Duke Mathematical Journal

Geometry of pseudodifferential algebra bundles and Fourier integral operators

Varghese Mathai and Richard B. Melrose

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study the geometry and topology of (filtered) algebra bundles ΨZ over a smooth manifold X with typical fiber ΨZ(Z;V), the algebra of classical pseudodifferential operators acting on smooth sections of a vector bundle V over the compact manifold Z and of integral order. First, a theorem of Duistermaat and Singer is generalized to the assertion that the group of projective invertible Fourier integral operators PG(FC(Z;V)) is precisely the automorphism group of the filtered algebra of pseudodifferential operators. We replace some of the arguments in their work by microlocal ones, thereby removing the topological assumption. We define a natural class of connections and B-fields on the principal bundle to which ΨZ 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 ΨZ/Ψ. Examples of pseudodifferential algebra bundles are given that are not associated to a finite-dimensional fiber bundle over X.

Article information

Duke Math. J., Volume 166, Number 10 (2017), 1859-1922.

Received: 23 January 2016
Revised: 22 October 2016
First available in Project Euclid: 21 March 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx]
Secondary: 53C08: Gerbes, differential characters: differential geometric aspects 53D22: Canonical transformations

pseudodifferential algebra bundles Fourier integral operators automorphisms of pseudodifferential operators derivations of pseudodifferential operators twisted (fiber) cosphere bundles regularized trace residue trace central extension gerbes Dixmier–Douady invariant


Mathai, Varghese; Melrose, Richard B. Geometry of pseudodifferential algebra bundles and Fourier integral operators. Duke Math. J. 166 (2017), no. 10, 1859--1922. doi:10.1215/00127094-0000013X. https://projecteuclid.org/euclid.dmj/1490061610

