18 research outputs found
Scalar--flat K\"ahler metrics with conformal Bianchi V symmetry
We provide an affirmative answer to a question posed by Tod \cite{Tod:1995b},
and construct all four-dimensional Kahler metrics with vanishing scalar
curvature which are invariant under the conformal action of Bianchi V group.
The construction is based on the combination of twistor theory and the
isomonodromic problem with two double poles. The resulting metrics are
non-diagonal in the left-invariant basis and are explicitly given in terms of
Bessel functions and their integrals. We also make a connection with the LeBrun
ansatz, and characterise the associated solutions of the SU(\infty) Toda
equation by the existence a non-abelian two-dimensional group of point
symmetries.Comment: Dedicated to Maciej Przanowski on the occasion of his 65th birthday.
Minor corrections. To appear in CQ
Twistor theory of hyper-K{\"a}hler metrics with hidden symmetries
We briefly review the hierarchy for the hyper-K\"ahler equations and define a
notion of symmetry for solutions of this hierarchy. A four-dimensional
hyper-K\"ahler metric admits a hidden symmetry if it embeds into a hierarchy
with a symmetry. It is shown that a hyper-K\"ahler metric admits a hidden
symmetry if it admits a certain Killing spinor. We show that if the hidden
symmetry is tri-holomorphic, then this is equivalent to requiring symmetry
along a higher time and the hidden symmetry determines a `twistor group' action
as introduced by Bielawski \cite{B00}. This leads to a construction for the
solution to the hierarchy in terms of linear equations and variants of the
generalised Legendre transform for the hyper-K\"ahler metric itself given by
Ivanov & Rocek \cite{IR96}. We show that the ALE spaces are examples of
hyper-K\"ahler metrics admitting three tri-holomorphic Killing spinors. These
metrics are in this sense analogous to the 'finite gap' solutions in soliton
theory. Finally we extend the concept of a hierarchy from that of \cite{DM00}
for the four-dimensional hyper-K\"ahler equations to a generalisation of the
conformal anti-self-duality equations and briefly discuss hidden symmetries for
these equations.Comment: Final version. To appear in the August 2003 special issue of JMP on
`Integrability, Topological Solitons, and Beyond
Harmonic functions, central quadrics, and twistor theory
Solutions to the -dimensional Laplace equation which are constant on a
central quadric are found. The associated twistor description of the case
is used to characterise Gibbons-Hawking metrics with tri-holomorphic SL(2,
\C) symmetry.Comment: Final version. To appear in CQ
Solitons and admissible families of rational curves in twistor spaces
It is well known that twistor constructions can be used to analyse and to
obtain solutions to a wide class of integrable systems. In this article we
express the standard twistor constructions in terms of the concept of an
admissible family of rational curves in certain twistor spaces. Examples of of
such families can be obtained as subfamilies of a simple family of rational
curves using standard operations of algebraic geometry. By examination of
several examples, we give evidence that this construction is the basis of the
construction of many of the most important solitonic and algebraic solutions to
various integrable differential equations of mathematical physics. This is
presented as evidence for a principal that, in some sense, all soliton-like
solutions should be constructable in this way.Comment: 15 pages, Abstract and introduction rewritten to clarify the
objectives of the paper. This is the final version which will appear in
Nonlinearit
On the central quadric ansatz: integrable models and Painleve reductions
It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley
(BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess
solutions whose level surfaces are central quadrics in the space of independent
variables (the so-called central quadric ansatz). It was demonstrated that
generic solutions of this type are described by Painleve equations PIII and
PII, respectively. The aim of our paper is threefold:
-- Based on the method of hydrodynamic reductions, we classify integrable
models possessing the central quadric ansatz. This leads to the five canonical
forms (including BF and dKP).
-- Applying the central quadric ansatz to each of the five canonical forms,
we obtain all Painleve equations PI - PVI, with PVI corresponding to the
generic case of our classification.
