FROM WAVE FUNCTIONS TO TAU-FUNCTIONS FOR THE VOLTERRA LATTICE HIERARCHY

doi:10.1007/s10473-024-0201-4

Abstract

Abstract: For an arbitrary solution to the Volterra lattice hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the matrix-resolvent method. In this paper, we define a pair of wave functions of the solution and use them to give an expression of the matrix resolvent; based on this we obtain a new formula for the $k$-point functions for the Volterra lattice hierarchy in terms of wave functions. As an application, we give an explicit formula of $k$-point functions for the even GUE (Gaussian Unitary Ensemble) correlators.

Key words: Volterra lattice hierarchy, matrix-resolvent method, wave functions, even GUE correlators

CLC Number:

37K10

Ang FU, Mingjin LI, Di YANG. FROM WAVE FUNCTIONS TO TAU-FUNCTIONS FOR THE VOLTERRA LATTICE HIERARCHY[J].Acta mathematica scientia,Series B, 2024, 44(2): 405-419.

Trendmd

References

[1] Adler M, van Moerbeke P. Matrix integrals, Toda symmetries, Virasoro constraints,orthogonal polynomials. Duke Math J, 1995, 80: 863-911
[2] Bertola M, Dubrovin B, Yang D. Correlation functions of the KdV hierarchy and applications to intersection numbers over $\overline{{\mathcal{M}}}_{g,n}$. Physica D, 2016, 327: 30-57
[3] Bertola M, Dubrovin B, Yang D. Simple Lie algebras and topological ODEs. Int Math Res Not, 2018, 2018: 1368-1410
[4] Bertola M, Dubrovin B, Yang D. Simple Lie algebras, Drinfeld-Sokolov hierarchies,multi-point correlation functions. Mosc Math J, 2021, 21: 233-270
[5] Bessis D, Itzykson C, Zuber J B. Quantum field theory techniques in graphical enumeration. Adv Appl Math, 1980, 1: 109-157
[6] Brézin E, Itzykson C, Parisi P, Zuber J B. Planar diagrams. Comm Math Phys, 1978, 59: 35-51
[7] Cafasso M, Yang D. Tau-functions for the Ablowitz-Ladik hierarchy: the matrix-resolvent method. J Phys A: Math Theor, 2022, 55: 204001
[8] Carlet G. The extended bigraded Toda hierarchy. J Phys A: Math Gen, 2006, 39: 9411-9435
[9] Carlet G, Dubrovin B, Zhang Y. The extended Toda hierarchy. Mosc Math J, 2004, 4: 313-332
[10] Deift P A.Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Providence, RI: American Mathematical Society, 1999
[11] Dubrovin B.Algebraic spectral curves over $\mathbb{Q}$ and their tau-functions//Donagi R, Shaska T. Integrable Systems and Algebraic Geometry. Cambridge: Cambridge University Press, 2020: 41-91
[12] Dubrovin B. Hamiltonian perturbations of hyperbolic PDEs: from classification results to the properties of solutions//Sidoravičius V. New Trends in Mathematical Physics. Dordrecht: Springer, 2009: 231-276
[13] Dubrovin B, Liu S Q, Yang D, Zhang Y. Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs. Adv Math, 2016, 293: 382-435
[14] Dubrovin B, Liu S Q, Yang D, Zhang Y. Hodge-GUE correspondence and the discrete KdV equation. Comm Math Phys, 2020, 379: 461-490
[15] Dubrovin B, Valeri D, Yang D. Affine Kac-Moody algebras and tau-functions for the Drinfeld-Sokolov hierarchies: the matrix-resolvent method. Symmetry Integrability Geom Methods Appl, 2022, 18: 077
[16] Dubrovin B, Yang D.Generating series for GUE correlators. Lett Math Phys, 2017, 107: 1971-2012
[17] Dubrovin B, Yang D. On cubic Hodge integrals and random matrices. Commun Number Theory Phys, 2017, 11: 311-336
[18] Dubrovin B, Yang D. Matrix resolvent and the discrete KdV hierarchy. Comm Math Phys, 2020, 377: 1823-1852
[19] Dubrovin B, Yang D, Zagier D. Gromov-Witten invariants of the Riemann sphere. Pure Appl Math Q, 2020, 16: 153-190
[20] Dubrovin B, Yang D, Zagier D. On tau-functions for the KdV hierarchy. Sel Math, 2021, 27: Art 12
[21] Dubrovin B, Zhang Y.Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov-Witten invariants. arXiv:math/0108160
[22] Fu A, Yang D. The matrix-resolvent method to tau-functions for the nonlinear Schrödinger hierarchy. J Geom Phys, 2022, 179: 104592
[23] Gerasimov A, Marshakov A, Mironov A, et al. Matrix models of two dimensional gravity and Toda theory. Nuclear Physics B, 1991, 357: 565-618
[24] Guo J, Yang D.On the large genus asymptotics of psi-class intersection numbers. Math Ann, 2022. DOI: 10.1007/s00208-022-02505-6
[25] Harer J, Zagier D. The Euler characteristic of the moduli space of curves. Invent Math, 1986, 85: 457-485
[26] 't Hooft G. A planar diagram theory for strong interactions. Nucl Phys B, 1974, 72: 461-473
[27] 't Hooft G. A two-dimensional model for mesons. Nucl Phys B, 1974, 75: 461-470
[28] Kazakov V, Kostov I, Nekrasov N. D-particles, matrix integrals and KP hierarchy. Nucl Phys B, 1999, 557: 413-442
[29] Mehta M L. Random Matrices.New York: Academic Press, 1991
[30] Morozov A, Shakirov S. Exact 2-point function in Hermitian matrix model. Journal of High Energy Physics, 2009, 12: Art 003
[31] Witten E. Two dimensional gravity and intersection theory on Moduli space. Surveys Diff Geom, 1991, 1: 243-310
[32] Yang D. On tau-functions for the Toda lattice hierarchy. Lett Math Phys, 2020, 110: 555-583
[33] Zhou J.Hermitian one-matrix model and KP hierarchy. arXiv:1809.07951