Duality of subregular \(\mathcal{W} \)-algebras and principal \(\mathcal{W} \)-superalgebras

DOI10.1016/J.AIM.2021.107685zbMATH Open1495.17038arXiv2005.10713OpenAlexW3026823774MaRDI QIDQ2020401

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Publication date: 23 April 2021

Published in: (Search for Journal in Brave)

Abstract: We prove Feigin-Frenkel type dualities between subregular W-algebras of type A, B and principal W-superalgebras of type

m a t h f r a k s l (1 | n), m a t h f r a k o s p (2 | 2 n)

. The type A case proves a conjecture of Feigin and Semikhatov. Let

(m a t h f r a k g_{1}, m a t h f r a k g_{2}) = (m a t h f r a k {s l}_{n + 1}, m a t h f r a k s l (1 | n + 1))

or

(m a t h f r a k {s o}_{2 n + 1}, m a t h f r a k o s p (2 | 2 n))

and let

r

be the lacity of

m a t h f r a k g_{1}

. Let k be a complex number and

e l l

defined by

r (k + h^{v} e e_{1}) (e l l + h^{v} e e_{2}) = 1

with

h^{v} e e_{i}

the dual Coxeter numbers of the

m a t h f r a k g_{i}

. Our first main result is that the Heisenberg cosets

m a t h c a l C^{k} (m a t h f r a k g_{1})

and

m a t h c a l C^{e} l l (m a t h f r a k g_{2})

of these W-algebras at these dual levels are isomorphic, i.e.

m a t h c a l C^{k} (m a t h f r a k g_{1}) s i m e q m a t h c a l C^{e} l l (m a t h f r a k g_{2})

for generic k. We determine the generic levels and furthermore establish analogous results for the cosets of the simple quotients of the W-algebras. Our second result is a novel Kazama-Suzuki type coset construction: We show that a diagonal Heisenberg coset of the subregular W-algebra at level

k

times the lattice vertex superalgebra

V_{m a t h b b Z}

is the principal W-superalgebra at the dual level

e l l

. Conversely a diagonal Heisenberg coset of the principal W-superalgebra at level

e l l

times the lattice vertex superalgebra

V_{s q r t - 1 m a t h b b Z}

is the subregular W-algebra at the dual level k. Again this is proven for the universal W-algebras as well as for the simple quotients. We show that a consequence of the Kazama-Suzuki type construction is that the simple principal W-superalgebra and its Heisenberg coset at level

e l l

are rational and/or C_2-cofinite if the same is true for the simple subregular W-algebra at dual level

e l l

. This gives many new C_2-cofiniteness and rationality results.

Full work available at URL: https://arxiv.org/abs/2005.10713

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