Algebro-Geometric Solutions of the Generalized Virasoro Constraints

Francisco José Plaza Martín

Symmetry, integrability and geometry: methods and applications, Tome 11 (2015)

Résumé

We will describe algebro-geometric solutions of the KdV hierarchy whose $\tau$-functions in addition satisfy a generalization of the Virasoro constraints (and, in particular, a generalization of the string equation). We show that these solutions are closely related to embeddings of the positive half of the Virasoro algebra into the Lie algebra of differential operators on the circle. Our results are tested against the case of Witten–Kontsevich $\tau$-function. As by-products, we exhibit certain links of our methods with double covers of the projective line equipped with a line bundle and with $\mathrm{Gl}(n)$-opers on the punctured disk.

Détail

Keywords: string equation; Virasoro constraints; KP hierarchy; $\mathrm{Gl}(n)$-opers; Sato Grassmannian; topological recursion.

Bibliographie

[1] Adler M., Morozov A., Shiota T., van Moerbeke P., “A matrix integral solution to $[P,Q]=P$ and matrix Laplace transforms”, Comm. Math. Phys., 180 (1996), 233–263, arXiv: hep-th/9610137 | DOI | MR | Zbl

[2] Alexandrov A., “Enumerative geometry, tau-functions and Heisenberg–Virasoro algebra”, Comm. Math. Phys., 338 (2015), 195–249, arXiv: 1404.3402 | DOI | MR | Zbl

[3] Álvarez Vázquez A., Muñoz Porras J. M., Plaza Martín F. J., “The algebraic formalism of soliton equation over arbitrary base fields”, Taller de Variedades Abelianas y Funciones Theta (Morelia, 1996), Aportaciones Matemáticas: Investigación, 13, eds. R. Rodríguez, J. M. Muñóz Porras, S. Recillas, Sociedad Matemática Mexicana, México, 1998, 3–40, arXiv: alg-geom/9606009 | MR | Zbl

[4] Beauville A., Narasimhan M. S., Ramanan S., “Spectral curves and the generalised theta divisor”, J. Reine Angew. Math., 398 (1989), 169–179 | DOI | MR | Zbl

[5] Beĭlinson A. A., Schechtman V. V., “Determinant bundles and Virasoro algebras”, Comm. Math. Phys., 118 (1988), 651–701 | DOI | MR

[6] Bouchard V., Mariño M., “Hurwitz numbers, matrix models and enumerative geometry”, From Hodge Theory to Integrability and TQFT $tt^*$-Geometry, Proc. Sympos. Pure Math., 78, Amer. Math. Soc., Providence, RI, 2008, 263–283, arXiv: 0709.1458 | DOI | MR | Zbl

[7] Dijkgraaf R., “Intersection theory, integrable hierarchies and topological field theory”, New Symmetry Principles in Quantum Field Theory (Cargèse, 1991), NATO Adv. Sci. Inst. Ser. B Phys., 295, Plenum, New York, 1992, 95–158, arXiv: hep-th/9201003 | MR | Zbl

[8] Dijkgraaf R., Verlinde H., Verlinde E., “Loop equations and Virasoro constraints in nonperturbative two-dimensional quantum gravity”, Nuclear Phys. B, 348 (1991), 435–456 | DOI | MR

[9] Douglas M. R., “Strings in less than one dimension and the generalized KdV hierarchies”, Phys. Lett. B, 238 (1990), 176–180 | DOI | MR

[10] Dubrovin B., Zhang Y., “Frobenius manifolds and Virasoro constraints”, Selecta Math. (N.S.), 5 (1999), 423–466, arXiv: math.AG/9808048 | DOI | MR | Zbl

[11] Eynard B., Mariño M., “A holomorphic and background independent partition function for matrix models and topological strings”, J. Geom. Phys., 61 (2011), 1181–1202, arXiv: 0810.4273 | DOI | MR | Zbl

[12] Eynard B., Orantin N., “Invariants of algebraic curves and topological expansion”, Commun. Number Theory Phys., 1 (2007), 347–452, arXiv: math-ph/0702045 | DOI | MR | Zbl

[13] Eynard B., Orantin N., Geometrical interpretation of the topological recursion, and integrable string theories, arXiv: 0911.5096

[14] Frenkel E., Langlands correspondence for loop groups, Cambridge Studies in Advanced Mathematics, 103, Cambridge University Press, Cambridge, 2007 | MR | Zbl

[15] Frenkel E., Ben-Zvi D., Vertex algebras and algebraic curves, Mathematical Surveys and Monographs, 88, 2nd ed., Amer. Math. Soc., Providence, RI, 2004 | DOI | MR | Zbl

[16] Frenkel E., Kac V., Radul A., Wang W., “${\mathcal W}_{1+\infty}$ and ${\mathcal W}({\mathfrak{gl}}_N)$ with central charge $N$”, Comm. Math. Phys., 170 (1995), 337–357, arXiv: hep-th/9405121 | DOI | MR | Zbl

