In this paper we put forward a systematic and unifying approach to construct gauge invariant composite fields out of connections. It relies on the existence in the theory of a group valued field with a prescribed gauge transformation. As an illustration, we detail some examples. Two of them are based on known results: the first one provides a reinterpretation of the symmetry breaking mechanism of the electroweak part of the Standard Model of particle physics; the second one is an application to Einstein's theory of gravity described as a gauge theory in terms of Cartan connections. The last example depicts a new situation: starting with a gauge field theory on Atiyah Lie algebroids, the gauge invariant composite fields describe massive vector fields. Some mathematical and physical discussions illustrate and highlight the relevance and the generality of this approach.Comment: 22 pages, revised version to appear in Int. J. of Geometric Methods in Modern Physic

Considering local conformal field theories on a Riemann surface by coupling conformal matter fields with a complex structure parametrized by a Beltrami differential, the local diffeomorphism cohomology modulo d of the Becchi–Rouet–Stora operator is computed directly by means of the spectral sequences method. Then, thanks to both power counting and locality principles of the Feynman algorithm, the local theory is analyzed. On the one hand, in the ghost number one sector, consistent anomalies turn out to be exactly, after elimination of a trace anomaly involving matter fields, the well‐known holomorphically split diffeomorphism anomaly due to the vacuum. On the other hand, in the ghost number zero sector, local observables of the theory, namely, the vertex operators, are generically calculated, as well as all possible classical actions for Lagrangian conformal model

We interpret the unimodularity condition in almost commutative geometries as central extensions of spin lifts. In Connes' formulation of the standard model this interpretation allows to compute the hypercharges of the fermions.