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.
Highlights
In this paper we study monodromy defects, working with scalar field theories as our main example.1 A monodromy defect is a codimension two defect, so q = 2
We study monodromy defects in O(N ) symmetric scalar field theories in d dimensions
We explicitly check the proposal that Ddecreases under defect RG flows in the examples we study, by calculating the defect expectation value on S1 × Hd−1
Summary
Consider an O(N ) symmetric theory of N free scalars in flat space. The most general monodromy defect that we can define imposes that the scalars satisfy [15]. The one-point function of a parity odd spin one operator in the presence of a defect is fixed by conformal symmetry [1]. The displacement operator does appear in the bulk defect OPE of the operator Φ Φ It can be seen from the defect channel decomposition of the two-point function of Φ Φ, which can be obtained by Wick contraction and contains the following term. We can compute the twisted free energy on the hyperbolic space as Ftwisted(θ) = dλD(λ) log λ + m2 + s2 s∈Z+v This can be used to calculate the expectation value of the defect, and it is natural to normalize it by the partition function of the untwisted theory. Using the round sphere setup below, we will obtain an expression for log D valid in continuous d
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