Renormalisation and locality: branched zeta values

IRMA Lect. in Math. and Theor. Phys. 32 (2020), 85-132 Multivariate renormalisation techniques are implemented in order to build, study and then renormalise at the poles, branched zeta functions associated with trees. For this purpose, we first prove algebraic results and develop analytic tools, which we then combine to study branched zeta functions. The algebraic aspects concern universal properties for locality algebraic structures, some of which had been discussed in previous work; we "branch/ lift" to trees operators acting on the decoration set of trees, and factorise branched maps through words by means of universal properties for words which we prove in the locality setup. The analytic tools are multivariate meromorphic germs of pseudodifferential symbols with linear poles which generalise the meromorphic germs of functions with linear poles studied in previous work. Multivariate meromorphic germs of pseudodifferential symbols form a locality algebra on which we build various locality maps in the framework of locality structures. We first show that the finite part at infinity defines a locality character from the latter symbol valued meromorphic germs to the scalar valued ones. We further equip the locality algebra of germs of pseudodifferential symbols with locality Rota-Baxter operators given by regularised sums and integrals. By means of the universal properties in the framework of locality structures we can lift Rota-Baxter operators to trees, and use the lifted discrete sums in order to build and study renormalised branched zeta values associated with trees. By construction these renormalised branched zeta values factorise on mutually independent (for the locality relation) trees.