-- We argue that solutions coming from the central quadric ansatz constitute
a subclass of two-phase solutions provided by the method of hydrodynamic
reductions.Comment: 12 page
Solvable vector nonlinear Riemann problems, exact implicit solutions of dispersionless PDEs and wave breaking
We have recently solved the inverse spectral problem for integrable PDEs in
arbitrary dimensions arising as commutation of multidimensional vector fields
depending on a spectral parameter . The associated inverse problem, in
particular, can be formulated as a non linear Riemann Hilbert (NRH) problem on
a given contour of the complex plane. The most distinguished examples
of integrable PDEs of this type, like the dispersionless
Kadomtsev-Petviashivili (dKP), the heavenly and the 2 dimensional
dispersionless Toda equations, are real PDEs associated with Hamiltonian vector
fields. The corresponding NRH data satisfy suitable reality and symplectic
constraints. In this paper, generalizing the examples of solvable NRH problems
illustrated in \cite{MS4,MS5,MS6}, we present a general procedure to construct
solvable NRH problems for integrable real PDEs associated with Hamiltonian
vector fields, allowing one to construct implicit solutions of such PDEs
parametrized by an arbitrary number of real functions of a single variable.
Then we illustrate this theory on few distinguished examples for the dKP and
heavenly equations. For the dKP case, we characterize a class of similarity
solutions, a class of solutions constant on their parabolic wave front and
breaking simultaneously on it, and a class of localized solutions breaking in a
point of the plane. For the heavenly equation, we characterize two
classes of symmetry reductions.Comment: 29 page
Spinor classification of the Weyl tensor in five dimensions
We investigate the spinor classification of the Weyl tensor in five
dimensions due to De Smet. We show that a previously overlooked reality
condition reduces the number of possible types in the classification. We
classify all vacuum solutions belonging to the most special algebraic type. The
connection between this spinor and the tensor classification due to Coley,
Milson, Pravda and Pravdov\'a is investigated and the relation between most of
the types in each of the classifications is given. We show that the black ring
is algebraically general in the spinor classification.Comment: 40 page
On the solutions of the second heavenly and Pavlov equations
We have recently solved the inverse scattering problem for one parameter
families of vector fields, and used this result to construct the formal
solution of the Cauchy problem for a class of integrable nonlinear partial
differential equations connected with the commutation of multidimensional
vector fields, like the heavenly equation of Plebanski, the dispersionless
Kadomtsev - Petviashvili (dKP) equation and the two-dimensional dispersionless
Toda (2ddT) equation, as well as with the commutation of one dimensional vector
fields, like the Pavlov equation. We also showed that the associated
Riemann-Hilbert inverse problems are powerfull tools to establish if the
solutions of the Cauchy problem break at finite time,to construct their
longtime behaviour and characterize classes of implicit solutions. In this
paper, using the above theory, we concentrate on the heavenly and Pavlov
equations, i) establishing that their localized solutions evolve without
breaking, unlike the cases of dKP and 2ddT; ii) constructing the longtime
behaviour of the solutions of their Cauchy problems; iii) characterizing a
distinguished class of implicit solutions of the heavenly equation.Comment: 16 pages. Submitted to the: Special issue on nonlinearity and
geometry: connections with integrability of J. Phys. A: Math. and Theor., for
the conference: Second Workshop on Nonlinearity and Geometry. Darboux day
Holographic renormalization and supersymmetry
Holographic renormalization is a systematic procedure for regulating
divergences in observables in asymptotically locally AdS spacetimes. For dual
boundary field theories which are supersymmetric it is natural to ask whether
this defines a supersymmetric renormalization scheme. Recent results in
localization have brought this question into sharp focus: rigid supersymmetry
on a curved boundary requires specific geometric structures, and general
arguments imply that BPS observables, such as the partition function, are
invariant under certain deformations of these structures. One can then ask if
the dual holographic observables are similarly invariant. We study this
question in minimal N = 2 gauged supergravity in four and five dimensions. In
four dimensions we show that holographic renormalization precisely reproduces
the expected field theory results. In five dimensions we find that no choice of
standard holographic counterterms is compatible with supersymmetry, which leads
us to introduce novel finite boundary terms. For a class of solutions
satisfying certain topological assumptions we provide some independent tests of
these new boundary terms, in particular showing that they reproduce the
expected VEVs of conserved charges.Comment: 70 pages; corrected typo