On two chain models for the gravity operad

Dupont, Clément; Horel, Geoffroy

doi:10.1090/proc/13874

On two chain models for the gravity operad

Authors: Clément Dupont and Geoffroy Horel
Journal: Proc. Amer. Math. Soc. 146 (2018), 1895-1910
MSC (2010): Primary 18D50, 32G15, 55P62; Secondary 55P42
DOI: https://doi.org/10.1090/proc/13874
Published electronically: December 12, 2017
MathSciNet review: 3767344
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we recall the construction of two chain level lifts of the gravity operad, one due to Getzler–Kapranov and one due to Westerland. We prove that these two operads are formal and that they indeed have isomorphic homology.

References

Yves André, Une introduction aux motifs (motifs purs, motifs mixtes, périodes), Panoramas et Synthèses [Panoramas and Syntheses], vol. 17, Société Mathématique de France, Paris, 2004 (French, with English and French summaries). MR 2115000

J. Alm and D. Petersen, Brown’s dihedral moduli space and freedom of the gravity operad, Annales scientifiques de l’ENS 50, fascicule 5 (2017), 1080-1122.

J. Cirici and G. Horel, Mixed Hodge structures and formality of symmetric monoidal functors, arXiv preprint arXiv:1703.06816 (2017).

Kevin Costello, Renormalization and effective field theory, Mathematical Surveys and Monographs, vol. 170, American Mathematical Society, Providence, RI, 2011. MR 2778558

Pierre Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57 (French). MR 498551

Benoit Fresse, Homotopy of operads and Grothendieck-Teichmüller groups. Part 1, Mathematical Surveys and Monographs, vol. 217, American Mathematical Society, Providence, RI, 2017. The algebraic theory and its topological background. MR 3643404

Benoit Fresse, Homotopy of operads and grothendieck–teichmüller groups: Part 2, American Mathematical Soc., 2017.

E. Getzler, Two-dimensional topological gravity and equivariant cohomology, Comm. Math. Phys. 163 (1994), no. 3, 473–489. MR 1284793

E. Getzler, Operads and moduli spaces of genus $0$ Riemann surfaces, The moduli space of curves (Texel Island, 1994) Progr. Math., vol. 129, Birkhäuser Boston, Boston, MA, 1995, pp. 199–230. MR 1363058, DOI https://doi.org/10.1007/978-1-4612-4264-2_8

E. Getzler and J. D. S. Jones, Operads, homotopy algebra, and iterated integrals for double loop spaces, arXiv:hep-th/9403055 (1994).

E. Getzler and M. M. Kapranov, Cyclic operads and cyclic homology, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 167–201. MR 1358617

E. Getzler and M. M. Kapranov, Modular operads, Compositio Math. 110 (1998), no. 1, 65–126. MR 1601666, DOI https://doi.org/10.1023/A%3A1000245600345

F. Guillén Santos, V. Navarro, P. Pascual, and A. Roig, Moduli spaces and formal operads, Duke Math. J. 129 (2005), no. 2, 291–335. MR 2165544, DOI https://doi.org/10.1215/S0012-7094-05-12924-6

G. Horel, Profinite completion of operads and the Grothendieck-Teichmüller group, Advances in Math. vol 321, (2017), pp 326-390.

John R. Klein, The dualizing spectrum of a topological group, Math. Ann. 319 (2001), no. 3, 421–456. MR 1819876, DOI https://doi.org/10.1007/PL00004441

Maxim Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), no. 1, 35–72. Moshé Flato (1937–1998). MR 1718044, DOI https://doi.org/10.1023/A%3A1007555725247

Maxim Kontsevich, Derived Grothendieck-Teichmüller group and graph complexes [after T. Willwacher], Séminaire Bourbaki, 69ème année (2016-2017), no. 1126, 2017.

Takashi Kimura, Jim Stasheff, and Alexander A. Voronov, On operad structures of moduli spaces and string theory, Comm. Math. Phys. 171 (1995), no. 1, 1–25. MR 1341693

Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659

Jacob Lurie, Higher algebra. 2016, Preprint, available at http://www. math. harvard. edu/˜ lurie (2016).

Pascal Lambrechts and Ismar Volić, Formality of the little $N$-disks operad, Mem. Amer. Math. Soc. 230 (2014), no. 1079, viii+116. MR 3220287

Dan Petersen, Minimal models, GT-action and formality of the little disk operad, Selecta Math. (N.S.) 20 (2014), no. 3, 817–822. MR 3217461, DOI https://doi.org/10.1007/s00029-013-0135-5

Frédéric Pham, Singularities of integrals, Universitext, Springer, London; EDP Sciences, Les Ulis, 2011. Homology, hyperfunctions and microlocal analysis; With a foreword by Jacques Bros; Translated from the 2005 French original; With supplementary references by Claude Sabbah. MR 2798679

Paolo Salvatore, Configuration spaces with summable labels, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 375–395. MR 1851264

Stefan Schwede and Brooke Shipley, Stable model categories are categories of modules, Topology 42 (2003), no. 1, 103–153. MR 1928647, DOI https://doi.org/10.1016/S0040-9383%2802%2900006-X

Dmitry E. Tamarkin, Formality of chain operad of little discs, Lett. Math. Phys. 66 (2003), no. 1-2, 65–72. MR 2064592, DOI https://doi.org/10.1023/B%3AMATH.0000017651.12703.a1

Craig Westerland, Equivariant operads, string topology, and Tate cohomology, Math. Ann. 340 (2008), no. 1, 97–142. MR 2349769, DOI https://doi.org/10.1007/s00208-007-0140-0