  We consider 2+1 dimensional conformal gauge theories coupled to additional
degrees of freedom which induce a spatially local but long-range in time
$1/(\tau-\tau')^2$ interaction between gauge-neutral local operators. Such
theories have been argued to describe the hole-doped cuprates near optimal
doping. We focus on a SU(2) gauge theory with $N_h$ flavors of adjoint Higgs
fields undergoing a quantum transition between Higgs and confining phases: the
$1/(\tau-\tau')^2$ interaction arises from a spectator large Fermi surface of
electrons. The large $N_h$ expansion leads to an effective action containing
fields which are bilocal in time but local in space. We find a strongly-coupled
fixed point at order $1/N_h$, with dynamic critical exponent $z > 1$. We show
that the entropy preserves hyperscaling, but nevertheless leads to a linear in
temperature specific heat with a co-efficient which has a finite enhancement
near the quantum critical point.
