Theories whose fluctuating degrees of freedom live on extended loops as opposed to points, are abundant in nature. One example is the action obtained upon eliminating the redundant gauge fields in a gauge theory. Formulating a Renormalization Group (RG) procedure for such a theory is an open problem. In this work, we outline a procedure that in principle computes the outcome of coarse-graining and rescaling of such a theory. We make estimates that lead to qualitative agreement with known results of phase transitions in gauge theories and the XY-model. For concreteness, we focus on compact pure U(1) gauge theory in 3+1 dimensions which is known to have a confinement-deconfinement phase transition. We start with a pedagogical review for a mapping from this theory to a dual sine Gordon-like theory of a Coulomb loop gas. We then illustrate our RG procedure on this theory by drawing parallels to known RG procedures for regular Sine-Gordon field theories, while at the same time emphasizing the loop nature of the theory. We show how our procedure readily generalizes to the XY model in 2+1 dimensions and to odd compact U(1) gauge theory in 3+1 dimensions. From a contemporary perspective, our work is an attempt at an RG procedure for a theory with one-form symmetries.

Pizza and drinks will be served after the seminar in ATL 2117.