Monodromy defects from hyperbolic space

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