Elliptic hypergeometric integrals in classical and quantum integrable systems
Project/Area Number  16F16318 
Research Category 
GrantinAid for JSPS Fellows

Allocation Type  Singleyear Grants 
Section  外国 
Research Field 
Basic analysis

Research Institution  The University of Tokyo 
Host Researcher 
国場 敦夫 東京大学, 大学院総合文化研究科, 教授 (70211886)

Foreign Research Fellow 
KELS ANDREW 東京大学, 総合文化研究科, 外国人特別研究員

Project Period (FY) 
20161107 – 20190331

Project Status 
Granted(Fiscal Year 2017)

Budget Amount *help 
¥1,400,000 (Direct Cost : ¥1,400,000)
Fiscal Year 2018 : ¥300,000 (Direct Cost : ¥300,000)
Fiscal Year 2017 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 2016 : ¥400,000 (Direct Cost : ¥400,000)

Keywords  YangBaxter equation / hypergeometric integral / gauge theory / duality / 可積分模型 / ヤン・バクスター方程式 / 超対称指標 
Outline of Annual Research Achievements 
1. We have mathematically proven new hypergeometric integral transformation formulas associated to root systems. These are the most general known formulas for hypergeometric functions, and contain many important results as limiting cases. We have shown how these formulas are relevant for both supersymmetric gauge theories, and integrable models of statistical mechanics. 2. I have proven a new generalisation of the Zinvariance property for exactly solved models of statistical mechanics. I showed that a square lattice model is invariant under "cubic flip" type deformations of the lattice, and gave the connection to classical integrable systems based on 3Dconsistency property. 3. We have shown how the lens elliptic gamma function solution of the startriangle relation, reduces to the chiral Potts and Kashiwara Miwa models in a root of unity limit. We also showed how the Q4 integrable quad equation arises in this limit, and proposed an interpretation of this limit for supersymmetric gauge theories. 4. I have shown how all 3Dconsistent integrable quad equations in the classification of Adler, Bobenko, and Suris, each arise from the classical limit of a counterpart startriangle relation. This work also provides an explicit connection between 3Dconsistent integrable equations, and hypergeometric beta integral formulas.

Current Status of Research Progress 
Current Status of Research Progress
2 : Research has progressed on the whole more than it was originally planned.
Reason
The research project is proceeding smoothly as planned. Much of the research of this project involves complicated mathematical calculations, and I have been able to successfully complete the calculations that have been necessary for making progress in this project. The results that I have obtained appear in three peerreviewed papers that have been published over this past year, and which appear in top journals of mathematical physics. This amount of publications over the past year is better than I have anticipated and is a good rate of publication for mathematical physics research. In addition to this, I have also uploaded a recent preprint to arXiv, which contains important results for this project, and is currently undergoing peerreview in the journal "Communications in Mathematical Physics". I have given several presentations of the results of this project, and I have received good interest and feedback on the research. The research that has been completed over the past year will provide a necessary basis for obtaining further results in this project.

Strategy for Future Research Activity 
The first of these projects involves mathematically proving new transformation formulas for hypergeometric integrals, obtaining associated new solutions of the YangBaxter equation, and interpreting these formulas in terms of duality of supersymmetric gauge theories. There are some calculations to be finalised and this project is expected to be completed in the coming months. The second of these projects involves obtaining new multicomponent 3Dconsistent integrable equations, through the use of the YBE/3Dconsistency correspondence. I have already obtained one new example of such integrable equation, and we (with Masahito Yamazaki) are currently seeing if my method can be extended to obtain other examples of multicomponent 3Dconsistent integrable system. I also expect to attend the Symmetries and Integrability of Difference Equations (SIDE) conference in Fukuoka in November to disseminate the results of this research. This is one of the major conferences in the area of this research project.

Report
(2results)
Research Products
(10results)