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

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