Skip to main content
SpringerLink
Log in

Cut-and-join operators in cohomological field theory and topological recursion

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We construct a cubic cut-and-join operator description for the partition function of the Chekhov–Eynard–Orantin topological recursion for a local spectral curve with simple ramification points. In particular, this class contains partition functions of all semi-simple cohomological field theories. The cut-and-join description leads to an algebraic version of topological recursion. For the same partition functions we also derive N families of the Virasoro constraints and prove that these constraints, supplemented by a deformed dimension constraint, imply the cut-and-join description.

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

Similar content being viewed by others

References

  1. Andersen, J.E., Borot, G., Chekhov, L.O., Orantin, N.: The ABCD of topological recursion (2017). arXiv:1703.03307

  2. Alexandrov, A.: Cut-and-join operator representation for Kontsevich–Witten tau-function. Modern Phys. Lett. A 26(29), 2193–2199 (2011). https://doi.org/10.1142/S0217732311036607

    Article  MathSciNet  Google Scholar 

  3. Alexandrov, A.: Cut-and-join description of generalized Brezin–Gross–Witten model. Adv. Theor. Math. Phys. 22(6), 1347–1399 (2018). https://doi.org/10.4310/ATMP.2018.v22.n6.a1

    Article  MathSciNet  Google Scholar 

  4. Alexandrov, A.: KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model. Anal. Math. Phys., 11(1):Paper No. 24, 82 (2021). https://doi.org/10.1007/s13324-020-00451-7

  5. Alexandrov, A.: KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions (2021). arXiv:2108.10023

  6. Alexandrov, A.: Cut-and-join operators for higher Weil–Petersson volumes. Bull. Lond. Math. Soc. 55(6), 3012–3028 (2023). https://doi.org/10.1112/blms.12907

    Article  MathSciNet  Google Scholar 

  7. Alexandrov, A., Mironov, A., Morozov, A.: Instantons and Merons in matrix models. Phys. D 235(1–2), 126–167 (2007). https://doi.org/10.1016/j.physd.2007.04.018

    Article  MathSciNet  Google Scholar 

  8. Alexandrov, A., Mironov, A., Morozov, A.: BGWM as second constituent of complex matrix model. J. High Energy Phys. 12, 053, 49 (2009). https://doi.org/10.1088/1126-6708/2009/12/053

    Article  MathSciNet  Google Scholar 

  9. Borot, G., Bouchard, V., Chidambaram, N.K., Creutzig, T., Noshchenko, D.: Higher Airy structures, W algebras and topological recursion (2018). arXiv:1812.08738

  10. Chidambaram, N.K., Garcia-Failde, E., Giacchetto, A.: Relations on \(\overline{\cal{M}}_{g,n}\) and the negative \(r\)-spin Witten conjecture (2022). arXiv:2205.15621

  11. Chekhov, L., Norbury, P.: Topological recursion with hard edges. Int. J. Math. 30(3), 1950014, 29 (2019). https://doi.org/10.1142/S0129167X19500149

    Article  MathSciNet  Google Scholar 

  12. Dunin-Barkowski, P., Orantin, N., Shadrin, S., Spitz, L.: Identification of the Givental formula with the spectral curve topological recursion procedure. Commun. Math. Phys. 328(2), 669–700 (2014). https://doi.org/10.1007/s00220-014-1887-2

    Article  MathSciNet  Google Scholar 

  13. Do, N., Norbury, P.: Topological recursion on the Bessel curve. Commun. Number Theory Phys. 12(1), 53–73 (2018). https://doi.org/10.4310/CNTP.2018.v12.n1.a2

    Article  MathSciNet  Google Scholar 

  14. Dubrovin, B.: Geometry of \(2\)D topological field theories. In: Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Volume 1620 of Lecture Notes in Mathematics, pp. 120–348. Springer, Berlin (1996). https://doi.org/10.1007/BFb0094793

  15. Dijkgraaf, R., Verlinde, H., Verlinde, E.: Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity. Nuclear Phys. B 348(3), 435–456 (1991). https://doi.org/10.1016/0550-3213(91)90199-8

    Article  MathSciNet  Google Scholar 

  16. Dubrovin, B., Zhang, Y.: Frobenius manifolds and Virasoro constraints. Sel. Math. (N.S.) 5(4), 423–466 (1999). https://doi.org/10.1007/s000290050053

    Article  MathSciNet  Google Scholar 

  17. Eynard, B., Orantin, N.: Invariants of algebraic curves and topological expansion. Commun. Number Theory Phys. 1(2), 347–452 (2007). https://doi.org/10.4310/CNTP.2007.v1.n2.a4

    Article  MathSciNet  Google Scholar 

  18. Eynard, B., Orantin, N.: Topological recursion in enumerative geometry and random matrices. J. Phys. A 42(29), 293001, 117 (2009). https://doi.org/10.1088/1751-8113/42/29/293001

    Article  MathSciNet  Google Scholar 

  19. Eynard, B.: Invariants of spectral curves and intersection theory of moduli spaces of complex curves. Commun. Number Theory Phys. 8(3), 541–588 (2014). https://doi.org/10.4310/CNTP.2014.v8.n3.a4

    Article  MathSciNet  Google Scholar 

  20. Fukuma, M., Kawai, H., Nakayama, R.: Continuum Schwinger–Dyson equations and universal structures in two-dimensional quantum gravity. Int. J. Modern Phys. A 6(8), 1385–1406 (1991). https://doi.org/10.1142/S0217751X91000733

