Skip to main content
SpringerLink
Log in

Generalized Cohomological Field Theories in the Higher Order Formalism

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In the classical Batalin–Vilkovisky formalism, the BV operator \(\Delta \) is a differential operator of order two with respect to the commutative product. In the differential graded setting, it is known that if the BV operator is homotopically trivial, then there is a tree level cohomological field theory induced on the homology; this is a manifestation of the fact that the homotopy quotient of the operad of BV algebras by \(\Delta \) is represented by the operad of hypercommutative algebras. In this paper, we study generalized Batalin–Vilkovisky algebras where the operator \(\Delta \) is of the given finite order. In that case, we unravel a new interesting algebraic structure on the homology whenever \(\Delta \) is homotopically trivial. We also suggest that the sequence of algebraic structures arising in the higher order formalism is a part of a “trinity” of remarkable mathematical objects, fitting the philosophy proposed by Arnold in the 1990s.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Notes

  1. In fact, the first two authors of the paper were present, still as high school students, at one of the first lectures he gave on this topic—or at least one of the first lectures of which there are lecture notes [2, 3].

References

  1. Akman, F.: On some generalizations of Batalin–Vilkovisky algebras. J. Pure Appl. Algebra 120(2), 105–141 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Arnold, V.I.: Symplectization, complexification and mathematical trinities. The Arnoldfest (Toronto, ON, 1997). Vol. 24. Fields Inst. Commun. Amer. Math. Soc., Providence, pp. 23–37 (1999)

  3. Arnold, V.I.: Mysterious mathematical trinities. Surveys in modern mathematics, vol. 321. London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, pp. 1–12 (2005)

  4. Barannikov, S., Kontsevich, M.: Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Notices 4, 201–215 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Bashkirov, D., Markl, M.: Calculus of multilinear differential operators, operator \(L_{\infty }\) -algebras and \(IBL_{\infty }\)-algebras. J. Geom. Phys. 173, 40, Paper No. 104431 (2022)

  6. Batalin, I.A., Bering, K., Damgaard, P.H.: Gauge independence of the Lagrangian path integral in a higher-order formalism. Phys. Lett. B 389(4), 673–676 (1996)

    ADS  MathSciNet  Google Scholar 

  7. Batalin, I.A., Bering, K.: Gauge independence in a higher-order Lagrangian formalism via change of variables in the path integral. Phys. Lett. B 742, 23–28 (2015)

    ADS  MathSciNet  MATH  Google Scholar 

  8. Bergeron, F., Labelle, G., Leroux, P.: Combinatorial species and tree-like structures, vol. 67. Encyclopedia of Mathematics and its Applications. Translated from the 1994 French original by Margaret Readdy, With a foreword by Gian-Carlo Rota. Cambridge University Press, Cambridge, pp. xx+457 (1998)

  9. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165(2), 311–427 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  10. Borjeson, K.: \(A_{\infty }\)-algebras derived from associative algebras with a non-derivation differential. J. Gen. Lie Theory Appl. 9(1), 5, Art. ID 1000214 (2015)

  11. Bremner, M.R., Dotsenko, V.: Algebraic Operads. An Algorithmic Companion, pp. xvii+365. CRC Press, Boca Raton (2016)

  12. Chapoton, F.: Les trinités remarquables. (2021). http://irma.math.unistra.fr/chapoton/trinites.html (visited on 09/07/2021)

  13. Chekeres, O., Losev, A.S., Mnev, P., Youmans, D.R.: Theory of holomorphic maps of two-dimensional complex manifolds to toric manifolds and type A multi-string theory. (2021). arXiv: 2111.06836

  14. Dechant, P.-P.: From the trinity (A3, B3, H3) to an ADE correspondence. Proc. A. 474(2220), 20180034, 19 (2018)

    MathSciNet  MATH  Google Scholar 

  15. Dotsenko, V.: Word operads and admissible orderings. Appl. Categ. Struct. 28(4), 595–600 (2020)

    MathSciNet  MATH  Google Scholar 

  16. Dotsenko, V., Griffin, J.: Cacti and filtered distributive laws. Algebr. Geom. Topol. 14(6), 3185–3225 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Dotsenko, V., Khoroshkin, A.: Gröbner bases for operads. Duke Math. J. 153(2), 363–396 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Dotsenko, V., Khoroshkin, A.: Quillen homology for operads via Gröbner bases. Doc. Math. 18, 707–747 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Dotsenko, V., Khoroshkin, A.: Shuffle algebras, homology, and consecutive pattern avoidance. Algebra Number Theory 7(3), 673–700 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Dotsenko, V., Shadrin, S., Vallette, B.: Givental group action on topological field theories and homotopy Batalin–Vilkovisky algebras. Adv. Math. 236, 224–256 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Dotsenko, V., Shadrin, S., Vallette, B.: De Rham cohomology and homotopy Frobenius manifolds. J. Eur. Math. Soc. (JEMS) 17(3), 535–547 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Dotsenko, V., Shadrin, S., Vallette, B.: Givental action and trivialisation of circle action. J. Éc. Polytech. Math. 2, 213–246 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Dotsenko, V., Shadrin, S., Vallette, B.: Pre-Lie deformation theory. Mosc. Math. J. 16(3), 505–543 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Dotsenko, V., Shadrin, S., Vallette, B.: The twisting procedure. (2018). arXiv:1810.02941

