937 research outputs found
Combinatorial quantisation of Euclidean gravity in three dimensions
In the Chern-Simons formulation of Einstein gravity in 2+1 dimensions the
phase space of gravity is the moduli space of flat G-connections, where G is a
typically non-compact Lie group which depends on the signature of space-time
and the cosmological constant. For Euclidean signature and vanishing
cosmological constant, G is the three-dimensional Euclidean group. For this
case the Poisson structure of the moduli space is given explicitly in terms of
a classical r-matrix. It is shown that the quantum R-matrix of the quantum
double D(SU(2)) provides a quantisation of that Poisson structure.Comment: cosmetic chang
Quantum gravity and non-commutative spacetimes in three dimensions: a unified approach
These notes summarise a talk surveying the combinatorial or Hamiltonian
quantisation of three dimensional gravity in the Chern-Simons formulation, with
an emphasis on the role of quantum groups and on the way the various physical
constants (c,G,\Lambda,\hbar) enter as deformation parameters. The classical
situation is summarised, where solutions can be characterised in terms of model
spacetimes (which depend on c and \Lambda), together with global
identifications via elements of the corresponding isometry groups. The quantum
theory may be viewed as a deformation of this picture, with quantum groups
replacing the local isometry groups, and non-commutative spacetimes replacing
the classical model spacetimes. This point of view is explained, and open
issues are sketched.Comment: Talk given at Geometry and Physics in Cracow, September 2010; 22
pages, 2 figure
On the existence of minima in the Skyrme model
Well-separated Skyrme solitons of arbitrary degree attract after a suitable
relative rotation in space and iso-space, provided the orders of the solitons'
leading multipoles do not differ by more than two. I summarise the derivation
of this result, obtained jointly with Manton and Singer, and discuss to what
extent its combination with earlier results of Esteban allows one to deduce the
existence of minima of the Skyrme energy functional.Comment: 11 pages amslatex, talk given at the workshop on Integrable theories,
Solitons and Duality, Sao Paulo, July 200
Adiabatic dynamics of instantons on
We define and compute the metric on the framed moduli space of circle
invariant 1-instantons on the 4-sphere. This moduli space is four dimensional
and our metric is symmetric. We study the behaviour of
generic geodesics and show that the metric is geodesically incomplete.
Circle-invariant instantons on the 4-sphere can also be viewed as hyperbolic
monopoles, and we interpret our results from this viewpoint. We relate our
results to work by Habermann on unframed instantons on the 4-sphere and, in the
limit where the radius of the 4-sphere tends to infinity, to results on
instantons on Euclidean 4-space.Comment: 49 pages, 11 figures. Significant improvements in the discussion of
framing in v
Taub-NUT Dynamics with a Magnetic Field
We study classical and quantum dynamics on the Euclidean Taub-NUT geometry
coupled to an abelian gauge field with self-dual curvature and show that, even
though Taub-NUT has neither bounded orbits nor quantum bound states, the
magnetic binding via the gauge field produces both. The conserved Runge-Lenz
vector of Taub-NUT dynamics survives, in a modified form, in the gauged model
and allows for an essentially algebraic computation of classical trajectories
and energies of quantum bound states. We also compute scattering cross sections
and find a surprising electric-magnetic duality. Finally, we exhibit the
dynamical symmetry behind the conserved Runge-Lenz and angular momentum vectors
in terms of a twistorial formulation of phase space.Comment: 36 pages, three figure
Unstable manifolds and Schroedinger dynamics of Ginzburg-Landau vortices
The time evolution of several interacting Ginzburg-Landau vortices according
to an equation of Schroedinger type is approximated by motion on a
finite-dimensional manifold. That manifold is defined as an unstable manifold
of an auxiliary dynamical system, namely the gradient flow of the
Ginzburg-Landau energy functional. For two vortices the relevant unstable
manifold is constructed numerically and the induced dynamics is computed. The
resulting model provides a complete picture of the vortex motion for arbitrary
vortex separation, including well-separated and nearly coincident vortices.Comment: 23 pages amslatex, 5 eps figures, minor typos correcte
S-duality in SU(3) Yang-Mills Theory with Non-abelian Unbroken Gauge Group
It is observed that the magnetic charges of classical monopole solutions in
Yang-Mills-Higgs theory with non-abelian unbroken gauge group are in
one-to-one correspondence with coherent states of a dual or magnetic group
. In the spirit of the Goddard-Nuyts-Olive conjecture this
observation is interpreted as evidence for a hidden magnetic symmetry of
Yang-Mills theory. SU(3) Yang-Mills-Higgs theory with unbroken gauge group U(2)
is studied in detail. The action of the magnetic group on semi-classical states
is given explicitly. Investigations of dyonic excitations show that electric
and magnetic symmetry are never manifest at the same time: Non-abelian magnetic
charge obstructs the realisation of electric symmetry and vice-versa. On the
basis of this fact the charge sectors in the theory are classified and their
fusion rules are discussed. Non-abelian electric-magnetic duality is formulated
as a map between charge sectors. Coherent states obey particularly simple
fusion rules, and in the set of coherent states S-duality can be formulated as
an SL(2,Z)-mapping between sectors which leaves the fusion rules invariant.Comment: 27 pages, harvmac, amssym, one eps figure; minor misprints corrected
and title amende
Classical r-matrices for the generalised Chern-Simons formulation of 3d gravity
We study the conditions for classical r-matrices to be compatible with the
generalised Chern-Simons action for 3d gravity. Compatibility means solving the
classical Yang-Baxter equations with a prescribed symmetric part for each of
the real Lie algebras and bilinear pairings arising in the generalised
Chern-Simons action. We give a new construction of r-matrices via a generalised
complexification and derive a non-linear set of matrix equations determining
the most general compatible r-matrix. We exhibit new families of solutions and
show that they contain known solutions for special parameter valuesComment: 20 pages, minor corrections and comments added in v
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