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

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. 1.

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

  2. 2.

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

  3. 3.

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

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Acknowledgements

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

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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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Keywords

  • Moduli space of curves
  • \(\kappa \) classes
  • \(\psi \) classes
  • Witten conjecture
  • Enumerative geometry
  • Intersection theory

Mathematics Subject Classification

  • 14N10
  • 14N35