  25. Dotsenko, V., Shadrin, S., Vallette, B.: Toric varieties of Loday’s associahedra and noncommutative cohomological field theories. J. Topol. 12(2), 463–535 (2019)

    MathSciNet  MATH  Google Scholar 

  26. Dotsenko, V., Vejdemo-Johansson, M.: Implementing Gröbner bases for operads. In: OPERADS 2009, vol. 26. Sémin, pp. 77–98. Congr. Soc. Math. France, Paris (2013)

  27. Drummond-Cole, G.C., Vallette, B.: The minimal model for the Batalin-Vilkovisky operad. Selecta Math. (N.S.) 19(1), 1–47 (2013)

    MathSciNet  MATH  Google Scholar 

  28. Dunin-Barkowski, P., Shadrin, S., Spitz, L.: Givental graphs and inversion symmetry. Lett. Math. Phys. 103(5), 533–557 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  29. Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. Special Volume, Part II. GAFA 2000 (Tel Aviv, 1999), pp. 560–673 (2000)

  30. Escobar, L.: Bott-Samelson varieties, subword complexes and brick polytopes. In: 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014). Discrete Math. Theor. Comput. Sci. Proc., AT. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, pp. 863–874 (2014)

  31. Gálvez-Carrillo, I., Tonks, A., Vallette, B.: Homotopy Batalin–Vilkovisky algebras. J. Noncommun. Geom. 6(3), 539–602 (2012)

    MathSciNet  MATH  Google Scholar 

  32. Getzler, E.: Batalin–Vilkovisky algebras and two-dimensional topological field theories. Commun. Math. Phys. 159(2), 265–285 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  33. Getzler, E.: Two-dimensional topological gravity and equivariant cohomology. Commun. Math. Phys. 163(3), 473–489 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  34. Getzler, E.: Operads and moduli spaces of genus 0 Riemann surfaces. The moduli space of curves (Texel Island, 1994), vol. 129, pp. 199–230. Progr. Math. Birkhäuser Boston, Boston (1995)

  35. Ginzburg, V., Kapranov, M.: Koszul duality for operads. Duke Math. J. 76(1), 203–272 (1994)

    MathSciNet  MATH  Google Scholar 

  36. Ginzburg, V., Schedler, T.: Differential operators and BV structures in noncommutative geometry. Selecta Math. (N.S.) 16(4), 673–730 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Hoffbeck, E.: A Poincaré-Birkhoff-Witt criterion for Koszul operads. Manuscr. Math. 131(1–2), 87–110 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Joyal, A.: Une théorie combinatoire des séries formelles. Adv. Math. 42(1), 1–82 (1981)

    MathSciNet  MATH  Google Scholar 

  39. Khoroshkin, A., Markarian, N., Shadrin, S.: Hypercommutative operad as a homotopy quotient of BV. Commun. Math. Phys. 322(3), 697–729 (2013)

    ADS  MathSciNet  MATH  Google Scholar 

  40. Khoroshkin, A.: PBW property for associative universal enveloping algebras over an operad. Int. Math. Res. Not. IMRN 4, 3106–3143 (2022)

    MathSciNet  MATH  Google Scholar 

  41. Kontsevich, M., Manin, Yu.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164(3), 525–562 (1994)

    ADS  MathSciNet  MATH  Google Scholar 

  42. Koszul, J.-L.: Crochet de Schouten-Nijenhuis et cohomologie. Numéro Hors Série. The mathematical heritage of Élie Cartan (Lyon, 1984), pp. 257–271 (1985)

  43. Kravchenko, O.: Deformations of Batalin–Vilkovisky algebras. Poisson geometry (Warsaw, 1998). Vol. 51. Banach Center Publ. Polish Acad. Sci. Inst. Math., Warsaw, pp. 131–139 (2000)

  44. Laubie, P.: The Losev–Manin twisted associative algebra. Internship report, University of Strasbourg (2021)

  45. Le Bruyn, L.: Arnold’s trinities. (2008). http://www.neverendingbooks.org/arnolds-trinities (visited on 09/07/2021)

  46. Le Bryun, L.: Arnold’s trinities version 2.0. (2008). http://www.neverendingbooks.org/arnolds-trinities-version-20 (visited on 09/07/2021)

  47. Loday, J.-L., Vallette, B.: Algebraic operads, vol. 346. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], pp. xxiv+634. Springer, Heidelberg (2012)

  48. Losev, A., Manin, Y.: New moduli spaces of pointed curves and pencils of flat connections, vol. 48. Dedicated to William Fulton on the occasion of his 60th birthday, pp. 443–472 (2000)

