Bergman space, Conformally flat 2-disk operads and affine Heisenberg vertex algebra

TL;DR

This paper explores the structure of holomorphic disk operads, introduces a suboperad with integrability conditions, and connects the Bergman space with the affine Heisenberg vertex algebra to produce geometric invariants.

Contribution

It defines a new suboperad of holomorphic disk embeddings, establishes a natural algebra structure on the Bergman space, and links it to the affine Heisenberg vertex algebra for geometric applications.

Findings

Bergman space forms a natural algebra over a suboperad of holomorphic disk embeddings.

Conformally flat factorization homology yields metric-dependent invariants of Riemannian surfaces.

The Bergman space is identified with the ind-Hilbert space completion of the affine Heisenberg vertex algebra.

Abstract

In this paper we consider the operad of holomorphic disk embeddings of the unit disk $\mathbb D \subset \mathbb C$. We introduce a suboperad $\mathbb{CE}_2^{HS}$ defined by square-integrability conditions and show that the symmetric algebra $\mathrm{Sym} A^{2}(\mathbb D)$ of the Bergman space carries a natural $\mathbb{CE}_2^{HS}$-algebra structure. Conformally flat factorization homology with coefficients in $\mathrm{Sym} A^{2}(\mathbb D)$ then yields metric-dependent invariants of two-dimensional Riemannian manifolds. Moreover, $\mathrm{Sym} A^{2}(\mathbb D)$ is identified with the ind-Hilbert space completion of the affine Heisenberg vertex operator algebra.