103 research outputs found
Hyper-K{\"a}hler Hierarchies and their twistor theory
A twistor construction of the hierarchy associated with the hyper-K\"ahler
equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in
four dimensions) is given. The recursion operator R is constructed and used to
build an infinite-dimensional symmetry algebra and in particular higher flows
for the hyper-K\"ahler equations. It is shown that R acts on the twistor data
by multiplication with a rational function. The structures are illustrated by
the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended
space-time is constructed whose extra dimensions correspond to
higher flows of the hierarchy. It is shown that is a moduli space of
rational curves with normal bundle in twistor
space and is canonically equipped with a Lax distribution for ASDVE
hierarchies. The space is shown to be foliated by four dimensional
hyper-K{\"a}hler slices. The Lagrangian, Hamiltonian and bi-Hamiltonian
formulations of the ASDVE in the form of the heavenly equations are given. The
symplectic form on the moduli space of solutions to heavenly equations is
derived, and is shown to be compatible with the recursion operator.Comment: 23 pages, 1 figur
Einstein supergravity amplitudes from twistor-string theory
This paper gives a twistor-string formulation for all tree amplitudes of
Einstein (super-)gravities for N=0 and 4. Formulae are given with and without
cosmological constant and with various possibilities for the gauging. The
formulae are justified by use of Maldacena's observation that conformal gravity
tree amplitudes with Einstein wave functions and non-zero cosmological constant
will correctly give the Einstein tree amplitudes. This justifies the
construction of Einstein gravity amplitudes at N=0 from twistor-string theory
and is extended to N=4 by requiring the standard relation between the
MHV-degree and the degree of the rational curve for Yang-Mills; this
systematically excludes the spurious conformal supergravity gravity
contributions. For comparison, BCFW recursion is used to obtain
twistor-string-like formulae at degree zero and one (anti-MHV and MHV) for
amplitudes with N=8 supersymmetry with and without cosmological constant.Comment: 20 pages. v2: minor corrections & clarification of relation to
formulae of Maldacena & Pimentel and Raju; v3: appendix on BCFW recursion
added, published version. v4: Full derivation for 3 point MHV amplitude now
include
Einstein Supergravity and New Twistor String Theories
A family of new twistor string theories is constructed and shown to be free
from world-sheet anomalies. The spectra in space-time are calculated and shown
to give Einstein supergravities with second order field equations instead of
the higher derivative conformal supergravities that arose from earlier twistor
strings. The theories include one with the spectrum of N=8 supergravity,
another with the spectrum of N=4 supergravity coupled to N=4 super-Yang-Mills,
and a family with supersymmetries with the spectra of self-dual
supergravity coupled to self-dual super-Yang-Mills. The non-supersymmetric
string with N=0 gives self-dual gravity coupled to self-dual Yang-Mills and a
scalar. A three-graviton amplitude is calculated for the N=8 and N=4 theories
and shown to give a result consistent with the cubic interaction of Einstein
supergravity.Comment: LaTeX, 69 pages, no figures; v2: minor corrections made, footnotes
and references adde
Einstein-Weyl geometry, the dKP equation and twistor theory
It is shown that Einstein-Weyl (EW) equations in 2+1 dimensions contain the
dispersionless Kadomtsev-Petviashvili (dKP) equation as a special case: If an
EW structure admits a constant weighted vector then it is locally given by , where satisfies the dKP
equation .
Linearised solutions to the dKP equation are shown to give rise to
four-dimensional anti-self-dual conformal structures with symmetries. All
four-dimensional hyper-K\"ahler metrics in signature for which the
self-dual part of the derivative of a Killing vector is null arise by this
construction. Two new classes of examples of EW metrics which depend on one
arbitrary function of one variable are given, and characterised.
A Lax representation of the EW condition is found and used to show that all
EW spaces arise as symmetry reductions of hyper-Hermitian metrics in four
dimensions. The EW equations are reformulated in terms of a simple and closed
two-form on the \CP^1-bundle over a Weyl space.
