Abstract
Dubrovin establishes a certain relationship between the GUE partition function and the partition function of Gromov–Witten invariants of the complex projective line. In this paper, we give a direct proof of Dubrovin’s result. We also present in a diagram the recent progress on topological gravity and matrix gravity.
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Adler, M., van Moerbeke, P.: Matrix integrals, Toda symmetries, Virasoro constraints, and orthogonal polynomials. Duke Math. J., 80, 863–911 (1995)
Alexandrov, A.: KP integrability of triple Hodge integrals. I. From Givental group to hierarchy symmetries. Commun. Number Theory Phys., 15, 615–650 (2021)
Alexandrov, A.: KP integrability of triple Hodge integrals. III. Cut-and-join description, KdV reduction, and topological recursions. arXiv:2108.10023 (2021)
Arakawa, T., Ibukiyama, T., Kaneko, M.: Bernoulli numbers and zeta functions (With an appendix by Don Zagier), Springer Monographs in Mathematics, Springer, Tokyo, 2014
Barnes, E. W.: The theory of the G-function. Q. J. Math., 31, 264–314 (1900)
Bleher, P. M., Deafio, A.: Topological expansion in the cubic random matrix model. IMRN, 2013, 2699–2755 (2013)
Bessis, D., Itzykson, C., Zuber, J.-B.: Quantum field theory techniques in graphical enumeration. Adv. Appl. Math., 1, 109–157 (1980)
Brezin, E., Itzykson, C., Parisi, P., et al.: Planar diagrams. Comm. Math. Phys., 59, 35–51 (1978)
Brini, A., Carlet, G., Romano, S., et al.: Rational reductions of the 2D-Toda hierarchy and mirror symmetry. J. Eur. Math. Soc., 19, 835–880 (2017)
Buryak, A., Posthuma, H., Shadrin, S.: A polynomial bracket for the Dubrovin-Zhang hierarchies. J. Differential Geom., 92, 153–185 (2012)
Carlet, G., Dubrovin, B., Zhang, Y.: The extended Toda hierarchy. Mosc. Math. J., 4, 313–332 (2004)
Carlet, G., van de Leur, J., Posthuma, H., et al.: Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy. Lett. Math. Phys., 111, Paper No. 63, 67 pp. (2021)
Deligne, P., Mumford, D.: The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math., 36, 75–109 (1969)
Deift, P.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, Courant Lecture Notes in Mathematics Vol. 3, American Mathematical Society, Providence, R.I., 1999
Di Francesco, P., Ginsparg, P., Zinn-Justin, J.: 2D gravity and random matrices. Phys. Reports, 254, 1–133 (1995)
Dijkgraaf, R., Witten, E.: Mean field theory, topological field theory, and multi-matrix models. Nucl. Phys. B, 342, 486–522 (1990)
Dubrovin, B.: Integrable systems and classification of 2D topological field theories. In: Babelon, O., Cartier, P., Kosmann-Schwarzbach, Y. (eds.) “Integrable Systems”, The J.-L.Verdier Memorial Conference, Actes du Colloque International de Luminy, Birkhäuser, 1993, 313–359
Dubrovin, B.: Geometry of 2D topological field theories. In: Francaviglia, M., Greco, S. (eds.) “Integrable Systems and Quantum Groups” (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 120–348
Dubrovin, B.: On Hamiltonian perturbations of hyperbolic systems of conservation laws. II. Universality of critical behaviour. Comm. Math. Phys., 267, 117–139 (2006)
Dubrovin, B.: On universality of critical behaviour in Hamiltonian PDEs. In: Geometry, Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 224, Adv. Math. Sci., Vol. 61, Amer. Math. Soc., Providence, RI, 2008, 59–109
Dubrovin, B.: Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions. In: Sidoravicius, V. (ed.) “New Trends in Mathematical Physics”, Springer, Dordrecht, 2009, 231–276
