21 research outputs found
A Dialogue of Multipoles: Matched Asymptotic Expansion for Caged Black Holes
No analytic solution is known to date for a black hole in a compact
dimension. We develop an analytic perturbation theory where the small parameter
is the size of the black hole relative to the size of the compact dimension. We
set up a general procedure for an arbitrary order in the perturbation series
based on an asymptotic matched expansion between two coordinate patches: the
near horizon zone and the asymptotic zone. The procedure is ordinary
perturbation expansion in each zone, where additionally some boundary data
comes from the other zone, and so the procedure alternates between the zones.
It can be viewed as a dialogue of multipoles where the black hole changes its
shape (mass multipoles) in response to the field (multipoles) created by its
periodic "mirrors", and that in turn changes its field and so on. We present
the leading correction to the full metric including the first correction to the
area-temperature relation, the leading term for black hole eccentricity and the
"Archimedes effect". The next order corrections will appear in a sequel. On the
way we determine independently the static perturbations of the Schwarzschild
black hole in dimension d>=5, where the system of equations can be reduced to
"a master equation" - a single ordinary differential equation. The solutions
are hypergeometric functions which in some cases reduce to polynomials.Comment: 47 pages, 12 figures, minor corrections described at the end of the
introductio
Holographic Correlators in a Flow to a Fixed Point
Using holographic renormalization, we study correlation functions throughout
a renormalization group flow between two-dimensional superconformal field
theories. The ultraviolet theory is an N=(4,4) CFT which can be thought of as a
symmetric product of U(2) super WZW models. It is perturbed by a relevant
operator which preserves one-quarter supersymmetry and drives the theory to an
infrared fixed point. We compute correlators of the stress-energy tensor and of
the relevant operators dual to supergravity scalars. Using the former, we put
together Zamolodchikov's C function, and contrast it with proposals for a
holographic C function. In passing, we address and resolve two puzzles also
found in the case of five-dimensional bulk supergravity.Comment: LaTeX2e, 48 pages, 4 figure
Soliton surfaces via zero-curvature representation of differential equations
The main aim of this paper is to introduce a new version of the
Fokas-Gel'fand formula for immersion of soliton surfaces in Lie algebras. The
paper contains a detailed exposition of the technique for obtaining exact forms
of 2D-surfaces associated with any solution of a given nonlinear ordinary
differential equation (ODE) which can be written in zero-curvature form. That
is, for any generalized symmetry of the zero-curvature condition of the
associated integrable model, it is possible to construct soliton surfaces whose
Gauss-Mainardi-Codazzi equations are equivalent to infinitesimal deformations
of the zero-curvature representation of the considered model. Conversely, it is
shown (Proposition 1) that for a given immersion function of a 2D-soliton
surface in a Lie algebra, it possible to derive the associated generalized
vector field in evolutionary form which characterizes all symmetries of the
zero-curvature condition. The theoretical considerations are illustrated via
surfaces associated with the Painlev\'e equations P1, P2 and P3, including
transcendental functions, the special cases of the rational and Airy solutions
of P2 and the classical solutions of P3.Comment: 28 pages, 2 figure