Monodromy defects from hyperbolic space

Abstract

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We study monodromy defects in <jats:italic>O</jats:italic>(<jats:italic>N</jats:italic>) symmetric scalar field theories in <jats:italic>d</jats:italic> dimensions. After a Weyl transformation, a monodromy defect may be described by placing the theory on <jats:italic>S</jats:italic><jats:sup>1</jats:sup> × <jats:italic>H</jats:italic><jats:sup><jats:italic>d−</jats:italic>1</jats:sup>, where <jats:italic>H</jats:italic><jats:sup><jats:italic>d−</jats:italic>1</jats:sup> is the hyperbolic space, and imposing on the fundamental fields a twisted periodicity condition along <jats:italic>S</jats:italic><jats:sup>1</jats:sup>. In this description, the codimension two defect lies at the boundary of <jats:italic>H</jats:italic><jats:sup><jats:italic>d−</jats:italic>1</jats:sup>. We first study the general monodromy defect in the free field theory, and then develop the large <jats:italic>N</jats:italic> expansion of the defect in the interacting theory, focusing for simplicity on the case of <jats:italic>N</jats:italic> complex fields with a one-parameter monodromy condition. We also use the <jats:italic>ϵ</jats:italic>-expansion in <jats:italic>d</jats:italic> = 4 <jats:italic>− ϵ</jats:italic>, providing a check on the large <jats:italic>N</jats:italic> 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 <jats:italic>S</jats:italic><jats:sup>1</jats:sup> × <jats:italic>H</jats:italic><jats:sup><jats:italic>d−</jats:italic>1</jats:sup>. 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 <jats:italic>H</jats:italic><jats:sup><jats:italic>d−</jats:italic>1</jats:sup>. We also show that, adapting standard techniques from the AdS/CFT literature, the <jats:italic>S</jats:italic><jats:sup>1</jats:sup> × <jats:italic>H</jats:italic><jats:sup><jats:italic>d−</jats:italic>1</jats:sup> 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.</jats:p>