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.
The factorization algebras developed by Costello-Gwilliam provide a formulation of quantum field theory [CG1,CG2]. For the quantum field theory known as the free scalar field, the associated factorization algebra assigns to a Riemannian manifold (M, g) a chain complex, called the BV complex, whose homology yields a symmetric monoidal functor from the symmetric monoidal category Mfld Riem d of d-dimensional Riemannian manifolds and isometric open embeddings to the category of (differentiable) vector spaces [CG1,Chapter 6.3]. In this paper, we study its analogue in conformal Riemannian geometry by considering the prefactorization algebra associated with the conformal Laplacian.
Fix d ≥ 2. Let Mfld CO d,emb denote the symmetric monoidal category whose objects are ddimensional oriented Riemannian manifolds without boundary and whose morphisms are orientation-preserving conformal open embeddings. On a Riemannian manifold (M, g), the conformal Laplacian (also known as the Yamabe operator) [Ya]
transforms under a change of metric ĝ = e 2ω g (ω : M → R a smooth function) as
Using this conformal covariance, we show that the prefactorization algebra associated with the conformal Laplacian defines a symmetric monoidal functor (Theorem 2.3)
Following Costello-Gwilliam, the vector space F CL (M, g) can be computed, whenever a Green’s function G(x, y) of the conformal Laplacian L g exists, via a linear isomorphism
Here C ∞ c (M) denotes the space of real valued smooth functions with compact support. On non-compact manifolds a Green’s function (i.e. a fundamental solution of L g ) typically depends on boundary conditions. In particular, for Ψ G to define a natural transformation with respect to conformal open embeddings, one needs a fundamental solution satisfying conformally invariant boundary conditions. In two dimensions, no such solution exists, and the failure of naturality of Ψ G gives rise to a central charge.
The main results of this paper are as follows:
(1) For an open subset U ⊂ R d of the flat Riemannian manifold (R d , g std ), we show that F CL (U) is isomorphic to SymH ′ (U) as vector spaces, where H ′ (U) is the topological dual of the space H(U) of harmonic functions on U.
(2) If d ≥ 3, then for the unit disk D d = {x ∈ R d | |x| < 1}, we show that F CL (D d ) embeds densely into the Hilbert Fock space ŜymH CFT , and we describe its image explicitly. Here H CFT is a Hilbert space carrying a unitary representation of SO + (d, 1). In dimension d = 2, the analogous embedding holds after restricting to a codimension-one subspace, reflecting the logarithmic features of the massless free scalar theory. (3) We explicitly describe the algebra structure on F CL (D d ) given by the restriction of F CL to the full monoidal subcategory Disk CO d,emb ⊂ Mfld CO d,emb whose objects are disjoint unions of disks ⊔ n D d (n ≥ 0). In particular, for d = 2, we show that the algebra structure admits a quantum correction identified with an explicit cocycle of harmonic functions H ϕ (the central charge).
We emphasize that, as already explained by , the isomorphism (0.1) is not merely a computational tool for homology, but rather encodes a choice of boundary conditions in quantum field theory. Accordingly, physical quantities such as correlation functions are defined only after choosing (0.1). In other mathematical formulations of quantum field theory, such as the Gårding-Wightman axioms and the Osterwalder-Schrader axioms, the theory is defined in terms of correlation functions [GW,OS1,OS2]. Thus, in comparing factorization algebras with these axiomatic frameworks, one must incorporate an appropriate choice of Ψ G .
In this paper, for a (non-compact) domain U ⊂ R d where the Green’s function is not unique, we study how the choice of such boundary conditions changes under conformal open embeddings. In particular, for d = 2, we rederive the central charge from the viewpoint of factorization algebras. Another key observation is that the value on the flat unit disk, F CL (D d ) ∈ Vect R , is closely related to the Hilbert Fock space ŜymH CFT appearing in axiomatic QFT. In particular, this observation raises the question of whether the algebra structure in (3), defined on a dense subspace, extends to a bounded operator on the Hilbert space ŜymH CFT . It turns out that the algebra structure on F CL (D d ) is not necessarily given by bounded operators [Mo2,Mo3].
A factorization algebra assigns a multiplication map to every isometric open embedding. In [Mo2,Mo3], conformal open embeddings that give rise to bounded operators are characterized in geometric or analytic terms, thereby providing a refinement of factorization algebras compatible with the Hilbert space structure. Our original motivation for focusing on the unit disk D d comes from the study of the relationship between vertex operator algebras and axiomatic or algebraic quantum field theory [CKLW, AGT, AMT]; see [Mo2,Mo3] for further background.
Hereafter, we will
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