## Conformally equivariant quantization

Citations: | 9 - 1 self |

### BibTeX

@MISC{Duval_conformallyequivariant,

author = {C. Duval and Université De La Méditerranée and V. Ovsienko},

title = {Conformally equivariant quantization},

year = {}

}

### OpenURL

### Abstract

Let (M,g) be a pseudo-Riemannian manifold and Fλ(M) the space of densities of degree λ on M. We study the space D2 λ,µ (M) of second-order differential operators from Fλ(M) to Fµ(M). If (M,g) is conformally flat with signature p −q, then D2 λ,µ (M) is viewed as a module over the group of conformal transformations of M. We prove that, for almost all values of µ − λ, the O(p+1,q+1)-modules D2 λ,µ (M) and the space of symbols (i.e., of second-order polynomials on T ∗M) are canonically isomorphic. This yields a conformally equivariant quantization for quadratic Hamiltonians. We furthermore show that this quantization map extends to arbitrary pseudo-Riemannian manifolds and depends only on the conformal class [g] of the metric. As an example, the quantization of the geodesic flow yields a novel conformally equivariant Laplace operator on half-densities, as well as the well-known Yamabe Laplacian. We also recover in this framework the multi-dimensional Schwarzian derivative of conformal transformations.

### Citations

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(Show Context)
Citation Context ...der first the space of polynomials C[ξ1, . . .,ξn] with the canonical action of the orthogonal Lie algebra o(p, q) with p+q = n, generated by Xij = ξi∂/∂ξ j −ξj∂/∂ξ i (cf. (3.1)). A classical theorem =-=[21, 4]-=- states that the commutant o(p, q) ! in the space End(C[ξ1, . . .,ξn]) is the associative algebra generated by: R = ξ i ξi, ∂ E = ξi ∂ξi + n ∂ , T = 2 ∂ξi ∂ ∂ξi (5.1) whose commutation relations are t... |

165 |
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Citation Context ...der first the space of polynomials C[ξ1, . . .,ξn] with the canonical action of the orthogonal Lie algebra o(p, q) with p+q = n, generated by Xij = ξi∂/∂ξ j −ξj∂/∂ξ i (cf. (3.1)). A classical theorem =-=[21, 4]-=- states that the commutant o(p, q) ! in the space End(C[ξ1, . . .,ξn]) is the associative algebra generated by: R = ξ i ξi, ∂ E = ξi ∂ξi + n ∂ , T = 2 ∂ξi ∂ ∂ξi (5.1) whose commutation relations are t... |

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(Show Context)
Citation Context ...r which its form is − 1 2 , 3 2 preserved by the action of Diff(S 1 ). (b) Again, in the one-dimensional case, the study of the modules Dk 1−k 1+k , 2 2 goes back to the pioneering work of Wilczynski =-=[22]-=-. (c) Yet another remarkable example is provided by the Yamabe-Laplace operator A = ∆ − (n − 2)/(4(n − 1)) R, where ∆ is the usual Laplace-Beltrami operator and R the scalar curvature on a (pseudo-)Ri... |

42 | V.: Conformally equivariant quantization: existence and uniqueness
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(Show Context)
Citation Context ...order. The singular values (1.2) of the shift δ = µ − λ are called resonances and lead to special and interesting modules. Theorem 1.1 is particular case of a more general result recently obtained in =-=[8]-=-. Theorem 1.2. For each resonant value of δ, there exist particular pairs (λ, µ) of weights such that the o(p + 1, q + 1)-modules S2 δ and D2 λ,µ are isomorphic, namely δ 2 n n+2 2n 1 n+1 n n+2 n λ n−... |

35 |
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Citation Context ...it, for a conformally equivariant quantization. 3Remark 1.4. In the particular case n = 1, the projective and conformal symmetries coincide; our results are in full accordance with those obtained in =-=[10, 9, 6]-=- and the , 2}. resonances are simply {1, 3 2 It turns out that this isomorphism Qλ,µ makes sense for an arbitrary pseudoRiemannian manifold (not necessarily conformally flat). The fundamental property... |

