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We consider 2+1 dimensional compact U(1) gauge theory at the Lifshitz pointwith dynamical critical exponent $z=2$. As in the usual $z=1$ theory, monopolesproliferate the vacuum for any value of the coupling, generating a mass scale.The theory of the dilute monopole gas is written in terms a non-relativisticSine-Gordon model with two real fields. While monopoles remove some of themassless poles of the perturbative field strength propagator, a gapless moderepresenting the incomplete screening of monopoles remains, and is protected bya shift invariance of the original theory. Timelike Wilson loops still obeyarea laws, implying that minimal charges are confined, but the action ofspacelike Wilson loops of linear size L goes instead as $L^3$.

Compact $z=2$ Electrodynamics in 2+1 dimensions: Confinement with gapless modes