Cluster Algebras and Scattering Amplitudes

          @misc{ pirsa_PIRSA:15050058,
            doi = {10.48660/15050058},
            url = {https://pirsa.org/15050058},
            author = {Spradlin, Marcus},
            keywords = {Mathematical physics},
            language = {en},
            title = {Cluster Algebras and Scattering Amplitudes},
            publisher = {Perimeter Institute},
            year = {2015},
            month = {may},
            note = {PIRSA:15050058 see, \url{https://pirsa.org}}
          }
          

Abstract

Supersymmetric gauge theory computes a very special class of (generalized) polylogarithm functions known as scattering amplitudes that have remarkable mathematical properties. In particular, there is a rich connection between these amplitudes and the G(4,n) Grassmannian cluster algebra. To explain this connection I will review some basic facts about the Hopf algebra of polylogarithms and cluster Poisson varieties. I will then define cluster polylogarithm functions which roughly speaking are polylogarithm functions whose arguments are cluster X-coordinates of some cluster algebra A. I will describe an additional property of certain scattering amplitudes, that they are "local" in the algebra A, and describe the classification of cluster polylogarithm functions with this property. The computation of new amplitudes can be greatly aided by knowledge of the class of functions in terms of which they may be expressed, as I will illustrate via an example.