Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 050, 16 pages      arXiv:2306.01501      https://doi.org/10.3842/SIGMA.2024.050

A Note on BKP for the Kontsevich Matrix Model with Arbitrary Potential

Gaëtan Borot ^a and Raimar Wulkenhaar ^b
a) Institut für Mathematik und Institut für Physik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany
^b) Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany

Received January 03, 2024, in final form June 01, 2024; Published online June 11, 2024

Abstract
We exhibit the Kontsevich matrix model with arbitrary potential as a BKP tau-function with respect to polynomial deformations of the potential. The result can be equivalently formulated in terms of Cartan-Plücker relations of certain averages of Schur $Q$-function. The extension of a Pfaffian integration identity of de Bruijn to singular kernels is instrumental in the derivation of the result.

Key words: BKP hierarchy; matrix models; classical integrability.

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References

1. Alexandrov A., KdV solves BKP, Proc. Natl. Acad. Sci. USA 118 (2021), e2101917118, 2 pages, arXiv:2012.10448.
2. Alexandrov A., Shadrin S., Elements of spin Hurwitz theory: closed algebraic formulas, blobbed topological recursion, and a proof of the Giacchetto-Kramer-Lewański conjecture, Selecta Math. (N.S.) 29 (2023), 26, 44 pages, arXiv:2105.12493.
3. Borot G., Eynard B., Orantin N., Abstract loop equations, topological recursion and new applications, Commun. Number Theory Phys. 9 (2015), 51-187, arXiv:1303.5808.
4. Borot G., Shadrin S., Blobbed topological recursion: properties and applications, Math. Proc. Cambridge Philos. Soc. 162 (2017), 39-87, arXiv:1502.00981.
5. Branahl J., Hock A., Wulkenhaar R., Blobbed topological recursion of the quartic Kontsevich model I: Loop equations and conjectures, Comm. Math. Phys. 393 (2022), 1529-1582, arXiv:2008.12201.
6. Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type, Phys. D 4 (1982), 343-365.
7. de Bruijn N.G., On some multiple integrals involving determinants, J. Indian Math. Soc. (N.S.) 19 (1955), 133-151.
8. Giacchetto A., Kramer R., Lewanski D., A new spin on Hurwitz theory and ELSV via theta characteristics, arXiv:2104.05697.
9. Grosse H., Hock A., Wulkenhaar R., Solution of all quartic matrix models, arXiv:1906.04600.
10. Harnad J., Balogh F., Tau functions and their applications, Cambridge Monogr. Math. Phys., Cambridge University Press, Cambridge, 2021.
11. Hock A., Wulkenhaar R., Blobbed topological recursion from extended loop equations, arXiv:2301.04068.
12. Isaacson E., Keller H.B., Analysis of numerical methods, Dover Publications, New York, 1994.
13. Itzykson C., Zuber J.B., The planar approximation. II, J. Math. Phys. 21 (1980), 411-421.
14. Kontsevich M., Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1-23.
15. Liu X., Yang C., Schur ${Q}$-polynomials and Kontsevich-Witten tau function, Peking Math. J., to appear, arXiv:2103.14318.
16. Mironov A., Morozov A., Superintegrability of Kontsevich matrix model, Eur. Phys. J. C 81 (2021), 270, 11 pages, arXiv:2011.12917.
17. Mironov A., Morozov A., Natanzon S., Orlov A., Around spin Hurwitz numbers, Lett. Math. Phys. 111 (2021), 124, 39 pages, arXiv:2012.09847.
18. Nimmo J.J.C., Hall-Littlewood symmetric functions and the BKP equation, J. Phys. A 23 (1990), 751-760.
19. Orlov A., Hypergeometric functions associated with Schur $Q$-polynomials, and the BKP equation, Theoret. and Math. Phys. 137 (2003), 1574-1589, arXiv:math-ph/0302011.
20. Schur J., Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math. 139 (1911), 155-250.
21. Schürmann J., Wulkenhaar R., An algebraic approach to a quartic analogue of the Kontsevich model, Math. Proc. Cambridge Philos. Soc. 174 (2023), 471-495, arXiv:1912.03979.