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

TL;DR

This paper develops a new geometric variant of factorization homology in conformally flat Riemannian geometry, linking it to conformal field theories and constructing explicit examples from unitary representations.

Contribution

It introduces conformally flat disk algebras as symmetric monoidal functors and proves their invariance, connecting geometric structures to conformal field theory partition functions.

Findings

The new invariant reproduces sphere partition functions under certain conditions.

Explicit examples are constructed from unitary representations of SO^+(d,1).

The framework generalizes factorization homology to conformally flat geometries.

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 $\mathrm{SO}^+(d,1)$.