866 research outputs found
Amenability of the Gauge Group
Let G be one of the local gauge groups C(X,U(n)), C^\infty(X,U(n)),
C(X,SU(n)) or C^\infty(X,SU(n)) where X is a compact Riemannian manifold. We
observe that G has a nontrivial group topology, coarser than its natural
topology, w.r.t. which it is amenable, viz the relative weak topology of
C(X,M(n)). This topology seems more useful than other known amenable topologies
for G. We construct a simple fermionic model containing an action of G,
continuous w.r.t. this amenable topology.Comment: 8 pages, Late
A gerbe obstruction to quantization of fermions on odd dimensional manifolds with boundary
We consider the canonical quantization of fermions on an odd dimensional
manifold with boundary, with respect to a family of elliptic hermitean boundary
conditions for the Dirac hamiltonian. We show that there is a topological
obstruction to a smooth quantization as a function of the boundary conditions.
The obstruction is given in terms of a gerbe and its Dixmier-Douady class is
evaluated.Comment: Improved introduction, minor corrections in the text, results
unchange
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
Index Theory, Gerbes, and Hamiltonian Quantization
We give an Atiyah-Patodi-Singer index theory construction of the bundle of
fermionic Fock spaces parametrized by vector potentials in odd space dimensions
and prove that this leads in a simple manner to the known Schwinger terms
(Faddeev-Mickelsson cocycle) for the gauge group action. We relate the APS
construction to the bundle gerbe approach discussed recently by Carey and
Murray, including an explicit computation of the Dixmier-Douady class. An
advantage of our method is that it can be applied whenever one has a form of
the APS theorem at hand, as in the case of fermions in an external
gravitational field.Comment: 16 pages, Plain TeX inputting AMSTe
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
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