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 $d\ge 2$. 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 recovers the sphere partition function of the associated conformal field theory. For $d\ge 3$, we construct explicit examples using unitary representations of ${\mathrm{SO}}^{+}(d,1)$ and harmonic analysis, and show that their operadic structures do not extend to bounded operations on the natural Hilbert space completions.

73 pages, 2 figures. v3: Added Sections 2.6 and 3.4, including a new theorem on the failure of Hilbert space completions and results on correlation functions