Invariants and TQFT's for cut cellular surfaces from finite groups

We introduce the notion of a cut cellular surface (CCS), being a surface with boundary, which is cut in a specified way to be represented in the plane, and is composed of 0-, 1- and 2-cells. We obtain invariants of CCS's under Pachner-like moves on the cellular structure, by counting colourings of the 1-cells with elements of a finite group, subject to a "flatness" condition for each 2-cell. These invariants are also described in a TQFT setting, which is not the same as the usual 2-dimensional TQFT framework. We study the properties of functions which arise in this context, associated to the disk, the cylinder and the pants surface, and derive general properties of these functions from topology, including properties which come from invariance under the Hatcher-Thurston moves on pants decompositions.Comment: 28 pages, 27 figures. Revised version, including a topological proof of the property: the number of conjugacy classes of a finite group G equals the commuting fraction of G times the order of G. To appear in Boletim da Sociedade Portuguesa de Matem\'atic

A Cubical Set Approach to 2-Bundles with Connection and Wilson Surfaces

In the context of non-abelian gerbes we define a cubical version of categorical group 2-bundles with connection over a smooth manifold. We define their two-dimensional parallel transport, study its properties, and define non-abelian Wilson surface functionals.Comment: Improvement on the exposition. Approximately 60 pages, 2 figure

One loop superstring effective actions and d=4 supergravity

We review our recent work on the existence of a new independent R^4 term, at one loop, in the type IIA and heterotic effective actions, after reduction to four dimensions, besides the usual square of the Bel-Robinson tensor. We discuss its supersymmetrization.Comment: 1+7 pages. Work presented at the XVI International Fall Workshop on Geometry and Physics, Lisbon, Portugal, September 200