Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP
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Author(s)
Bergére, Michel
Borot, Gaetan
Eynard, Bertrand
Date Issued
January 2015
Journal
Annales Henri Poincaré
Publisher
Springer Basel
Citation
Bergére, Michel et al. “Rational Differential Systems, Loop Equations, and Application to the Qth Reductions of KP.” Annales Henri Poincaré 16, 12 (January 2015): 2713–2782 © 2015 Springer Nature
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Author's final manuscript
Abstract
To any solution of a linear system of differential equations, we associate a matrix kernel, correlators satisfying a set of loop equations, and in the presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight ħ per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non-trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the qth reductions of KP—which contain the (p, q) models as a specialization.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1007/s00023-014-0391-8
