Givental action and trivialisation of circle action
Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 213-246.

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 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.

Received:
Accepted:
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
Vladimir Dotsenko 1; Sergey Shadrin 2; Bruno Vallette 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
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@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/

[Cos05] K. J. Costello - “The Gromov-Witten potential associated to a TCFT” (2005), arXiv:math/0509264

[DC14] G. C. Drummond-Cole - “Homotopically trivializing the circle in the framed little disks”, J. Topology 7 (2014) no. 3, p. 641-676 | DOI | MR | Zbl

[DCV13] G. C. Drummond-Cole & B. Vallette - “The minimal model for the Batalin–Vilkovisky operad”, Selecta Math. (N.S.) 19 (2013) no. 1, p. 1-47 | DOI | MR | Zbl

[DK10] V. Dotsenko & A. Khoroshkin - “Gröbner bases for operads”, Duke Math. J. 153 (2010) no. 2, p. 363-396 | DOI | Zbl

[DSV13] V. Dotsenko, S. Shadrin & B. Vallette - “Givental group action on topological field theories and homotopy Batalin–Vilkovisky algebras”, Advances in Math. 236 (2013), p. 224-256 | DOI | MR | Zbl

[DSV15a] V. Dotsenko, S. Shadrin & B. Vallette - “De Rham cohomology and homotopy Frobenius manifolds”, J. Eur. Math. Soc. (JEMS) 17 (2015), p. 535-547 | DOI | MR

[DSV15b] V. Dotsenko, S. Shadrin & B. Vallette - “Pre-Lie deformation theory” (2015), arXiv:1502.03280 | Zbl

[GCTV12] I. Galvez-Carrillo, A. Tonks & B. Vallette - “Homotopy Batalin–Vilkovisky algebras”, J. Noncommut. Geom. 6 (2012) no. 3, p. 539-602 | DOI | MR | Zbl

[Get95] E. Getzler - “Operads and moduli spaces of genus 0 Riemann surfaces”, in The moduli space of curves (Texel Island, 1994), Progress in Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, p. 199-230 | DOI | MR | Zbl

[Get09] E. Getzler - “Lie theory for nilpotent L -algebras”, Ann. of Math. (2) 170 (2009) no. 1, p. 271-301 | DOI | MR | Zbl

[Giv01a] A. B. Givental - “Gromov-Witten invariants and quantization of quadratic Hamiltonians”, Moscow Math. J. 1 (2001) no. 4, p. 551-568 | DOI | MR | Zbl

[Giv01b] A. B. Givental - “Semisimple Frobenius structures at higher genus”, Internat. Math. Res. Notices (2001) no. 23, p. 1265-1286 | DOI | MR | Zbl

[GM88] W. M. Goldman & J. J. Millson - “The deformation theory of representations of fundamental groups of compact Kähler manifolds”, Publ. Math. Inst. Hautes Études Sci. 67 (1988), p. 43-96 | DOI | Numdam

[Hof10] E. Hoffbeck - “A Poincaré-Birkhoff-Witt criterion for Koszul operads”, Manuscripta Math. 131 (2010) no. 1-2, p. 87-110 | DOI | MR | Zbl

[KM94] M. Kontsevich & Y. I. Manin - “Gromov-Witten classes, quantum cohomology, and enumerative geometry”, Comm. Math. Phys. 164 (1994) no. 3, p. 525-562 | MR | Zbl

[KMS13] A. Khoroshkin, N. Markarian & S. Shadrin - “Hypercommutative operad as a homotopy quotient of BV”, Comm. Math. Phys. 322 (2013) no. 3, p. 697-729 | DOI | MR | Zbl

[Lee09] Y.-P. Lee - “Invariance of tautological equations. II. Gromov-Witten theory”, J. Amer. Math. Soc. 22 (2009) no. 2, p. 331-352, With an appendix by Y. Iwao and the author | DOI | MR | Zbl

[LV12] J.-L. Loday & B. Vallette - Algebraic operads, Grundlehren Math. Wiss., vol. 346, Springer-Verlag, Berlin, 2012 | MR | Zbl

[Man99] Y. I. Manin - Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Colloq. Publ., vol. 47, American Mathematical Society, Providence, RI, 1999 | MR | Zbl

[MV09] S. Merkulov & B. Vallette - “Deformation theory of representations of prop(erad)s. I”, J. reine angew. Math. 634 (2009), p. 51-106 | MR | Zbl

[Tel12] C. Teleman - “The structure of 2D semi-simple field theories”, Invent. Math. 188 (2012) no. 3, p. 525-588 | MR | Zbl

[VdL02] P. Van der Laan - “Operads up to homotopy and deformations of operad maps” (2002), arXiv:math.QA/0208041

Cited by Sources: