One-Loop One-Point Functions in Gauge-Gravity Dualities with Defects

We initiate the calculation of loop corrections to correlation functions in 4D defect conformal field theories (dCFTs). More precisely, we consider $\mathcal{N}=4$ SYM theory with a codimension-one defect separating two regions of space, $x_3>0$ and $x_3<0$, where the gauge group is $SU(N)$ and $SU(N-k)$, respectively. This setup is made possible by some of the real scalar fields acquiring a nonvanishing and $x_3$-dependent vacuum expectation value for $x_3>0$. The holographic dual is the $D3–D5$ probe brane system where the $D5$-brane geometry is ${\mathrm{AdS}}_4\times {\mathrm{S}}^2$ and a background gauge field has $k$ units of flux through the ${\mathrm{S}}^2$. We diagonalize the mass matrix of the dCFT making use of fuzzy-sphere coordinates and we handle the $x_3$ dependence of the mass terms in the 4D Minkowski space propagators by reformulating these as standard massive ${\mathrm{AdS}}_4$ propagators. Furthermore, we show that only two Feynman diagrams contribute to the one-loop correction to the one-point function of any single-trace operator and we explicitly calculate this correction in the planar limit for the simplest chiral primary. The result of this calculation is compared to an earlier string-theory computation in a certain double scaling limit and perfect agreement is found. Finally, we discuss how to generalize our calculation to any single-trace operator, to finite $N$, and to other types of observables such as Wilson loops.

Figure 1

Tree-level (a) and one-loop [(b) tadpole and (c) lollipop] contributions to one-point functions. A cross stands for the insertion of the classical solution, while the operator is depicted as a dot.