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Blobbed topological recursion: properties and applications

Published online by Cambridge University Press:  27 May 2016

GAËTAN BOROT  and
SERGEY SHADRIN
GAËTAN BOROT
Affiliation:
Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany. e-mail: gborot@mpim-bonn.mpg.de
SERGEY SHADRIN
Affiliation:
Korteweg de Vries Instituut voor Wiskunde, Universiteit van Amsterdam, P.O. Box 94248, 1090 GE Amsterdam, Netherlands. e-mail: s.shadrin@uva.nl
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Abstract

We study the set of solutions (ωg,n )g⩾0,n⩾1 of abstract loop equations. We prove that ωg,n is determined by its purely holomorphic part: this results in a decomposition that we call “blobbed topological recursion”. This is a generalisation of the theory of the topological recursion, in which the initial data (ω0,1, ω0,2) is enriched by non-zero symmetric holomorphic forms in n variables (φg,n )2g−2+n>0. In particular, we establish for any solution of abstract loop equations: (1) a graphical representation of ωg,n in terms of φg,n ; (2) a graphical representation of ωg,n in terms of intersection numbers on the moduli space of curves; (3) variational formulas under infinitesimal transformation of φg,n ; (4) a definition for the free energies ωg,0 = Fg respecting the variational formulas. We discuss in detail the application to the multi-trace matrix model and enumeration of stuffed maps.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 1 , January 2017 , pp. 39 - 87
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Arbarello, E., Cornalba, M. and Griffiths, P. II Geometry of algebraic curves. Grundlehren der mathematischen Wissenschaften. vol. 268 (Springer).Google Scholar
[2] Borot, G. Formal multidimensional integrals, stuffed maps, and topological recursion. Ann. Institut Poincaré D. 1 (2) (2014), 225–264. math-ph/1307.4957.Google Scholar
[3] Borot, G. and Eynard, B. Geometry of spectral curves and all order dispersive integrable system. SIGMA. 8 (100) (2012). math-ph/1110.4936.Google Scholar
[4] Borot, G. and Eynard, B. Root systems, spectral curves, and analysis of a Chern–Simons matrix model for Seifert fibered spaces. (2016). To appear in Selecta Mathematica. math-ph/1407.4500.Google Scholar
[5] Borot, G. and Eynard, B. All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials. Quantum Topology. (2015), math-ph/1205.2261.Google Scholar
[6] Borot, G., Eynard, B. and Orantin, N. Abstract loop equations, topological recursion, and applications. Commun. Number Theory and Physics. 9 (1) (2015), 51–187.Google Scholar
[7] Borot, G. and Guionnet, A. Asymptotic expansion of β matrix models in the one-cut regime. Commun. Math. Phys. 317 (2) (2013), 447–483. math-PR/1107.1167.Google Scholar
[8] Borot, G., Guionnet, A. and Kozlowski, K. Large-N asymptotic expansion for mean field models with Coulomb gas interaction. Inft. Math. Res. Nst. (2015). math-ph/1312.6664.Google Scholar
[9] Bouchard, V., Klemm, A., Mariño, M. and Pasquetti, S. Remodeling the B-model. Commun. Math. Phys. 287 (2009), 117–178. hep-th/0709.1453.Google Scholar
[10] Brézin, É., Itzykson, C., Parisi, G. and Zuber, J.-B. Planar diagrams. Commun. Math. Phys. 59 (1978), 35–51.Google Scholar
[11] Brini, A., Eynard, B. and Mariño, M. Torus knots and mirror symmetry. Ann. Henri Poincaré. (2012), hep-th/1105.2012.Google Scholar
[12] Dijkgraaf, R., Fuji, H. and Manabe, M. The volume conjecture, perturbative knot invariants, and recursion relations for topological strings. Nucl. Phys. B. 849 (2011), 166–211. hep-th/1010.4542.Google Scholar
[13] Dunin-Barkowski, P., Orantin, N., Shadrin, S. and Spitz, L. Identification of the Givental formula with the spectral curve topological recursion procedure. Commun. Math. Phys. 328 (2) (2014), 669–700. math-ph/1211.4021.Google Scholar
[14] Eynard, B. Recursion between Mumford volumes of moduli spaces. Ann. Henri Poincaré. 12 (8) (2011), 1431–1447. math.AG/0706.4403.Google Scholar
[15] Eynard, B. Invariants of spectral curves and intersection theory of moduli spaces of complex curves. Commun. Number Theory and Physics. 8 (3) (2014), math-ph/1110.2949.Google Scholar
[16] Eynard, B. Counting surfaces: combinatorics, matrix models and algebraic geometry. Progress in Mathematical Physics. vol. 114 (Birkhäuser, Basel, 2016). http://eynard.bertrand.voila.net/TOCbook.htm.Google Scholar
[17] Eynard, B. and Orantin, N. Invariants of algebraic curves and topological expansion. Commun. Number Theory and Physics. 1 (2) (2007), math-ph/0702045.Google Scholar
[18] Eynard, B. and Orantin, N. Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models. (2007), math-ph/0705.3600.Google Scholar
[19] Eynard, B. and Orantin, N. Topological expansion of mixed correlations in the hermitian 2 matrix model and x-y symmetry of the Fg invariants. J. Phys. A: Math. Theor. 41 (2008), math-ph/0705.0958.Google Scholar
[20] Eynard, B. and Orantin, N. Topological recursion in random matrices and enumerative geometry. J. Phys. A: Mathematical and Theoretical. 42 (29) (2009), math-ph/0811.3531.Google Scholar
[21] Eynard, B. and Orantin, N. Computation of open Gromov–Witten invariants for toric Calabi–Yau 3-folds by topological recursion, a proof of the BKMP conjecture. (2012), math-ph/1205.1103.Google Scholar
[22] Givental, A. B. Gromov-Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J. 1 (4) (2001), 551–568, 645.Google Scholar
[23] Harer, J. and Zagier, D. The Euler characteristics of the moduli space of curves. Invent. Math. 85 (1986), 457–485.Google Scholar
[24] Kontsevich, M. Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147 (1992), 1–23.Google Scholar
[25] Mulase, M. and Dumitrescu, O. Quantum curves for Hitchin fibrations and the Eynard-Orantin theory. Lett. Math. Phys. 104 (2014), 635–671. math.AG/1310.6022.Google Scholar
[26] Norbury, P. and Scott, N. Gromov–Witten invariants of P 1 and Eynard-Orantin invariants (2011). math.AG/1106.1337.Google Scholar
[27] Witten, E. Two dimensional gravity and intersection theory on moduli space. Surveys in Diff. Geom. 1 (1991), 243–310.Google Scholar
[28] Zhou, J. Local mirror symmetry for the topological vertex (2009), math.AG/0911.2343.Google Scholar