Prefactorization algebras for the conformal Laplacian: Central charge and Hilbert Fock space

Abstract

Let $d \geq 2$. We consider the symmetric monoidal category of oriented Riemannian $d$-manifolds with conformal open embeddings. The prefactorization algebra associated with the conformal Laplacian defines a symmetric monoidal functor from this category to real vector spaces. For Euclidean domains $U\subset\mathbb{R}^d$, the value of this functor is identified, via the Green function, with the symmetric algebra on the topological dual of the space of harmonic functions. For $d \geq 3$ this identification is natural under all conformal transformations, while in dimension two, its failure of naturality is governed by a harmonic cocycle, which plays the role of a central charge. For the unit disk, the resulting vector space carries an algebra structure over the operad of conformal disk embeddings and admits a canonical dense embedding into the Hilbert Fock space. In dimension two, this statement holds after restricting to a codimension-one subspace, as suggested by logarithmic CFT.