Conformally flat factorization homology in Ind-Hilbert spaces and Conformal field theory

TL;DR

This work develops a metric-dependent analogue of factorization homology for conformally flat geometry by introducing the conformally flat d-disk operad ${\mathbb{CE}}_{d}$ and its disk subcategory ${\rm Disk}_{d}^{\mathrm{CO}}$, with manifolds modeled as germs equipped with conformal open embeddings. A conformally flat $d$-disk algebra valued in ${\rm Ind}\underline{\mathrm{Hilb}}$ yields a left Kan extension Lan_A giving metric-dependent invariants of conformally flat manifolds, and under positivity/continuity assumptions this framework recovers sphere partition functions. The paper provides an explicit construction of nontrivial ${\rm Ind}\underline{\mathrm{Hilb}}$-valued conformally flat $d$-disk algebras for $d\ge3$ from unitary $SO^{+}(d,1)$ representations, realized via harmonic polynomials and reproducing-kernel Hilbert spaces, with a Wick-contraction formalism ensuring bounded higher products. Furthermore, Lan_A on spheres produces a unique conformally invariant state reflecting the conformal bootstrap-like data, while the framework relates to Segal-type functorial CFT and Beilinson–Drinfeld style factorization via a geometric-to-analytic bridge between local Hilbert-space data and global geometric invariants. This approach aims to unify the geometric content of CFT with analytic Hilbert-space methods in an operadic, homological setting.

Abstract

We propose a metric-dependent geometric variant of factorization homology in conformally flat Riemannian geometry for $d \geq 2$. We introduce a symmetric monoidal category of germs of d-dimensional Riemannian manifolds and orientation-preserving conformal open embeddings, and its full monoidal subcategory generated by flat disks. A conformally flat $d$-disk algebra is a symmetric monoidal functor from this disk category to a target category; in this paper we take the target to be $\mathrm{IndHilb}$, the ind-category of Hilbert spaces, which provides a mathematical formulation of $d$-dimensional conformal field theories. The (1-categorical) left Kan extension of an $\mathrm{IndHilb}$-valued conformally flat $d$-disk algebra defines a metric-dependent invariant of conformally flat manifolds. Under suitable positivity and continuity assumptions, we prove that its value on the standard sphere $(S^d,g_{\mathrm{std}})$ reproduces the sphere partition function of the associated conformal field theory. For $d>2$, we construct nontrivial $\mathrm{IndHilb}$-valued conformally flat $d$-disk algebras from unitary representations of $\mathrm{SO}^+(d,1)$.

Figures (2)

• Figure 1: $\mathbb{CE}_d$-operad
• Figure 2: Kan extension

Theorems & Definitions (156)

• Theorem 1
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Remark 1.4
• Proposition 1.5
• Proposition 1.6
• Lemma 1.7
• Lemma 1.8
• proof : proof of Proposition \ref{['prop_faithful']}