An algebraic locality principle to renormalise higher zeta functions

          @misc{ pirsa_PIRSA:18080078,
            doi = {10.48660/18080078},
            url = {https://pirsa.org/18080078},
            author = {Paycha, Sylvie},
            keywords = {Mathematical physics},
            language = {en},
            title = { An algebraic locality principle to renormalise higher zeta functions },
            publisher = {Perimeter Institute},
            year = {2018},
            month = {aug},
            note = {PIRSA:18080078 see, \url{https://pirsa.org}}
          }
          

Abstract

According to the principle of locality in physics, events taking place at different locations should behave independently of each other, a feature expected to be reflected in the measurements. We propose an algebraic locality framework to keep track of the independence, where sets are equipped with a binary symmetric relation we call a locality relation on the set, this giving rise to a locality set category. In this algebraic locality setup, we implement a multivariate regularisation, which gives rise to multivariate meromorphic functions. In this case, independence of events is reflected in the fact that the multivariate meromorphic functions involve independent sets of variables. A minimal subtraction scheme defined in terms of a projection map onto the holomorphic part then yields renormalised values. This multivariate approach can be implemented to renormalise at poles, various higher multizeta functions such as conical zeta functions (discrete sums on convex cones) and branched zeta functions (discrete sums associated with rooted trees). This renormalisation scheme strongly relies on the fact that the maps we are renormalizing can be viewed as locality algebra morphisms. This talk is based on joint work with Pierre Clavier, Li Guo and Bin Zhang.