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

Abstract

We introduce a metric-dependent geometric variant of factorization homology in conformally flat Riemannian geometry for $d \geq 2$. Its coefficients are symmetric monoidal functors from a disk category in conformal Riemannian geometry to the ind-category of Hilbert spaces, which we call conformally flat $d$-disk algebras. We prove that their left Kan extensions define symmetric monoidal invariants of conformally flat manifolds. Under suitable positivity and continuity assumptions, the value on the standard sphere reproduces the sphere partition function of the associated conformal field theory. For $d>2$, we construct explicit examples from unitary representations of $\mathrm{SO}^+(d,1)$.