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BCOV theory via Givental group action on cohomological field theories

S. Shadrin

Abstract

In a previous paper (arXiv:0704.1001), Losev, me, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, Ooguri, and Vafa. In the present paper, we give an interpretation of this full descendant potential in terms of Givental group action on cohomological field theories. In particular, the fact that it satisfies all tautological equations becomes a trivial observation.

BCOV theory via Givental group action on cohomological field theories

Abstract

In a previous paper (arXiv:0704.1001), Losev, me, and Shneiberg constructed a full descendant potential associated to an arbitrary cyclic Hodge dGBV algebra. This contruction extended the construction of Barannikov and Kontsevich of solution of the WDVV equation, based on the earlier paper of Bershadsky, Cecotti, Ooguri, and Vafa. In the present paper, we give an interpretation of this full descendant potential in terms of Givental group action on cohomological field theories. In particular, the fact that it satisfies all tautological equations becomes a trivial observation.

Paper Structure

This paper contains 42 sections, 3 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. BCOV theory
  3. Givental group action
  4. Plan of the paper
  5. Remarks
  6. Acknowledgements
  7. Basic notions
  8. The moduli space of curves
  9. Natural mappings
  10. Strata in $\overline{\mathcal{M}}_{g,n}$
  11. Tautological classes
  12. Additive generators of the tautological ring
  13. Tautological relations
  14. Gromov-Witten theory
  15. CohFT
  16. ...and 27 more sections

Key Result

Proposition 1

The Leibniz rule for $Q$eq:leibniz is equivalent to the equation $QZ^{\circ}=0$. The $7$-term relation eq:7-term and $1/12$-axiom eq:1/12-axiom for $G_-$ are equivalent to $(G_-z)\hat{\ }Z^{\circ}=0$.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3