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Categorical enumerative invariants of the ground field

Junwu Tu

Abstract

For an $S^1$-framed modular operad $P$, we introduce its "Feynman compactification" denoted by $FP$ which is a modular operad. Let $\{\mathbb{M}^{\sf fr}(g,n)\}_{(g,n)}$ be the $S^1$-framed modular operad defined using moduli spaces of smooth curves with framings along punctures. We prove that the homology operad of $F\mathbb{M}^{\sf fr}$ is isomorphic to $H_*(\overline{M})$, the homology operad of the Deligne-Mumford operad. Using this isomorphism, we obtain an explicit formula of the fundamental class of $[\overline{M}_{g,n}/S_n]$ in terms of Sen-Zwiebach's string vertices. As an immediate application, under mild assumptions, we prove that Costello's categorical enumerative invariants of the ground field match with the Gromov-Witten invariants of a point.

Categorical enumerative invariants of the ground field

Abstract

For an -framed modular operad , we introduce its "Feynman compactification" denoted by which is a modular operad. Let be the -framed modular operad defined using moduli spaces of smooth curves with framings along punctures. We prove that the homology operad of is isomorphic to , the homology operad of the Deligne-Mumford operad. Using this isomorphism, we obtain an explicit formula of the fundamental class of in terms of Sen-Zwiebach's string vertices. As an immediate application, under mild assumptions, we prove that Costello's categorical enumerative invariants of the ground field match with the Gromov-Witten invariants of a point.

Paper Structure

This paper contains 34 sections, 164 equations, 2 figures.

Table of Contents

  1. Introduction
  2. CEI of the algebra $\mathbb{Q}$.
  3. Basic constructions and notations.
  4. String vertices.
  5. Feynman compactification of $\mathbb{M}^{\sf fr}$.
  6. Geometric understanding of string vertices.
  7. Organization of the paper.
  8. Notations and Conventions.
  9. Acknowledgments.
  10. Feynman compactifications
  11. Recollections of modular operads.
  12. $hS^1$-equivariant modular operads.
  13. A first example.
  14. A second example.
  15. Twisted sewing operations.
  16. ...and 19 more sections

Figures (2)

  • Figure 1: Higher homotopies
  • Figure 2: Illustration of $\mathcal{K}_{(G,f)}(\gamma_1,\gamma_2,\gamma_3)$

Theorems & Definitions (12)

  • remark 1
  • remark 2
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 2 more