Dubrovin, B.: Hamiltonian PDEs: deformations, integrability, solutions. J. Phys. A, 43, 434002, 20 pp. (2010)
Dubrovin, B.: Gromov-Witten invariants and integrable hierarchies of topological type. In: Topology, Geometry, Integrable Systems, and Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 234, Adv. Math. Sci., Vol. 67, Amer. Math. Soc., Providence, RI, 2014, 141–171
Dubrovin, B., Grava, T., Klein, C., Moro, A.: On critical behaviour in systems of Hamiltonian partial differential equations. J. Nonlinear Sci., 25, 631–707 (2015)
Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs. Adv. Math., 293, 382–435 (2016)
Dubrovin, B., Liu, S.-Q., Yang, D., Zhang, Y.: Hodge-GUE correspondence and the discrete KdV equation. Comm. Math. Phys., 379, 461–490 (2020)
Dubrovin, B., Yang, D.: Generating series for GUE correlators. Lett. Math. Phys., 107, 1971–2012 (2017)
Dubrovin, B., Yang, D.: On cubic Hodge integrals and random matrices. Commun. Number Theory Phys., 11, 311–336 (2017)
Dubrovin, B., Yang, D.: Remarks on intersection numbers and integrable hierarchies.I.Quasi-triviality. Adv. Theor. Math. Phys., 24, 1055–1085 (2020)
Dubrovin, B., Yang, D.: Matrix resolvent and the discrete KdV hierarchy. Comm. Math. Phys., 377, 1823–1852 (2020)
Dubrovin, B., Yang, D., Zagier, D.: Classical Hurwitz numbers and related combinatorics. Mosc. Math. J., 17, 601–633 (2017)
Dubrovin, B., Zhang, Y.: Frobenius manifolds and Virasoro constraints. Selecta Math. (N.S.), 5, 423–466 (1999)
Dubrovin, B., Zhang, Y.: Normal forms of hierarchies of integrable PDEs, Frobenius Manifolds and Gromov-Witten invariants. arXiv:math/0108160 (2001)
Dubrovin, B., Zhang, Y.: Virasoro symmetries of the extended Toda hierarchy. Comm. Math. Phys., 250, 161–193 (2004)
Eguchi, T., Yamada, Y., Yang, S.-K.: On the genus expansion in the topological string theory. Rev. Math. Phys., 7, 279–309 (1995)
Eguchi, T., Yang, S.-K.: The topological CP1 model and the large-N matrix integral. Modern Physics Letters A, 9, 2893–2902 (1994)
Ercolani, N. M., McLaughlin, K. D. T.-R., Pierce, V. U.: Random matrices, graphical enumeration and the continuum limit of Toda lattices. Comm. Math. Phys., 278, 31–81 (2008)
Faber, C., Pandharipande, R.: Hodge integrals and Gromov-Witten theory. Invent. Math., 139, 173–199 (2000)
Ferreira, C., Lopez, J. L.: An asymptotic expansion of the double gamma function. J. Approx. Theory, 111, 298–314 (2001)
Flaschka, H.: On the Toda lattice. II. Inverse-scattering solution. Progr. Theoret. Phys., 51, 703–716 (1974)
Frenkel, E.: Deformations of the KdV hierarchy and related soliton equations. IMRN, 1996, 55–76 (1996)
Fu, A., Yang, D.: The matrix-resolvent method to tau-functions for the nonlinear Schrödinger hierarchy. J. Geom. Phys., 179, Paper No. 104592 (2022)
Getzler, E.: The Toda conjecture. In: Symplectic Geometry and Mirror Symmetry (KIAS, Seoul, 2000), World Scientific, Singapore, 2001, 51–79
Getzler, E.: The jet-space of a Frobenius manifold and higher-genus Gromov-Witten invariants. In: Frobenius Manifolds, Aspects Math., Vol. E36, Friedr. Vieweg, Wiesbaden, 2004, 45–89
Givental, A.: Semisimple Frobenius structures at higher genus. Intern. Math. J., 48, 295–304 (2000)
Givental, A.: Gromov–Witten invariants and quantization of quadratic Hamiltonians. Mosc. Math. J., 1, 1–23 (2001)
Harer, J., Zagier, D.: The Euler characteristic of the moduli space of curves. Invent. Math., 85, 457–485 (1986)
