## Periodic Reporting for period 1 - INTSYS (Algebraic, Geometric, and Field-Theoretic Aspects of Integrable Many-Body Systems)

Reporting period: 2018-09-01 to 2020-08-31

General context:

This research project studies a very active area of modern mathematical physics, which is concerned with a particular kind of dynamical systems called integrable systems. Loosely speaking, in such systems there are many conserved quantities, besides energy and momentum. This restricts the dynamics and the underlying equations can be solved exactly. In certain cases, solutions can be expressed in terms of integrals, hence the name.

Applications in physics:

Integrable systems are extremely useful in describing various natural phenomena. Perhaps the most famous example is the gravitational two-body problem, which was solved by Newton. Newton's solution explained Kepler's laws of planetary motion. The harmonic oscillator and the hydrogen atom are also invaluable integrable systems in physics. More recently, a treasure trove of integrable partial differential equations with so-called soliton solutions has been discovered. In particular, the sine-Gordon model, which is used by particle physicists as a theoretical laboratory for testing new ideas. Other remarkable examples are given by the Korteweg-de Vries and Kadomtsev-Petviashvili equations, which describe waves in shallow water or plasma. The non-linear Schrödinger equation is another famous example with applications in the theory of optical fibres.

Mathematical connections:

The mathematical features of integrable systems are exceptionally rich and tie in with most branches of mathematics, including algebra, analysis, geometry and probability theory. For example, the previously mentioned conservation laws correspond to the presence of hidden symmetries, which can be analysed using algebraic methods (group theory, representation theory). The study of quantum integrable systems involves Hilbert space functional analysis, while the natural framework for classical systems is that of symplectic geometry. Connections to stochastic processes and random matrix theory have also been established.

This Fellowship:

The main focus of my project is the celebrated Calogero-Ruijsenaars family of integrable many-body systems. These models describe the pairwise interaction of equal-mass particles moving on a line or circle. The strength of particle interaction is regulated by a (real) number, the so-called coupling parameter. Setting this parameter to zero means no interaction, i.e. free particles, while non-zero parameter values result in a complicated motion. This is due to the non-linear nature of particle interaction, of which we distinguish four types, named rational, hyperbolic, trigonometric, and elliptic. The particles can be thought of as either non-relativistic bodies obeying the laws of Newtonian mechanics or relativistic point masses with an upper speed limit (given by the speed of light). Integrable quantum mechanical versions also exist. In addition, Calogero-Ruijsenaars type systems have several generalisations preserving integrability, such as models in external fields (involving multiple couplings) or particles with spin (internal degrees of freedom). This profusion of variants enhances the importance of these systems.

In fact, Calogero-Ruijsenaars type models are intimately related to various integrable systems of seemingly different character. These include soliton equations, lattice models (e.g. Toda model), solvable spin and vertex models (e.g. the Heisenberg XYZ model and the 8-vertex model).

This research project is structured around three major themes aimed at finding new models, structures, and connections. The most important outcomes are the following:

1. Discovery of quantum and classical Lax pairs for hyperbolic, trigonometric, and elliptic relativistic models containing multiple couplings.

2. Solution of the classical and quantum dynamics of new compactified trigonometric relativistic systems.

3. Finding new and extending already existing links to quiver gauge theory and topological quantum field theory.

This research project studies a very active area of modern mathematical physics, which is concerned with a particular kind of dynamical systems called integrable systems. Loosely speaking, in such systems there are many conserved quantities, besides energy and momentum. This restricts the dynamics and the underlying equations can be solved exactly. In certain cases, solutions can be expressed in terms of integrals, hence the name.

Applications in physics:

Integrable systems are extremely useful in describing various natural phenomena. Perhaps the most famous example is the gravitational two-body problem, which was solved by Newton. Newton's solution explained Kepler's laws of planetary motion. The harmonic oscillator and the hydrogen atom are also invaluable integrable systems in physics. More recently, a treasure trove of integrable partial differential equations with so-called soliton solutions has been discovered. In particular, the sine-Gordon model, which is used by particle physicists as a theoretical laboratory for testing new ideas. Other remarkable examples are given by the Korteweg-de Vries and Kadomtsev-Petviashvili equations, which describe waves in shallow water or plasma. The non-linear Schrödinger equation is another famous example with applications in the theory of optical fibres.

Mathematical connections:

The mathematical features of integrable systems are exceptionally rich and tie in with most branches of mathematics, including algebra, analysis, geometry and probability theory. For example, the previously mentioned conservation laws correspond to the presence of hidden symmetries, which can be analysed using algebraic methods (group theory, representation theory). The study of quantum integrable systems involves Hilbert space functional analysis, while the natural framework for classical systems is that of symplectic geometry. Connections to stochastic processes and random matrix theory have also been established.

This Fellowship:

The main focus of my project is the celebrated Calogero-Ruijsenaars family of integrable many-body systems. These models describe the pairwise interaction of equal-mass particles moving on a line or circle. The strength of particle interaction is regulated by a (real) number, the so-called coupling parameter. Setting this parameter to zero means no interaction, i.e. free particles, while non-zero parameter values result in a complicated motion. This is due to the non-linear nature of particle interaction, of which we distinguish four types, named rational, hyperbolic, trigonometric, and elliptic. The particles can be thought of as either non-relativistic bodies obeying the laws of Newtonian mechanics or relativistic point masses with an upper speed limit (given by the speed of light). Integrable quantum mechanical versions also exist. In addition, Calogero-Ruijsenaars type systems have several generalisations preserving integrability, such as models in external fields (involving multiple couplings) or particles with spin (internal degrees of freedom). This profusion of variants enhances the importance of these systems.

In fact, Calogero-Ruijsenaars type models are intimately related to various integrable systems of seemingly different character. These include soliton equations, lattice models (e.g. Toda model), solvable spin and vertex models (e.g. the Heisenberg XYZ model and the 8-vertex model).

This research project is structured around three major themes aimed at finding new models, structures, and connections. The most important outcomes are the following:

1. Discovery of quantum and classical Lax pairs for hyperbolic, trigonometric, and elliptic relativistic models containing multiple couplings.

2. Solution of the classical and quantum dynamics of new compactified trigonometric relativistic systems.

3. Finding new and extending already existing links to quiver gauge theory and topological quantum field theory.

During the fellowship three Work Packages [WP1,WP2,WP3] have been carried out. These consisted of Conferences [C1–C7], Dissemination [D1–D10], Seminars [S1–S7], Public Engagement [P1–P7], Research Visits [V1–V4], and a Workshop [W1]. These points reflect the various activities described in Part B of the INTSYS research proposal.

In the case of Objective 2, we went far beyond what anyone previously expected by solving not just the original problem, but a much harder version of it which has been unsolved for more than 30 years. This is the most significant research outcome of the fellowship. The results of the fellowship have a real potential of shaping future scientific developments, because they opened new avenues of research which are of great interest to many mathematicians and physicists. These potential users of the project's outcomes include researchers working in the fields of integrable systems, orthogonal polynomials, special functions, quantum field theories, statistical physics, condensed matter physics, string theory, and many more.