Аннотация:
We propose a definition for a Tau function and a spinor kernel (closely related to Baker–Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large.
The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between “correlators”, the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.
Ключевые слова:
topological recursion; Tau function; Sato formula; Hirota equations; Whitham equations.
Поступила:14 ноября 2011 г.; в окончательном варианте 11 декабря 2012 г.; опубликована 18 декабря 2012 г.
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\by Г.~Боро, Bertrand~Eynard
\paper Geometry of Spectral Curves and All Order Dispersive Integrable System
\jour SIGMA
\yr 2012
\vol 8
\papernumber 100
\totalpages 53
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Эта публикация цитируется в следующих 22 статьяx:
Gaëtan Borot, Vincent Bouchard, Nitin K. Chidambaram, Thomas Creutzig, “Whittaker vectors for W-algebras from topological recursion”, Sel. Math. New Ser., 30:2 (2024)
Bertrand Eynard, Elba Garcia-Failde, Olivier Marchal, Nicolas Orantin, “Quantization of Classical Spectral Curves via Topological Recursion”, Commun. Math. Phys., 405:5 (2024)
Matijn François, Alba Grassi, “Painlevé Kernels and Surface Defects at Strong Coupling”, Ann. Henri Poincaré, 2024
Gaëtan Borot, Thomas Buc-D"Alche, “Fay Identities of Pfaffian Type for Hyperelliptic Curves”, SIGMA, 20 (2024), 054, 38 pp.
Marco Bertola, Dmitry Korotkin, Ramtin Sasani, “Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices”, SIGMA, 19 (2023), 104, 22 pp.
Marchal O., Orantin N., “Quantization of Hyper-Elliptic Curves From Isomonodromic Systems and Topological Recursion”, J. Geom. Phys., 171 (2022), 104407
Marchal O., Orantin N., “Isomonodromic Deformations of a Rational Differential System and Reconstruction With the Topological Recursion: the Sl2 Case”, J. Math. Phys., 61:6 (2020)
Kohei Iwaki, “2-Parameter τ-Function for the First Painlevé Equation: Topological Recursion and Direct Monodromy Problem via Exact WKB Analysis”, Commun. Math. Phys., 377:2 (2020), 1047
Marino M., Zakany S., “Wavefunctions, Integrability, and Open Strings”, J. High Energy Phys., 2019, no. 5, 014
R. Belliard, B. Eynard, O. Marchal, “Loop equations from differential systems on curves”, Ann. Henri Poincare, 19:1 (2018), 141–161
K. Iwaki, O. Marchal, A. Saenz, “Painlevé equations, topological type property and reconstruction by the topological recursion”, J. Geom. Phys., 124 (2018), 16–54
N. Do, P. Norbury, “Topological recursion on the Bessel curve”, Commun. Number Theory Phys., 12:1 (2018), 53–73
V. Bouchard, N. K. Chidambaram, T. Dauphinee, “Quantizing weierstrass”, Commun. Number Theory Phys., 12:2 (2018), 253–303
G. Borot, S. Shadrin, “Blobbed topological recursion: properties and applications”, Math. Proc. Camb. Philos. Soc., 162:1 (2017), 39–87
R. Belliard, B. Eynard, O. Marchal, “Integrable differential systems of topological type and reconstruction by the topological recursion”, Ann. Henri Poincare, 18:10 (2017), 3193–3248
M. Marino, S. Zakany, “Exact eigenfunctions and the open topological string”, J. Phys. A-Math. Theor., 50:32 (2017), 325401
V. Bouchard, B. Eynard, “Reconstructing WKB from topological recursion”, Journal de l'Ecole Polytechnique - Mathematiques, 4 (2017), 845-908
Bergere M., Borot G., Eynard B., “Rational Differential Systems, Loop Equations, and Application To the Qth Reductions of Kp”, Ann. Henri Poincare, 16:12 (2015), 2713–2782
Borot, G.; Eynard, B., “All order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials”, Quantum Topology, 6:1 (2015), 39-138
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