Orbifolds and Cosets of Minimal W -Algebras - Arakawa, Tomoyuki; Creutzig, Thomas; Kawasetsu, Kazuya; Linshaw, Andrew R. (Communications in Mathematical Physics, 2017)
Let g be a simple, finite-dimensional Lie (super)algebra equipped with an embedding of sl2 inducing the minimal gradation on g. The corresponding minimal W-algebra Wk(g, e-θ) introduced by Kac and Wakimoto has strong generators in weights 1 , 2 , 3 / 2 , and all operator product expansions are known explicitly. The weight one subspace generates an affine vertex (super)algebra Vk′(g♮), where g♮⊂ g denotes the centralizer of sl2. Therefore, Wk(g, e-θ) has an action of a connected Lie group G0♮ with Lie algebra g0♮, where g0♮ denotes the even part of g♮. We show that for any reductive subgroup G⊂G0♮, and for any reductive Lie algebra g′⊂ g♮, the orbifold Ok=Wk(g,e-θ)G and the coset Ck= Com (V(g′) , Wk(g, e-θ)) are strongly finitely generated for generic values of k. Here V(g′) denotes the affine vertex algebra associated to g′. We find explicit minimal strong generating sets for Ck when g′= g♮ and g is either sln, sp2n, sl(2 | n) for n≠ 2 , psl(2 | 2) , or osp(1 | 4). Finally, we conjecture some surprising coincidences among families of cosets Ck which are the simple quotients of Ck, and we prove several cases of our conjecture.
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