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Givental Action and Trivialisation of Circle Action

Vladimir Dotsenko, Sergey Shadrin, Bruno Vallette

Abstract

In this paper, we show that the Givental group action on genus zero cohomological field theories, also known as formal Frobenius manifolds or hypercommutative algebras, naturally arises in the deformation theory of Batalin--Vilkovisky algebras. We prove that the Givental action is equal to an action of the trivialisations of the trivial circle action. This result relies on the equality of two Lie algebra actions coming from two apparently remote domains: geometry and homotopical algebra.

Givental Action and Trivialisation of Circle Action

Abstract

In this paper, we show that the Givental group action on genus zero cohomological field theories, also known as formal Frobenius manifolds or hypercommutative algebras, naturally arises in the deformation theory of Batalin--Vilkovisky algebras. We prove that the Givental action is equal to an action of the trivialisations of the trivial circle action. This result relies on the equality of two Lie algebra actions coming from two apparently remote domains: geometry and homotopical algebra.

Paper Structure

This paper contains 23 sections, 19 theorems, 124 equations, 1 figure.

Table of Contents

  1. Recollections
  2. Hypercommutative algebras and cohomological field theories
  3. Intersection theory on moduli spaces
  4. Givental action on CohFTs
  5. Trees
  6. Homotopy Lie algebra
  7. Convolution algebras
  8. Gauge symmetries in homotopy Lie algebras
  9. The homotopy Lie algebra encoding skeletal homotopy BV-algebras
  10. Homotopy BV-algebras and skeletal homotopy BV-algebras
  11. The three convolution dg Lie algebras
  12. The convolution homotopy Lie algebra of skeletal homotopy BV-algebras
  13. Gauge interpretation of infinitesimal Givental action
  14. The main theorem
  15. Gauge symmetries restrict to hypercommutative algebras
  16. ...and 8 more sections

Key Result

Theorem 1

For any hypercommutative algebra structure on a graded vector space $A$ encoded by a Maurer--Cartan element $\alpha\in\mathfrak{g}_{\mathop{\mathrm{HyperCom}}\nolimits}$ and for any degree $0$ element $r(z)\in\mathfrak{g}_{\Delta}$, the infinitesimal Givental action of $r(z)$ on $\alpha$ is equal to

Figures (1)

  • Figure 1: Example of a shuffle tree

Theorems & Definitions (39)

  • Theorem : Thm. \ref{['thm:MainGiv=LinftyAction']}
  • Theorem : Thm. \ref{['thm:MainII']}
  • Definition 1: Hypercommutative algebra
  • Definition 2: Homotopy hypercommutative algebras
  • Definition 3: Genus $0$ CohFT KontsevichManin94
  • Definition 4: $\psi$-classes
  • Proposition 1
  • Definition 5: $L_\infty$-algebra
  • Theorem \oldthetheorem: Homotopy Transfer Theorem, see LodayVallette12
  • proof
  • ...and 29 more