Abstract
We introduce a new formulation of the so-called topological recursion, that is defined globally on a compact Riemann surface. We prove that it is equivalent to the generalized recursion for spectral curves with arbitrary ramification. Using this global formulation, we also prove that the correlation functions constructed from the recursion for curves with arbitrary ramification can be obtained as suitable limits of correlation functions for curves with only simple ramification. It then follows that they both satisfy the properties that were originally proved only for curves with simple ramification.
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G. Borot and B. Eynard, All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, arXiv:1205.2261 [INSPIRE].
G. Borot, B. Eynard, M. Mulase and B. Safnuk, A Matrix model for simple Hurwitz numbers and topological recursion, J. Geom. Phys. 61 (2011) 522 [arXiv:0906.1206] [INSPIRE].
V. Bouchard, A. Catuneanu, O. Marchal and P. Sulkowski, The Remodeling conjecture and the Faber-Pandharipande formula, Lett. Math. Phys. 103 (2013) 59 [arXiv:1108.2689] [INSPIRE].
V. Bouchard, J. Hutchinson, P. Loliencar, M. Meiers and M. Rupert, A generalized topological recursion for arbitrary ramification, arXiv:1208.6035 [INSPIRE].
V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [INSPIRE].
V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Topological open strings on orbifolds, Commun. Math. Phys. 296 (2010) 589 [arXiv:0807.0597] [INSPIRE].
V. Bouchard and M. Mariño, Hurwitz numbers, matrix models and enumerative geometry, Proc. Symp. Pure Math. 78 (2008) 263 [arXiv:0709.1458] [INSPIRE].
V. Bouchard and P. Sulkowski, Topological recursion and mirror curves, arXiv:1105.2052 [INSPIRE].
A. Brini, B. Eynard and M. Mariño, Torus knots and mirror symmetry, Annales Henri Poincaré 13 (2012) 1873 [arXiv:1105.2012] [INSPIRE].
L. Chen, Bouchard-Klemm-Marino-Pasquetti Conjecture for \( {{\mathbb{C}}^3} \), arXiv:0910.3739 [INSPIRE].
R. Dijkgraaf, H. Fuji and M. Manabe, The Volume Conjecture, Perturbative Knot Invariants and Recursion Relations for Topological Strings, Nucl. Phys. B 849 (2011) 166 [arXiv:1010.4542] [INSPIRE].
O. Dumitrescu, M. Mulase, B. Safnuk and A. Sorkin, The spectral curve of the Eynard-Orantin recursion via the Laplace transform, arXiv:1202.1159.
B. Eynard, Intersection numbers of spectral curves, arXiv:1104.0176 [INSPIRE].
B. Eynard, Invariants of spectral curves and intersection theory of moduli spaces of complex curves, arXiv:1110.2949 [INSPIRE].
B. Eynard, M. Mulase and B. Safnuk, The Laplace transform of the cut-and-join equation and the Bouchard-Mariño conjecture on Hurwitz numbers, arXiv:0907.5224 [INSPIRE].
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, Commun. Num. Theor. Phys. 1 (2007) 347 [math-ph/0702045] [INSPIRE].
B. Eynard and N. Orantin, Topological expansion of mixed correlations in the hermitian 2 Matrix Model and x − y symmetry of the F g invariants, arXiv:0705.0958 [INSPIRE].
B. Eynard and N. Orantin, Algebraic methods in random matrices and enumerative geometry, arXiv:0811.3531 [INSPIRE].
B. Eynard and N. Orantin, Computation of open Gromov-Witten invariants for toric Calabi-Yau 3-folds by topological recursion, a proof of the BKMP conjecture, arXiv:1205.1103 [INSPIRE].
P.A. Griffiths, Translation of Mathematical Monographs. Vol. 76: Introduction to Algebraic Curves, American Mathematical Society, Providence U.S.A. (1989).
S. Gukov and P. Sulkowski, A-polynomial, B-model and quantization, JHEP 02 (2012) 070 [arXiv:1108.0002] [INSPIRE].
M.-x. Huang and A. Klemm, Holomorphicity and modularity in Seiberg-Witten theories with matter, JHEP 07 (2010) 083 [arXiv:0902.1325] [INSPIRE].
M. Mariño, Open string amplitudes and large order behavior in topological string theory, JHEP 03 (2008) 060 [hep-th/0612127] [INSPIRE].
R. Miranda, Graduate Studies in Mathematics. Vol. 5: Algebraic Curves and Riemann Surfaces, American Mathematical Society, Providence U.S.A. (1995).
M. Mulase and P. Sulkowski, Spectral curves and the Schroedinger equations for the Eynard-Orantin recursion, arXiv:1210.3006 [INSPIRE].
P. Norbury and N. Scott, Polynomials representing Eynard-Orantin invariants, arXiv:1001.0449.
P. Norbury and N. Scott, Gromov-Witten invariants of \( {{\mathbb{P}}^1} \) and Eynard-Orantin invariants, arXiv:1106.1337 [INSPIRE].
A.P. Ferrer, New recursive residue formulas for the topological expansion of the Cauchy Matrix Model, JHEP 10 (2010) 090 [arXiv:0912.2984] [INSPIRE].
J. Zhou, Local Mirror Symmetry for One-Legged Topological Vertex, arXiv:0910.4320 [INSPIRE].
J. Zhou, Local Mirror Symmetry for the Topological Vertex, arXiv:0911.2343.
S. Zhu, On a proof of the Bouchard-Sulkowski conjecture, arXiv:1108.2831 [INSPIRE].
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ArXiv ePrint: 1211.2302
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Bouchard, V., Eynard, B. Think globally, compute locally. J. High Energ. Phys. 2013, 143 (2013). https://doi.org/10.1007/JHEP02(2013)143
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DOI: https://doi.org/10.1007/JHEP02(2013)143