It is proved that complex solutions to the dKP equations, modulo a certain
coordinate freedom, are in a one-to-one correspondence with minitwistor spaces
(two-dimensional complex manifolds containing a rational curve with
normal bundle \O(2)) that admit a section of , where
is the canonical bundle of . Real solutions are obtained if the
minitwistor space also admits an anti-holomorphic involution with fixed points
together with a rational curve and section of that are
invariant under the involution.Comment: 22 pages, 1 figur
Twistor theory at fifty: from contour integrals to twistor strings.
We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space-time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold-the twistor space. After giving an elementary construction of this space, we demonstrate how solutions to linear and nonlinear equations of mathematical physics-anti-self-duality equations on Yang-Mills or conformal curvature-can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang-Mills and gravitational instantons, which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of anti-self-dual (ASD) Yang-Mills equations, and Einstein-Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally, we discuss the Newtonian limit of twistor theory and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function
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
Amplitudes at Weak Coupling as Polytopes in AdS_5
We show that one-loop scalar box functions can be interpreted as volumes of
geodesic tetrahedra embedded in a copy of AdS_5 that has dual conformal
space-time as boundary. When the tetrahedron is space-like, it lies in a
totally geodesic hyperbolic three-space inside AdS_5, with its four vertices on
the boundary. It is a classical result that the volume of such a tetrahedron is
given by the Bloch-Wigner dilogarithm and this agrees with the standard physics
formulae for such box functions. The combinations of box functions that arise
in the n-particle one-loop MHV amplitude in N=4 super Yang-Mills correspond to
the volume of a three-dimensional polytope without boundary, all of whose
vertices are attached to a null polygon (which in other formulations is
interpreted as a Wilson loop) at infinity.Comment: 16 pages, 5 figure
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
Activated mutant NRasQ61K drives aberrant melanocyte signaling, survival, and invasiveness via a rac1-Dependent mechanism
Around a fifth of melanomas exhibit an activating mutation in the oncogene NRas that confers constitutive signaling to proliferation and promotes tumor initiation. NRas signals downstream of the major melanocyte tyrosine kinase receptor c-kit and activated NRas results in increased signaling via the extracellular signal–regulated kinase (ERK)/MAPK/ERK kinase/mitogen-activated protein kinase (MAPK) pathways to enhance proliferation. The Ras oncogene also activates signaling via the related Rho GTPase Rac1, which can mediate growth, survival, and motility signaling. We tested the effects of activated NRasQ61K on the proliferation, motility, and invasiveness of melanoblasts and melanocytes in the developing mouse and ex vivo explant culture as well as in a melanoma transplant model. We find an important role for Rac1 downstream of NRasQ61K in mediating dermal melanocyte survival in vivo in mouse, but surprisingly NRasQ61K does not appear to affect melanoblast motility or proliferation during mouse embryogenesis. We also show that genetic deletion or pharmacological inhibition of Rac1 in NRasQ61K induced melanoma suppresses tumor growth, lymph node spread, and tumor cell invasiveness, suggesting a potential value for Rac1 as a therapeutic target for activated NRas-driven tumor growth and invasiveness
Absolute polarization angle calibration using polarized diffuse Galactic emission observed by BICEP
We present a method of cross-calibrating the polarization angle of a
polarimeter using BICEP Galactic observations. \bicep\ was a ground based
experiment using an array of 49 pairs of polarization sensitive bolometers
observing from the geographic South Pole at 100 and 150 GHz. The BICEP
polarimeter is calibrated to +/-0.01 in cross-polarization and less than +/-0.7
degrees in absolute polarization orientation. BICEP observed the temperature
and polarization of the Galactic plane (R.A= 100 degrees ~ 270 degrees and Dec.
= -67 degrees ~ -48 degrees). We show that the statistical error in the 100 GHz
BICEP Galaxy map can constrain the polarization angle offset of WMAP Wband to
0.6 degrees +\- 1.4 degrees. The expected 1 sigma errors on the polarization
angle cross-calibration for Planck or EPIC are 1.3 degrees and 0.3 degrees at
100 and 150 GHz, respectively. We also discuss the expected improvement of the
BICEP Galactic field observations with forthcoming BICEP2 and Keck
observations.Comment: 13 pages, 10 figures and 2 tables. To appear in Proceedings of SPIE
Astronomical Telescopes and Instrumentation 201
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