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

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.

💡 Research Summary

The paper develops a prefactorization algebra associated with the conformal Laplacian on oriented Riemannian manifolds of dimension $d\ge 2$, organized in the symmetric monoidal category $\mathrm{Mfld}_{d,\mathrm{emb}}^{\mathrm{CO}}$ whose morphisms are conformal open embeddings. For each object $U$ the authors consider the space $\mathcal H(U)$ of smooth harmonic functions (i.e. $\Delta_g h=0$) and its continuous dual $H’(U)=\mathcal H(U)^{\prime}$. Using the Green’s function $G_U$ of the conformal Laplacian, they construct a natural identification \