Home Weyl 𝑛-algebras and the Swiss cheese operad
Article
Licensed
Unlicensed Requires Authentication

Weyl 𝑛-algebras and the Swiss cheese operad

  • Nikita Markarian EMAIL logo
Published/Copyright: February 2, 2021

Abstract

We apply Weyl 𝑛-algebras to prove formality theorems for higher Hochschild cohomology. We present two approaches: via propagators and via the factorization complex. It is shown that the second approach is equivalent to the first one taken with a new family of propagators we introduce.

MSC 2010: 18D50; 16E40; 55N35

Funding statement: The study has been funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project “5-100”.

Acknowledgements

I am grateful to D. Calaque, V. Dotsenko, B. Feigin, A. Khoroshkin, S. Merkulov, B. Shoikhet, D. Tamarkin and A. Voronov for fruitful discussions. I thank the referee for useful recommendations and for pointing out many relevant references.

  1. Communicated by: Frederick R. Cohen

References

[1] S. Axelrod and I. M. Singer, Chern–Simons perturbation theory, Proceedings of the 20th International Conference on Differential Geometric Methods in Theoretical Physics. Vol. 1, 2 (New York 1991), World Scientific, River Edge (1992), 3–45. 10.4310/jdg/1214454681Search in Google Scholar

[2] D. Ayala, J. Francis and H. L. Tanaka, Factorization homology of stratified spaces, Selecta Math. (N. S.) 23 (2017), no. 1, 293–362. 10.1007/s00029-016-0242-1Search in Google Scholar

[3] C. Berger, Combinatorial models for real configuration spaces and En-operads, Operads: Proceedings of Renaissance Conferences (Hartford/Luminy 1995), Contemp. Math. 202, American Mathematical Society, Providence (1997), 37–52. 10.1090/conm/202/02582Search in Google Scholar

[4] S. Chmutov, S. Duzhin and J. Mostovoy, Introduction to Vassiliev Knot Invariants, Cambridge University, Cambridge, 2012. 10.1017/CBO9781139107846Search in Google Scholar

[5] J. Francis, The tangent complex and Hochschild cohomology of En-rings, Compos. Math. 149 (2013), no. 3, 430–480. 10.1112/S0010437X12000140Search in Google Scholar

[6] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. (2) 78 (1963), 267–288. 10.2307/1970343Search in Google Scholar

[7] E. Getzler and J. D. S. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces, preprint (1994), https://arxiv.org/abs/hep-th/9403055. Search in Google Scholar

[8] G. Ginot, Notes on factorization algebras, factorization homology and applications, Mathematical Aspects of Quantum Field Theories, Math. Phys. Stud., Springer, Cham (2015), 429–552. 10.1007/978-3-319-09949-1_13Search in Google Scholar

[9] G. Ginot, T. Tradler and M. Zeinalian, Higher Hochschild cohomology, Brane topology and centralizers of En-algebra maps, preprint (2012), https://arxiv.org/abs/1205.7056. Search in Google Scholar

[10] G. Ginot, T. Tradler and M. Zeinalian, Higher Hochschild homology, topological chiral homology and factorization algebras, Comm. Math. Phys. 326 (2014), no. 3, 635–686. 10.1007/s00220-014-1889-0Search in Google Scholar

[11] V. Ginzburg and M. Kapranov, Koszul duality for operads, Duke Math. J. 76 (1994), no. 1, 203–272. 10.1215/S0012-7094-94-07608-4Search in Google Scholar

[12] E. Hoefel, M. Livernet and A. Quesney, On the deformation complex of homotopy affine actions, Adv. Math. 358 (2019), Article ID 106857. 10.1016/j.aim.2019.106857Search in Google Scholar

[13] R. M. Kaufmann, Moduli space actions on the Hochschild co-chains of a Frobenius algebra. II. Correlators, J. Noncommut. Geom. 2 (2008), no. 3, 283–332. 10.4171/JNCG/22Search in Google Scholar

[14] R. M. Kaufmann, Open/closed string topology and moduli space actions via open/closed Hochschild actions, SIGMA Symmetry Integrability Geom. Methods Appl. 6 (2010), Paper No. 036. 10.3842/SIGMA.2010.036Search in Google Scholar

[15] M. Kontsevich, Operads and motives in deformation quantization, Lett. Math. Phys. 48 (1999), 35–72. 10.1023/A:1007555725247Search in Google Scholar

[16] M. Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. 10.1023/B:MATH.0000027508.00421.bfSearch in Google Scholar

[17] M. Kontsevich and Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I, Math. Phys. Stud. 21, Kluwer Academic, Dordrecht (2000), 255–307. Search in Google Scholar

[18] J.-L. Loday, Cyclic Homology, 2nd ed., Grundlehren Math. Wiss. 301, Springer, Berlin, 1998. 10.1007/978-3-662-11389-9Search in Google Scholar

[19] J. Lurie, Higher algebra, preprint (2017), http://www.math.harvard.edu/~lurie/papers/HigherAlgebra.pdf. Search in Google Scholar

[20] N. Markarian, Weyl 𝑛-algebras and the Kontsevich integral of the unknot, J. Knot Theory Ramifications 25 (2016), no. 12, Article ID 1642008. 10.1142/S0218216516420086Search in Google Scholar

[21] N. Markarian, Weyl 𝑛-algebras, Comm. Math. Phys. 350 (2017), no. 2, 421–442. 10.1007/s00220-017-2835-8Search in Google Scholar

[22] N. Markarian and H. L. Tanaka, Factorization homology in 3-dimensional topology, Mathematical Aspects of Quantum Field Theories, Math. Phys. Stud., Springer, Cham (2015), 213–231. 10.1007/978-3-319-09949-1_7Search in Google Scholar

[23] M. Markl, A compactification of the real configuration space as an operadic completion, J. Algebra 215 (1999), no. 1, 185–204. 10.1006/jabr.1998.7709Search in Google Scholar

[24] C. A. Rossi and T. Willwacher, P. Etingof’s conjecture about Drinfeld associators, preprint (2014), https://arxiv.org/abs/1404.2047. Search in Google Scholar

[25] P. Salvatore, Configuration spaces with summable labels, Cohomological Methods in Homotopy Theory (Bellaterra 1998), Progr. Math. 196, Birkhäuser, Basel (2001), 375–395. 10.1007/978-3-0348-8312-2_23Search in Google Scholar

[26] P. Salvatore and N. Wahl, Framed discs operads and Batalin–Vilkovisky algebras, Q. J. Math. 54 (2003), no. 2, 213–231. 10.1093/qmath/hag012Search in Google Scholar

[27] J. Thomas, Kontsevich’s Swiss cheese conjecture, Geom. Topol. 20 (2016), no. 1, 1–48. 10.2140/gt.2016.20.1Search in Google Scholar

[28] A. A. Voronov, The Swiss-cheese operad, Homotopy Invariant Algebraic Structures (Baltimore 1998), Contemp. Math. 239, American Mathematical Society, Providence (1999), 365–373. 10.1090/conm/239/03610Search in Google Scholar

Received: 2020-06-17
Revised: 2020-11-17
Published Online: 2021-02-02
Published in Print: 2021-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 17.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2020-0158/html?lang=en
Scroll to top button