Homotopification and categorification of Leibniz conformal algebras

Abstract

Bakalov, Kac and Voronov introduced Leibniz conformal algebras (and their cohomology) as a non-commutative analogue of Lie conformal algebras. Leibniz conformal algebras are closely related to field algebras which are non-skew-symmetric generalizations of vertex algebras. In this paper, we first introduce $Lei{b}_{\infty }$-conformal algebras (also called strongly homotopy Leibniz conformal algebras) where the Leibniz conformal identity holds up to homotopy. We give some equivalent descriptions of $Lei{b}_{\infty }$-conformal algebras and characterize some particular classes of $Lei{b}_{\infty }$-conformal algebras in terms of the cohomology of Leibniz conformal algebras and crossed modules of Leibniz conformal algebras. On the other hand, we also introduce Leibniz conformal $2$-algebras that can be realized as the categorification of Leibniz conformal algebras. Finally, we observe that the category of Leibniz conformal $2$-algebras is equivalent to the category of $2$-term $Lei{b}_{\infty }$-conformal algebras.

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