  The classical cubic dimer model has a columnar ordering transition that is
continuous and described by a critical Anderson--Higgs theory containing an
SU(2)-symmetric complex field minimally coupled to a noncompact U(1) gauge
theory. Defects in the dimer constraints correspond to monopoles of the gauge
theory, with charge determined by the deviation from unity of the dimer
occupancy. By introducing such defects into Monte Carlo simulations of the
dimer model at its critical point, we determine the scaling dimensions $y_2 =
1.48\pm0.07$ and $y_3 = 0.20 \pm 0.03$ for the operators corresponding to
defects of charge $q=2$ and $3$ respectively. These results, which constitute
the first direct determination of the scaling dimensions, shed light on the
deconfined critical point of spin-1/2 quantum antiferromagnets, thought to
belong to the same universality class. In particular, the positive value of
$y_3$ implies that the transition in the JQ model on the honeycomb lattice is
of first order.
