We study pure noncommutative U(1) gauge theory representing its one-loop effective action in terms of a phase space worldline path integral. We write the quadratic action using the background field method to keep explicit gauge invariance, and then employ the worldline formalism to write the one-loop effective action, singling out UV-divergent parts and finite (planar and non-planar) parts, and study renormalization properties of the theory. This amounts to employ worldline Feynman rules for the phase space path integral, that nicely incorporate the Fadeev-Popov ghost contribution and efficiently separate planar and non-planar contributions. We also show that the effective action calculation is independent of the choice of the worldline Green’s function, that corresponds to a particular way of factoring out a particle zero-mode. This allows to employ homogeneous string-inspired Feynman rules that greatly simplify the computation.

We analyze the renormalization group (RG) flow of the temperature independent term of the entropy in the high temperature limit β/a ≪ 1 of a massive field theory in 3-dimensional spherical spaces, M ₃, with constant curvature 6/a ². For masses lower than \( \frac{2\pi }{\beta } \) , this term can be identified with the free energy of the same theory on M ₃ considered as a 3-dimensional Euclidean space-time. The non-extensive part of this free energy, S _hol, is generated by the holonomy of the spatial metric connection. We show that for homogeneous spherical spaces the holonomy entropy S _hol decreases monotonically when the RG scale flows to the infrared. At the conformal fixed points the values of the holonomy entropy do coincide with the genuine topological entropies recently introduced. The monotonic behavior of the RG flow leads to an inequality between the topological entropies of the conformal field theories connected by such flow, i.e. S _top ^UV > S _top ^IR . From a 3-dimensional viewpoint the same term arises in the 3-dimensional Euclidean effective action and has the same monotonic behavior under the RG group flow. We conjecture that such monotonic behavior is generic, which would give rise to a 3-dimensional generalization of the c-theorem, along the lines of the 2-dimensional c-theorem and the 4-dimensional a-theorem. The conjecture is related to recent formulations of the F -theorem. In particular, the holonomy entropy on lens spaces is directly related to the topological Rényi entanglement entropy on disks of 2-dimensional flat spaces.

We present a detailed analytic study on the three-dimensional sphere of the most popular candidates for C -functions, both for Dirac and scalar free massive fields. We discuss to which extent the effective action, the Rényi entanglement entropy and the renormalized entanglement entropy fulfill the conditions expected from C -functions. In view of the absence of a good candidate in the case of the scalar field, we introduce a new candidate, which we call the modified effective action, and analyze its pros and cons.