1210.0990 (Varghese Mathai et al.)

Geometry of Pseudodifferential algebra bundles and Fourier Integral
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Varghese Mathai, R. B. Melrose

We study the geometry and topology of (filtered) algebra-bundles $\bfPsi^\bbZ$ over a smooth manifold $X$ with typical fibre $\Psi^\bbZ(Z; V)$, the algebra of classical pseudodifferential operators of integral order on the compact manifold $Z$ acting on smooth sections of a vector bundle $V$. First a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators $\PGL(\cF^\bullet(Z; V))$, is precisely the automorphism group, $\Aut(\Psi^\bbZ(Z; V)),$ of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their paper by microlocal ones, thereby removing the topological assumption well as extending their result to sections of a vector bundle. We define a natural class of connections and B-fields the principal bundle to which $\bfPsi^\bbZ$ is associated and obtain a de Rham representative of the Dixmier-Douady class, in terms of the outer derivation on the Lie algebra and the residue trace of Guillemin and Wodzicki; the resulting formula only depends on the formal symbol algebra $\bfPsi^\bbZ/\bfPsi^{-\infty}.$ Some examples of pseudodifferential bundles with non-torsion Dixmier-Douady class are given; in general such a bundle is not associated to a finite dimensional fibre bundle over $X.$

View original: http://arxiv.org/abs/1210.0990