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

Abstract

Let $d\geq 2$ and let $\mathrm{Mfld}_{d,\mathrm{emb}}^{\mathrm{CO}}$ be the symmetric monoidal category of oriented Riemannian $d$-manifolds and conformal open embeddings. The prefactorization algebra of the conformal Laplacian defines a symmetric monoidal functor $F_{\mathrm{CL}}:\mathrm{Mfld}_{d,\mathrm{emb}}^{\mathrm{CO}}\rightarrow \mathrm{Vect}_\mathbb{R}$. For Euclidean domains $U\subset\mathbb{R}^d$, $F_{\mathrm{CL}}(U)$ is identified with $\mathrm{Sym} H'(U)$ via the Green function, where $H'(U)$ is the continuous dual of harmonic functions on $U$. For $d\geq 3$ this identification is natural under all conformal transformations, while in $d=2$ the failure of naturality is governed by an explicit harmonic cocycle (the central charge). For the unit disk $\mathbb{D}$, $F_{\mathrm{CL}}(\mathbb D)$ carries an algebra structure over the operad of conformal disk embeddings and admits a canonical dense embedding into the Hilbert Fock space; in $d=2$ the latter holds after restricting to a codimension-one subspace, as suggested by logarithmic CFT.

Figures (1)

• Figure 1: $\mathbb{CE}_d^{\mathrm{emb}}$-operad

Theorems & Definitions (59)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Proposition 1.4
• Theorem 1.5
• Proposition 1.6
• Proposition 2.1
• Lemma 2.2
• proof
• Theorem 2.3