Algebroid Yang-Mills Theories

We present a framework for constructing gauge theories where the structural Lie algebra is generalized to a Lie algebroid (and, in the integrable case, the structural Lie group to the corresponding Lie groupoid). Besides examples such as the Poisson sigma model, we will present a generalization of ordinary YM theories in arbitrary spacetime dimension. From the point of view physics, the resulting theory turns out to effectively describe a bundle of ordinary Yang-Mills theories, defined over a finite dimensional moduli space. The gluing of different fibers is dictated by the originally chosen Lie algebroid, such as the gluing of fibers over a base manifold in a fiber bundle is prescribed by the global construction of the total space---with the exception that there is no typical fiber: critical orbits in the Lie algebroid give rise to points of symmetry enhancement in the moduli space, i.e. to a YM theory defined over the cricitcal point in the moduli space which has a higher dimensional structure group.