Geodesic
Topological recursion for Gaussian means and cohomological field
Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 3, pp. 371-409 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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, Kontsevich–Penner matrix model, discrete volume, Deligne–Mumford compactification.
Mots-clés : Givental decomposition, moduli space 
@article{TMF_2015_185_3_a0,
     author = {J. E. Andersen and L. O. Chekhov and P. Norbury and R. C. Penner},
     title = {Topological recursion for {Gaussian} means and cohomological field},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {371--409},
     year = {2015},
     volume = {185},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2015_185_3_a0/}
}
TY  - JOUR
AU  - J. E. Andersen
AU  - L. O. Chekhov
AU  - P. Norbury
AU  - R. C. Penner
TI  - Topological recursion for Gaussian means and cohomological field
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2015
SP  - 371
EP  - 409
VL  - 185
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2015_185_3_a0/
LA  - ru
ID  - TMF_2015_185_3_a0
ER  - 
%0 Journal Article
%A J. E. Andersen
%A L. O. Chekhov
%A P. Norbury
%A R. C. Penner
%T Topological recursion for Gaussian means and cohomological field
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2015
%P 371-409
%V 185
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2015_185_3_a0/
%G ru
%F TMF_2015_185_3_a0
J. E. Andersen; L. O. Chekhov; P. Norbury; R. C. Penner. Topological recursion for Gaussian means and cohomological field. Teoretičeskaâ i matematičeskaâ fizika, Tome 185 (2015) no. 3, pp. 371-409. http://geodesic.mathdoc.fr/item/TMF_2015_185_3_a0/

[1] J. Harer, D. Zagier, Invent. Math., 85:3 (1986), 457–485 | DOI | MR | Zbl

[2] É. Brezin, S. Hikami, Commun. Math. Phys., 283:2 (2008), 507–521, arXiv: 0708.2210 | DOI | MR | Zbl

[3] A. Morozov, Sh. Shakirov, From Brezin–Hikami to Harer–Zagier formulas for Gaussian correlators, arXiv: 1007.4100

[4] L. Chekhov, B. Eynard, JHEP, 03 (2006), 014, 18 pp., arXiv: hep-th/0504116 | DOI | MR

[5] B. Eynard, N. Orantin, Commun. Number Theory Phys., 1:2 (2007), 347–452 | DOI | MR | Zbl

[6] S. Gukov, P. Sułkowski, JHEP, 2 (2012), 070, 56 pp., arXiv: 1108.0002 | DOI | MR | Zbl

[7] M. Mulase, P. Sułkowski, Spectral curves and the Schrödinger equations for the Eynard–Orantin recursion, arXiv: 1210.3006

[8] O. Dumitrescu, M. Mulase, Quantization of spectral curves for meromorphic Higgs bundles through topological recursion, arXiv: 1411.1023

[9] J. E. Andersen, L. O. Chekhov, P. Norbury, R. C. Penner, Models of discretized moduli spaces, cohomological field theories, and Gaussian means, arXiv: 1501.05867 | MR

[10] L. Chekhov, Yu. Makeenko, Modern Phys. Lett. A, 7:14 (1992), 1223–1236, arXiv: hep-th/9201033 | DOI | MR | Zbl

[11] J. Ambjørn, L. Chekhov, C. F. Kristjansen, Yu. Makeenko, Nucl. Phys. B, 404:1–2 (1993), 127–172, arXiv: ; Erratum, 449:3 (1995), 681 hep-th/9302014 | DOI | MR | DOI | Zbl

[12] Y. Manin, Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces, American Mathematical Society Colloquium Publications, 47, AMS, Providence, RI, 1999 | MR | Zbl

[13] L. Chekhov, Yu. Makeenko, Phys. Lett. B, 278:3 (1992), 271–278, arXiv: hep-th/9202006 | DOI | MR

[14] A. Marshakov, A. Mironov, A. Morozov, Phys. Lett. B, 265:1–2 (1991), 99–107 | DOI | MR

[15] P. Norbury, Quantum curves and topological recursion, arXiv: 1502.04394 | MR

[16] L. Chekhov, J. Geom. Phys., 12:3 (1993), 153–164, arXiv: hep-th/9205106 | DOI | MR | Zbl

[17] P. Norbury, Trans. Amer. Math. Soc., 365:4 (2013), 1687–1709 | DOI | MR | Zbl

[18] M. Mulase, M. Penkava, Adv. Math., 230:3 (2012), 1322–1339, arXiv: 1009.2135 | DOI | MR | Zbl

