A manifestly gauge invariant exact renormalization group

In these lectures we describe the construction of a gauge invariant renormalization group equation for pure non-Abelian gauge theory. In the process, a non-perturbative gauge invariant continuum Wilsonian effective action is precisely defined. The formulation makes sense without gauge fixing and thus manifest gauge invariance may be preserved at all stages. In the large N limit (of SU(N) gauge theory) the effective action simplifies: it may be expressed through a path integral for a single particle whose trajectory describes a Wilson loop. Regularization is achieved with the help of a set of Pauli-Villars fields whose formulation follows naturally in this picture. Finally, we show how the one loop beta function was computed, for the first time without any gauge fixing.