,4, and big N = 4 superconformal algebras. I will explain classification of unitary representations of W-algebras, associated to nilpotent orbits of minimal dimension in the even part of 𝔀, which cover all the above examples.

Vera Serganova: Invariant integration on homogeneous superspaces

1st September, 2023

We use Berezin integral in the category of CS-manifolds to construct an invariant integral for the ring of regular functions on a homogeneous affine supervariety G/K. This construction has several applications in representation theory of G. We will explain how it is used in the proof of projectivity detection for support varieties and for description of stable categories for defect 1 supergroups. We also see how this integral can be used to generalize some classical statements from modular representation theory of finite groups to supergroups in characteristic zero.

Peng Shan: Modularity for W-algebras and affine Springer fibres

31st August, 2023

We will explain a bijection between admissible representations of affine Kac-Moody algebras and fixed points in affine Springer fibres. We will also explain how to match the modular group action on the characters with the one defined by Cherednik in terms of double affine Hecke algebras, and extensions of these relations to representations of W-algebras. This is based on joint work with Dan Xie and Wenbin Yan.

Jethro van Ekeren: Modular tensor categories from exceptional W-algebras

31st August, 2023

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.

Ana Balibanu: Moment maps, multiplicative reduction, and Dirac geometry

31st August, 2023

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.

Andrew Linshaw: Universal vertex algebras beyond the W∞-algebras

31st August, 2023

The principal W-algebras Wk(𝔰𝔩n), as well as the cosets Com(Vk(𝔀𝔩n), Vk(𝔰𝔩n+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(𝔰𝔭2n), Vk(𝔰𝔭2n+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.

Anne Moreau: Functorial constructions of double Poisson vertex algebras

30th August, 2023

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.

Ben Webster: Finite W-algebras as Coulomb branches

29th August, 2023

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.

Naoki Genra: Reduction by stages on W-algebras

29th August, 2023

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 (𝔀,π’ͺ) is obtained by a reduction of the Slodowy slice associated to (𝔀,π’ͺ’) for a simple Lie algebra 𝔀 and nilpotent orbits π’ͺ, π’ͺ’ such that π’ͺ > π’ͺ’ with some conditions. The quantum cases imply that the finite/affine W-algebras associated to (𝔀,π’ͺ) are obtained by W-algebras associated to (𝔀,π’ͺ’), which proves a conjecture of Morgan in finite cases and gives a conjectural generalization of results of Madsen and Ragoucy in affine cases.

Uhi Rinn Suh: N=2 primary superconformal structure of the classical SUSY W-algebra W(𝔰𝔩(n+1|n))

29th August, 2023

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 𝔰𝔩(n+1|n) and its odd principal nilpotent fpr is relatively simple. In particular, 𝔰𝔩(n+1|n) is the only simple basic Lie superalgebra which admits principal 𝔰𝔩(2|1)-embedding and it gives rise to an N=2 primary superconformal structure of the classical SUSY W-algebra for 𝔰𝔩(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 𝔰𝔩(n+1|n) and f