24 research outputs found
Krein spectral triples and the fermionic action
Motivated by the space of spinors on a Lorentzian manifold, we define Krein
spectral triples, which generalise spectral triples from Hilbert spaces to
Krein spaces. This Krein space approach allows for an improved formulation of
the fermionic action for almost-commutative manifolds. We show by explicit
calculation that this action functional recovers the correct Lagrangians for
the cases of electrodynamics, the electro-weak theory, and the Standard Model.
The description of these examples does not require a real structure, unless one
includes Majorana masses, in which case the internal spaces also exhibit a
Krein space structure.Comment: 17 page
Indefinite Kasparov modules and pseudo-Riemannian manifolds
We present a definition of indefinite Kasparov modules, a generalisation of
unbounded Kasparov modules modelling non-symmetric and non-elliptic (e.g.
hyperbolic) operators. Our main theorem shows that to each indefinite Kasparov
module we can associate a pair of (genuine) Kasparov modules, and that this
process is reversible. We present three examples of our framework: the Dirac
operator on a pseudo-Riemannian spin manifold (i.e. a manifold with an
indefinite metric), the harmonic oscillator, and the construction via the
Kasparov product of an indefinite spectral triple from a family of spectral
triples. This last construction corresponds to a foliation of a globally
hyperbolic spacetime by spacelike hypersurfaces.Comment: 24 pages, Annales Henri Poincar\'e, online version 201
Electrodynamics from Noncommutative Geometry
Within the framework of Connes' noncommutative geometry, the notion of an
almost commutative manifold can be used to describe field theories on compact
Riemannian spin manifolds. The most notable example is the derivation of the
Standard Model of high energy physics from a suitably chosen almost commutative
manifold. In contrast to such a non-abelian gauge theory, it has long been
thought impossible to describe an abelian gauge theory within this framework.
The purpose of this paper is to improve on this point. We provide a simple
example of a commutative spectral triple based on the two-point space, and show
that it yields a U(1) gauge theory. Then, we slightly modify the spectral
triple such that we obtain the full classical theory of electrodynamics on a
curved background manifold.Comment: 16 page
Homotopy equivalence in unbounded KK-theory
We propose a new notion of unbounded -cycle, mildly generalising
unbounded Kasparov modules, for which the direct sum is well-defined. To a pair
of -unital -algebras, we can then associate a semigroup
of homotopy equivalence classes of unbounded cycles,
and we prove that this semigroup is in fact an abelian group. In case is
separable, our group is isomorphic to Kasparov's
-theory group via the bounded transform. We also discuss
various notions of degenerate cycles, and we prove that the homotopy relation
on unbounded cycles coincides with the relation generated by
operator-homotopies and addition of degenerate cycles.Comment: 33 page
The APS-index and the spectral flow
We study the Atiyah-Patodi-Singer (APS) index, and its equality to the
spectral flow, in an abstract, functional analytic setting. More precisely, we
consider a (suitably continuous or differentiable) family of self-adjoint
Fredholm operators on a Hilbert space, parametrised by in a finite
interval. We then consider two different operators, namely (the abstract analogue of a Riemannian Dirac operator) and (the abstract analogue of a Lorentzian Dirac operator). The
latter case is inspired by a recent index theorem by B\"ar and Strohmaier
(Amer.\ J.\ Math. 141 (2019), 1421--1455) for a Lorentzian Dirac operator
equipped with APS boundary conditions. In both cases, we prove that Fredholm
index of the operator equipped with APS boundary conditions is equal to the
spectral flow of the family
Lorentzian geometry and physics in Kasparov's theory
We study two geometric themes, Lorentzian geometry and gauge theory, from the
perspective of Connes’ noncommutative geometry and (the unbounded version of)
Kasparov’s KK-theory. Lorentzian geometry is the mathematical framework underlying
Einstein’s description of gravity. The geometric formulation of a gauge theory
(in terms of principal bundles) offers a classical description for the interactions
between particles. The underlying motivation is the hope that this noncommutative
approach may lead to a unified description of gauge theories coupled with
gravity on a Lorentzian manifold.
The main objects in noncommutative geometry are spectral triples, which encompass
and generalise Riemannian spin manifolds. A spectral triple defines a
class in K-homology, via which one can access the topology of the (noncommutative)
manifold. In this thesis we present two possible definitions for ‘Lorentian
spectral triples’, which offer noncommutative generalisations of Lorentzian manifolds
as well. We will prove that both definitions preserve the link with analytic
K-homology. We will describe under which conditions Lorentzian (or pseudo-
Riemannian) manifolds satisfy these definitions. Another main example is the
harmonic oscillator, which in particular shows that our framework allows to deal
with more than just metrics of indefinite signature.
In the context of noncommutative geometry, the description of a gauge theory
can be obtained from so-called almost-commutative manifolds. While the usual approach
yields by default a topologically trivial gauge theory (in the sense that the
corresponding principal fibre bundle is globally trivial), we show in this thesis that
the framework can be adapted, using the internal unbounded Kasparov product,
to allow for globally non-trivial gauge theories as well.
Finally, we combine the two themes of Lorentzian geometry and gauge theory,
and we define Krein spectral triples, which generalise spectral triples from Hilbert
spaces to Krein spaces. We use this definition to construct almost-commutative
Lorentzian manifolds. Furthermore, we propose a Lorentzian alternative for the
fermionic action, which allows to derive (the fermionic part of) the Lagrangian of
a gauge theory. We show that our alternative action recovers exactly the correct
physical Lagrangian
On globally non-trivial almost-commutative manifolds
Within the framework of Connes' noncommutative geometry, we define and study
globally non-trivial (or topologically non-trivial) almost-commutative
manifolds. In particular, we focus on those almost-commutative manifolds that
lead to a description of a (classical) gauge theory on the underlying base
manifold. Such an almost-commutative manifold is described in terms of a
'principal module', which we build from a principal fibre bundle and a finite
spectral triple. We also define the purely algebraic notion of 'gauge modules',
and show that this yields a proper subclass of the principal modules. We
describe how a principal module leads to the description of a gauge theory, and
we provide two basic yet illustrative examples.Comment: 34 pages, minor revision