Û(1) Kac-Moody gauge fields have the infinite dimensional Û(1) Kac-Moody group as their gauge group. The pure gauge sector, unlike the usual U(1) Maxwell Lagrangian, is nonlinear and nonlocal; the Euclidean theory is defined on a ($d+1\mathrm{}$)-dimensional manifold ${\mathcal{R}}_d\times {\mathcal{S}}^1$ and, hence, is also asymmetric. We quantize this theory using the background field method and examine its renormalizability at one loop by analyzing all the relevant diagrams. We find that, for a suitable choice of the gauge field propagators, this theory is one-loop renormalizable in 3 + 1 dimensions. This pure Û(1) Kac-Moody gauge theory in 3 + 1 dimensions has only one running coupling constant and the theory is asymptotically free. When fermions are added the number of independent couplings increases and a richer structure is obtained. Finally, we note some features of the theory which suggest its possible relevance to the study of anisotropic condensed matter systems, in particular, that of high-temperature superconductors.

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