Givental action and trivialisation of circle action

Dotsenko, Vladimir; Shadrin, Sergey; Vallette, Bruno

doi:10.5802/jep.23

Givental action and trivialisation of circle action
[Action de Givental et trivialisation de l’action du cercle]

Dotsenko, Vladimir ¹ ; Shadrin, Sergey ² ; Vallette, Bruno ³

¹ School of Mathematics, Trinity College Dublin 2, Ireland
² Korteweg-de Vries Institute for Mathematics, University of Amsterdam P. O. Box 94248, 1090 GE Amsterdam, The Netherlands
³ Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Sorbonne Paris Cité, CNRS, UMR 7539 93430 Villetaneuse, France

Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 213-246.

Résumé
Abstract

Dans cet article, nous montrons que l’action du groupe de Givental sur les théories cohomologiques des champs de genre $0$ , aussi appelées variétés de Frobenius formelles ou algèbres hypercommutatives, naît naturellement de la théorie de la déformation des algèbres de Batalin-Vilkovisky. Nous démontrons que l’action de Givental est égale à une action provenant des trivialisations des actions du cercle. Ce résultat repose sur l’égalité des actions de deux algèbres de Lie apparentant a priori à deux domaines distincts : la géométrie et l’algèbre homotopique.

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.

DOI : 10.5802/jep.23

Classification : 18G55, 18D50, 53D45
Keywords: Givental action, circle action, cohomological field theory, Batalin–Vilkovisky algebra, homotopy Lie algebras
Mot clés : Action de Givental, action du cercle, théories cohomologiques des champs, algèbres de Batalin-Vilkovisky, algèbres de Lie à homotopie près

Affiliations des auteurs :

Dotsenko, Vladimir ¹ ; Shadrin, Sergey ² ; Vallette, Bruno ³

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     title = {Givental action and trivialisation of circle~action},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Dotsenko, Vladimir; Shadrin, Sergey; Vallette, Bruno. Givental action and trivialisation of circle action. Journal de l’École polytechnique — Mathématiques, Tome 2 (2015), pp. 213-246. doi : 10.5802/jep.23. http://www.numdam.org/articles/10.5802/jep.23/

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