[17] Givental A. B., “Gromov–Witten invariants and quantization of quadratic Hamiltonians”, Mosc. Math. J., 1 (2001), 551–568, arXiv: math.AG/0108100 | MR | Zbl

[18] Gómez González E., Hernández Serrano D., Muñoz Porras J. M., Plaza Martín F. J., “Geometric approach to Kac–Moody and Virasoro algebras”, J. Geom. Phys., 62 (2012), 1984–1997 | DOI | MR | Zbl

[19] Kac V. G., Peterson D. H., “Spin and wedge representations of infinite-dimensional Lie algebras and groups”, Proc. Nat. Acad. Sci. USA, 78, 1981, 3308–3312 | MR | Zbl

[20] Kac V. G., Raina A. K., Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, Advanced Series in Mathematical Physics, 2, World Scientific Publishing Co., Inc., Teaneck, NJ, 1987 | MR | Zbl

[21] Kac V. G., Schwarz A., “Geometric interpretation of the partition function of $2$D gravity”, Phys. Lett. B, 257 (1991), 329–334 | DOI | MR

[22] Katsura T., Shimizu Y., Ueno K., “Complex cobordism ring and conformal field theory over $\mathbf{Z}$”, Math. Ann., 291 (1991), 551–571 | DOI | MR | Zbl

[23] Kazarian M., “KP hierarchy for Hodge integrals”, Adv. Math., 221 (2009), 1–21, arXiv: 0809.3263 | DOI | MR | Zbl

[24] Kirillov A. A., “Geometric approach to discrete series of unirreps for Vir”, J. Math. Pures Appl., 77 (1998), 735–746 | DOI | MR | Zbl

[25] 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

[26] Laine I., Nevanlinna theory and complex differential equations, de Gruyter Studies in Mathematics, 15, Walter de Gruyter Co., Berlin, 1993 | DOI | MR

[27] Liu K., Xu H., “Recursion formulae of higher Weil–Petersson volumes”, Int. Math. Res. Not., 2009 (2009), 835–859, arXiv: 0708.0565 | DOI | MR | Zbl

[28] Mironov A., Morozov A., “Virasoro constraints for Kontsevich–Hurwitz partition function”, J. High Energy Phys., 2009:2 (2009), 024, 52 pp., arXiv: 0807.2843 | DOI | MR | Zbl

[29] Mirzakhani M., “Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces”, Invent. Math., 167 (2007), 179–222 | DOI | MR | Zbl

[30] Mirzakhani M., “Weil–Petersson volumes and intersection theory on the moduli space of curves”, J. Amer. Math. Soc., 20 (2007), 1–23 | DOI | MR | Zbl

[31] Mulase M., “Algebraic theory of the KP equations”, Perspectives in mathematical physics, Conf. Proc. Lecture Notes Math. Phys., III, Int. Press, Cambridge, MA, 1994, 151–217 | MR | Zbl

[32] Mulase M., Safnuk B., “Mirzakhani's recursion relations, Virasoro constraints and the KdV hierarchy”, Indian J. Math., 50 (2008), 189–218, arXiv: math.QA/0601194 | MR | Zbl

[33] Mulase M., Zhang N., “Polynomial recursion formula for linear Hodge integrals”, Commun. Number Theory Phys., 4 (2010), 267–293, arXiv: 0908.2267 | DOI | MR | Zbl

[34] Muñoz Porras J. M., Plaza Martín F. J., “Automorphism group of $k((t))$: applications to the bosonic string”, Comm. Math. Phys., 216 (2001), 609–634, arXiv: hep-th/9903250 | DOI | MR | Zbl

[35] Plaza Martín F. J., “Arithmetic infinite Grassmannians and the induced central extensions”, Collect. Math., 61 (2010), 107–129, arXiv: 0810.0354 | DOI | MR | Zbl

[36] Plaza Martín F. J., “Representations of the Witt algebra and $\mathrm{Gl}(n)$-opers”, Lett. Math. Phys., 103 (2013), 1079–1101 | DOI | MR | Zbl

[37] Schwarz A., “On solutions to the string equation”, Modern Phys. Lett. A, 6 (1991), 2713–2725, arXiv: hep-th/9109015 | DOI | MR | Zbl

[38] Schwarz A., “Quantum curves”, Comm. Math. Phys., 338 (2015), 483–500, arXiv: 1401.1574 | DOI | MR | Zbl

[39] Segal G., Wilson G., “Loop groups and equations of KdV type”, Inst. Hautes Études Sci. Publ. Math., 1985, 5–65 | MR | Zbl

[40] Witten E., “Two-dimensional gravity and intersection theory on moduli space”, Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, 243–310 | MR