29 |
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Citation Context ... the space of differential n-forms: F1 = Ω n (M). Definition 2.1. An operator A : Fλ → Fµ is called a local operator on M if for all φ ∈ Fλ one has Supp(A(φ)) ⊂ Supp(φ). It is a classical result (see =-=[19]-=-) that such operators are in fact locally given by differential operators. The space Dλ,µ of differential operators from λ-densities to µ-densities on M is naturally a Diff(M)-module. There is a filtr... |

29 |
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(Show Context)
Citation Context ...-Laplace operator A = ∆g − (n − 2)/(4(n − 1)) Rg, where ∆g is the usual Laplace-Beltrami 3operator and Rg the scalar curvature on a (pseudo-)Riemannian manifold (M, g) of dimension n ≥ 2. (See, e.g. =-=[1]-=-.) This operator has been extensively used in the mathematical and physical literature because of its characteristic property of being invariant under conformal changes of metrics. It is well known th... |

20 |
Space of second order linear differential operators as a module over the Lie algebra of vector fields
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(Show Context)
Citation Context ...meter family of Diff(M)-modules, Sδ(M), where δ = µ − λ. The modules Dλ,µ(M) have already been considered in the classic literature on differential operators and, more recently, in a series of papers =-=[7, 14, 10, 15, 9, 17]-=-. The general problem of classification of these Diff(M)-modules has been solved in these articles. We will be considering the modules of second-order operators, D2 λ,µ symbols, S2 δ (M). The main pur... |

19 |
Comparison of some modules of the Lie algebra of vector fields
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(Show Context)
Citation Context ...meter family of Diff(M)-modules, Sδ(M), where δ = µ − λ. The modules Dλ,µ(M) have already been considered in the classic literature on differential operators and, more recently, in a series of papers =-=[7, 14, 10, 15, 9, 17]-=-. The general problem of classification of these Diff(M)-modules has been solved in these articles. We will be considering the modules of second-order operators, D2 λ,µ symbols, S2 δ (M). The main pur... |

17 |
The Schwarzian derivative and conformal mapping of Riemannian mainfolds
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(Show Context)
Citation Context ... interesting generalization of the Schwarzian derivative for conformal diffeomorphisms in the multi-dimensional case. In the situation (3.12) with F = e 2f , the Schwarzian derivative of ϕ is defined =-=[18, 5]-=- as the symmetric twice-covariant tensor S(ϕ) such that S(ϕ)(X, Y ) = X(Y f) − (∇XY )f − (Xf)(Y f) + 1 2 ‖df‖2 g g(X, Y ) (3.13) for any X, Y ∈ Vect(M). In our notation, it reads S(ϕ) = 1 2F 3 1 ∇dF −... |

15 |
On the cohomology of sl(m + 1, R) acting on differential operators and sl(m + 1, R)-equivariant symbol
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(Show Context)
Citation Context ...quivariant symbol calculus and quantization was introduced if M is endowed with a flat projective structure. In this case the group of (local) symmetries is G = SL(n + 1, R) with n = dim(M). See also =-=[13]-=- for a cohomological treatment of this subject. Bearing in mind that the best-known geometries associated with a local and maximal (see Section 8) symmetry group are the projective and conformal geome... |

14 | Space of linear differential operators on the real line as a module over the Lie algebra of vector fields
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(Show Context)
Citation Context ...meter family of Diff(M)-modules, Sδ(M), where δ = µ − λ. The modules Dλ,µ(M) have already been considered in the classic literature on differential operators and, more recently, in a series of papers =-=[7, 14, 10, 15, 9, 17]-=-. The general problem of classification of these Diff(M)-modules has been solved in these articles. We will be considering the modules of second-order operators, D2 λ,µ symbols, S2 δ (M). The main pur... |

14 | Lie Algebras of Vector Fields - González-López, Kamran, et al. - 1992 |

11 |
Lorentz surfaces, Walter de Gruyter
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(Show Context)
Citation Context ...iffeomorphism of M, and F ∈ C ∞ (M, R ∗ + ), and g0 is a metric of constant curvature. Let us emphasize that this weaker form of the uniformization theorem still holds in the Lorentz case (see, e.g., =-=[20]-=-). There exists in the recent literature an interesting generalization of the Schwarzian derivative for conformal diffeomorphisms in the multi-dimensional case. In the situation (3.12) with F = e 2f ,... |