    Article  MathSciNet  Google Scholar 

  21. Givental, A.B.: Gromov–Witten invariants and quantization of quadratic Hamiltonians. vol. 1, pp. 551–568, 645 (2001). Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. https://doi.org/10.17323/1609-4514-2001-1-4-551-568

  22. Givental, A.B.: Semisimple Frobenius structures at higher genus. Int. Math. Res. Not. 23, 1265–1286 (2001). https://doi.org/10.1155/S1073792801000605

    Article  MathSciNet  Google Scholar 

  23. Gross, D.J., Newman, M.J.: Unitary and Hermitian matrices in an external field. II. The Kontsevich model and continuum Virasoro constraints. Nuclear Phys. B 380(1–2), 168–180 (1992). https://doi.org/10.1016/0550-3213(92)90520-L

    Article  MathSciNet  Google Scholar 

  24. Kontsevich, M., Manin, Yu.: Gromov–Witten classes, quantum cohomology, and enumerative geometry. Commun. Math. Phys. 164(3), 525–562 (1994). https://doi.org/10.1007/BF02101490

    Article  MathSciNet  Google Scholar 

  25. Kazarian, M., Norbury, P.: Polynomial relations among kappa classes on the moduli space of curves. Int. Math. Res. Not. (2023). https://doi.org/10.1093/imrn/rnad061

    Article  Google Scholar 

  26. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147(1), 1–23 (1992). https://doi.org/10.1007/BF02099526

    Article  MathSciNet  Google Scholar 

  27. Kontsevich, M., Soibelman, Y.: Airy structures and symplectic geometry of topological recursion. In: Topological Recursion and Its influence in Analysis, Geometry, and Topology, volume 100 of Proceedings of Symposia in Pure Mathematics, pp. 433–489. American Mathematics Society, Providence, RI (2018). https://doi.org/10.1090/pspum/100/01765

  28. Manin, Y.I.: Frobenius manifolds, quantum cohomology, and moduli spaces, volume 47 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence, RI (1999). https://doi.org/10.1090/coll/047

  29. Milanov, T.: The Eynard–Orantin recursion for the total ancestor potential. Duke Math. J. 163(9), 1795–1824 (2014). https://doi.org/10.1215/00127094-2690805

    Article  MathSciNet  Google Scholar 

  30. Mironov, A., Morozov, A., Semenoff, G.W.: Unitary matrix integrals in the framework of the generalized Kontsevich model. Int. J. Modern Phys. A 11(28), 5031–5080 (1996). https://doi.org/10.1142/S0217751X96002339

    Article  MathSciNet  Google Scholar 

  31. Morozov, A., Shakirov, Sh.: Generation of matrix models by \(\hat{W}\)-operators. J. High Energy Phys. (4):064, 33 (2009). https://doi.org/10.1088/1126-6708/2009/04/064

  32. Norbury, P.: Enumerative geometry via the moduli space of super Riemann surfaces (2020). arXiv:2005.04378

  33. Norbury, P.: A new cohomology class on the moduli space of curves. Geom. Topol. 27(7), 2695–2761 (2023). https://doi.org/10.2140/gt.2023.27.2695

    Article  MathSciNet  Google Scholar 

  34. Pandharipande, R.: Cohomological field theory calculations. In: Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, pp. 869–898. World Scientific Publishing, Hackensack (2018). https://doi.org/10.1142/9789813272880_0031

  35. Pandharipande, R., Pixton, A., Zvonkine, D.: Relations on \(\overline{\cal{M} }_{g, n}\) via \(3\)-spin structures. J. Am. Math. Soc. 28(1), 279–309 (2015). https://doi.org/10.1090/S0894-0347-2014-00808-0

    Article  MathSciNet  Google Scholar 

  36. Shadrin, S.: BCOV theory via Givental group action on cohomological fields theories. Mosc. Math. J. 9(2), 411–429 (2009). https://doi.org/10.17323/1609-4514-2009-9-2-411-429

    Article  MathSciNet  Google Scholar 

  37. Teleman, C.: The structure of 2D semi-simple field theories. Invent. Math. 188(3), 525–588 (2012). https://doi.org/10.1007/s00222-011-0352-5

    Article  MathSciNet  Google Scholar 

  38. Witten, E.: Two-dimensional gravity and intersection theory on moduli space. In: Surveys in Differential Geometry (Cambridge, MA, 1990), pp. 243–310. Lehigh University, Bethlehem, PA (1991). https://doi.org/10.4310/SDG.1990.v1.n1.a5

Download references

Acknowledgements

The author is grateful to Sergey Shadrin and Gaëtan Borot for the inspiring discussions and to the anonymous referee of the paper for the useful remarks. This work was supported by the Institute for Basic Science (IBS-R003-D1). The author would like to thank IHES, where the part of this project was done, for the kind hospitality.

Author information

Authors and Affiliations

  1. Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang, 37673, Korea

    Alexander Alexandrov

Authors
  1. Alexander Alexandrov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Alexandrov.

Additional information

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

Alexandrov, A. Cut-and-join operators in cohomological field theory and topological recursion. Sel. Math. New Ser. 30, 45 (2024). https://doi.org/10.1007/s00029-024-00933-7

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00029-024-00933-7

Keywords

Mathematics Subject Classification

Search

Navigation