Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 089, 12 pages      arXiv:0707.3164      https://doi.org/10.3842/SIGMA.2007.089
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

The Symmetric Tensor Lichnerowicz Algebra and a Novel Associative Fourier-Jacobi Algebra

Karl Hallowell and Andrew Waldron
Department of Mathematics, University of California, Davis CA 95616, USA

Received July 21, 2007; Published online September 13, 2007

Abstract
Lichnerowicz's algebra of differential geometric operators acting on symmetric tensors can be obtained from generalized geodesic motion of an observer carrying a complex tangent vector. This relation is based upon quantizing the classical evolution equations, and identifying wavefunctions with sections of the symmetric tensor bundle and Noether charges with geometric operators. In general curved spaces these operators obey a deformation of the Fourier-Jacobi Lie algebra of sp(2,R). These results have already been generalized by the authors to arbitrary tensor and spinor bundles using supersymmetric quantum mechanical models and have also been applied to the theory of higher spin particles. These Proceedings review these results in their simplest, symmetric tensor setting. New results on a novel and extremely useful reformulation of the rank 2 deformation of the Fourier-Jacobi Lie algebra in terms of an associative algebra are also presented. This new algebra was originally motivated by studies of operator orderings in enveloping algebras. It provides a new method that is superior in many respects to common techniques such as Weyl or normal ordering.

Key words: symmetric tensors; Fourier-Jacobi algebras; higher spins; operator orderings.

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References

  1. Alvarez-Gaume L., Witten E., Gravitational anomalies, Nuclear Phys. B 234 (1984), 269-330.
  2. Witten E., Supersymmetry and Morse theory, J. Differential Geom. 17 (1982), 661-692.
  3. Hallowell K., Waldron A., Supersymmetric quantum mechanics and super-Lichnerowicz algebras, Comm. Math. Phys., to appear, hep-th/0702033.
  4. Olver P.J., Differential hyperforms I, University of Minnesota, Report 82-101, 1982, 118 pages.
    Olver P.J., Invariant theory and differential equations, in Invariant Theory, Editor S. Koh, Lecture Notes in Math., Vol. 1278, Springer-Verlag, Berlin, 1987, 62-80.
  5. Dubois-Violette M., Henneaux M., Generalized cohomology for irreducible tensor fields of mixed Young symmetry type, Lett. Math. Phys. 49 (1999), 245-252, math.QA/9907135.
    Dubois-Violette M., Henneaux M., Tensor fields of mixed Young symmetry type and N complexes, Comm. Math. Phys. 226 (2002), 393-418, math.QA/0110088.
  6. Edgar S.B., Senovilla J.M.M., A weighted de Rham operator acting on arbitrary tensor fields and their local potentials, J. Geom. Phys. 56 (2006), 2153-2162.
  7. Bekaert X., Boulanger N., Tensor gauge fields in arbitrary representations of GL(D,R). Duality and Poincaré lemma, Comm. Math. Phys. 245 (2004), 27-67, hep-th/0208058.
    Bekaert X., Boulanger N., On geometric equations and duality for free higher spins, Phys. Lett. B 561 (2003), 183-190, hep-th/0301243.
    Bekaert X., Boulanger N., Tensor gauge fields in arbitrary representations of GL(D,R), Comm. Math. Phys. 271 (2007), 723-773, hep-th/0606198.
  8. de Medeiros P., Hull C., Geometric second order field equations for general tensor gauge fields, JHEP 2003 (2003), no. 5, 019, 26 pages, hep-th/0303036.
    de Medeiros P., Hull C., Exotic tensor gauge theory and duality, Comm. Math. Phys. 235 (2003), 255-273, hep-th/0208155.
  9. Lichnerowicz A., Propagateurs et commutateurs en relativité générale, Inst. Hautes Études Sci. Publ. Math. (1961), no. 10, 56 pages.
    Lichnerowicz A., Champs spinoriels et propagateurs en relativité générale, Bull. Soc. Math. France 92 (1964), 11-100.
  10. Hallowell K., Waldron A., Constant curvature algebras and higher spin action generating functions, Nuclear Phys. B 724 (2005), 453-486, hep-th/0505255.
  11. Duval C., Lecomte P., Ovsienko V., Conformally equivariant quantization: existence and uniqueness, Ann. Inst. Fourier (Grenoble) 49 (1999), 1999-2029, math.DG/9902032.
    Duval C., Ovsienko V., Conformally equivariant quantum Hamiltonians, Selecta Math. (N.S.) 7 (2001), 291-320.
  12. Labastida J.M.F., Massless particles in arbitrary representations of the Lorentz group, Nuclear Phys. B 322 (1989), 185-209.
  13. Vasiliev M.A., Equations of motion of interacting massless fields of all spins as a free differential algebra Phys. Lett. B 209 (1988), 491-497.
  14. Bekaert X., Cnockaert S., Iazeolla C., Vasiliev M.A., Nonlinear higher spin theories in various dimensions, hep-th/0503128.
  15. Deser S., Waldron A., Gauge invariances and phases of massive higher spins in (A)dS, Phys. Rev. Lett. 87 (2001), 031601, 4 pages, hep-th/0102166.
    Deser S., Waldron A., Partial masslessness of higher spins in (A)dS, Nuclear Phys. B 607 (2001), 577-604, hep-th/0103198.
    Deser S., Waldron A., Stability of massive cosmological gravitons, Phys. Lett. B 508 (2001), 347-353, hep-th/0103255.
    Deser S., Waldron A., Null propagation of partially massless higher spins in (A)dS and cosmological constant speculations, Phys. Lett. B 513 (2001), 137-141, hep-th/0105181.
    Deser S., Waldron A., Arbitrary spin representations in de Sitter from dS/CFT with applications to dS supergravity, Nuclear Phys. B 662 (2003), 379-392, hep-th/0301068.
    Deser S., Waldron A., Conformal invariance of partially massless higher spins, Phys. Lett. B 603 (2004), 30-34, hep-th/0408155.
  16. Deser A., Waldron A., Einstein tensors for mixed symmetry higher spins, in preparation.
  17. Cherney D., Hallowell D., Hodge A., Shaukat A., Waldron A., Higher rank Fourier-Jacobi algebras, in preparation.

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