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 $\beta$. We first compute these in free massless and massive theories, recovering in the former case the known $\sin (e\pi \beta )$ dependence and obtaining in the latter a ${\sin }^2(e\pi \beta )$ dependence. We then turn to holography and compute 1-point functions for operators $O$ of charge $J=\Delta$ 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 $\Delta$ we recover the structure of standard and anchored saddles previously found in the literature, finding that, to subleading order in $1/\Delta$, 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 $O^{†}O$, finding a smooth ${\sin }^2(J\pi \beta )$ behavior.

31 pages