Cosets of affine vertex algebras inside larger structures

By T. Creutzig, A. Linshaw. Journal of Algebra, 2014

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Abstract Given a finite-dimensional reductive Lie algebra g equipped with a nondegenerate, invariant, symmetric bilinear form B, let V k ( g , B ) denote the universal affine vertex algebra associated to g and B at level k. Let A k be a vertex (super)algebra admitting a homomorphism V k ( g , B ) → A k . Under some technical conditions on A k , we characterize the coset Com ( V k ( g , B ) , A k ) for generic values of k. We establish the strong finite generation of this coset in full generality in the following cases: A k = V k ( g ′ , B ′ ) , A k = V k − l ( g ′ , B ′ ) ⊗ F , and A k = V k − l ( g ′ , B ′ ) ⊗ V l ( g ″ , B ″ ) . Here g ′ and g ″ are finite-dimensional Lie (super)algebras containing g , equipped with nondegenerate, invariant, (super)symmetric bilinear forms B ′ and B ″ which extend B, l ∈ C is fixed, and F is a free field algebra admitting a homomorphism V l ( g , B ) → F . Our approach is essentially constructive and leads to minimal strong finite generating sets for many interesting examples. As an application, we give a new proof of the rationality of the simple N = 2 superconformal algebra with c = 3 k k + 2 for all positive integers k.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/68e674f679447752d17373d0d062dc33a7594a2a