Givental action and trivialisation of circle action
[Action de Givental et trivialisation de l’action du cercle]
Journal de l’École polytechnique — Mathématiques, Tome 2 (2015) , pp. 213-246.

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 : https://doi.org/10.5802/jep.23
Classification : 18G55,  18D50,  53D45
Mots clés : Action de Givental, action du cercle, théories cohomologiques des champs, algèbres de Batalin-Vilkovisky, algèbres de Lie à homotopie près
@article{JEP_2015__2__213_0,
     author = {Dotsenko, Vladimir and Shadrin, Sergey and Vallette, Bruno},
     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},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jep.23/}
}
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://archive.numdam.org/articles/10.5802/jep.23/

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

[DC14] Drummond-Cole, G. C. Homotopically trivializing the circle in the framed little disks, J. Topology, Volume 7 (2014) no. 3, pp. 641-676 | Article | MR 3252959 | Zbl 1301.55005

[DCV13] Drummond-Cole, G. C.; Vallette, B. The minimal model for the Batalin–Vilkovisky operad, Selecta Math. (N.S.), Volume 19 (2013) no. 1, pp. 1-47 | MR 3029946 | Zbl 1264.18010

[DK10] Dotsenko, V.; Khoroshkin, A. Gröbner bases for operads, Duke Math. J., Volume 153 (2010) no. 2, pp. 363-396 | MR 2667136 | Zbl 1208.18007

[DSV13] Dotsenko, V.; Shadrin, S.; Vallette, B. Givental group action on topological field theories and homotopy Batalin–Vilkovisky algebras, Advances in Math., Volume 236 (2013), pp. 224-256 | MR 3019721 | Zbl 1294.14019

[DSV15a] Dotsenko, V.; Shadrin, S.; Vallette, B. De Rham cohomology and homotopy Frobenius manifolds, J. Eur. Math. Soc. (JEMS), Volume 17 (2015), pp. 535-547 | EuDML 277486 | MR 3323198 | Zbl 1317.58003

[DSV15b] Dotsenko, V.; Shadrin, S.; Vallette, B. Pre-Lie deformation theory (2015) (arXiv:1502.03280) | MR 3323198

[GCTV12] Galvez-Carrillo, I.; Tonks, A.; Vallette, B. Homotopy Batalin–Vilkovisky algebras, J. Noncommut. Geom., Volume 6 (2012) no. 3, pp. 539-602 | MR 2956319 | Zbl 1258.18005

[Get09] Getzler, E. Lie theory for nilpotent L -algebras, Ann. of Math. (2), Volume 170 (2009) no. 1, pp. 271-301 | MR 2521116 | Zbl 1246.17025

[Get95] Getzler, E. Operads and moduli spaces of genus 0 Riemann surfaces, The moduli space of curves (Texel Island, 1994) (Progress in Math.), Volume 129, Birkhäuser Boston, Boston, MA, 1995, pp. 199-230 | MR 1363058 | Zbl 0851.18005

[Giv01a] Givental, A. B. Gromov-Witten invariants and quantization of quadratic Hamiltonians, Moscow Math. J., Volume 1 (2001) no. 4, pp. 551-568 | MR 1901075 | Zbl 1008.53072

[Giv01b] Givental, A. B. Semisimple Frobenius structures at higher genus, Internat. Math. Res. Notices (2001) no. 23, pp. 1265-1286 | MR 1866444 | Zbl 1074.14532

[GM88] Goldman, W. M.; Millson, J. J. The deformation theory of representations of fundamental groups of compact Kähler manifolds, Publ. Math. Inst. Hautes Études Sci., Volume 67 (1988), pp. 43-96 | Numdam | MR 972343 | Zbl 0678.53059

[Hof10] Hoffbeck, E. A Poincaré-Birkhoff-Witt criterion for Koszul operads, Manuscripta Math., Volume 131 (2010) no. 1-2, pp. 87-110 | MR 2574993 | Zbl 1207.18009

[KM94] Kontsevich, M.; Manin, Yu. I. Gromov-Witten classes, quantum cohomology, and enumerative geometry, Comm. Math. Phys., Volume 164 (1994) no. 3, pp. 525-562 | MR 1291244 | Zbl 0853.14020

[KMS13] Khoroshkin, A.; Markarian, N.; Shadrin, S. Hypercommutative operad as a homotopy quotient of BV, Comm. Math. Phys., Volume 322 (2013) no. 3, pp. 697-729 | MR 3079329 | Zbl 1281.55011

[Lee09] Lee, Y.-P. Invariance of tautological equations. II. Gromov-Witten theory, J. Amer. Math. Soc., Volume 22 (2009) no. 2, pp. 331-352 (With an appendix by Y. Iwao and the author) | MR 2476776 | Zbl 1206.14078

[LV12] Loday, J.-L.; Vallette, B. Algebraic operads, Grundlehren Math. Wiss., 346, Springer-Verlag, Berlin, 2012, pp. xviii+512 | MR 2954392 | Zbl 1260.18001

[Man99] Manin, Yu. I. Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Colloq. Publ., 47, American Mathematical Society, Providence, RI, 1999, pp. xiv+303 | MR 1702284 | Zbl 0952.14032

[MV09] Merkulov, S.; Vallette, B. Deformation theory of representations of prop(erad)s. I, J. reine angew. Math., Volume 634 (2009), pp. 51-106 | MR 2560406 | Zbl 1187.18006

[Tel12] Teleman, C. The structure of 2D semi-simple field theories, Invent. Math., Volume 188 (2012) no. 3, pp. 525-588 | MR 2917177 | Zbl 1248.53074

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