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Witten’s conjecture and recursions for \(\kappa \) classes

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Abstract

We construct a countable number of differential operators \(\widehat{L}_n\) that annihilate a generating function for intersection numbers of \(\kappa \) classes on \(\overline{{\mathscr {M}}}_g\) (the \(\kappa \)-potential). This produces recursions among intersection numbers of \(\kappa \) classes which determine all such numbers from a single initial condition. The starting point of the work is a combinatorial formula relating intersection numbers of \(\psi \) and \(\kappa \) classes. Such a formula produces an exponential differential operator acting on the Gromov–Witten potential to produce the \(\kappa \)-potential; after restricting to a hyperplane, we have an explicit change of variables relating the two generating functions, and we conjugate the “classical” Virasoro operators to obtain the operators \(\widehat{L}_n\).

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Notes

  1. If \(g_i = 0\), we require \(|P_i| \hbox {\,\,\char 062\,}2\).

  2. We are not concerned with issues of convergence here; see [7] for a discussion.

  3. It is important to note that the coefficients of the vector field are functions in variables that commute with the \(\partial _{t_i}\).

References

  1. Alexeev, V., Guy, G.M.: Moduli of weighted stable maps and their gravitational descendants. J. Inst. Math. Jussieu 7(3), 425–456 (2008)

    Article  MathSciNet  Google Scholar 

  2. Arbarello, E., Cornalba, M.: Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves. J. Algebr. Geom. 5(4), 705–749 (1996)

    MathSciNet  MATH  Google Scholar 

  3. Arbogast, L.F.A.: Du calcul des Derivations. Levrault, Strasbourg (1855)

    Google Scholar 

  4. Bell, E.T.: Partition polynomials. Ann. Math. 29(1–4), 38–46 (1927/1928)

  5. Blankers, V., Cavalieri, R.: Intersections of \(\omega \) classes in \(\overline{{\mathscr {M}}}_{g,n}\). In: Akbulut, S., et al. (eds.) Proceedings of the Gökova Geometry-Topology Conference 2017, pp. 37–52. Inernational Press, Somerville (2017)

  6. Blankers, V., Cavalieri, R.: Wall-crossings for Hassett descendant potentials. Int. Math. Res. Not. IMRN # Rnaa077 (2020). https://doi.org/10.1093/imrn/rnaa077

  7. Coates, T., Iritani, H.: On the convergence of Gromov–Witten potentials and Givental’s formula. Mich. Math. J. 64(3), 587–631 (2015)

    Article  MathSciNet  Google Scholar 

  8. Faa di Bruno, F.: Sullo sviluppo delle funzioni. Annali di Scienze Matematiche e Fisiche (1855)

  9. Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer, New York (1998)

  10. Hori, K., Katz, S., Klemm, A., Pandharipande, R., Thomas, R., Vafa, C., Vakil, R., Zaslow, E.: Mirror Symmetry. Clay Mathematics Monographs, vol. 1. American Mathematical Society, Providence (2003)

  11. Kaufmann, R., Manin, Y., Zagier, D.: Higher Weil–Petersson volumes of moduli spaces of stable \(n\)-pointed curves. Commun. Math. Phys. 181(3), 763–787 (1996)

    Article  MathSciNet  Google Scholar 

  12. Knudsen, F.F.: The projectivity of the moduli space of stable curves. II. The stacks \(M_{g, n}\). Math. Scand. 52(2), 161–199 (1983)

    Article  MathSciNet  Google Scholar 

  13. Knudsen, F.F.: The projectivity of the moduli space of stable curves. III. The line bundles on \(M_{g, n}\), and a proof of the projectivity of \({\overline{M}}_{g, n}\) in characteristic \(0\). Math. Scand. 52(2), 200–212 (1983)

    Article  MathSciNet  Google Scholar 

  14. Kock, J.: Notes on psi classes (2001). http://mat.uab.es/~kock/GW/notes/psi-notes.pdf

  15. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys. 147(1), 1–23 (1992)

    Article  MathSciNet  Google Scholar 

  16. Madsen, I., Weiss, M.: The stable moduli space of Riemann surfaces: Mumford’s conjecture. Ann. Math. (2) 165(3), 843–941 (2007)

    Article  MathSciNet  Google Scholar 

  17. Manin, YuI, Zograf, P.: Invertible cohomological field theories and Weil–Petersson volumes. Ann. Inst. Fourier (Grenoble) 50(2), 519–535 (2000)

    Article  MathSciNet  Google Scholar 

  18. Mumford, D.: Toward an enumerative geometry of the moduli space of curves. In: Artin, M., Tate, J. (eds.) Arithmetic and Geometry, vol. II(36). Progress in Mathematics, vol. 36, pp. 271–326. Birkhäuser, Boston (1983)

  19. Pandharipande, R.: The \(\varkappa \) ring of the moduli of curves of compact type. Acta Math. 208(2), 335–388 (2012)

    Article  MathSciNet  Google Scholar 

  20. Pandharipande, R.: A calculus for the moduli space of curves. In: de Fernex, T., et al. (eds.) Algebraic Geometry: Salt Lake City 2015. Proceedings of Symposia in Pure Mathematics, vol. 97.1, pp. 459–487. American Mathematical Society, Providence (2018)

  21. Vakil, R.: The moduli space of curves and Gromov-Witten theory. In: Behrend, K., Manetti, M. (eds.) Enumerative Invariants in Algebraic Geometry and String Theory. Lecture Notes in Mathematics, vol. 1947, pp. 143–198. Springer, Berlin (2008)

  22. 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 (1991)

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Acknowledgements

We would like to thank Nick Ercolani and Hiroshi Iritani for helpful conversations.

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  1. Department of Mathematics, Northeastern University, Boston, MA, 02115-5005, USA

    Vance Blankers

  2. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523-1874, USA

    Renzo Cavalieri

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  1. Vance Blankers
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  2. Renzo Cavalieri
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Correspondence to Renzo Cavalieri.

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R.C. acknowledges support from Simons Foundation Collaboration Grant 420720.

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Blankers, V., Cavalieri, R. Witten’s conjecture and recursions for \(\kappa \) classes. European Journal of Mathematics 7, 309–339 (2021). https://doi.org/10.1007/s40879-020-00430-z

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