  We study monodromy defects in $O(N)$ symmetric scalar field theories in $d$
dimensions. After a Weyl transformation, a monodromy defect may be described by
placing the theory on $S^1\times H^{d-1}$, where $H^{d-1}$ is the hyperbolic
space, and imposing on the fundamental fields a twisted periodicity condition
along $S^1$. In this description, the codimension two defect lies at the
boundary of $H^{d-1}$. We first study the general monodromy defect in the free
field theory, and then develop the large $N$ expansion of the defect in the
interacting theory, focusing for simplicity on the case of $N$ complex fields
with a one-parameter monodromy condition. We also use the $\epsilon$-expansion
in $d=4-\epsilon$, providing a check on the large $N$ approach. When the defect
has spherical geometry, its expectation value is a meaningful quantity, and it
may be obtained by computing the free energy of the twisted theory on
$S^1\times H^{d-1}$. It was conjectured that the logarithm of the defect
expectation value, suitably multiplied by a dimension dependent sine factor,
should decrease under a defect RG flow. We check this conjecture in our
examples, both in the free and interacting case, by considering a defect RG
flow that corresponds to imposing alternate boundary conditions on one of the
low-lying Kaluza-Klein modes on $H^{d-1}$. We also show that, adapting standard
techniques from the AdS/CFT literature, the $S^1\times H^{d-1}$ setup is well
suited to the calculation of the defect CFT data, and we discuss various
examples, including one-point functions of bulk operators, scaling dimensions
of defect operators, and four-point functions of operator insertions on the
defect.