  49. Losev, A., Nekrasov, N., Shatashvili, S.: Issues in topological gauge theory. Nuclear Phys. B 534(3), 549–611 (1998)

    ADS  MathSciNet  MATH  Google Scholar 

  50. Losev, A., Polyubin, I.: Commutativity equations and dressing transformations. JETP Lett. 77, 53–57 (2003)

    ADS  Google Scholar 

  51. Losev, A., Shadrin, S.: From Zwiebach invariants to Getzler relation. Commun. Math. Phys. 271(3), 649–679 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  52. Losev, A., Shadrin, S., Shneiberg, I.: Tautological relations in Hodge field theory. Nuclear Phys. B 786(3), 267–296 (2007)

    ADS  MathSciNet  MATH  Google Scholar 

  53. Losev, A., Moore, G., Nekrasov, N., Shatashvili, S.: Four-dimensional avatars of two-dimensional RCFT. Nuclear Phys. B Proc. Suppl. 46, 130–145 (1996)

    ADS  MathSciNet  MATH  Google Scholar 

  54. Lysov, V.: Anticommutativity equation in topological quantum mechanics. JETP Lett. 76, 724–727 (2002)

    ADS  Google Scholar 

  55. Markl, M.: Higher braces via formal (non)commutative geometry. Geometric methods in physics. Trends Math, pp. 67–81. Birkhäuser/Springer, Cham (2015)

  56. Markl, M.: On the origin of higher braces and higher-order derivations. J. Homotopy Relat. Struct. 10(3), 637–667 (2015)

    MathSciNet  MATH  Google Scholar 

  57. Markl, M.: Permutads via operadic categories, and the hidden associahedron. J. Combin. Theory Ser. A 175, 105277, 40 (2020)

  58. Méndez, M.A.: Koszul duality for monoids and the operad of enriched rooted trees. Adv. Appl. Math. 44(3), 261–297 (2010)

    MathSciNet  MATH  Google Scholar 

  59. Merkulov, S.A.: Formality of canonical symplectic complexes and Frobenius manifolds. Int. Math. Res. Notices 14, 727–733 (1998)

    MathSciNet  MATH  Google Scholar 

  60. Positsel’skiĭ, L.E.: Nonhomogeneous quadratic duality and curvature. Funktsional. Anal. i Prilozhen. 27(3) , 57.66, 96 (1993)

  61. Priddy, S.B.: Koszul resolutions. Trans. Am. Math. Soc. 152, 39–60 (1970)

    MathSciNet  MATH  Google Scholar 

  62. Procesi, C.: The Toric Variety Associated to Weyl Chambers, pp. 153–161. Mots. Lang. Raison. Calc. Hermès, Paris (1990)

    MATH  Google Scholar 

  63. Shadrin, S.: BCOV theory via Givental group action on cohomological fields theories. Mosc. Math. J. 9(2), 411–429 (2009), back matter

  64. Shadrin, S., Zvonkine, D.: A group action on Losev-Manin cohomological field theories. Ann. Inst. Fourier (Grenoble) 61(7), 2719–2743 (2011)

    MathSciNet  MATH  Google Scholar 

  65. Soukhanov, L.: An operad from the secondary polytope (2015). arXiv:1505.08157

  66. Tamaroff, P.: The cohomology of coalgebras in species. Commun. Algebra 50(7), 2811–2830 (2022)

    MathSciNet  MATH  Google Scholar 

  67. Vallette, B.: Homotopy theory of homotopy algebras. Ann. Inst. Fourier (Grenoble) 70(2), 683–738 (2020)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Basile Coron, Paul Laubie, Andrey Losev, Sergei Merkulov and Bruno Vallette for useful discussions. V. D. was supported by Institut Universitaire de France, by University of Strasbourg Institute for Advanced Study (USIAS) through the Fellowship USIAS-2021-061 within the French national program “Investment for the future” (IdEx-Unistra), and by the French national research agency project ANR-20-CE40-0016. S. S. was supported by the Netherlands Organization for Scientific Research. S. S. was also partially supported by International Laboratory of Cluster Geometry NRU HSE, RF Government grant, ag. No 075-15-2021-608 dated 08.06.2021

Author information

Authors and Affiliations

  1. Institut de Recherche Mathématique Avancée, UMR 7501, Université de Strasbourg et CNRS, 7 Rue René-Descartes, 67000, Strasbourg, France

    Vladimir Dotsenko

  2. Korteweg-de Vriesinstituut voor Wiskunde, Universiteit van Amsterdam, Postbus 94248, 1090GE, Amsterdam, The Netherlands

    Sergey Shadrin

  3. Institut für Mathematik Johann von Neumann-Haus, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Pedro Tamaroff

Authors
  1. Vladimir Dotsenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sergey Shadrin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pedro Tamaroff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergey Shadrin.

Additional information

Communicated by H.-T. Yau.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dotsenko, V., Shadrin, S. & Tamaroff, P. Generalized Cohomological Field Theories in the Higher Order Formalism. Commun. Math. Phys. 399, 1439–1500 (2023). https://doi.org/10.1007/s00220-022-04577-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-022-04577-6

Search

Navigation