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DUBROVIN’S SUPERPOTENTIAL AS A GLOBAL SPECTRAL CURVE

Published online by Cambridge University Press:  17 April 2017

P. Dunin-Barkowski
Affiliation:
Faculty of Mathematics, National Research University Higher School of Economics, Usacheva 6, 119048 Moscow, Russia (ptdbar@hse.ru)
P. Norbury
Affiliation:
School of Mathematics and Statistics, University of Melbourne, 3010 Australia (pnorbury@ms.unimelb.edu.au)
N. Orantin
Affiliation:
Département de mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland (nicolas.orantin@epfl.ch)
A. Popolitov
Affiliation:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Postbus 94248, 1090 GE Amsterdam, The Netherlands (A.Popolitov@uva.nl; S.Shadrin@uva.nl) ITEP, Moscow, Russia
S. Shadrin
Affiliation:
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Postbus 94248, 1090 GE Amsterdam, The Netherlands (A.Popolitov@uva.nl; S.Shadrin@uva.nl)

Abstract

We apply the spectral curve topological recursion to Dubrovin’s universal Landau–Ginzburg superpotential associated to a semi-simple point of any conformal Frobenius manifold. We show that under some conditions the expansion of the correlation differentials reproduces the cohomological field theory associated with the same point of the initial Frobenius manifold.

Type
Research Article
Copyright
© Cambridge University Press 2017 

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References

Alexandrov, A., Mironov, A. and Morozov, A., Partition functions of matrix models: first special functions of string theory, Internat. J. Modern Phys. A 19(24) (2004), 41274163.Google Scholar
Andersen, J., Chekhov, L., Norbury, P. and Penner, R., Models of discretized moduli spaces, cohomological field theories, and Gaussian means, J. Geom. Phys. 98 (2015), 312329.Google Scholar
Bouchard, V. and Eynard, B., Think globally, compute locally, JHEP (02) (2013), 143, 34pp.Google Scholar
Bouchard, V. and Eynard, B., Private communication.Google Scholar
Do, N. and Manescu, D., Quantum curves for the enumeration of ribbon graphs and hypermaps, Commun. Number Theory Phys. 8(4) (2014), 677701.Google Scholar
Dubrovin, B., Geometry of 2D topological field theories, in Integrable Systems and Quantum Groups (Authors: R. Donagi, B. Dubrovin, E. Frenkel, E. Previato) (ed. Francaviglia, M. and Greco, S.), Springer Lecture Notes in Mathematics, 1620, pp. 120348 (Springer, Berlin, 1996).Google Scholar
Dubrovin, B., Painlevé transcendents and two-dimensional topological field theory, in The Painlevé Property: One Century Later (ed. Conte, R.), pp. 287412 (Springer, New York, 1999).Google Scholar
Dubrovin, B., On almost duality for Frobenius manifolds, in Geometry, Topology, and Mathematical Physics, American Mathematical Society Translation Series 2, 212, pp. 75132 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Dumitrescu, O., Mulase, M., Safnuk, B. and Sorkin, A., The spectral curve of the Eynard–Orantin recursion via the Laplace transform, in Algebraic and Geometric Aspects of Integrable Systems and Random Matrices (ed. Dzhamay, Maruno and Pierce), Contemporary Mathematics, 593, pp. 263315 (Amer. Math. Soc., Providence, RI, 2013).Google Scholar
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 (2014), 669700.Google Scholar
Dunin-Barkowski, P., Orantin, N., Popolitov, A. and Shadrin, S., Combinatorics of loop equations for branched covers of sphere, preprint, 2014, arXiv:1412.1698.Google Scholar
Dunin-Barkowski, P., Lewanski, D., Popolitov, A. and Shadrin, S., Polynomiality of orbifold Hurwitz numbers, spectral curve, and a new proof of the Johnson–Pandharipande–Tseng formula, preprint, 2015, arXiv:1504.07440.Google Scholar
Dunin-Barkowski, P., Shadrin, S. and Spitz, L., Givental graphs and inversion symmetry, Lett. Math. Phys. 103(5) (2013), 533557.Google Scholar
Eynard, B., Invariants of spectral curves and intersection theory of moduli spaces of complex curves, Commun. Number Theory Phys. 8(3) (2014), 541588.Google Scholar
Eynard, B., A short overview of the ‘Topological recursion’, preprint, 2014,arXiv:1412.3286.Google Scholar
Eynard, B. and Orantin, N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1(2) (2007), 347452.Google Scholar
Eynard, B. and Orantin, N., Topological recursion in enumerative geometry and random matrices, J. Phys. A: Math. Theor. 42 (2009), 293001 (117pp).Google Scholar
Faber, C., Shadrin, S. and Zvonkine, D., Tautological relations and the r-spin Witten conjecture, Ann. Sci. Éc. Norm. Supér. (4) 43(4) (2010), 621658.Google Scholar
Fang, B., Liu, C.-C. M. and Zong, Z., The Eynard–Orantin recursion and equivariant mirror symmetry for the projective line, Geom. Topol. to appear arXiv:1411.3557.Google Scholar
Frobenius, G. and Stickelberger, L., Über die Differentiation der elliptischen Functionen nach den Perioden und Invarianten, J. Reine Angew. Math. 92 (1882), 311327.Google Scholar
Givental, A., Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1(4) (2001), 551568.Google Scholar
Givental, A., A n-1 singularities and n-KdV hierarchies, Mosc. Math. J. 3(2) (2003), 475505.Google Scholar
Kontsevich, M. and Manin, Y., Gromov–Witten classes, quantum cohomology, and enumerative geometry, Commun. Math. Phys. 164(3) (1994), 525562.Google Scholar
Lewanski, D., Popolitov, A., Shadrin, S. and Zvonkine, D., Chiodo formulas for the $r$ -th roots and topological recursion, preprint, 2015, arXiv:1504.07439.Google Scholar
Milanov, T., The Eynard–Orantin recursion for the total ancestor potential, Duke Math. J. 163(9) (2014), 17951824.Google Scholar
Milanov, T., The Eynard–Orantin recursion for simple singularities, Commun. Number Theory Phys. 9 (2015), 707739.Google Scholar
Norbury, P., Counting lattice points in the moduli space of curves, Math. Res. Lett. 17 (2010), 467481.Google Scholar
Pandharipande, R., Pixton, A. and Zvonkine, D., Relations on M g, n via 3-spin structures, J. Amer. Math. Soc. 28(1) (2015), 279309.Google Scholar
Shadrin, S., BCOV theory via Givental group action on cohomological field theories, Mosc. Math. J. 9(2) (2009), 411429.Google Scholar
Teleman, C., The structure of 2D semi-simple field theories, Invent. Math. 188(3) 525588.Google Scholar
Witten, E., Algebraic geometry associated with matrix models of two-dimensional gravity, in Topological Methods in Modern Mathematics (Stony Brook, NY, 1991), pp. 235269 (Publish or Perish, Houston, TX, 1993).Google Scholar