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Licensed Unlicensed Requires Authentication Published by De Gruyter November 25, 2014

Quantum spectral curve for the Gromov–Witten theory of the complex projective line

  • Petr Dunin-Barkowski EMAIL logo , Motohico Mulase , Paul Norbury , Alexander Popolitov and Sergey Shadrin

Abstract

We construct the quantum curve for the Gromov–Witten theory of the complex projective line.

Award Identifier / Grant number: DMS-1104734

Award Identifier / Grant number: DMS-1309298

Funding statement: P. Dunin-Barkowski is supported by a free competition grant of the NWO, M. Mulase is supported by NSF grants DMS-1104734 and DMS-1309298, P. Norbury is supported by ARC grant DP1094328, A. Popolitov and S. Shadrin are supported by a Vici grant of the NWO, and S. Shadrin is supported by a Vidi grant of the NWO.

References

[1] Aganagic M., Dijkgraaf R., Klemm A., Mariño M. and Vafa C., Topological strings and integrable hierarchies, Comm. Math. Phys. 261 (2006), 451–516. 10.1007/s00220-005-1448-9Search in Google Scholar

[2] Chekhov L., Eynard B. and Orantin N., Free energy topological expansion for the 2-matrix model, J. High Energy Phys. 2006 (2006), no. 12, Article 053. 10.1088/1126-6708/2006/12/053Search in Google Scholar

[3] Dijkgraaf R., Hollands L. and Sułkowski P., Quantum curves and 𝒟-modules, J. High Energy Phys. 2009 (2009), no. 11, Article 047. Search in Google Scholar

[4] Dijkgraaf R., Hollands L., Sułkowski P. and Vafa C., Supersymmetric gauge theories, intersecting branes and free fermions, J. High Energy Phys. 2008 (2008), no. 2, Article 106. 10.1088/1126-6708/2008/02/106Search in Google Scholar

[5] Dumitrescu O. and Mulase M., Quantum curves for Hitchin fibrations and the Eynard–Orantin theory, Lett. Math. Phys. 104 (2014), no. 6, 635–671. 10.1007/s11005-014-0679-0Search in Google Scholar

[6] Dumitrescu O., Mulase M., Safnuk B. and Sorkin A., The spectral curve of the Eynard–Orantin recursion via the Laplace transform, Algebraic and geometric aspects of integrable systems and random matrices, Contemp. Math. 593, American Mathematical Society, Providence (2013), 263–315. 10.1090/conm/593/11867Search in Google Scholar

[7] Dunin-Barkowski P., Kazarian M., Orantin N., Shadrin S. and Spitz L., Polynomiality of Hurwitz numbers, Bouchard–Mariño conjecture, and a new proof of the ELSV formula, preprint 2013, http://arxiv.org/abs/1307.4729. 10.1016/j.aim.2015.03.016Search in Google Scholar

[8] Dunin-Barkowski P., Orantin N., Shadrin S. and Spitz L., Identification of the Givental formula with the spectral curve topological recursion procedure, Comm. Math. Phys. 328 (2014), no. 2, 669–700. 10.1007/s00220-014-1887-2Search in Google Scholar

[9] Eynard B. and Mariño M., A holomorphic and background independent partition function for matrix models and topological strings, J. Geom. Phys. 61 (2011), 1181–1202. 10.1016/j.geomphys.2010.11.012Search in Google Scholar

[10] Eynard B. and Orantin N., Invariants of algebraic curves and topological expansion, Commun. Number Theory Phys. 1 (2007), 347–452. 10.4310/CNTP.2007.v1.n2.a4Search in Google Scholar

[11] Givental A., Gromov–Witten invariants and quantization of quadratic Hamiltonians, Mosc. Math. J. 1 (2001), no. 4, 551–568. 10.17323/1609-4514-2001-1-4-551-568Search in Google Scholar

[12] Gukov S. and Sułkowski P., A-polynomial, B-model, and quantization, J. High Energy Phys. 2012 (2012), Article 70. 10.1007/978-3-319-06514-4_4Search in Google Scholar

[13] Han G.-N., Hook lengths and shifted parts of partitions, Ramanujan J. 23 (2010), no. 1–3, 127–135. 10.1007/s11139-009-9170-4Search in Google Scholar

[14] Hollands L., Topological strings and quantum curves, Ph.D. thesis, University of Amsterdam, 2009, http://arxiv.org/abs/0911.341310.5117/9789085550204Search in Google Scholar

[15] Norbury P. and Scott N., Gromov–Witten invariants of 1 and Eynard–Orantin invariants, Geom. Topol. 18 (2014), no. 4, 1865–1910. 10.2140/gt.2014.18.1865Search in Google Scholar

[16] Okounkov A. and Pandharipande R., Gromov–Witten theory, Hurwitz theory, and completed cycles, Ann. of Math. (2) 163 (2006), no. 2, 517–560. 10.4007/annals.2006.163.517Search in Google Scholar

[17] Shadrin S., Spitz L. and Zvonkine D., On double Hurwitz numbers with completed cycles, J. Lond. Math. Soc. (2) 86 (2012), no. 2, 407–432. 10.1112/jlms/jds010Search in Google Scholar

Received: 2014-2-11
Revised: 2014-8-3
Published Online: 2014-11-25
Published in Print: 2017-5-1

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