The aim of this paper is to take some further steps in this direction by considering
the holographic realization of defect conformal field theories arising from brane systems.
Generally speaking, these are CFTs defined on a defect hypersurface within the background
of a higher-dimensional bulk CFT [1–6]. From the point of view of this “mother” theory, the
presence of the defect is realized through a deformation associated to a position-dependent
coupling. This deformation turns out to partially break conformal invariance in the bulk,
while only preserving the conformal transformations leaving the defect CFT intact. As
an immediate consequence, the one-point correlation functions are no longer vanishing,
and a non-trivial displacement operator appears. This a sign of the fact that the energy-
momentum tensor needs not be conserved in the presence of the defect.
The first realizations of defect CFTs in string theory were constructed in [7]. Then
many other examples and applications followed (for a non-exhaustive list of references on
conformal defects in string theory and holography see [8–36]). The key idea is to let defect
CFTs emerge from some particular supersymmetric brane configurations in which some
“defect branes” end on a given brane system, which is known to give rise to an AdS vacuum
in the near-horizon limit. The main effect of these intersections is to break partially the
isometry group of the AdS vacuum of the original brane system and to produce a lower-
dimensional warped AdS solution. The defect CFT describes the boundary conditions
defining the intersection with the defect branes and the warping of the corresponding
background describes the backreaction of the defect onto the bulk geometry. This may be
viewed as the supergravity picture associated to the position-dependent deformation of the
“mother” SCFT, dual to the original higher-dimensional AdS vacuum.
More concretely, let us consider a SUSY AdS
d
closed string vacuum associated with the
near-horizon of some brane system, where we furthermore assume the existence of a con-
sistent truncation linking the 10d (or 11d) picture to a solution in a d-dimensional gauged
supergravity describing the excitations around the AdS
d
vacuum. If some defect branes
end on this system, then we have a bound state with a (p + 1)-dimensional worldvolume
whose physics is captured by a d-dimensional Janus-type background
ds
2
d
= e
2U(r)
ds
2
AdS
p+2
+ e
2V (r)
dr
2
+ e
2W (r)
ds
2
d−p−3
. (1.1)
The d-dimensional background is thus characterized by a AdS
p+2
slicing and an asymp-
totic region locally described by the AdS
d
vacuum.
1
The solutions like (1.1) can be then
consistently uplifted producing warped geometries of the type AdS
p+2
×M
d−p−2
×Σ
D−d
,
where M
d−p−2
is realized as a fibration of the (d −p −3)-dimensional transverse manifold
over the interval I
r
and Σ
D−d
is the internal manifold of the truncation with D = 10 or 11.
1
We point out that the main difference between this case and the one of RG flows across dimensions
can be observed by considering the “radial” coordinates giving rise to the AdS vacua respectively in the
UV and IR. In a conformal defect, the radii of the AdS
p+2
and AdS
d
are different, while in a supergravity
solution describing an RG flow across dimensions, the AdS backrounds arising in the UV and in the IR are
described by the same radial coordinate. Conformal defects and RG flows across dimensions are somehow
two complementary descriptions. For example one may guess the existence of more general flows involving
r as well as the radial coordinate of AdS
p+2
describing a geometry where the metric (1.1) is replaced by an
R
1,p
slicing of the d-dimensional background.
– 2 –