8 |
Intertwining operators between some spaces of differential operators on a manifold
- Mathonet
- 1999
(Show Context)
Citation Context |

8 |
Projectively invariant symbol map and cohomology of vector fields Lie algebras intervening in quantization
- Lecomte, Ovsienko
(Show Context)
Citation Context ...uantization of the geodesic flow yields a novel conformally equivariant Laplace operator on half-densities, as well as the well-known Yamabe Laplacian. This article is a continuation of previous work =-=[13]-=- dealing with SL(n + 1, R)-equivariant quantization; it constitutes the second volet of the study of quantization constrained to equivariance with respect to the automorphisms of G-structures. Keyword... |

6 |
A remark about the Lie algebra of infinitesimal conformal transformations of the Euclidean space
- Boniver, Lecomte
(Show Context)
Citation Context ...nformal algebra which was of central importance in our work. The Lie algebra o(p+1, q+1) is a maximal Lie subalgebra of Vect(R n ) in the sense that any larger subalgebra is infinite-dimensional (see =-=[3]-=-). This property implied the uniqueness of the isomorphisms of the modules of differential operators and symbols under study. Recall that the same is true for the projective Lie algebra sl(n + 1, R). ... |

5 |
Sur la géométrie des operateurs différentiels linéaires sur R, Preprint CPT
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- 1997
(Show Context)
Citation Context ...pling in the same framework. We have also chosen to put aside the cohomological content of many aspects of the problem. It should be stressed that Lie-algebra cohomology proved useful in earlier work =-=[7, 15, 9, 13]-=- on the modules of differential operators. The resonances appearing in (1.2) should thus certainly hide non-trivial o(p + 1, q + 1)-cohomology classes. Let us finish by mentioning a crucial property o... |

5 |
Gauge invariant quantization on Riemannian manifolds
- Zh-J, Qian
- 1992
(Show Context)
Citation Context ...oof will be Theorem 3.2. We also use, in the sequel, the notation Ric = Rij dx i ⊗dx j for the Ricci tensor. Remark 3.8. Another quantization formula for second-order polynomials has been proposed in =-=[16]-=- using a (pseudo-)Riemannian metric on M and the local identification of T ∗ M with R 2n endowed with its standard sp(2n, R) action. 113.7 Lower-dimensional cases and Schwarzian derivatives The gener... |

4 |
The Schwarzian derivative for conformal maps
- Carne
- 1990
(Show Context)
Citation Context ... interesting generalization of the Schwarzian derivative for conformal diffeomorphisms in the multi-dimensional case. In the situation (3.12) with F = e 2f , the Schwarzian derivative of ϕ is defined =-=[18, 5]-=- as the symmetric twice-covariant tensor S(ϕ) such that S(ϕ)(X, Y ) = X(Y f) − (∇XY )f − (Xf)(Y f) + 1 2 ‖df‖2 g g(X, Y ) (3.13) for any X, Y ∈ Vect(M). In our notation, it reads S(ϕ) = 1 2F 3 1 ∇dF −... |

2 |
Closed algebras of differential operators
- Kirillov
- 1996
(Show Context)
Citation Context ...2 is true: our conjecture is that (sl(2, R)⋉h1) ! = U(e(p, q)) for n ≥ 3; in other words is it true that U(e(p, q)) !! = U(e(p, q)) ? Similar problems have recently been investigated by A.A. Kirillov =-=[11]-=-. 216 Equation characterizing conformal equivariance 6.1 Equivariance with respect to the affine subalgebra We first consider, for the sake of completeness, the case of the whole affine Lie subalgebr... |

2 |
Quantization and representation theory, in “Proc
- Blattner
- 1974
(Show Context)
Citation Context ...is the Schwarzian derivative and ϕ the diffeomorphism which defines the metric g = ϕ∗ (dx2). (d) The special module D1 1 , has been introduced in the context of geometric 2 2 quantization by Blattner =-=[2]-=- and Kostant [10]. This module will also naturally arise in our quantization procedure. equals ∆g − 1 2g 2.3 The modules Fλ and Dλ,µ as deformations If M is orientable, Fλ can be identified with C∞ (M... |