Symmetries: Algebra and Physics

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Symmetries are important in Science and especially in Physics. Models that can be solved exactly form the backbone of much theoretical understanding and it is observed that the presence of symmetries is the hallmark of such systems. Symmetries are described mathematically by algebras and because of this connection, the discovery of various algebraic structures has most often led to advances in theoretical physics. My research bears on the topics of the virtuous circle composed by symmetries, algebra, representation theory, special functions and physical models. I will determine the entanglement of quantum many body systems, design models related to tasks in quantum information and develop the mathematics of symmetries with an eye to special functions. This program comprises five related parts. Here are some details. 1. Heun operators. The standard Heun operator defines the differential equation with four regular singularities; it arises in many problems such as generalized tops or Gaudin magnets. I will transcend the current picture by introducing many new operators of Heun type with the help of the tridiagonalization method. The Bethe ansatz will be used to diagonalize these operators and applications in physics will be developed. I will show that from these constructs one can obtain algebras of Sklyanin type that are central in quantum integrable models. 2. Entanglement. This is a chief feature of quantum theories which is described by entropies. I will use the analogy between the time and band limiting problem in signal processing and the characterization of the entanglement of systems of fermions (on graphs), together with the key role that Heun operators play in the former context, to make headways in the latter. In the process I will advance long-standing questions in algebraic combinatorics. 3. Fractional revival. This is the phenomenon where an excitation periodically resurges at different locations simultaneously; in the case of only one location, it is Perfect Transfer. I will use orthogonal polynomials to design analytic mass-spring chains that possess this feature exactly or approximately. This will relate to the body of work on devices performing quantum information tasks. 4. Rational functions. To each family of orthogonal polynomials of the Askey scheme there is an algebra that encodes their properties. I will extend the Askey scheme to rational functions. This will involve generalized eigenvalue problems and introducing meta-algebras encompassing the ones associated to polynomials. Applications to ASEP models will be explored. 5. Askey-Wilson algebras, topological theories. I will obtain the reflection matrices of the quantum double of the algebras of finite groups and determine their roles in the Kitaev quantum error correcting codes with boundaries. I will develop many generalizations of the algebra associated to the Askey-Wilson polynomials and find its realization in Chern-Simons field theories that embody links invariants.