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

Yuto Moriwaki

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.

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

Abstract

In this paper we consider the operad of holomorphic disk embeddings of the unit disk . We introduce a suboperad defined by square-integrability conditions and show that the symmetric algebra of the Bergman space carries a natural -algebra structure. Conformally flat factorization homology with coefficients in then yields metric-dependent invariants of two-dimensional Riemannian manifolds. Moreover, is identified with the ind-Hilbert space completion of the affine Heisenberg vertex operator algebra.
Paper Structure (11 sections, 31 theorems, 159 equations, 1 figure, 1 table)

This paper contains 11 sections, 31 theorems, 159 equations, 1 figure, 1 table.

Table of Contents

  1. Preliminary
  2. Conformally flat 2-disk operad and its complex conjugate
  3. Bergman spaces and their tensor products
  4. Harmonic cocycles and Hilbert-Schmidt conditions
  5. Conformally flat 2-disk algebra on the Bergman space
  6. Classical and quantum monoid action
  7. Construction of geometric wick contraction
  8. Construction of conformally flat disk 2-algebra
  9. Relation with affine Heisenberg vertex algebra
  10. Review on affine Heisenberg vertex algebra
  11. Relation between $\mathbb{CE}_2$-algebra and affine Heisenberg vertex algebra

Key Result

Theorem A

The ind-Hilbert space $\mathrm{Sym} A^2(\mathbb{D})$ inherits the structure of a $\mathbb{CE}_2^{\mathrm{HS}}$-algebra. The restriction on the suboperad $\mathbb{CE}_2\subset \mathbb{CE}_2^{\mathrm{HS}}$ yields a symmetric monoidal functor $\mathrm{Disk}_2^{\mathrm{CO}} \rightarrow {\mathrm{Ind}} {\ is symmetric monoidal. Moreover, $\mathrm{Sym} A^2(\mathbb{D})$ has no proper closed ideal as a $\m

Figures (1)

  • Figure :

Theorems & Definitions (56)

  • Theorem A: Theorem \ref{['thm_CF']}, Theorem \ref{['thm_CF_algebra']}, and Corollary \ref{['cor_simple']}
  • Definition 1.1
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • Proposition 1.7
  • ...and 46 more