Combinatorics

Physics

Conformal geometric algebra

Filtered algebra

Loop (graph theory)

Differential algebraic equation

Pure mathematics

Mathematics

Mathematical physics

Statistical physics

Chiral Potts curve

Algebra over a field

Mathematical analysis

Geometric algebra

Differential algebraic geometry

Geometry

Theoretical physics

Ising model

Quantum mechanics

Algebraic geometry

Differential equation

Potts model

Ordinary differential equation

Algebraic number

Journal of High Energy Physics

We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of the affine Temperley-Lieb algebra, and the algebraic structure of solutions to the Bethe equations of the XXZ spin chain. We study the solution of Bethe equations analytically by computational algebraic geometry, and find that the solution space encodes rich information about the representation theory of Temperley-Lieb algebra. Using these connections, we compute the partition function of the completely-packed loop model and of the closely related random-cluster Potts model, on medium-size lattices with toroidal boundary conditions, by two quite different methods. We consider the partial thermodynamic limit of infinitely long tori and analyze the corresponding condensation curves of the zeros of the partition functions. Two components of these curves are obtained analytically in the full thermodynamic limit.

Geometric algebra and algebraic geometry of loop and Potts models