Monodromy defects from hyperbolic space

Abstract

Abstract 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 S1 × Hd−1, where Hd−1 is the hyperbolic space, and imposing on the fundamental fields a twisted periodicity condition along S1. In this description, the codimension two defect lies at the boundary of Hd−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 ϵ-expansion in d = 4 − ϵ, 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 S1 × Hd−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 Hd−1. We also show that, adapting standard techniques from the AdS/CFT literature, the S1 × Hd−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.