Export citation


  • [1] M. Adams, T. Ratiu, and R. Schmid, A Lie group structure for Fourier integral operators, Math. Ann. 276 (1986), 19–41.
  • [2] O. Alvarez, I. M. Singer, and B. Zumino, Gravitational anomalies and the family’s index theorem, Comm. Math. Phys. 96 (1984), 409–417.
  • [3] M. F. Atiyah and I. M. Singer, The index of elliptic operators, I, Ann. of Math. (2) 87 (1968), 484–530.
  • [4] M. F. Atiyah and I. M. Singer, Dirac operators coupled to vector potentials, Proc. Natl. Acad. Sci. USA 81 (1984), 2597–2600.
  • [5] A. Banyaga, “The group of diffeomorphisms preserving a regular contact form” in Topology and Algebra (Zürich, 1977), Monogr. Enseign. Math. 26, Univ. Genève, Geneva, 1978, 47–53.
  • [6] R. Beals, A general calculus of pseudodifferential operators, Duke Math. J. 42 (1975), 1–42.
  • [7] R. Beals, Characterization of pseudodifferential operators and applications, Duke Math. J. 44 (1977), 45–57. Correction, Duke Math. J. 46 (1979), 215.
  • [8] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Progr. Math. 107, Birkhäuser, Boston, 1993.
  • [9] J.-L. Brylinski and E. Getzler, The homology of algebras of pseudodifferential symbols and the noncommutative residue, $K$-Theory 1 (1987), 385–403.
  • [10] J.-L. Brylinski and D. A. McLaughlin, The geometry of degree-four characteristic classes and of line bundles on loop spaces, I, Duke Math. J. 75 (1994), 603–638.
  • [11] J. Davis and P. Kirk, Lecture Notes in Algebraic Topology, Grad. Stud. Math. 35, Amer. Math. Soc., Providence, 2001.
  • [12] J. Dixmier and A. Douady, Champs continus d’espaces hilbertiens et de C$^{*}$-algèbres, Bull. Soc. Math. France 91 (1963), 227–284.
  • [13] J. Duistermaat, Fourier Integral Operators, Progr. Math. 130, Birkhäuser, Boston, 1996.
  • [14] J. Duistermaat and I. M. Singer, Order-preserving isomorphisms between algebras of pseudo-differential operators, Comm. Pure Appl. Math. 29 (1976), 39–47.
  • [15] C. J. Earle and J. Eells, A fibre bundle description of Teichmüller theory, J. Differential Geometry 3 (1969), 19–43.
  • [16] M. Eidelheit, On isomorphisms of rings of linear operators, Studia Math. 9 (1940), 97–105.
  • [17] H. Emerson and R. Meyer, Euler characteristics and Gysin sequences for group actions on boundaries, Math. Ann. 334 (2006), 853–904.
  • [18] C. Epstein and R. B. Melrose, Contact degree and the index of Fourier integral operators, Math. Res. Lett. 5 (1998), 363–381.
  • [19] D. S. Freed, Determinants, torsion, and strings, Comm. Math. Phys. 107 (1986), 483–513.
  • [20] D. S. Freed, M. J. Hopkins, and C. Teleman, Loop groups and twisted $K$-theory, II, J. Amer. Math. Soc. 26 (2013), 595–644.
  • [21] A. Gramain, Le type d’homotopie du groupe des difféomorphismes d’une surface compacte, Ann. Sci. Éc. Norm. Supér. (4) 6 (1973), 53–66.
  • [22] V. Guillemin, A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. in Math. 55 (1985), 131–160.
  • [23] V. Guillemin, Residue traces for certain algebras of Fourier integral operators, J. Funct. Anal. 115 (1993), 391–417.
  • [24] J. Harer, The second homology group of the mapping class group of an orientable surface, Invent. Math. 72 (1983), 221–239.
  • [25] L. Hörmander, Fourier integral operators, I, Acta Math. 127 (1971), 79–183.
  • [26] L. Hörmander, The Analysis of Linear Partial Differential Operators, III: Pseudodifferential Operators, Grundlehren Math. Wiss. 274, Springer, Berlin, 1985.
  • [27] L. Hörmander, The Analysis of Linear Partial Differential Operators, IV: Fourier Integral Operators, Grundlehren Math. Wiss. 275, Springer, Berlin, 1985.
  • [28] A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories, Prog. Phys. 2, Birkhäuser, Boston, 1980.
  • [29] M. Kontsevich and S. Vishik, “Geometry of determinants of elliptic operators” in Functional Analysis on the Eve of the 21st Century, Vol. 1 (New Brunswick, 1993), Progr. Math. 131, Birkhäuser, Boston, 1995, 173–197.
  • [30] M. Kontsevich and S. Vishik, Determinants of elliptic pseudo-differential operators, preprint, arXiv:hep-th/9404046v1.
  • [31] A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, Math. Surveys Monogr. 53, Amer. Math. Soc., Providence, 1997.
  • [32] E. Leichtnam, R. Nest, and B. Tsygan, Local formula for the index of a Fourier integral operator, J. Differential Geom. 59 (2001), 269–300.
  • [33] C. Maclachlan, Modulus space is simply-connected, Proc. Amer. Math. Soc. 29 (1971), 85–86.
  • [34] V. Mathai, R. B. Melrose, and I. M. Singer, The index of projective families of elliptic operators, Geom. Topol. 9 (2005), 341–373.
  • [35] V. Mathai, R. B. Melrose, and I. M. Singer, Fractional analytic index, J. Differential Geom. 74 (2006), 265–292.
  • [36] V. Mathai, R. B. Melrose, and I. M. Singer, Equivariant and fractional index of projective elliptic operators, J. Differential Geom. 78 (2008), 465–473.
  • [37] V. Mathai, R. B. Melrose, and I. M. Singer, The index of projective families of elliptic operators: The decomposable case, Astérisque 328 (2009), 255–296.
  • [38] R. B. Melrose and V. Nistor, Homology of pseudodifferential operators, I: Manifolds with boundary, preprint, arXiv:funct-an/9606005v2.
  • [39] R. B. Melrose and F. Rochon, “Eta forms and the odd pseudodifferential families index” in Surveys in Differential Geometry, Volume XV: Perspectives in Mathematics and Physics, Surv. Differ. Geom. 15, International Press, Somerville, Mass., 2011, 279–322.
  • [40] S. Morita, Geometry of Characteristic Classes, Transl. Math. Monogr. 199, Amer. Math. Soc., Providence, 2001.
  • [41] M. K. Murray and D. Stevenson, Bundle gerbes: Stable isomorphism and local theory, J. London Math. Soc. (2) 62 (2000), 925–937.
  • [42] M. K. Murray and D. Stevenson, Higgs fields, bundle gerbes and string structures, Comm. Math. Phys. 243 (2003), 541–555.
  • [43] H. Omori, Infinite-Dimensional Lie Groups, Transl. Math. Monogr. 158, Amer. Math. Soc., Providence, 1997.
  • [44] S. Paycha, Chern–Weil calculus extended to a class of infinite dimensional manifolds, preprint, arXiv:0706.2554v1 [math.DG].
  • [45] S. Paycha and S. Rosenberg, Curvature on determinant bundles and first Chern forms, J. Geom. Phys. 45 (2003), 393–429.
  • [46] J. Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978), no. 3, 347–350.
  • [47] F. Rochon, Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre $0$, Ann. Inst. Fourier (Grenoble) 58 (2008), 29–62.
  • [48] R. T. Seeley, “Complex powers of an elliptic operator” in Singular Integrals (Chicago, Ill., 1966), Amer. Math. Soc., Providence, 1967, 288–307.
  • [49] R. Vozzo, Loop groups, Higgs fields and generalised string classes, Ph.D. dissertation, University of Adelaide, Adelaide, Australia, 2009.
  • [50] M. Wodzicki, Local invariants of spectral asymmetry, Invent. Math. 75 (1984), 143–177.
  • [51] M. Wodzicki, Excision in cyclic homology and in rational algebraic K-theory, Ann. of Math. (2) 129 (1989), 591–639.