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 Mosc. Math. J., 2016, Volume 16, Number 1, Pages 27–44 (Mi mmj593)

Open intersection numbers and the wave function of the KdV hierarchy

A. Buryak

Department of Mathematics, ETH Zurich, Ramistrasse 101 8092, HG G 27.1, Zurich, Switzerland

Abstract: Recently R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers is a specific solution of a system of PDEs, which they called the open KdV equations. In this paper we show that the open KdV equations are closely related to the equations for the wave function of the KdV hierarchy. This allows us to give an explicit formula for the specific solution in terms of Witten's generating series of the intersection numbers on the moduli space of stable curves.

Key words and phrases: Riemann surfaces with boundary, moduli space, KdV equations.

 Funding Agency Grant Number Eidgenösische Technische Hochschule Zürich ERC-2012-AdG-320368-MCSK Ministry of Education and Science of the Russian Federation 11.634.31.0005NSh-4850.2012.1 Russian Foundation for Basic Research 13-01-00755 This work was supported by grant ERC-2012-AdG-320368-MCSK in the group of R. Pandharipande at ETH Zurich, by the Russian Federation Government grant no. 2010-220-01-077 (ag. no. 11.634.31.0005), the grants RFFI 13-01-00755 and NSh-4850.2012.1.

DOI: https://doi.org/10.17323/1609-4514-2016-16-1-27-44

Full text: http://www.mathjournals.org/.../2016-016-001-002.html
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Bibliographic databases:

MSC: Primary 35Q53; Secondary 14H10
Received: October 16, 2014; in revised form April 23, 2015
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Citation: A. Buryak, “Open intersection numbers and the wave function of the KdV hierarchy”, Mosc. Math. J., 16:1 (2016), 27–44

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mmj/v16/i1/p27

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Buryak A., Janda F., Pandharipande R., “the Hypergeometric Functions of the Faber-Zagier and Pixton Relations”, Pure Appl. Math. Q., 11:4, 1, SI (2015), 591–631
2. A. Alexandrov, A. Buryak, R. J. Tessler, “Refined open intersection numbers and the Kontsevich–Penner matrix model”, J. High Energy Phys., 2017, no. 3, 123
3. K. Aleshkin, V. Belavin, Ch. Rim, “Minimal gravity and Frobenius manifolds: bulk correlation on sphere and disk”, J. High Energy Phys., 2017, no. 11, 169
4. A. Alexandrov, “Open intersection numbers and free fields”, Nucl. Phys. B, 922 (2017), 247–263
5. A. Buryak, R. J. Tessler, “Matrix models and a proof of the open analog of Witten's conjecture”, Commun. Math. Phys., 353:3 (2017), 1299–1328
6. A. Bawane, H. Muraki, Ch. Rim, “Open KdV hierarchy and minimal gravity on disk”, Phys. Lett. B, 783 (2018), 183–185
7. L. Chekhov, M. Mazzocco, “Colliding holes in Riemann surfaces and quantum cluster algebras”, Nonlinearity, 31:1 (2018), 54–107