Periodic Reporting for period 1 - ELLIS-SDI (Elliptic Integrable Systems: solutions, deformations and integrability)
Reporting period: 2021-09-01 to 2023-08-31
The main topic of this project is the study of new integrable systems, the links between seemingly different integrable systems, and the properties of exact solutions given by special functions. Integrable systems is a distinct class of dynamical systems characterized by predictable behavior due to the existence of sufficiently many conservation laws, such as conservation of energy, conservation of momenta, and numerous others that are not explicitly named. Among integrable systems, there is a class called `exactly solvable models´ where a complete set of exact solutions can be obtained. A prominent examples is the Kepler problem of planetary motion which has exact solutions expressed in terms of integrals and hence the name integrable systems.
Integrable systems have made significant contributions to our understanding of both mathematics and physics, such as 1) the so-called sine-Gordon equation which appear in the context of propagation in junctions between two superconductors (i.e. Josephson junctions) and scattering data for relativistic particles in physics context, and differential geometry in mathematical context and 2) the Calogero-Moser-Sutherland (CMS) models that appear in physical systems such as the fractional quantum Hall effect and topological insulators, and in mathematical subjects as combinatorics and representation theory.
In the beginning of the 1980s, Ruijsenaars found that behaviour of the solutions of the sine-Gordon could also be obtained from the solutions of relativistic generalization of the CMS models, which gave rise to a new family of integrable systems. Ruijsenaars found that the interaction of the models has rational, trigonometric, hyperbolic, and elliptic versions, as was known for the CMS models, and it was later shown that there is an underlying root system of a Lie algebra associated to the model. While Liouville integrability has been established for all these relativistic models, little is known about the exact solutions in the elliptic cases. The most general of these models is the so-called van Diejen model which is a relativistic version of the elliptic CMS model associated to the (non-reduced) BC root system.
Another mathematically natural generalizations of the CMS model were discovered by Chalykh, Feigin, Sergeev, and Veselov (CFSV) which was related to root systems of Lie super-algebras. These generalizations of CFSV type are often referred to as deformed CMS models. The deformed model also has relativistic generalizations of Ruijsenaars and van Diejen type that have been discovered in the last decade. These models are known to be Liouville integrable for the non-elliptic cases, where an algebraic approach using so-called double affine Hecke algebras are applicable, and exactly solvable in the trigonometric case related to A-type root system.
The Painlevé equations are a set of non-linear differential equations with the so-called Painlevé property. At the end of the 19th century, it was shown that the Painlevé VI equation can be mapped to either an isomonodromic deformation of a linear ordinary differential equation or to a non-autonomous version of the one-variable elliptic CMS model of BC-type. This is commonly referred to as the Painlevé-Calogero correspondence.
The objectives of this project are to construct and study the exact solutions of the elliptic CMS models, prove the integrability of the elliptic deformed CMS models, and to extend the Painlevé-Calogero correspondence to the multivariate CMS models.
Integrable systems have made significant contributions to our understanding of both mathematics and physics, such as 1) the so-called sine-Gordon equation which appear in the context of propagation in junctions between two superconductors (i.e. Josephson junctions) and scattering data for relativistic particles in physics context, and differential geometry in mathematical context and 2) the Calogero-Moser-Sutherland (CMS) models that appear in physical systems such as the fractional quantum Hall effect and topological insulators, and in mathematical subjects as combinatorics and representation theory.
In the beginning of the 1980s, Ruijsenaars found that behaviour of the solutions of the sine-Gordon could also be obtained from the solutions of relativistic generalization of the CMS models, which gave rise to a new family of integrable systems. Ruijsenaars found that the interaction of the models has rational, trigonometric, hyperbolic, and elliptic versions, as was known for the CMS models, and it was later shown that there is an underlying root system of a Lie algebra associated to the model. While Liouville integrability has been established for all these relativistic models, little is known about the exact solutions in the elliptic cases. The most general of these models is the so-called van Diejen model which is a relativistic version of the elliptic CMS model associated to the (non-reduced) BC root system.
Another mathematically natural generalizations of the CMS model were discovered by Chalykh, Feigin, Sergeev, and Veselov (CFSV) which was related to root systems of Lie super-algebras. These generalizations of CFSV type are often referred to as deformed CMS models. The deformed model also has relativistic generalizations of Ruijsenaars and van Diejen type that have been discovered in the last decade. These models are known to be Liouville integrable for the non-elliptic cases, where an algebraic approach using so-called double affine Hecke algebras are applicable, and exactly solvable in the trigonometric case related to A-type root system.
The Painlevé equations are a set of non-linear differential equations with the so-called Painlevé property. At the end of the 19th century, it was shown that the Painlevé VI equation can be mapped to either an isomonodromic deformation of a linear ordinary differential equation or to a non-autonomous version of the one-variable elliptic CMS model of BC-type. This is commonly referred to as the Painlevé-Calogero correspondence.