’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B, 72, 461–473 (1974)
’t Hooft, G.: A two-dimensional model for mesons. Nucl. Phys. B, 75, 461–470 (1974)
Itzykson, C., Zuber, J.-B.: Matrix integration and combinatorics of modular groups. Comm. Math. Phys., 134, 197–207 (1990)
Kazakov, V., Kostov, I., Nekrasov, N.: D-particles, matrix integrals and KP hierarchy. Nucl. Phys. B, 557, 413–442 (1999)
Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Comm. Math. Phys., 147, 1–23 (1992)
Kontsevich, M., Manin, Yu.: Gromov-Witten classes, quantum cohomology, and enumerative geometry. Comm. Math. Phys., 164, 525–562 (1994)
Liu, C.-C. M., Liu, K., Zhou, J.: A proof of a conjecture of Marino-Vafa on Hodge integrals. J. Differential Geom., 65, 289–340 (2003)
Liu, S.-Q., Yang, D., Zhang, Y., Zhou, C.: The Hodge-FVH correspondence. J. Reine Angew. Math., 775, 259–300 (2021)
Liu, X., Tian, G.: Virasoro constraints for quantum cohomology. J. Differential Geom., 50, 537–590 (1998)
Manakov, S. V.: Complete integrability and stochastization of discrete dynamical systems. J. Experiment. Theoret. Phys., 67, 543–555 (in Russian). English translation in: Soviet Physics JETP, 40, 269–274 (1974)
Mehta, M. L.: Random Matrices, 2nd edition. Academic Press, Boston, 1991
Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz theory, and completed cycles. Annals of Mathematics, 163, 517–560 (2006)
Okounkov, A., Pandharipande, R.: The equivariant Gromov–Witten theory of P1. Annals of Mathematics, 163, 561–605 (2006)
Okounkov, A., Pandharipande, R.: Hodge integrals and invariants of the unknot. Geom. Topol., 8, 675–699 (2004)
Teleman, C.: The structure of 2D semi-simple field theories. Invent. Math., 188, 525–588 (2012)
Tsarev, S. P.: The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method. Math. USSR Izvestiya, 37, 397–419 (1991)
Whittaker, E. T., Watson, G. N.: A Course of Modern Analysis, 4th edn., Cambridge University Press, Cambridge, 1963
Witten, E.: Two-dimensional gravity and intersection theory on moduli space, In: Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, 243–310
Yang, D.: On tau-functions for the Toda lattice hierarchy. Lett. Math. Phys., 110, 555–583 (2020)
Yang, D., Zagier, D.: Mapping partition functions. In preparation
Yang, D., Zhang, Q.: On the Hodge-BGW correspondence, arXiv:2112.12736 (2021)
Yang, D., Zhou, J.: Grothendieck’s dessins d’enfants in a web of dualities. III. J. Phys. A, 56, 055201, 34 pp. (2023)
Zhang, Y.: On the CP1 topological sigma model and the Toda lattice hierarchy. J. Geom. Phys., 40, 215–232 (2002)
Zhou, J.: Genus expansions of Hermitian one-matrix models: fat graphs vs. thin graphs. arXiv:1809.10870 (2018)
Zhou, J.: Grothendieck’s dessins d’enfants in a web of dualities. arXiv:1905.10773 (2019)
Acknowledgements
The author is grateful to Boris Dubrovin, Don Zagier, Youjin Zhang and Jian Zhou for their advice.
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Supported by NSFC (Grant No. 1206113101)
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Yang, D. GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back. Acta. Math. Sin.-English Ser. 40, 383–405 (2024). https://doi.org/10.1007/s10114-024-2258-3
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DOI: https://doi.org/10.1007/s10114-024-2258-3
Keywords
- Frobenius manifold
- Dubrovin–Zhang hierarchy
- GUE
- Toda lattice hierarchy
- jet space
- topological gravity
- matrix gravity