The proposal uses insights gained from models used in mathematical physics in order to understand algebraic structures found in pure mathematics. The models are from a special class of models, called integrable models. This means that due to their high degree of symmetry, they have a sufficient number of conservation laws -- generalizing conservation of energy -- that they can be solved exactly. Such models can be found in statistical mechanics (discrete, possibly finite systems) and in quantum field theory (continuous, infinite-dimensional systems). There are many structures in mathematics that can be studied by using techniques and results from the solutions of such systems. They appear in combinatorics, number theory, representations of non-commutative algebras, and geometry, to name a few. The results of this project will advance understanding in all these areas.
Frequently, integrable systems such as quantum spin chains in statistical mechanics, and conformal field theories and their massive deformations, can be described in representation-theoretical terms. For example, the transfer matrix of the Heisenberg spin chains can be given a meaning as a q-character of a finite-dimensional module of quantum affine algebras. The Hilbert space of integrable quantum field theories can be expressed as an infinite-dimensional modules of extensions of the (deformed) Virasoro algebra. Transfer matrices, and the characters of the Virasoro modules, satisfy difference equations or equations which can be shown to be discrete integrable equations. The transfer matrices satisfy a discrete Hirota-type equation which have interpretations as cluster algebra mutations. The following projects are proposed here: (1) The study of the difference equations satisfied by the non-commutative generating functions of graded conformal blocks of WZW theories (generalizations of Demazure modules). These generating functions are expressed in terms of the cluster variables in the quantum cluster algebra corresponding to Q-systems for characters of Kirillov-Reshetikhin modules; (2) The explicit solutions of these equations as fermionic character formulas; (3) The integrable structure of the difference equations; (4) The stabilized limits of these functions which give characters of affine algebra modules or Virasoro modules; (5) Solutions of discrete integrable equations known as T-systems and quantum T-systems (in the sense of quantum cluster algebras) and their relation to Nakajima's t,q-characters; (6) Higher-dimensional integrable difference equations, generalizing the T-systems viewed as Plucker relations, expressing the discrete structure of the higher-dimensional analogs of the pentagram maps in algebraic terms; and (7) Statistical path models which give explicit solutions for Whittaker vectors, functions and quantum Toda Hamiltonians for finite, affine and quantum algebras.
