Let $G$ be an arbitrary compact Lie group. In this work we apply the method of the analytic continuation of traces in order to compute the Wodzicki residue for a classical pseudo-differential operator on $G$ in terms of its matrix-valued symbol (which is globally defined on the non-commutative phase space $G\times \widehat{G},$ with $\widehat{G}$ being the unitary dual of $G$). Our main theorem is complementary to the results in [2], where we remove the ellipticity hypothesis when the operators belong to the Hörmander classes on $G$ defined by local coordinate systems.
For compact real manifolds, a new double conformal invariant is constructed using the Wodzicki residue and the $d$ operator in the framework of Connes. In the flat case, we compute this double conformal invariant, and in some special cases, we also compute this double conformal invariants. For complex manifolds, a new double conformal invariant is constructed using the Wodzicki residue and the $\bar{\partial}$ operator in the same way, and this double conformal invariant is computed in the flat case.
For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical {\sf GJMS} operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue of a pseudo-differential operator of order $-2,$ originally defined by A. Connes, acting on middle dimension forms.
\begin{abstract} By following the paradigm of the global quantisation, instead of the analysis under changes of coordinates, in this work we establish a global analysis for the explicit computation of the Dixmier trace and the Wodzicki residue of (elliptic and subelliptic) pseudo-differential operators on compact Lie groups. The regularised determinant for the Dixmier trace is also computed. We obtain these formulae in terms of the global symbol of the corresponding operators. In particular, our approach links the Dixmier trace and Wodzicki residue to the representation theory of the group. Al…
In [3], Connes found a conformal invariant using Wodzicki's 1-density and computed it in the case of 4-dimensional manifold without boundary. In [14], Ugalde generalized the Connes' result to $n$-dimensional manifold without boundary. In this paper, we generalize the results of [3] and [14] to the case of manifolds with boundary.
For a pseudodifferential operator $S$ of order 0 acting on sections of a vector bundle $B$ on a compact manifold $M$ without boundary, we associate a differential form of order dimension of $M$ acting on $C^\infty(M)\times C^\infty(M)$. This differential form $Ω_{n,S}$ is given in terms of the Wodzicki 1-density $\wres([S,f][S,h])$. In the particular case of an even dimensional, compact, conformal manifold without boundary, we study this differential form for the case $(B,S)=(\cH,F)$, that is, the Fredholm module associated by A. Connes to the manifold $M.$ We give its explicit expression in t…
The Wodzicki residue and the cut-off integral extend to classical symbol-valued forms. We show that they obey a Stokes' type property and that the extended Wodzicki residue can be interpreted as a complex residue like the ordinary one. In the case of cut-off integrals, Stokes' property (i.e. vanishing on exact forms) only holds for non-integer order symbol-valued forms and leads to an integration by parts formula and translation invariance for cut-off integrals on non-integer order classical symbols. The extended Wodzicki residue yields an even residue cycle on classical symbols and an odd coc…
Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the resolvent trace coefficients, the residues of the spectral zeta function as well as certain Wodzicki residues. We then use the Wodzicki residue to obtain results about the local counting function of operators of Dirac and Laplace type. In particular, we express the second term of the…
The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $ζ$-invariants using lifted defect formulae which express discrepancies of $ζ$-regularised traces in terms of Wodzicki residues. We derive Atiyah's $L^2$-index theorem as an instance of the $\mathbb Z_2$-graded generalisation of the canonical lift of spectral $ζ$-invariants and we show that certain lifted s…
In this paper we give formulae for the Dixmier trace and the noncommutative residue (also called Wodzicki's residue) of pseudo-differential operators by using the notion of global symbol. We consider both cases, compact manifolds with or without boundary. Our analysis on the Dixmier trace of invariant pseudo-differential operators on closed manifolds will be based on the Fourier analysis associated to every elliptic and positive operator and the quantization process developed by Delgado and Ruzhansky. In particular, for compact Lie groups this can be done by using the representation theory of …
A well known result on pseudodifferential operators states that the noncommutative residue (Wodzicki residue) of a pseudodifferential projection vanishes. This statement is non-local and implies the regularity of the eta invariant at zero of Dirac type operators. We prove that in a filtered algebra the value of a projection under any residual trace depends only on the principal part of the projection. This general, purely algebraic statement applied to the algebra of projective pseudodifferential operators implies that the noncommutative residue factors to a map from the twisted K-theory of th…
We provide local expressions for Chern-Weil type forms built from superconnections associated with families of Dirac operators previously investigated in work by S. Scott and later work by S. Scott and the second author. When the underlying fibration of manifolds is trivial, the even degree forms can be interpreted as renormalised Chern-Weil forms in as far as they coincide with regularised Chern-Weil forms up to residue correction terms. Similarly, a new formula for the curvature of the local fermionic vacuum line bundles is derived using a residue correction term added to the naive curvature…
We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the identity. The leading order Chern classes are nonzero in general, and they detect elements of the de Rham cohomology groups of the classifying space BZ.
