|
SIGMA 10 (2014), 016, 26 pages arXiv:1308.1046
https://doi.org/10.3842/SIGMA.2014.016
Second Order Symmetries of the Conformal Laplacian
Jean-Philippe Michel a, Fabian Radoux a and Josef Šilhan b
a) Department of Mathematics of the University of Liège, Grande Traverse 12, 4000 Liège, Belgium
b) Department of Algebra and Geometry of the Masaryk University in Brno, Janàčkovo nàm. 2a, 662 95 Brno, Czech Republic
Received October 25, 2013, in final form February 05, 2014; Published online February 14, 2014
Abstract
Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3.
We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order.
They are constructed from conformal Killing 2-tensors satisfying a natural and conformally invariant
condition. As a consequence, we get also the classification of the second order symmetries of the conformal
Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein
manifolds respectively. We illustrate our results on two families of examples in dimension three.
Key words:
Laplacian; quantization; conformal geometry; separation of variables.
pdf (503 kb)
tex (35 kb)
References
- Bailey T.N., Eastwood M.G., Gover A.R., Thomas's structure bundle for
conformal, projective and related structures, Rocky Mountain J.
Math. 24 (1994), 1191-1217.
- Ballesteros Á., Enciso A., Herranz F.J., Ragnisco O., Riglioni D., Quantum
mechanics on spaces of nonconstant curvature: the oscillator problem and
superintegrability, Ann. Physics 326 (2011), 2053-2073,
arXiv:1102.5494.
- Benenti S., Chanu C., Rastelli G., Remarks on the connection between the
additive separation of the Hamilton-Jacobi equation and the
multiplicative separation of the Schrödinger equation. II. First
integrals and symmetry operators, J. Math. Phys. 43 (2002),
5223-5253.
- Boe B.D., Collingwood D.H., A comparison theory for the structure of induced
representations, J. Algebra 94 (1985), 511-545.
- Boe B.D., Collingwood D.H., A comparison theory for the structure of induced
representations. II, Math. Z. 190 (1985), 1-11.
- Bonanos S., Riemannian geometry and tensor calculus (Mathematica package),
Version 3.8.5, 2012, available at
http://www.inp.demokritos.gr/~sbonano/RGTC/.
- Boyer C.P., Kalnins E.G., Miller Jr. W., Symmetry and separation of variables
for the Helmholtz and Laplace equations, Nagoya Math. J.
60 (1976), 35-80.
- Boyer C.P., Kalnins E.G., Miller Jr. W., R-separable coordinates for
three-dimensional complex Riemannian spaces, Trans. Amer. Math.
Soc. 242 (1978), 355-376.
- Cap A., Šilhan J., Equivariant quantizations for AHS-structures,
Adv. Math. 224 (2010), 1717-1734, arXiv:0904.3278.
- Čap A., Slovák J., Souček V., Bernstein-Gelfand-Gelfand
sequences, Ann. of Math. 154 (2001), 97-113,
math.DG/0001164.
- Carter B., Killing tensor quantum numbers and conserved currents in curved
space, Phys. Rev. D 16 (1977), 3395-3414.
- 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,
math.DG/9801122.
- Duval C., Valent G., Quantum integrability of quadratic Killing tensors,
J. Math. Phys. 46 (2005), 053516, 22 pages,
math-ph/0412059.
- Eastwood M., Higher symmetries of the Laplacian, Ann. of Math.
161 (2005), 1645-1665, hep-th/0206233.
- Eastwood M., Leistner T., Higher symmetries of the square of the Laplacian,
in Symmetries and Overdetermined Systems of Partial Differential Equations,
IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 319-338,
math.DG/0610610.
- Fegan H.D., Conformally invariant first order differential operators,
Quart. J. Math. Oxford (2) 27 (1976), 371-378.
- Gover A.R., Šilhan J., Higher symmetries of the conformal powers of the
Laplacian on conformally flat manifolds, J. Math. Phys.
53 (2012), 032301, 26 pages, arXiv:0911.5265.
- Kalnins E.G., Miller Jr. W., Intrinsic characterisation of orthogonal R
separation for Laplace equations, J. Phys. A: Math. Gen.
15 (1982), 2699-2709.
- Kolář I., Michor P.W., Slovák J., Natural operations in
differential geometry, Springer-Verlag, Berlin, 1993, available at
http://www.emis.de/monographs/KSM/.
- Kozaki H., Koike T., Ishihara H., Exactly solvable strings in Minkowski
spacetime, Classical Quantum Gravity 27 (2010), 105006,
10 pages, arXiv:0907.2273.
- Lecomte P.B.A., Towards projectively equivariant quantization, Progr.
Theoret. Phys. Suppl. 144 (2001), 125-132.
- Loubon Djounga S.E., Conformally invariant quantization at order three,
Lett. Math. Phys. 64 (2003), 203-212.
- Mathonet P., Radoux F., On natural and conformally equivariant quantizations,
J. Lond. Math. Soc. 80 (2009), 256-272,
arXiv:0707.1412.
- Mathonet P., Radoux F., Existence of natural and conformally invariant
quantizations of arbitrary symbols, J. Nonlinear Math. Phys.
17 (2010), 539-556, arXiv:0811.3710.
- Michel J.-P., Higher symmetries of Laplacian via quantization, Ann.
Inst. Fourier (Grenoble), to appear, arXiv:1107.5840.
- Penrose R., Rindler W., Spinors and space-time. Vol. 1. Two-spinor calculus
and relativistic fields, Cambridge Monographs on Mathematical Physics,
Cambridge University Press, Cambridge, 1984.
- Perelomov A.M., Integrable systems of classical mechanics and Lie algebras.
Vol. I, Birkhäuser Verlag, Basel, 1990.
- Radoux F., An explicit formula for the natural and conformally invariant
quantization, Lett. Math. Phys. 89 (2009), 249-263,
arXiv:0902.1543.
- Šilhan J., Conformally invariant quantization - towards the complete
classification, Differential Geom. Appl. 33 (2014), suppl.,
162-176, arXiv:0903.4798.
- Vlasáková Z., Symmetries of CR sub-Laplacian, arXiv:1201.6219.
|
|