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.
Dans cet article, nous montrons que l’action du groupe de Givental sur les théories cohomologiques des champs de genre , 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.
Accepted:
DOI: 10.5802/jep.23
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
Vladimir Dotsenko 1; Sergey Shadrin 2; Bruno Vallette 3
@article{JEP_2015__2__213_0, author = {Vladimir Dotsenko and Sergey Shadrin and Bruno Vallette}, title = {Givental action and trivialisation of circle~action}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {213--246}, publisher = {\'Ecole polytechnique}, volume = {2}, year = {2015}, doi = {10.5802/jep.23}, mrnumber = {3426748}, zbl = {1331.18010}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.23/} }
TY - JOUR AU - Vladimir Dotsenko AU - Sergey Shadrin AU - Bruno Vallette TI - Givental action and trivialisation of circle action JO - Journal de l’École polytechnique — Mathématiques PY - 2015 SP - 213 EP - 246 VL - 2 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.23/ DO - 10.5802/jep.23 LA - en ID - JEP_2015__2__213_0 ER -
%0 Journal Article %A Vladimir Dotsenko %A Sergey Shadrin %A Bruno Vallette %T Givental action and trivialisation of circle action %J Journal de l’École polytechnique — Mathématiques %D 2015 %P 213-246 %V 2 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.23/ %R 10.5802/jep.23 %G en %F JEP_2015__2__213_0
Vladimir Dotsenko; Sergey Shadrin; Bruno Vallette. Givental action and trivialisation of circle action. Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 213-246. doi : 10.5802/jep.23. https://jep.centre-mersenne.org/articles/10.5802/jep.23/
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