Flat Structure on the Space of Isomonodromic Deformations
Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently, the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as a Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Ok...
Saved in:
| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Date: | 2020 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2020
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211010 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Flat Structure on the Space of Isomonodromic Deformations. Mitsuo Kato, Toshiyuki Mano and Jiro Sekiguchi. SIGMA 16 (2020), 110, 36 pages |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862736003687514112 |
|---|---|
| author | Kato, Mitsuo Mano, Toshiyuki Sekiguchi, Jiro |
| author_facet | Kato, Mitsuo Mano, Toshiyuki Sekiguchi, Jiro |
| citation_txt | Flat Structure on the Space of Isomonodromic Deformations. Mitsuo Kato, Toshiyuki Mano and Jiro Sekiguchi. SIGMA 16 (2020), 110, 36 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently, the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as a Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of system of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of a Frobenius manifold. As a consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation.
|
| first_indexed | 2026-04-17T16:31:24Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211010 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-04-17T16:31:24Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Kato, Mitsuo Mano, Toshiyuki Sekiguchi, Jiro 2025-12-22T09:27:04Z 2020 Flat Structure on the Space of Isomonodromic Deformations. Mitsuo Kato, Toshiyuki Mano and Jiro Sekiguchi. SIGMA 16 (2020), 110, 36 pages 1815-0659 2020 Mathematics Subject Classification: 34M56; 33E17; 35N10; 32S25 arXiv:1511.01608 https://nasplib.isofts.kiev.ua/handle/123456789/211010 https://doi.org/10.3842/SIGMA.2020.110 Flat structure was introduced by K. Saito and his collaborators at the end of 1970's. Independently, the WDVV equation arose from the 2D topological field theory. B. Dubrovin unified these two notions as a Frobenius manifold structure. In this paper, we study isomonodromic deformations of an Okubo system, which is a special kind of system of linear differential equations. We show that the space of independent variables of such isomonodromic deformations can be equipped with a Saito structure (without a metric), which was introduced by C. Sabbah as a generalization of a Frobenius manifold. As a consequence, we introduce flat basic invariants of well-generated finite complex reflection groups and give explicit descriptions of Saito structures (without metrics) obtained from algebraic solutions to the sixth Painlevé equation. Professor Yoshishige Haraoka taught the first author (M.K.) that integrable systems in three variables are useful to derive the Painleve VI solutions. This is the starting point of our work. The authors would like to thank Professor Haraoka for his advice. After a preprint of this paper was written, the authors received helpful comments, including information on the papers [1, 3, 7, 14, 19, 43, 44, 47] from Professors B. Dubrovin, Y. Konishi, C. Hertling, P. Boalch, A. Arsie, P. Lorenzoni, J. Michel, and H. Terao. The authors express their sincere gratitude to these people. The authors thank anonymous referees for their useful comments and suggestions in order to improve the manuscript. This work was partially supported by JSPS KAKENHI Grant Numbers 25800082, 17K05335, 26400111, and 17K05269. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Flat Structure on the Space of Isomonodromic Deformations Article published earlier |
| spellingShingle | Flat Structure on the Space of Isomonodromic Deformations Kato, Mitsuo Mano, Toshiyuki Sekiguchi, Jiro |
| title | Flat Structure on the Space of Isomonodromic Deformations |
| title_full | Flat Structure on the Space of Isomonodromic Deformations |
| title_fullStr | Flat Structure on the Space of Isomonodromic Deformations |
| title_full_unstemmed | Flat Structure on the Space of Isomonodromic Deformations |
| title_short | Flat Structure on the Space of Isomonodromic Deformations |
| title_sort | flat structure on the space of isomonodromic deformations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211010 |
| work_keys_str_mv | AT katomitsuo flatstructureonthespaceofisomonodromicdeformations AT manotoshiyuki flatstructureonthespaceofisomonodromicdeformations AT sekiguchijiro flatstructureonthespaceofisomonodromicdeformations |