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In this paper, we obtain bounds on the Wilson loop expectations in 4D $U(1)$lattice gauge theory which quantify the effect of topological defects. In thecase of a Villain interaction, by extending the non-perturbative techniqueintroduced in [GS20a], we obtain the following estimate for a large loop$\gamma$ at low temperatures: \[ |\langle W_\gamma\rangle_{\beta}| \leq \exp\left(-\frac{C_{GFF}} {2\beta}(1+C \beta e^{- 2\pi^2 \beta})(|\gamma|+o(|\gamma|)) \right)\,. \] Our result is in the line of recent works[Cha20, Cao20, FLV20, For21] which analyze the case where the gauge group isdiscrete. In the present case where the gauge group is continuous and Abelian,the fluctuations of the gauge field decouple into a Gaussian part, related tothe so-called {\em free electromagnetic wave} [Gro83, Dri87], and a gas of {\emtopological defects}. As such, our work gives new quantitative bounds on thefluctuations of the latter which complement the works by Guth andFr\"ohlich-Spencer [Gut80, FS82].  Finally, we improve, also in a non-perturbative way, the correction term from$e^{-2\pi^2\beta}$ to $e^{-\pi^2\beta}$ in the case of the free-energy of thesystem. This provides a matching lower-bound with the prediction of Guth[Gut80] based on renormalization group techniques.

Improved spin-wave estimate for Wilson loops in $U(1)$ lattice gauge theory