We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras…
On a 6-dimensional, conformal, oriented, compact manifold $M$ without boundary, we compute a whole family of differential forms $Ω_6(f,h)$ of order 6, with $f,h \in C^\infty(M).$ Each of these forms will be symmetric on $f,$ and $h,$ conformally invariant, and such that $\int_M f_0 Ω_6(f_1,f_2)$ defines a Hochschild 2-cocycle over the algebra $C^\infty(M).$ In the particular 6-dimensional conformally flat case, we compute the unique one satisfying $\Wres(f_0[F,f][F,h]) = \int_M f_0Ω_6(f,h)$ for $(\cH,F)$ the Fredholm module associated by A. Connes \cite{Con1} to the manifold $M,$ and $\Wres$ t…
We study various aspects of the noncommutative residue for an algebra of pseudodifferential operators whose symbols have an expansion $a\sim \sum_{j=0}^\infty a_{m-j}, a_{m-j}(x,ξ)=\sum_{l=0}^k a_{m-j,l}(x,ξ) \log^l|ξ|,$ where $a_{m-j,l}$ is homogeneous in $ξ$ of degree $m-j$. We will explain why this algebra of pseudodifferential operators is natural. For a pseudodifferential operator in this class, $A$, and a classical elliptic pseudodifferential operator, $P$, we show that the generalized zeta-function $\Tr(AP^{-s})$ has a meromorphic continuation to the whole complex plane, however possibl…
For operators on a compact manifold $X$ with boundary $\partial X$, the basic zeta coefficient $C_0(B, P_{1,T})$ is the regular value at $s=0$ of the zeta function $\Tr(B P_{1,T}^{-s})$, where $B=P_++G$ is a pseudodifferential boundary operator (in the Boutet de Monvel calculus) -- for example the solution operator of a classical elliptic problem -- and $P_{1,T}$ is a realization of an elliptic differential operator $P_1$, having a ray free of eigenvalues. Relative formulas (e.g. for the difference between the constants with two different choices of $P_{1,T}$) have been known for some time and…
These are notes on seminal work of Freed, and subsequent developments, on the curvature properties of (Sobolev Lie) groups of maps from a Riemannian manifold into a compact Lie group. We are mainly interested in critical cases which are relevant to quantum field theory. For example Freed showed that, in a necessarily qualified sense, the quotient space $W^{1/2}(S^1,K)/K$ is a (positive constant) Einstein `manifold' with respect to the essentially unique PSU(1,1) invariant metric, where $W^{s}$ denotes maps of $L^2$ Sobolev order s. In a similarly qualified sense, and in addition making use of …
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 $4$ 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 …
The homotopy groups of the (stabilized) group of invertible pseudodifferential operators of order zero acting on a closed manifold X are computed in terms of the K-theory of the cosphere bundle S*X. At the same time, we show that the subgroup of invertible compact perturbations of the identity is weakly retractable inside this group. These results are also adapted to the case of suspended operators. Some applications in index theory and for the residue determinant of Simon Scott are also given.
