Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP
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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Springer Basel
2020
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| Online Access: | https://hdl.handle.net/1721.1/125209 |
| _version_ | 1811072553973710848 |
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| author | Bergére, Michel Borot, Gaetan Eynard, Bertrand |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Bergére, Michel Borot, Gaetan Eynard, Bertrand |
| author_sort | Bergére, Michel |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T09:07:46Z |
| format | Article |
| id | mit-1721.1/125209 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:07:46Z |
| publishDate | 2020 |
| publisher | Springer Basel |
| record_format | dspace |
| spelling | mit-1721.1/1252092022-09-26T10:40:32Z Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP Bergére, Michel Borot, Gaetan Eynard, Bertrand Massachusetts Institute of Technology. Department of Mathematics 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. 2020-05-13T15:37:12Z 2020-05-13T15:37:12Z 2015-01 2013-12 2019-02-02T04:46:00Z Article http://purl.org/eprint/type/JournalArticle 1424-0637 1424-0661 https://hdl.handle.net/1721.1/125209 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 en https://doi.org/10.1007/s00023-014-0391-8 Annales Henri Poincaré 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. Springer Basel application/pdf Springer Basel Springer Basel |
| spellingShingle | Bergére, Michel Borot, Gaetan Eynard, Bertrand Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP |
| title | Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP |
| title_full | Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP |
| title_fullStr | Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP |
| title_full_unstemmed | Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP |
| title_short | Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP |
| title_sort | rational differential systems loop equations and application to the qth reductions of kp |
| url | https://hdl.handle.net/1721.1/125209 |
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