[19] P. Norbury, Math. Res. Lett., 17:3 (2010), 467–481 | DOI | MR | Zbl

[20] L. Chekhov, Acta Appl. Math., 48:1 (1997), 33–90, arXiv: hep-th/9509001 | DOI | MR | Zbl

[21] A. B. Givental, Mosc. Math. J., 1:4 (2001), 551–568 | MR | Zbl

[22] B. Eynard, Commun. Number Theory Phys., 8:3 (2014), 541–588, arXiv: 1110.2949 | DOI | MR | Zbl

[23] P. Dunin-Barkowski, N. Orantin, S. Shadrin, L. Spitz, Commun. Math. Phys., 328:2 (2014), 669–700, arXiv: 1211.4021 | DOI | MR | Zbl

[24] B. Eynard, JHEP, 11 (2004), 031, 35 pp., arXiv: hep-th/0407261 | DOI | MR

[25] L. Chekhov, B. Eynard, N. Orantin, JHEP, 12 (2006), 053, 31 pp., arXiv: math-ph/0603003 | DOI | MR | Zbl

[26] A. Alexandrov, A. Mironov, A. Morozov, Internat. J. Modern Phys. A, 19:24 (2004), 4127–4165, arXiv: hep-th/0310113 | DOI | MR

[27] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable Systems and Quantum groups (Montecatini Terme, Italy, June 14–22, 1993), Lecture Notes in Mathematics, 1620, eds. M. Francaviglia, S. Greco, Springer, Berlin, 1996, 120–348, arXiv: hep-th/9407018 | DOI | MR | Zbl

[28] P. Dunin-Barkowski, P. Norbury, N. Orantin, P. Popolitov, S. Shadrin, “Superpotentials and the Eynard–Orantin topological recursion”, In preparation

[29] J. E. Andersen, R. C. Penner, C. M. Reidys, M. S. Waterman, J. Math. Biol., 67:5 (2013), 1261–1278 | DOI | MR | Zbl

[30] J. Harer, Invent. Math., 84:1 (1986), 157–176 | DOI | MR | Zbl

[31] K. Strebel, Quadratic Differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, Springer, Berlin, 1984 | MR | Zbl

[32] R. C. Penner, Commun. Math. Phys., 113:2 (1987), 299–339 | DOI | MR | Zbl

[33] R. C. Penner, J. Differential Geom., 27:1 (1988), 35–53 | DOI | MR | Zbl

[34] M. L. Kontsevich, Commun. Math. Phys., 147:1 (1992), 1–23 | DOI | MR | Zbl

[35] S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, A. Zabrodin, Phys. Lett. B, 275:3–4 (1992), 311–314, arXiv: hep-th/9111037 | DOI | MR

[36] J. Ambjørn, L. Chekhov, Ann. Inst. Henri Poincaré D, 1:3 (2014), 337–361, arXiv: 1404.4240 | DOI | MR | Zbl

[37] A. Alexandrov, A. Mironov, A. Morozov, S. Natanzon, JHEP, 11 (2014), 080, 30 pp., arXiv: 1405.1395 | DOI | MR | Zbl

[38] J. E. Andersen, L. O. Chekhov, C. M. Reidys, R. C. Penner, P. Sułkowski, Nucl. Phys. B, 866:3 (2013), 414–443, arXiv: 1205.0658 | DOI | MR | Zbl

[39] S. Keel, Trans. Amer. Math. Soc., 330:2 (1992), 545–574 | DOI | MR | Zbl

[40] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable Systems and Quantum Groups (Montecatini Terme, Italy, June 14–22, 1993), Lecture Notes in Mathematics, 1620, eds. M. Francaviglia, S. Greco, Springer, Berlin, 1996, 120–348 | DOI | MR | Zbl

[41] C. Teleman, Invent. Math., 188:3 (2012), 525–588 | DOI | MR | Zbl

[42] U. Haagerup, S. Thorbjørnsen, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 15:1 (2012), 1250003, 41 pp. | DOI | MR | Zbl

[43] R. J. Milgram, R. C. Penner, “Riemann's moduli space and the symmetric group”, Mapping Class Groups and Moduli Spaces of Riemann Surfaces (Göttingen, Germany, June 24–28, 1991; Seattle, WA, USA, August 6–10, 1991), Contemporary Mathematics, 150, eds. C.-F. Bödigheimer, R. M. Hain, AMS, Providence, RI, 1993, 247–290 | DOI | MR | Zbl

[44] B. Fang, C.-C. M. Liu, Z. Zong, The Eynard–Orantin recursion and equivariant mirror symmetry for the projective line, arXiv: 1411.3557 | MR