79 research outputs found
Weighted trace cochains; a geometric setup for anomalies
We extend formulae which measure discrepancies for regularized traces on
classical pseudodifferential operators to regularized trace cochains,
regularized traces corresponding to 0-regularized trace cochains. This
extension from 0-cochains to -cochains is appropriate to handle
simultaneously algebraic and geometric discrepancies/anomalies. Algebraic
anomalies are Hochschild coboundaries of regularized trace cochains on a fixed
algebra of pseudodifferential operators weighted by a fixed classical
pseudodifferential operator with positive order and positive scalar leading
symbol. In contrast, geometric anomalies arise when considering families of
pseudodifferential operators associated with a smooth fibration of manifolds.
They correspond to covariant derivatives (and possibly their curvature) of
smooth families of regularized trace cochains, the weight being here an
elliptic operator valued form on the base manifold. Both types of discrepancies
can be expressed as finite linear combinations of Wodzicki residues.We apply
the formulae obtained in the family setting to build Chern-Weil type weighted
trace cochains on one hand, and on the other hand, to show that choosing the
curvature of a Bismut-Quillen type super connection as a weight, provides
covariantly closed weighted trace cochains in which case the geometric
discrepancies vanish
A Canonical Trace Associated with Certain Spectral Triples
In the abstract pseudodifferential setup of Connes and Moscovici, we prove a
general formula for the discrepancies of zeta-regularised traces associated
with certain spectral triples, and we introduce a canonical trace on operators,
whose order lies outside (minus) the dimension spectrum of the spectral triple
The logarithmic residue density of a generalised Laplacian
We show that the residue density of the logarithm of a generalised Laplacian
on a closed manifold defines an invariant polynomial valued differential form.
We express it in terms of a finite sum of residues of classical
pseudodifferential symbols. In the case of the square of a Dirac operator,
these formulae provide a pedestrian proof of the Atiyah-Singer formula for a
pure Dirac operator in dimension and for a twisted Dirac operator on a flat
space of any dimension. These correspond to special cases of a more general
formula by S. Scott and D. Zagier announced in \cite{Sc2} and to appear in
\cite{Sc3}. In our approach, which is of perturbative nature, we use either a
Campbell-Hausdorff formula derived by Okikiolu or a non commutative Taylor type
formula.Comment: 24 pages, no figure
Nested sums of symbols and renormalised multiple zeta functions
We define discrete nested sums over integer points for symbols on the real
line, which obey stuffle relations whenever they converge. They relate to Chen
integrals of symbols via the Euler-MacLaurin formula. Using a suitable
holomorphic regularisation followed by a Birkhoff factorisation, we define
renormalised nested sums of symbols which also satisfy stuffle relations. For
appropriate symbols they give rise to renormalised multiple zeta functions
which satisfy stuffle relations at all arguments. The Hurwitz multiple zeta
functions fit into the framework as well. We show the rationality of multiple
zeta values at nonpositive integer arguments, and a higher-dimensional analog
is also investigated.Comment: Two major changes : improved treatment of the Hurwitz multiple zeta
functions, and more conceptual (and shorter) approach of the multidimensional
cas
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