243 research outputs found
Poincare duality for K-theory of equivariant complex projective spaces
We make explicit Poincare duality for the equivariant K-theory of equivariant complex projective spaces. The case of the trivial group provides a new approach to the K-theory orientation
The probability that and commute in a compact group
We show that a compact group has finite conjugacy classes, i.e., is an
FC-group if and only if its center is open if and only if its commutator
subgroup is finite. Let denote the Haar measure of the set of all
pairs in for which ; this, formally, is the
probability that two randomly picked elements commute. We prove that is
always rational and that it is positive if and only if is an extension of
an FC-group by a finite group. This entails that is abelian by finite. The
proofs involve measure theory, transformation groups, Lie theory of arbitrary
compact groups, and representation theory of compact groups. Examples and
references to the history of the discussion are given at the end of the paper.Comment: 17 pages; we have cut some points ; to appear in Math. Proc.
Cambridge Phil. So
On the localized phase of a copolymer in an emulsion: supercritical percolation regime
In this paper we study a two-dimensional directed self-avoiding walk model of
a random copolymer in a random emulsion. The copolymer is a random
concatenation of monomers of two types, and , each occurring with
density 1/2. The emulsion is a random mixture of liquids of two types, and
, organised in large square blocks occurring with density and ,
respectively, where . The copolymer in the emulsion has an energy
that is minus times the number of -matches minus times the
number of -matches, where without loss of generality the interaction
parameters can be taken from the cone . To make the model mathematically tractable, we assume that the
copolymer is directed and can only enter and exit a pair of neighbouring blocks
at diagonally opposite corners.
In \cite{dHW06}, it was found that in the supercritical percolation regime , with the critical probability for directed bond percolation on
the square lattice, the free energy has a phase transition along a curve in the
cone that is independent of . At this critical curve, there is a transition
from a phase where the copolymer is fully delocalized into the -blocks to a
phase where it is partially localized near the -interface. In the present
paper we prove three theorems that complete the analysis of the phase diagram :
(1) the critical curve is strictly increasing; (2) the phase transition is
second order; (3) the free energy is infinitely differentiable throughout the
partially localized phase.Comment: 43 pages and 10 figure
How many invariant polynomials are needed to decide local unitary equivalence of qubit states?
Given L-qubit states with the fixed spectra of reduced one-qubit density
matrices, we find a formula for the minimal number of invariant polynomials
needed for solving local unitary (LU) equivalence problem, that is, problem of
deciding if two states can be connected by local unitary operations.
Interestingly, this number is not the same for every collection of the spectra.
Some spectra require less polynomials to solve LU equivalence problem than
others. The result is obtained using geometric methods, i.e. by calculating the
dimensions of reduced spaces, stemming from the symplectic reduction procedure.Comment: 22 page
Graded infinite order jet manifolds
The relevant material on differential calculus on graded infinite order jet
manifolds and its cohomology is summarized. This mathematics provides the
adequate formulation of Lagrangian theories of even and odd variables on smooth
manifolds in terms of the Grassmann-graded variational bicomplex.Comment: 30 page
The orientation-preserving diffeomorphism group of S^2 deforms to SO(3) smoothly
Smale proved that the orientation-preserving diffeomorphism group of S^2 has
a continuous strong deformation retraction to SO(3). In this paper, we
construct such a strong deformation retraction which is diffeologically smooth.Comment: 16 page
A new invariant on hyperbolic Dehn surgery space
In this paper we define a new invariant of the incomplete hyperbolic
structures on a 1-cusped finite volume hyperbolic 3-manifold M, called the
ortholength invariant. We show that away from a (possibly empty) subvariety of
excluded values this invariant both locally parameterises equivalence classes
of hyperbolic structures and is a complete invariant of the Dehn fillings of M
which admit a hyperbolic structure. We also give an explicit formula for the
ortholength invariant in terms of the traces of the holonomies of certain loops
in M. Conjecturally this new invariant is intimately related to the boundary of
the hyperbolic Dehn surgery space of M.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-23.abs.htm
Abelian gauge theories on compact manifolds and the Gribov ambiguity
We study the quantization of abelian gauge theories of principal torus
bundles over compact manifolds with and without boundary. It is shown that
these gauge theories suffer from a Gribov ambiguity originating in the
non-triviality of the bundle of connections whose geometrical structure will be
analyzed in detail. Motivated by the stochastic quantization approach we
propose a modified functional integral measure on the space of connections that
takes the Gribov problem into account. This functional integral measure is used
to calculate the partition function, the Greens functions and the field
strength correlating functions in any dimension using the fact that the space
of inequivalent connections itself admits the structure of a bundle over a
finite dimensional torus. The Greens functions are shown to be affected by the
non-trivial topology, giving rise to non-vanishing vacuum expectation values
for the gauge fields.Comment: 33 page
Orbits of quantum states and geometry of Bloch vectors for -level systems
Physical constraints such as positivity endow the set of quantum states with
a rich geometry if the system dimension is greater than two. To shed some light
on the complicated structure of the set of quantum states, we consider a
stratification with strata given by unitary orbit manifolds, which can be
identified with flag manifolds. The results are applied to study the geometry
of the coherence vector for n-level quantum systems. It is shown that the
unitary orbits can be naturally identified with spheres in R^{n^2-1} only for
n=2. In higher dimensions the coherence vector only defines a non-surjective
embedding into a closed ball. A detailed analysis of the three-level case is
presented. Finally, a refined stratification in terms of symplectic orbits is
considered.Comment: 15 pages LaTeX, 3 figures, reformatted, slightly modified version,
corrected eq.(3), to appear in J. Physics
Kinematic Orbits and the Structure of the Internal Space for Systems of Five or More Bodies
The internal space for a molecule, atom, or other n-body system can be
conveniently parameterised by 3n-9 kinematic angles and three kinematic
invariants. For a fixed set of kinematic invariants, the kinematic angles
parameterise a subspace, called a kinematic orbit, of the n-body internal
space. Building on an earlier analysis of the three- and four-body problems, we
derive the form of these kinematic orbits (that is, their topology) for the
general n-body problem. The case n=5 is studied in detail, along with the
previously studied cases n=3,4.Comment: 38 pages, submitted to J. Phys.
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