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Janus solutions in six-dimensional gauged supergravity
Motivated by an analysis of the sub-superalgebras of the five-dimensional
superconformal algebra , we search for the holographic duals to
co-dimension one superconformal defects in 5d CFTs which have bosonic symmetry. In particular, we look for domain wall solutions to
six-dimensional gauged supergravity coupled to a single vector
multiplet. It is found that supersymmetric domain wall solutions do not exist
unless there is a non-trivial profile for one of the vector multiplet scalars
which is charged under the gauged R-symmetry. This non-trivial profile
breaks the to , thus matching expectations from the superalgebra
analysis. A consistent set of BPS equations is then obtained and solved
numerically. While the numerical solutions are generically singular and thought
to be dual to boundary CFTs, it is found that for certain fine-tuned choices of
parameters regular Janus solutions may be obtained.Comment: 35 pages, pdf-latex, 9 figures. v2: minor corrections, reference
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Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D
Let be a potential on \RR^3 that is smooth everywhere except at a
discrete set \maS of points, where it has singularities of the form
, with for close to and continuous on
\RR^3 with for p \in \maS. Also assume that and
are smooth outside \maS and is smooth in polar coordinates around each
singular point. We either assume that is periodic or that the set \maS is
finite and extends to a smooth function on the radial compactification of
\RR^3 that is bounded outside a compact set containing \maS. In the
periodic case, we let be the periodicity lattice and define \TT :=
\RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the
eigenfunctions of the Schr\"odinger-type operator acting on
L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by
restricting the action of to Bloch waves. Under some additional
assumptions, we extend these regularity and solvability results to the
non-periodic case. We sketch some applications to approximation of
eigenfunctions and eigenvalues that will be studied in more detail in a second
paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55
(103), no. 2/201
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