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

DOI : https://doi.org/10.5802/jep.23
Classification:  18G55,  18D50,  53D45
Keywords: Givental action, circle action, cohomological field theory, Batalin–Vilkovisky algebra, homotopy Lie algebras
@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'\'Ecole polytechnique - Math\'ematiques},
     publisher = {Ecole polytechnique},
     volume = {2},
     year = {2015},
     pages = {213-246},
     doi = {10.5802/jep.23},
     language = {en},
     url = {http://www.numdam.org/item/JEP_2015__2__213_0}
}
Dotsenko, Vladimir; Shadrin, Sergey; Vallette, Bruno. Givental action and trivialisation of circle action. Journal de l’École polytechnique - Mathématiques, Volume 2 (2015) , pp. 213-246. doi : 10.5802/jep.23. http://www.numdam.org/item/JEP_2015__2__213_0/

[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, Tome 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.), Tome 19 (2013) no. 1, pp. 1-47 | MR 3029946 | Zbl 1264.18010

[DK10] Dotsenko, V.; Khoroshkin, A. Gröbner bases for operads, Duke Math. J., Tome 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., Tome 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), Tome 17 (2015), pp. 535-547 | 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., Tome 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), Tome 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), Birkhäuser Boston, Boston, MA (Progress in Math.) Tome 129 (1995), pp. 199-230 | MR 1363058 | Zbl 0851.18005

[Giv01a] Givental, A. B. Gromov-Witten invariants and quantization of quadratic Hamiltonians, Moscow Math. J., Tome 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., Tome 67 (1988), pp. 43-96 | Numdam | MR 972343 | Zbl 0678.53059

[Hof10] Hoffbeck, E. A Poincaré-Birkhoff-Witt criterion for Koszul operads, Manuscripta Math., Tome 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., Tome 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., Tome 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., Tome 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, Springer-Verlag, Berlin, Grundlehren Math. Wiss., Tome 346 (2012), pp. xviii+512 | MR 2954392 | Zbl 1260.18001

[Man99] Manin, Yu. I. Frobenius manifolds, quantum cohomology, and moduli spaces, American Mathematical Society, Providence, RI, Amer. Math. Soc. Colloq. Publ., Tome 47 (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., Tome 634 (2009), pp. 51-106 | MR 2560406 | Zbl 1187.18006

[Tel12] Teleman, C. The structure of 2D semi-simple field theories, Invent. Math., Tome 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)