2019 Fiscal Year Annual Research Report
The tt* equations: a bridge between the differential geometry of moduli spaces and classical isomonodromy theory
| Project/Area Number |
18H03668
|
|---|
| Research Institution | Waseda University |
|---|
Principal Investigator |
Guest Martin 早稲田大学, 理工学術院, 教授 (10295470)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
細野 忍 学習院大学, 理学部, 教授 (60212198)
大仁田 義裕 大阪市立大学, 大学院理学研究科, 教授 (90183764)
|
|---|
| Project Period (FY) |
2018-04-01 – 2023-03-31
|
| Keywords | Integrable systems / Geometry / Quantum cohomology / tt* equations / Isomonodromy |
|---|
| Outline of Annual Research Achievements |
Progress was made by the Principal Investigator (Guest) mainly on the Lie-theoretic structure of the monodromy data of the tt*-Toda and related equations. This was reported at several seminar talks. Joint research was carried out with Peter Crooks (Northeastern University, USA), Alexander Its (IUPUI, USA), Nan-Kuo-Ho (NTHU, Taiwan), and Ian McIntosh (York, UK). Progress made by the Co-Investigators (Hosono, Ohnita) was as follows. Hosono continued his collaborations on double cover families of Calabi-Yau varieties with Bong Lian (Brandeis, USA), S.T.Yau (Harvard, USA) and Tsung-Ju Lee (Harvard, USA). Ohnita continued his research on homogenous spaces and submanifold theory from the viewpoint of special geometry, in particular Lagrangian aspects of R-spaces.
Several research activities were partially supported. Guest and Ohnita co-organised the 2nd Taiwan-Japan Joint Conference on Differential Geometry at the NCTS, National Taiwan University, Taipei in November 2019 and also the 3rd International Workshop on the Geometry of Submanifolds and Integrable Systems in December 2019 at OCAMI, Osaka City University. Guest organised lectures on Poisson Geometry, Moduli Spaces, and Applications by Yvette Kosmann-Schwarzbach (Ecole Polytechnique) and Nan-Kuo Ho (National Tsing-Hua University in June 2019 at Waseda University. Guest and Otofuji organised a workshop in the series “Koriyama Geometry and Physics Days” at Nihon University (Koriyama) in February 2020.
|
| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
Investigation of the symplectic structure of moduli spaces of meromorphic bundles (focusing on those related to the tt*-Toda equations) was continued by Guest and N. K. Ho. Research on the loop group approach to the tt*-Toda equations was continued by Guest and J. Dorfmeister. Progress on understanding the Lie-theoretic structure of the monodromy data of the tt*-Toda equations was made by Guest. Based on this, a new application to the soliton structure of supersymmetric quantum field theories was initiated. Joint work of Guest with C.S. Lin and A. Its on the Riemann-Hilbert approach to Painleve-type equations was continued. In his joint work with B. Lian, S.T. Yau and T.J. Lee, Hosono found a certain mirror duality for classes of double cover Calabi-Yau varieties, which generalize the previous results in 2018 for double cover family of K3 surfaces. Results are published in a preprint arXiv:2003.07148. Hosono started to study the Kaehler geometry on the moduli spaces, which is a special case of the tt* geometry. Ohnita showed that any R-space, canonically embedded in a Kahler C-space, is a globally tight Lagrangian submanifold, extending work of M. Takeuchi and S. Kobayashi, and of H. Tasaki and M. Tanaka. He began to study totally complex submanifolds of quaternionic Kahler manifolds from the same point of view.
|
| Strategy for Future Research Activity |
A joint article with N.K. Ho on the symplectic structure of the space of solutions to the tt*-Toda equations, based on the Lie-theoretic approach, is in preparation. A joint article with R. Sinclair on the numerics of solutions to the tt*-Toda equations is in preparation. It is anticipated that the original physics aspects of the tt* equations will play a greater role in the project from now on, with the intention of making contact with quantum aspects of tt* theory.
Activities planned for this year include: (1) lectures by Ian McIntosh (York) on integrable systems
at Waseda University in September 2020; (2) a conference on Special Geometry and Integrable Systems at Waseda University in December 2020, (3) a UK-Japan Winter School on Vertex Algebras in Mathematics and Physics at Glasgow University in January 2021; (4) a workshop in the series “Koriyama Geometry and Physics Days” at Nihon University (Koriyama).
|
