23 research outputs found
On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for -Dynamical Systems
The analog of the Chern-Gauss-Bonnet theorem is studied for a -dynamical
system consisting of a -algebra equipped with an ergodic action of a
compact Lie group . The structure of the Lie algebra of
is used to interpret the Chevalley-Eilenberg complex with coefficients in the
smooth subalgebra as noncommutative differential forms
on the dynamical system. We conformally perturb the standard metric, which is
associated with the unique -invariant state on , by means of a Weyl
conformal factor given by a positive invertible element of the algebra, and
consider the Hermitian structure that it induces on the complex. A Hodge
decomposition theorem is proved, which allows us to relate the Euler
characteristic of the complex to the index properties of a Hodge-de Rham
operator for the perturbed metric. This operator, which is shown to be
selfadjoint, is a key ingredient in our construction of a spectral triple on
and a twisted spectral triple on its opposite algebra. The
conformal invariance of the Euler characteristic is interpreted as an
indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral
triples encoding the conformally perturbed metrics are shown to enjoy the same
spectral summability properties as the unperturbed case
A twisted local index formula for curved noncommutative two tori
We consider the Dirac operator of a general metric in the canonical conformal
class on the noncommutative two torus, twisted by an idempotent (representing
the -theory class of a general noncommutative vector bundle), and derive a
local formula for the Fredholm index of the twisted Dirac operator. Our
approach is based on the McKean-Singer index formula, and explicit heat
expansion calculations by making use of Connes' pseudodifferential calculus. As
a technical tool, a new rearrangement lemma is proved to handle challenges
posed by the noncommutativity of the algebra and the presence of an idempotent
in the calculations in addition to a conformal factor.Comment: 27 page
Bell Polynomials and Brownian Bridge in Spectral Gravity Models on Multifractal Robertson–Walker Cosmologies
We obtain an explicit formula for the full expansion of the spectral action on Robertson–Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman–Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson–Walker spacetimes along an Apollonian sphere packing. Using Mellin transforms, we show that the asymptotic expansion of the spectral action contains the same terms as in the case of a single Robertson–Walker spacetime, but with zeta-regularized coefficients, given by values at integers of the zeta function of the fractal string of the radii of the sphere packing, as well as additional log-periodic correction terms arising from the poles (off the real line) of this zeta function