|
Abstract |
In this talk, I will discuss how to apply methods of modern computational
algebraic geometry to Bethe ansatz. I will show that algebraic geometry
provides natural mathematical languages and powerful tools to understand the
structure of solutions of Bethe ansatz equations (BAE). In particular, I will
present new methods to count the number of physical solutions with fixed
quantum numbers based on Gr\"obner basis and quotient ring. This method can be
applied to study the completeness of Bethe ansatz. I will also discuss an
analytical method to compute the sum of on-shell physical quantities over all
physical solutions without explicitly solving BAE. This method has important
applications in calculating the sum rules of OPE coefficients in
$\mathcal{N}=4$ super-Yang-Mills theory. |
