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Advances in Mathematics

We prove Feigin-Frenkel type dualities between subregular W-algebras of type A,B and principal W-superalgebras of type sl(1|n),osp(2|2n). The type A case proves a conjecture of Feigin and Semikhatov. Let (g1,g2)=(sln+1,sl(1|n+1)) or (so2n+1,osp(2|2n)) and let r be the lacity of g1. Let k be a complex number and ℓ defined by r(k+h1∨)(ℓ+h2∨)=1 with hi∨ the dual Coxeter numbers of the gi. Our first main result is that the Heisenberg cosets Ck(g1) and Cℓ(g2) of these W-algebras at these dual levels are isomorphic, i.e. Ck(g1)≃Cℓ(g2) 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 VZ is the principal W-superalgebra at the dual level ℓ. Conversely a diagonal Heisenberg coset of the principal W-superalgebra at level ℓ times the lattice vertex superalgebra V−1Z 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 ℓ are rational and/or C2-cofinite if the same is true for the simple subregular W-algebra at dual level ℓ. This gives many new C2-cofiniteness and rationality results.

Duality of subregular <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">W</mml:mi></mml:math>-algebras and principal <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="script">W</mml:mi></mml:math>-superalgebras

Duality of subregular W-algebras and principal W-superalgebras