Duality for local fields and sheaves on the category of fields

Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly p-torsion) finite flat group schemes was obtained by Begueri, Bester and Kato. In this paper, we give another formulation and proof of this result. We use the category of fields and a Grothendieck topology on it. This simplifies the formulation and proof and reduces the duality to classical results on Galois cohomology. A key point is that the resulting site correctly captures extension groups between algebraic groups.Comment: 63 pages. Revision after the second referee repor

Some remarks on the local class field theory of Serre and Hazewinkel

We give a new approach for the local class field theory of Serre and Hazewinkel. We also discuss two-dimensional local class field theory in this framework.Comment: 23 pages; final versio

Massless two-loop "master" and three-loop two point function in NDIM

NDIM (Negative Dimensional Integration Method) is a technique for evaluating Feynman integrals based on the concept of analytic continuation. The method has been successfully applied to many diagrams in covariant and noncovariant gauge field interactions and has shown its utility as a powerful technique to handle Feynman loop integrations in quantum field theories. In principle NDIM can handle any loop calculation; however, in practical terms, the resulting multiseries with several variables in general cannot be summed up conveniently and its analytic properties are generally unknown. The alternative then is to use order by order (loop by loop) integration in which the first integral is of the triangle diagram type. However, the na\"{i}ve momentum integration of this leads to wrong results. Here we use the shortened version for the triangle in NDIM that is suitable for a loop by loop calculation and show that it leads (after appropriate analytic continuation to positive dimension) to agreement with the known result for the two-loop master diagram. From it, a three-loop is then calculated and shown again its consistency with the already published result for such a diagram.Comment: 15 pages, 2 figure