The objectives of this project are to construct and study the exact solutions of the elliptic CMS models, prove the integrability of the elliptic deformed CMS models, and to extend the Painlevé-Calogero correspondence to the multivariate CMS models.
During the fellowship, I have made several important steps to constructing exact solutions of the van Diejen model. In particular, great progress has been made on constructing exact solutions of the van Diejen model using a perturbative method through a kernel function approach. Furthermore, I showed that there are explicit transformations for solutions of the van Diejen model that can be used to generate new solutions. Among these generators, we find several that can be used to generate the action of the Weyl group corresponding to the exceptional Lie algebra E8 on the parameter space. We have also worked on constructing the higher order symmetries of the van Diejen model explicitly, which are not clearly found in the literature (although existence and commutativity have been established).
As mentioned previously, the CMS models and their various generalizations are related to the root systems of classical Lie algebras. Recently, it appeared that there are integrable models of CMS type related to paticular hyperplane arrangements. During the fellowship, we have also worked on constructing a relativistic generalization of Ruijsenaars’ model related to an A-type hyperplane arrangements. More specifically, we have constructed certain exact polynomial eigenfunctions of the trigonometric version, worked on proving Liouville integrability, constructing (quantum) Lax pairs, and finding kernel function identities for this generalized model.
Our work on the multivariate Painlevé-Calogero correspondence included a more algebraic approach using Dunkl operators and Cherednik algebras where we have investigated the relation between these and a suitable space of isomonodromic deformations for the multivariate Painlevé equations.
During the fellowship, I have organized workshops, disseminated the results of the project in various international conferences and workshops, and had two outreach events where I explained the contributions of integrable systems and special functions.
As mentioned previously, the CMS models and their various generalizations are related to the root systems of classical Lie algebras. Recently, it appeared that there are integrable models of CMS type related to paticular hyperplane arrangements. During the fellowship, we have also worked on constructing a relativistic generalization of Ruijsenaars’ model related to an A-type hyperplane arrangements. More specifically, we have constructed certain exact polynomial eigenfunctions of the trigonometric version, worked on proving Liouville integrability, constructing (quantum) Lax pairs, and finding kernel function identities for this generalized model.
Our work on the multivariate Painlevé-Calogero correspondence included a more algebraic approach using Dunkl operators and Cherednik algebras where we have investigated the relation between these and a suitable space of isomonodromic deformations for the multivariate Painlevé equations.
During the fellowship, I have organized workshops, disseminated the results of the project in various international conferences and workshops, and had two outreach events where I explained the contributions of integrable systems and special functions.
The results obtain has not only advanced our understanding of exact solutions for elliptic integrable systems and their symmetries, but also developed novel methods for constructing solutions of other integrable systems explicitly.
Our results for the van Diejen model have given (to our knowledge) the first proof of the conjectured E8 symmetry of the model, but also shown that the actual symmetries is larger than expected. The results are now being used to construct Hilbert-Schmidt integral operators and prove the Hilbert space aspects of the solutions as well as raising/lowering operators for the model. The methods used to obtain these results are also now being used to construct solutions of other integrable generalizations. We also expect that the results can be used for constructing Lax pairs and tau functions for multivariate generalization of Sakai’s elliptic Painlevé equation and elliptic q-Garnier systems.
Our results on the generalized Ruijsenaars models have given a new contribution to the field of integrable many-particle system and the theory of special functions, while our approach to the Painlevé-Calogero correspondence suggests new approaches to multivariate Painlevé equations.
It is expected that the results obtained in this fellowship will be useful for mathematicians working not only with integrable systems and the theory of special functions, but also in fields such as representation theory of elliptic quantum groups and (affine) Lie super-algebras, and for mathematical physicists working in fields such as condensed matter physics, quantum field theories, and string theory.
Our results for the van Diejen model have given (to our knowledge) the first proof of the conjectured E8 symmetry of the model, but also shown that the actual symmetries is larger than expected. The results are now being used to construct Hilbert-Schmidt integral operators and prove the Hilbert space aspects of the solutions as well as raising/lowering operators for the model. The methods used to obtain these results are also now being used to construct solutions of other integrable generalizations. We also expect that the results can be used for constructing Lax pairs and tau functions for multivariate generalization of Sakai’s elliptic Painlevé equation and elliptic q-Garnier systems.
Our results on the generalized Ruijsenaars models have given a new contribution to the field of integrable many-particle system and the theory of special functions, while our approach to the Painlevé-Calogero correspondence suggests new approaches to multivariate Painlevé equations.
It is expected that the results obtained in this fellowship will be useful for mathematicians working not only with integrable systems and the theory of special functions, but also in fields such as representation theory of elliptic quantum groups and (affine) Lie super-algebras, and for mathematical physicists working in fields such as condensed matter physics, quantum field theories, and string theory.