Connes' tangent groupoid and strict quantization

Cariñena, José F.; Clemente-Gallardo, Jesús; Follana, E.; Gracia‐Bondía, José M.; Rivero, Alejandro; Várilly, Joseph C.

doi:10.1016/s0393-0440(98)00028-x

Cited by 20 publications

(26 citation statements)

References 10 publications

(32 reference statements)

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“…by means of a chart of the form, onto an open neighborhood in Ᏻ of the boundary T M (see [Connes 1994], [Hilsum and Skandalis 1987], [Cariñena et al 1999]). …”

Section: Differentiable Groupoidsmentioning

confidence: 99%

The tangent grupoid of a Heisenberg manifold

Ponge

2006

Pacific J. Math.

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RAPHAËL PONGEAs a step towards proving an index theorem for hypoelliptic operators on Heisenberg manifolds, including for those on CR and contact manifolds, we construct an analogue for Heisenberg manifolds of Connes' tangent groupoid of a manifold. As is well known for a Heisenberg manifold (M, H) the relevant notion of tangent bundle is rather that of a Lie group bundle of graded 2-step nilpotent Lie groups G M. We define the tangent groupoid of (M, H) as a differentiable groupoid Ᏻ H M encoding the smooth deformation of M × M to G M. In particular, this construction makes a crucial use of a refined notion of privileged coordinates and of a tangent-approximation result for Heisenberg diffeomorphisms. IntroductionA somewhat long standing open question is the existence of an index theorem for geometric operators on contact and CR manifolds. In this context the operators are not elliptic, so we cannot apply the classical index theorem of Atiyah-Singer [1968a;1968b]. The natural pseudodifferential tool to deal with hypoelliptic operators on contact and CR manifolds is provided by the Heisenberg calculus of BealsGreiner [1988] and Taylor [1984]. The latter holds in full generality for Heisenberg manifolds, that is, manifolds M together with a distinguished hyperplane bundle H ⊂ T M. This definition includes that of CR and contact manifolds, as well as that of codimension one foliations and confoliations. Therefore, what we would like to have is an analogue of the Atiyah-Singer theorem for hypoelliptic operators on Heisenberg manifolds.There are various proofs of the Atiyah-Singer index theorem. A simple and fairly general proof is that of Connes [1994, Sect. II.5]. A salient feature in Connes' proof is the use of the tangent groupoid of a manifold, that is, the differentiable groupoid encoding the smooth deformation of M × M to T M (see [Connes 1994;Hilsum and Skandalis 1987]).In this paper, as a step towards proving an index theorem for hypoelliptic operators on Heisenberg manifolds, we construct an analogue for Heisenberg manifolds MSC2000: primary 58H05; secondary 53C10, 53D10, 32V05.

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“…by means of a chart of the form, onto an open neighborhood in Ᏻ of the boundary T M (see [Connes 1994], [Hilsum and Skandalis 1987], [Cariñena et al 1999]). …”

Section: Differentiable Groupoidsmentioning

confidence: 99%

The tangent grupoid of a Heisenberg manifold

Ponge

2006

Pacific J. Math.

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“…In this way in Section 4 we show that the F ⋆ product can be produced by a convolution over T R n . This convolution is generated by a version of the Connes' tangential groupoid [26,27,28] but with an additional rapidly oscillating factor represented by the electromagnetic flux. Because of the rapid oscillations, this type of star-product is outside the framework of the formal deformation quantization method.…”

Section: Introduction and Overviewmentioning

confidence: 99%

Symplectic areas, quantization, and dynamics in electromagnetic fields

Karasev

Osborn

2002

Journal of Mathematical Physics

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A gauge invariant quantization in a closed integral form is developed over a linear phase space endowed with an inhomogeneous Faraday electromagnetic tensor. An analog of the Groenewold product formula (corresponding to Weyl ordering) is obtained via a membrane magnetic area, and extended to the product of N symbols. The problem of ordering in quantization is related to different configurations of membranes: a choice of configuration determines a phase factor that fixes the ordering and controls a symplectic groupoid structure on the secondary phase space. A gauge invariant solution of the quantum evolution problem for a charged particle in an electromagnetic field is represented in an exact continual form and in the semiclassical approximation via the area of dynamical membranes.

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“…The integration in (3.12) is taken with respect to the measure dm r(n ′ ) = dm l (n ′′ ) over the manifold M. Formula (3.12) belongs to the class of Connes' type tangential groupoid quantization formulas [27,31,4]. Note that in the convolution integrand (3.12) we have an additional groupoid cocycle…”

Section: Mmentioning

confidence: 99%

“…Our formulas (3.5), (3.9) in this case are similar to formulas (7) from [31], but a non-trivial recalculation of the Jacobian in the cochain (3.6) is required in order to bring it into the form used in [31].…”

Section: Whose Momentum Fourier Image Belongs To D(t M)mentioning

confidence: 99%

“…The recent mathematical investigation of this problem has been carried out in [25][26][27][28][29][30][31][32][33] including the case where the metric and the magnetic field are both present on M [25,26,33]. There is also a large literature, beginning with the paper by Widom [34], where this problem was studied from the perspective of pseudodifferential and Fourier integral operators.…”

Section: Introductionmentioning

confidence: 99%

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Cotangent bundle quantization: entangling of metric and magnetic field

Karasev

Osborn

2005

J. Phys. A: Math. Gen.

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For manifolds M of noncompact type endowed with an affine connection (for example, the Levi-Civita connection) and a closed 2-form (magnetic field) we define a Hilbert algebra structure in the space L 2 (T * M) and construct an irreducible representation of this algebra in L 2 (M). This algebra is automatically extended to polynomial in momenta functions and distributions. Under some natural conditions this algebra is unique. The non-commutative product over T * M is given by an explicit integral formula. This product is exact (not formal) and is expressed in invariant geometrical terms. Our analysis reveals this product has a front, which is described in terms of geodesic triangles in M. The quantization of δ-functions induces a family of symplectic reflections in T * M and generates a magneto-geodesic connection Γ on T * M. This symplectic connection entangles, on the phase space level, the original affine structure on M and the magnetic field. In the classical approximation, the 2 -part of the quantum product contains the Ricci curvature of Γ and a magneto-geodesic coupling tensor.

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Connes' tangent groupoid and strict quantization

Cited by 20 publications

References 10 publications

The tangent grupoid of a Heisenberg manifold

The tangent grupoid of a Heisenberg manifold

Symplectic areas, quantization, and dynamics in electromagnetic fields

Cotangent bundle quantization: entangling of metric and magnetic field

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