Holographic Renormalization of Probe D-Branes in AdS/CFT

TL;DR

The paper establishes that holographic renormalization for SUSY probe D-branes in $AdS_5\times S^5$ uses the same counterterms as a free scalar in the corresponding $AdS_m$ factor, plus the standard AdS volume counterterms, across four embeddings: D7, two D5s, and D3. For the D7 system, a finite counterterm is fixed by supersymmetry, ensuring the vev vanishes on the SUSY background despite nontrivial subleading asymptotics. The D5 systems illustrate mass and vev deformations of defect CFTs, with vevs arising in the vev-deformation case and the energy behavior reflecting brane dynamics and defect separation. The D3/D3 system, where the slipping mode saturates the BF bound, exhibits a reversed interpretation between mass and vev and yields explicit two-point functions consistent with the holographic renormalization framework. Together, these results clarify the renormalization structure for probe branes, illuminate how vevs and masses map in defect setups, and provide concrete prescriptions for calculating renormalized one- and two-point functions in these holographic defect theories.

Abstract

We perform holographic renormalization for probe branes in AdS_5 x S^5. We show that for four known probe D-branes wrapping an AdS_m x S^n, the counterterms needed to render the action finite are identical to those for the free, massive scalar in AdS_m plus counterterms for the renormalization of the volume of AdS_m. The four cases we consider are the probe D7, two different probe D5's and a probe D3. In the D7 case there are scheme-dependent finite counterterms that can be fixed by supersymmetry.