16 research outputs found
Quantisation of twistor theory by cocycle twist
We present the main ingredients of twistor theory leading up to and including
the Penrose-Ward transform in a coordinate algebra form which we can then
`quantise' by means of a functorial cocycle twist. The quantum algebras for the
conformal group, twistor space CP^3, compactified Minkowski space CMh and the
twistor correspondence space are obtained along with their canonical quantum
differential calculi, both in a local form and in a global *-algebra
formulation which even in the classical commutative case provides a useful
alternative to the formulation in terms of projective varieties. We outline how
the Penrose-Ward transform then quantises. As an example, we show that the
pull-back of the tautological bundle on CMh pulls back to the basic instanton
on S^4\subset CMh and that this observation quantises to obtain the
Connes-Landi instanton on \theta-deformed S^4 as the pull-back of the
tautological bundle on our \theta-deformed CMh. We likewise quantise the
fibration CP^3--> S^4 and use it to construct the bundle on \theta-deformed
CP^3 that maps over under the transform to the \theta-deformed instanton.Comment: 68 pages 0 figures. Significant revision now has detailed formulae
for classical and quantum CP^
Is there a Jordan geometry underlying quantum physics?
There have been several propositions for a geometric and essentially
non-linear formulation of quantum mechanics. From a purely mathematical point
of view, the point of view of Jordan algebra theory might give new strength to
such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of
the algebra of observables, in the same way as Lie groups belong to the Lie
part. Both the Lie geometry and the Jordan geometry are well-adapted to
describe certain features of quantum theory. We concentrate here on the
mathematical description of the Jordan geometry and raise some questions
concerning possible relations with foundational issues of quantum theory.Comment: 30 page
Twistors and Black Holes
Motivated by black hole physics in N=2, D=4 supergravity, we study the
geometry of quaternionic-Kahler manifolds M obtained by the c-map construction
from projective special Kahler manifolds M_s. Improving on earlier treatments,
we compute the Kahler potentials on the twistor space Z and Swann space S in
the complex coordinates adapted to the Heisenberg symmetries. The results bear
a simple relation to the Hesse potential \Sigma of the special Kahler manifold
M_s, and hence to the Bekenstein-Hawking entropy for BPS black holes. We
explicitly construct the ``covariant c-map'' and the ``twistor map'', which
relate real coordinates on M x CP^1 (resp. M x R^4/Z_2) to complex coordinates
on Z (resp. S). As applications, we solve for the general BPS geodesic motion
on M, and provide explicit integral formulae for the quaternionic Penrose
transform relating elements of H^1(Z,O(-k)) to massless fields on M annihilated
by first or second order differential operators. Finally, we compute the exact
radial wave function (in the supergravity approximation) for BPS black holes
with fixed electric and magnetic charges.Comment: 47 pages, v2: typos corrected, reference added, v3: minor change
Quantum Attractor Flows
Motivated by the interpretation of the Ooguri-Strominger-Vafa conjecture as a
holographic correspondence in the mini-superspace approximation, we study the
radial quantization of stationary, spherically symmetric black holes in four
dimensions. A key ingredient is the classical equivalence between the radial
evolution equation and geodesic motion of a fiducial particle on the moduli
space M^*_3 of the three-dimensional theory after reduction along the time
direction. In the case of N=2 supergravity, M^*_3 is a para-quaternionic-Kahler
manifold; in this case, we show that BPS black holes correspond to a particular
class of geodesics which lift holomorphically to the twistor space Z of M^*_3,
and identify Z as the BPS phase space. We give a natural quantization of the
BPS phase space in terms of the sheaf cohomology of Z, and compute the exact
wave function of a BPS black hole with fixed electric and magnetic charges in
this framework. We comment on the relation to the topological string amplitude,
extensions to N>2 supergravity theories, and applications to automorphic black
hole partition functions.Comment: 43 pages, 6 figures; v2: typos and references added; v3: published
version, minor change
The trade-off between tidal-turbine array yield and environmental impact: a habitat suitability modelling approach
In the drive towards a carbon-free society, tidal energy has the potential to become a valuable part of the UK energy supply. Developments are subject to intense scrutiny, and potential environmental impacts must be assessed. Unfortunately many of these impacts are still poorly understood, including the implications that come with altering the hydrodynamics. Here, methods are proposed to quantify ecological impact and to incorporate its minimisation into the array design process. Four tidal developments in the Pentland Firth are modelled with the array optimisation tool OpenTidalFarm, that designs arrays to generate the maximum possible profit. Maximum entropy modelling is used to create habitat suitability maps for species that respond to changes in bedshear stress. Changes in habitat suitability caused by an altered tidal regime are assessed. OpenTidalFarm is adapted to simultaneously optimise array design to maximise both this habitat suitability and to maximise the profit of the array. The problem is thus posed as a multi-objective optimisation problem, and a set of Pareto solutions found, allowing trade-offs between these two objectives to be identified. The methods proposed generate array designs that have reduced negative impact, or even positive impact, on the habitat suitability of specific species or habitats of interest