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

Gaëtan Borot ; Raimar Wulkenhaar

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

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.

Détail

Keywords: BKP hierarchy, matrix models, classical integrability.

Bibliographie

[1] Alexandrov A., “KdV solves BKP”, Proc. Natl. Acad. Sci. USA, 118 (2021), e2101917118, 2 pp., arXiv: 2012.10448 | DOI | MR

[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 pp., arXiv: 2105.12493 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[4] Borot G., Shadrin S., “Blobbed topological recursion: properties and applications”, Math. Proc. Cambridge Philos. Soc., 162 (2017), 39–87, arXiv: 1502.00981 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[7] de Bruijn N.G., “On some multiple integrals involving determinants”, J. Indian Math. Soc. (N.S.), 19 (1955), 133–151 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | MR

[13] Itzykson C., Zuber J.B., “The planar approximation. II”, J. Math. Phys., 21 (1980), 411–421 | DOI | MR | Zbl

[14] Kontsevich M., “Intersection theory on the moduli space of curves and the matrix Airy function”, Comm. Math. Phys., 147 (1992), 1–23 | DOI | MR | Zbl

[15] Liu X., Yang C., “Schur ${Q}$-polynomials and Kontsevich–Witten tau function”, Peking Math. J. (to appear) , arXiv: 2103.14318 | DOI

[16] Mironov A., Morozov A., “Superintegrability of Kontsevich matrix model”, Eur. Phys. J. C, 81 (2021), 270, 11 pp., arXiv: 2011.12917 | DOI | MR

[17] Mironov A., Morozov A., Natanzon S., Orlov A., “Around spin Hurwitz numbers”, Lett. Math. Phys., 111 (2021), 124, 39 pp., arXiv: 2012.09847 | DOI | MR | Zbl

[18] Nimmo J.J.C., “Hall–Littlewood symmetric functions and the BKP equation”, J. Phys. A, 23 (1990), 751–760 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[20] Schur J., “Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen”, J. Reine Angew. Math., 139 (1911), 155–250 | DOI | MR

[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 | DOI | MR | Zbl