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Topological recursion for Gaussian means and cohomological field theories

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Abstract

We introduce explicit relations between genus-filtrated s-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM), which is the generating function for volumes of discretized (open) moduli spaces M disc g,s (discrete volumes). Using these relations, we express Gaussian means in all orders of the genus expansion as polynomials in special times weighted by ancestor invariants of an underlying cohomological field theory. We translate the topological recursion of the Gaussian model into recurrence relations for the coefficients of this expansion, which allows proving that they are integers and positive. We find the coefficients in the first subleading order for M g,1 for all g in three ways: using the refined Harer–Zagier recursion, using the Givental-type decomposition of the KPMM, and counting diagrams explicitly.

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Author information

Authors and Affiliations

  1. Center for Quantum Geometry of Moduli Spaces, Århus University, Copenhagen, Denmark

    J. E. Andersen & R. C. Penner

  2. California Institute of Technology, Pasadena, California, USA

    J. E. Andersen

  3. Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

    L. O. Chekhov

  4. University of Melbourne, Melbourne, Australia

    P. Norbury

  5. Institut des Hautes Études Scientifique, Bures-sur-Yvette, France

    R. C. Penner

Authors
  1. J. E. Andersen
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  2. L. O. Chekhov
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  3. P. Norbury
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  4. R. C. Penner
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Corresponding author

Correspondence to L. O. Chekhov.

Additional information

Sections 2, 3, and 6.2 were written by L. O. Chekhov, and Secs. 1, 4, 5, and 7 and also the other parts of Sec. 6 were written by J. E. Andersen, P. Norbury, and R. C. Penner.

The research of L. O. Chekhov was funded by a grant from the Russian Science Foundation (Project No. 14-50-00005) and was performed in Steklov Mathematical Institute of Russian Academy of Sciences.

Prepared from an English manuscript submitted by the authors; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 185, No. 3, pp. 371–409, December, 2015.

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Andersen, J.E., Chekhov, L.O., Norbury, P. et al. Topological recursion for Gaussian means and cohomological field theories. Theor Math Phys 185, 1685–1717 (2015). https://doi.org/10.1007/s11232-015-0373-0

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