55 research outputs found
Families index theorem in supersymmetric WZW model and twisted K-theory: The SU(2) case
The construction of twisted K-theory classes on a compact Lie group is
reviewed using the supersymmetric Wess-Zumino-Witten model on a cylinder. The
Quillen superconnection is introduced for a family of supercharges parametrized
by a compact Lie group and the Chern character is explicitly computed in the
case of SU(2). For large euclidean time, the character form is localized on a
D-brane.Comment: Version 2: Essentially simplified proof of the main result using a
map from twisted K-theory to gerbes modulo the twisting gerbe; references
added + minor correction
Anomalies and Schwinger terms in NCG field theory models
We study the quantization of chiral fermions coupled to generalized Dirac
operators arising in NCG Yang-Mills theory. The cocycles describing chiral
symmetry breaking are calculated. In particular, we introduce a generalized
locality principle for the cocycles. Local cocycles are by definition
expressions which can be written as generalized traces of operator commutators.
In the case of pseudodifferential operators, these traces lead in fact to
integrals of ordinary local de Rham forms. As an application of the general
ideas we discuss the case of noncommutative tori. We also develop a gerbe
theoretic approach to the chiral anomaly in hamiltonian quantization of NCG
field theory.Comment: 30 page
Schwinger Terms and Cohomology of Pseudodifferential Operators
We study the cohomology of the Schwinger term arising in second quantization
of the class of observables belonging to the restricted general linear algebra.
We prove that, for all pseudodifferential operators in 3+1 dimensions of this
type, the Schwinger term is equivalent to the ``twisted'' Radul cocycle, a
modified version of the Radul cocycle arising in non-commutative differential
geometry. In the process we also show how the ordinary Radul cocycle for any
pair of pseudodifferential operators in any dimension can be written as the
phase space integral of the star commutator of their symbols projected to the
appropriate asymptotic component.Comment: 19 pages, plain te
Quantization of Magnetic Poisson Structures:LMS/EPSRC Durham Symposium on Higher Structures in M-Theory
We describe three perspectives on higher quantization, using the example of
magnetic Poisson structures which embody recent discussions of nonassociativity
in quantum mechanics with magnetic monopoles and string theory with
non-geometric fluxes. We survey approaches based on deformation quantization of
twisted Poisson structures, symplectic realization of almost symplectic
structures, and geometric quantization using 2-Hilbert spaces of sections of
suitable bundle gerbes. We compare and contrast these perspectives, describing
their advantages and shortcomings in each case, and mention many open avenues
for investigation.Comment: 13 pages, Contribution to Proceedings of LMS/EPSRC Durham Symposium
Higher Structures in M-Theory, August 201
Unitary Representations of Unitary Groups
In this paper we review and streamline some results of Kirillov, Olshanski
and Pickrell on unitary representations of the unitary group \U(\cH) of a
real, complex or quaternionic separable Hilbert space and the subgroup
\U_\infty(\cH), consisting of those unitary operators for which g - \1
is compact. The Kirillov--Olshanski theorem on the continuous unitary
representations of the identity component \U_\infty(\cH)_0 asserts that they
are direct sums of irreducible ones which can be realized in finite tensor
products of a suitable complex Hilbert space. This is proved and generalized to
inseparable spaces. These results are carried over to the full unitary group by
Pickrell's Theorem, asserting that the separable unitary representations of
\U(\cH), for a separable Hilbert space \cH, are uniquely determined by
their restriction to \U_\infty(\cH)_0. For the classical infinite rank
symmetric pairs of non-unitary type, such as (\GL(\cH),\U(\cH)), we
also show that all separable unitary representations are trivial.Comment: 42 page
Open-closed string correspondence: D-brane decay in curved space
This paper analyzes the effect of curved closed string backgrounds on the
stability of D-branes within boundary string field theory. We identify the
non-local open string background that implements shifts in the closed string
background and analyze the tachyonic sector off-shell. The renormalization
group flow reveals some characteristic properties, which are expected for a
curved background, like the absence of a stable space-filling brane. In
3-dimensions we describe tachyon condensation processes to lower-dimensional
branes, including a curved 2-dimensional brane. We argue that this 2-brane is
perturbatively stable. This is in agreement with the known maximally symmetric
WZW-branes and provides further support to the bulk-boundary factorization
approach to open-closed string correspondence.Comment: 23 pages, harvma
Gauged diffeomorphisms and hidden symmetries in Kaluza-Klein theories
We analyze the symmetries that are realized on the massive Kaluza-Klein modes
in generic D-dimensional backgrounds with three non-compact directions. For
this we construct the unbroken phase given by the decompactification limit, in
which the higher Kaluza-Klein modes are massless. The latter admits an
infinite-dimensional extension of the three-dimensional diffeomorphism group as
local symmetry and, moreover, a current algebra associated to SL(D-2,R)
together with the diffeomorphism algebra of the internal manifold as global
symmetries. It is shown that the `broken phase' can be reconstructed by gauging
a certain subgroup of the global symmetries. This deforms the three-dimensional
diffeomorphisms to a gauged version, and it is shown that they can be governed
by a Chern-Simons theory, which unifies the spin-2 modes with the Kaluza-Klein
vectors. This provides a reformulation of D-dimensional Einstein gravity, in
which the physical degrees of freedom are described by the scalars of a gauged
non-linear sigma model based on SL(D-2,R)/SO(D-2), while the metric appears in
a purely topological Chern-Simons form.Comment: 23 pages, minor changes, v3: published versio
Covariant Schwinger terms
There exist two versions of the covariant Schwinger term in the literature.
They only differ by a sign. However, we shall show that this is an essential
difference. We shall carefully (taking all signs into account) review the
existing quantum field theoretical computations for the covariant Schwinger
term in order to determine the correct expression.Comment: 26 pages, Latex, some references adde
Central extensions of groups of sections
If q : P -> M is a principal K-bundle over the compact manifold M, then any
invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a
Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms
modulo exact forms. In the present paper we analyze the integrability of this
extension to a Lie group extension for non-connected, possibly
infinite-dimensional Lie groups K. If K has finitely many connected components
we give a complete characterization of the integrable extensions. Our results
on gauge groups are obtained by specialization of more general results on
extensions of Lie groups of smooth sections of Lie group bundles. In this more
general context we provide sufficient conditions for integrability in terms of
data related only to the group K.Comment: 54 pages, revised version, to appear in Ann. Glob. Anal. Geo
Gerbes, M5-Brane Anomalies and E_8 Gauge Theory
Abelian gerbes and twisted bundles describe the topology of the NS-NS 3-form
gauge field strength H. We review how they have been usefully applied to study
and resolve global anomalies in open string theory. Abelian 2-gerbes and
twisted nonabelian gerbes describe the topology of the 4-form field strength G
of M-theory. We show that twisted nonabelian gerbes are relevant in the study
and resolution of global anomalies of multiple coinciding M5-branes. Global
anomalies for one M5-brane have been studied by Witten and by Diaconescu, Freed
and Moore. The structure and the differential geometry of twisted nonabelian
gerbes (i.e. modules for 2-gerbes) is defined and studied. The nonabelian
2-form gauge potential living on multiple coinciding M5-branes arises as
curving (curvature) of twisted nonabelian gerbes. The nonabelian group is in
general , the central extension of the E_8 loop group. The
twist is in general necessary to cancel global anomalies due to the
nontriviality of the 11-dimensional 4-form G field strength and due to the
possible torsion present in the cycles the M5-branes wrap. Our description of
M5-branes global anomalies leads to the D4-branes one upon compactification of
M-theory to Type IIA theory.Comment: 19 page
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