460 research outputs found
Loop groups, anyons and the Calogero-Sutherland model
The positive energy representations of the loop group of U(1) are used to
construct a boson-anyon correspondence. We compute all the correlation
functions of our anyon fields and study an anyonic W-algebra of unbounded
operators with a common dense domain. This algebra contains an operator with
peculiar exchange relations with the anyon fields. This operator can be
interpreted as a second quantised Calogero-Sutherland (CS) Hamiltonian and may
be used to solve the CS model. In particular, we inductively construct all
eigenfunctions of the CS model from anyon correlation functions, for all
particle numbers and positive couplings.Comment: 34 pages, Late
The universal gerbe, Dixmier-Douady class, and gauge theory
We clarify the relation between the Dixmier-Douady class on the space of self
adjoint Fredholm operators (`universal B-field') and the curvature of
determinant bundles over infinite-dimensional Grassmannians. In particular, in
the case of Dirac type operators on a three dimensional compact manifold we
obtain a simple and explicit expression for both forms.Comment: 13 pages, no figure
Spectral flow of monopole insertion in topological insulators
Inserting a magnetic flux into a two-dimensional one-particle Hamiltonian
leads to a spectral flow through a given gap which is equal to the Chern number
of the associated Fermi projection. This paper establishes a generalization to
higher even dimension by inserting non-abelian monopoles of the Wu-Yang type.
The associated spectral flow is then equal to a higher Chern number. For the
study of odd spacial dimensions, a new so-called `chirality flow' is introduced
which, for the insertion of a monopole, is then linked to higher winding
numbers. This latter fact follows from a new index theorem for the spectral
flow between two unitaries which are conjugates of each other by a self-adjoint
unitary.Comment: title changed; final corrections before publication; to appear in
Commun. Math. Phy
Higher spectral flow and an entire bivariant JLO cocycle
Given a smooth fibration of closed manifolds and a family of generalised
Dirac operators along the fibers, we define an associated bivariant JLO
cocycle. We then prove that, for any , our bivariant JLO cocycle
is entire when we endow smoooth functions on the total manifold with the
topology and functions on the base manifold with the
topology. As a by-product of our theorem, we deduce that the bivariant JLO
cocycle is entire for the Fr\'echet smooth topologies. We then prove that our
JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher
spectral flow
Spectral flow and Dixmier traces
We obtain general theorems which enable the calculation of the Dixmier trace
in terms of the asymptotics of the zeta function and of the heat operator in a
general semi-finite von Neumann algebra. Our results have several applications.
We deduce a formula for the Chern character of an odd -summable Breuer-Fredholm module in terms of a Hochschild
1-cycle. We explain how to derive a Wodzicki residue for pseudo-differential
operators along the orbits of an ergodic \IR^n action on a compact space .
Finally we give a short proof an index theorem of Lesch for generalised
Toeplitz operators
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