Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes

In the paper arXiv:0810.4291 we have shown, in the context of type II superstring theory, the classification of the allowed B-field and A-field configurations in the presence of anomaly-free D-branes, the mathematical framework being provided by the geometry of gerbes. Here we complete the discussion considering in detail the case of a stack of D-branes, carrying a non-abelian gauge theory, which was just sketched in the previous paper. In this case we have to mix the geometry of abelian gerbes, describing the B-field, with the one of higher-rank bundles, ordinary or twisted. We describe in detail the various cases that arise according to such a classification, as we did for a single D-brane, showing under which hypoteses the A-field turns out to be a connection on a canonical gauge bundle. We also generalize to the non-abelian setting the discussion about "gauge bundles with non integral Chern classes", relating them to twisted bundles with connection. Finally, we analyze the geometrical nature of the Wilson loop for each kind of gauge theory on a D-brane or stack of D-branes.Comment: 29 page

A Sampling Theorem for Rotation Numbers of Linear Processes in R2{\R}^{2}

We prove an ergodic theorem for the rotation number of the composition of a sequence os stationary random homeomorphisms in $S^{1}$. In particular, the concept of rotation number of a matrix $g\in Gl^{+}(2,{\R})$ can be generalized to a product of a sequence of stationary random matrices in $Gl^{+}(2,{\R})$. In this particular case this result provides a counter-part of the Osseledec's multiplicative ergodic theorem which guarantees the existence of Lyapunov exponents. A random sampling theorem is then proved to show that the concept we propose is consistent by discretization in time with the rotation number of continuous linear processes on ${\R}^{2}.