arXiv subjects
Publications and source records attributed to Sylvie Paycha.
For a holomorphic family of classical pseudodifferential operators on a closed manifold we give exact formulae for all coefficients in the Laurent expansion of its Kontsevich-Vishik canonical trace. This generalizes a known result identifying the Wodzicki residue with the pole at zero to all higher order terms.
This paper has been withdrawn by the authors, as it contains some errors. The core of it has been entirely rewritten in math.FA/0510454.
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 $n$-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.
We construct Chern-Weil classes on infinite dimensional vector bundles with structure group contained in the algebra $\cl[\leq 0](M, E)$ of non-positive order classical pseudo-differential operators acting on a finite rank vector bundle $E$ over a closed manifold $M$. Mimicking the finite dimensional Chern-Weil construction, we replace the ordinary trace on matrices by linear functionals on $\cl[\leq 0] (M, E)$ built from the leading symbols of the operators. The corresponding Chern classes vanish for loop groups, but a weighted trace construction yields a non-zero class perviously constructed by Freed. For loop spaces, the structure group reduces to a gauge group of bundle automorphisms, and we produce non-vanishing universal Chern classes in all degrees, using a universal connection theorem for these bundles.
zeta-regularized traces, resp. super-traces, are defined on a classical pseudo-differential operator A by: tr^Q(A):= f.p.tr(A Q^{-z})_{|_{z=0}}, resp. str^Q(A):= f.p.str(A Q^{-z})_{|_{z=0}}, where f.p. refers to the finite part and Q is an (invertible and admissible) elliptic reference operator with positive order. They are widly used in quantum field theory in spite of the fact that, unlike ordinary traces on matrices, they are neither cyclic nor do they commute with exterior differentiation, thus giving rise to tracial anomalies. The purpose of this article is to show, on two examples, how tracial anomalies can lead to anomalous phenomena in quantum field theory.
The Quillen-Bismut-Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut-Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues. We illustrate these results with a string theory computation. There is a natural super vector bundle over the manifold of smooth almost complex structures on a Riemannian surface. The Bismut-Freed superconnection is identified with classical Teichmuller theory connections, and its curvature and regularized first Chern form are computed.