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Models of discretized moduli spaces, cohomological field theories, and Gaussian means

Andersen, Jørgen Ellegaard and Chekhov, Leonid O. and Norbury, Paul and Penner, Robert C. (2015) Models of discretized moduli spaces, cohomological field theories, and Gaussian means. Journal of Geometry and Physics, 98 . pp. 312-339. ISSN 0393-0440.

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We prove combinatorially the explicit relation between genus filtrated ss-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich–Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces M_(g,s)^(disc) given by N_(g,s)(P_1,…,P_s) for (P_1,…,P_s)∈Z_+^s. This generating function therefore enjoys the topological recursion, and we prove that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition. We use another Givental-type decomposition obtained for this model by the second authors in 1995 in terms of special times related to the discretization of moduli spaces thus representing its asymptotic expansion terms (and therefore those of the Gaussian means) as finite sums over graphs weighted by lower-order monomials in times thus giving another proof of (quasi)polynomiality of the discrete volumes. As an application, we find the coefficients in the first subleading order for M_(g,1) in two ways: by using the refined Harer–Zagier recursion and by exploiting the above Givental-type transformation. We put forward the conjecture that the above graph expansions can be used for probing the reduction structure of the Deligne–Mumford compactification M_(g,s) of moduli spaces of punctured Riemann surfaces.

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Additional Information:© 2015 Elsevier B.V. Received 22 July 2015, Accepted 19 August 2015, Available online 28 August 2015. This paper has been started in 2011 and during the years of writing it we had numerous useful discussions with B. Eynard, A.D. Mironov, S. Shadrin, P. Dunin-Barkovsky, M. Mulase, C. Reidys, P. Sułkowski, and Don Zagier to all of whom we are deeply grateful. The work of J.E.A., L.O.Ch. and R.C.P. was partially supported by the center of excellence grant "Centre for Quantum Geometry of Moduli Spaces" from the Danish National Research Foundation (DNRF95). The work of L.O.Ch. was supported by the Russian Foundation for Basic Research (Grant Nos. 14-01-00860-a and 13-01-12405-ofi-m2). The work of P.N. was partially supported by the Australian Research Council grant DP1094328.
Funding AgencyGrant Number
Danish National Research FoundationDNRF95
Russian Foundation for Basic Research14-01-00860-a
Russian Foundation for Basic Research13-01-12405-ofi-m2
Australian Research CouncilDP1094328
Subject Keywords:Chord diagram; Kontsevich–Penner matrix model; Discrete volumes; Harer–Zagier recursion; Givental decomposition; Ancestor invariants
Record Number:CaltechAUTHORS:20151204-094841968
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Official Citation:Jørgen Ellegaard Andersen, Leonid O. Chekhov, Paul Norbury, Robert C. Penner, Models of discretized moduli spaces, cohomological field theories, and Gaussian means, Journal of Geometry and Physics, Volume 98, December 2015, Pages 312-339, ISSN 0393-0440, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:62610
Deposited By: Ruth Sustaita
Deposited On:04 Dec 2015 18:13
Last Modified:03 Oct 2019 09:20

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