Boundary Layers and One-point Functions in the Presence of Monodromy Defects

Abstract

We study one-point functions of composites of charge $e$ operators in the presence of a monodromy defect for a $U(1)$ global symmetry with monodromy $β$. We first compute these in free massless and massive theories, recovering in the former case the known $\sin(eπβ)$ dependence and obtaining in the latter a $\sin^2(eπβ)$ dependence. We then turn to holography and compute 1-point functions for operators $O$ of charge $J=Δ$ in $\mathfrak{su}(N)$ $\mathcal{N}=4$ SYM in the presence of a monodromy defect for a $U(1)\in SO(6)_R$. From a WKB analysis in large $Δ$ we recover the structure of standard and anchored saddles previously found in the literature, finding that, to subleading order in $1/Δ$, the anchored regime is resolved by a boundary layer effect. Finally, using heat kernel methods, we determine the monodromy dependence of the induced 1-point function for the composite $O^{\dagger}O$, finding a smooth $\sin^2(Jπβ)$ behavior.