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 TMF, 2015, Volume 185, Number 3, Pages 371–409 (Mi tmf8951)

Topological recursion for Gaussian means and cohomological field theories

J. E. Andersenab, L. O. Chekhovc, P. Norburyd, R. C. Pennereb

a Center for Quantum Geometry of Moduli Spaces, Århus University, Denmark
b California Institute of Technology, Pasadena, USA
c Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia
d University of Melbourne, Melbourne, Australia
e Institut des Hautes Études Scientifiques, Bures-sur-Yvette, France

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 (openm) moduli spaces $M_{g,s}^\mathrm{disc}$ (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 $mathcal M_{g,1}$
for all $g$ in three ways: using the refined Harer–Zagier recursion, using the Givental-type decomposition of the KPMM, and counting diagrams explicitly.

Keywords: chord diagram, Givental decomposition, Kontsevich–Penner matrix model, discrete volume, moduli space, Deligne–Mumford compactification

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 The research of L. O. Chekhov (the results in Secs. 2, 3, and 6.2) was funded by a grant from the Russian Science Foundation (Project No. 14-50-00005).

DOI: https://doi.org/10.4213/tmf8951

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English version:
Theoretical and Mathematical Physics, 2015, 185:3, 1685–1717

Bibliographic databases:

Citation: J. E. Andersen, L. O. Chekhov, P. Norbury, R. C. Penner, “Topological recursion for Gaussian means and cohomological field theories”, TMF, 185:3 (2015), 371–409; Theoret. and Math. Phys., 185:3 (2015), 1685–1717

Citation in format AMSBIB
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\paper Topological recursion for Gaussian means and cohomological field
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\pages 371--409
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3438626}
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\jour Theoret. and Math. Phys.
\yr 2015
\vol 185
\issue 3
\pages 1685--1717
\crossref{https://doi.org/10.1007/s11232-015-0373-0}
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• http://mi.mathnet.ru/eng/tmf8951
• https://doi.org/10.4213/tmf8951
• http://mi.mathnet.ru/eng/tmf/v185/i3/p371

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This publication is cited in the following articles:
1. Borot G., Garcia-Failde E., “Simple Maps, Hurwitz Numbers, and Topological Recursion”, Commun. Math. Phys.
2. Chekhov L.O., “The Harer–Zagier recursion for an irregular spectral curve”, J. Geom. Phys., 110 (2016), 30–43
3. M. Kontsevich, Ya. Soibelman, “Airy structures and symplectic geometry of topological recursion”, Topological Recursion and Its Influence in Analysis, Geometry, and Topology, Proceedings of Symposia in Pure Mathematics, 100, eds. C. Liu, M. Mulase, Amer. Math. Soc., 2018, 433–489
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