Star products and local line bundles

Richard Melrose[1]

  • [1] Massachusetts Institute of Technology, Department of Mathematics (USA)

Annales de l’institut Fourier (2004)

  • Volume: 54, Issue: 5, page 1581-1600
  • ISSN: 0373-0956

Abstract

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The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products of Lecomte and DeWilde ([3]) see also Fedosov's construction in ([7]). This also shows that the trace on the star algebra is identified with the residue trace of Wodzicki ([18]) and Guillemin ([10]).

How to cite

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Melrose, Richard. "Star products and local line bundles." Annales de l’institut Fourier 54.5 (2004): 1581-1600. <http://eudml.org/doc/116153>.

@article{Melrose2004,
abstract = {The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products of Lecomte and DeWilde ([3]) see also Fedosov's construction in ([7]). This also shows that the trace on the star algebra is identified with the residue trace of Wodzicki ([18]) and Guillemin ([10]).},
affiliation = {Massachusetts Institute of Technology, Department of Mathematics (USA)},
author = {Melrose, Richard},
journal = {Annales de l’institut Fourier},
keywords = {deformation quantization; star product; Toeplitz algebra; local line bundle; gerbe; Szeghö projection; contact manifold; index formula; real cohomology; Szegö projection},
language = {eng},
number = {5},
pages = {1581-1600},
publisher = {Association des Annales de l'Institut Fourier},
title = {Star products and local line bundles},
url = {http://eudml.org/doc/116153},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Melrose, Richard
TI - Star products and local line bundles
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1581
EP - 1600
AB - The notion of a local line bundle on a manifold, classified by 2-cohomology with real coefficients, is introduced. The twisting of pseudodifferential operators by such a line bundle leads to an algebroid with elliptic elements with real-valued index, given by a twisted variant of the Atiyah-Singer index formula. Using ideas of Boutet de Monvel and Guillemin the corresponding twisted Toeplitz algebroid on any compact symplectic manifold is shown to yield the star products of Lecomte and DeWilde ([3]) see also Fedosov's construction in ([7]). This also shows that the trace on the star algebra is identified with the residue trace of Wodzicki ([18]) and Guillemin ([10]).
LA - eng
KW - deformation quantization; star product; Toeplitz algebra; local line bundle; gerbe; Szeghö projection; contact manifold; index formula; real cohomology; Szegö projection
UR - http://eudml.org/doc/116153
ER -

References

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  1. R. Beals, P. Greiner, Calculus on Heisenberg manifolds, vol. 119 (1988), Princeton University Press, Princeton, NJ Zbl0654.58033MR953082
  2. L. Boutet de Monvel, V. Guillemin, The spectral theory of Toeplitz operators, vol. 99 (1981), Princeton University Press Zbl0469.47021MR620794
  3. M. De Wilde, P.B.A. Lecomte, Star-produits et déformations formelles associées aux variétés symplectiques exactes, C.R. Acad. Sci. Paris Sér. I Math 296 (1983), 825-828 Zbl0525.58040MR711841
  4. Ch. Epstein, R. Melrose, Contact degree and the index of Fourier integral operators, Math. Res. Lett 5 (1998), 363-381 Zbl0929.58012MR1637844
  5. C.L. Epstein, R. B. Melrose, The Heisenberg algebra, index theory and homology Zbl0929.58012
  6. C.L. Epstein, R. B. Melrose, G. Mendoza, The Heisenberg algebra, index theory and homology (in preparation) 
  7. B.V. Fedosov, Deformation quantization and asymptotic operator representation, Funktsional. Anal. i Prilozhen 25 (1991), 24-36 Zbl0737.47042MR1139872
  8. D. Geller, Analytic pseudodifferential operators for the Heisenberg group and local solvability, (1990), Princeton University Press, Princeton, NJ Zbl0695.47051MR1030277
  9. V. Guillemin, Star products on compact pre-quantizable symplectic manifolds, Lett. Math. Phys 35 (1995), 85-89 Zbl0842.58041MR1346047
  10. V.W. Guillemin, A new proof of Weyl's formula on the asymptotic distribution of eigenvalues, Adv. Math 55 (1985), 131-160 Zbl0559.58025MR772612
  11. L. Hörmander, The analysis of linear partial differential operators, vol. 3 (1985), Springer-Verlag, Berlin, Heidelberg, New York, Tokyo Zbl0601.35001MR404822
  12. V. Mathai, R.B. Melrose, I.M. Singer, Fractional analytic index Zbl1115.58021
  13. R.B. Melrose, V. Nistor, K -theory of C * -algebras of b -pseudodifferential operators, Geom. Funct. Anal 8 (1998), 88-122 Zbl0898.46060MR1601850
  14. R.B. Melrose, The Atiyah-Patodi-Singer index theorem, (1993), A K Peters Ltd., Wellesley, MA Zbl0796.58050MR1348401
  15. M.K. Murray, Bundle gerbes, J. London Math. Soc 54 (1996), 403-416 Zbl0867.55019MR1405064
  16. R.T. Seeley, Complex powers of an elliptic operator, Singular Integrals (Proc. Sympos. Pure Math. Chicago, III, 1966) (1967), 288-307, Amer. Math. Soc., Providence, R.I Zbl0159.15504
  17. M.E. Taylor, Noncommutative microlocal analysis. I, vol. 313 (1984), AMS Zbl0554.35025MR764508
  18. M. Wodzicki, Noncommutative residue. I. Fundamentals, -theory, arithmetic and geometry (Moscow, 1984–1986) (1987), 320-399, Springer, Berlin-New York Zbl0649.58033

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