$\hbar$-Vertex algebras and chiralization of star products

Abstract

We develop the theory of $\hbar$-vertex algebras, algebraic structures closely related to vertex algebras but with a deformed translation covariance axiom. We establish their structure theory, including analogues of Goddard's Uniqueness Theorem, the Reconstruction Theorem, Borcherds Identity, and the OPE Expansion Formula, and introduce the associated notions of $\hbar$-Lie conformal and $\hbar$-Poisson vertex algebras. The formalism provides a natural and simplified construction of the Zhu algebra. The main application is to the chiralization of classical star-products: we show that every star-product on the symmetric algebra of a Lie algebra (or its central extensions) admits a chiralization, and we derive explicit formulae for these chiral star-products, including the Moyal-Weyl and Gutt star-products. Setting $\hbar=0$ recovers explicit deformation quantizations of a broad class of Poisson vertex algebras, including the classical limits of free-boson, $βγ$-system, affine, and Virasoro vertex algebras.