Deformation theory and quantization. I. Deformations of symplectic structures

Bayen, F.; Flato, M.; Frønsdal, C.; Lichnerowicz, André; Sternheimer, Daniel

doi:10.1016/0003-4916(78)90224-5

Cited by 1,332 publications

(893 citation statements)

References 16 publications

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“…There is no a priori reference to states, or any probabilistic interpretation, so the quantum theory presents a smooth deviation away from the classical Liouville equation [93,94,95,96].…”

Section: Phase-space Approachmentioning

confidence: 99%

Brane partons and singleton strings

2006

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We examine p-branes in AdS D in two limits where they exhibit partonic behavior: rotating branes with energy concentrated to cusp-like solitons; tensionless branes with energy distributed over singletonic partons on the Dirac hypercone. Evidence for a smooth transition from cusps to partons is found. First, each cusp yields D − 2 normal-coordinate bound states with protected frequencies (for p > 2 there are additional bound states); and can moreover be related to a short open p-brane whose tension diverges at the AdS boundary leading to a decoupled singular CFT at the "brane at the end-of-the-universe". Second, discretizing the closed p-brane and keeping the number N of discrete partons finite yields an sp(2N )-gauged phase-space sigma model giving rise to symmetrized Ntupletons of the minimal higher-spin algebra ho 0 (D − 1, 2) ⊃ so(D − 1, 2). The continuum limit leads to a 2d chiral sp(2)-gauged sigma model which is critical in D = 7; equivalentá la Bars-Vasiliev to an su(2)-gauged spinor string; and furthermore dual to a WZW model in turn containing a topological so(6, 2) −2 /( so(6) ⊕ so(2)) −2 coset model with a chiral ring generated by singleton-valued weight-0 spin fields. Moreover, the two-parton truncation can be linked via a reformulationá la Cattaneo-Felder-Kontsevich to a topological open string on the phase space of the D-dimensional Dirac hypercone. We present evidence that a suitable deformation of the open string leads to the Vasiliev equations based on vector oscillators and weak sp(2)-projection. Geometrically, the bi-locality reflects broken boundary-singleton worldlines, while Vasiliev's intertwiner κ can be seen to relate T and R-ordered deformations of the boundary and the bulk of the worldsheet, respectively.

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“…There is no a priori reference to states, or any probabilistic interpretation, so the quantum theory presents a smooth deviation away from the classical Liouville equation [93,94,95,96].…”

Section: Phase-space Approachmentioning

confidence: 99%

Brane partons and singleton strings

2006

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“…A possible alternative to quantize Nambu bracket by deformation quantization [2], [3] was discussed in [20] (see Sect. 2.2 for a brief review on star-products).…”

Section: Difficulties With Usual Quantizationsmentioning

confidence: 99%

“…For completeness we give here a brief review on deformation quantization and star-products; a full treatment can be found in [2], [3] and a recent review in [10]. Let M be a Poisson manifold.…”

Section: Deformation Quantizationmentioning

confidence: 99%

Deformation quantization and Nambu Mechanics

et al. 1997

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Starting from deformation quantization (star-products), the quantization problem of Nambu Mechanics is investigated. After considering some impossibilities and pushing some analogies with field quantization, a solution to the quantization problem is presented in the novel approach of Zariski quantization of fields (observables, functions, in this case polynomials). This quantization is based on the factorization over R of polynomials in several real variables. We quantize the infinite-dimensional algebra of fields generated by the polynomials by defining a deformation of this algebra which is Abelian, associative and distributive. This procedure is then adapted to derivatives (needed for the Nambu brackets), which ensures the validity of the Fundamental Identity of Nambu Mechanics also at the quantum level. Our construction is in fact more general than the particular case considered here: it can be utilized for quite general defining identities and for much more general star-products.

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“…where {·, ·} is the Poisson bracket defined by w. For some earlier results on star products on symplectic manifolds, see Bayen et al [1], De Wilde-Lecomte [5], Fedosov [6]. In general a Z-graded commutative algebra may have a deformation by Z 2 -graded commutative algebras, e.g.…”

Section: Huai-dong Cao and Jian Zhou (Communicated By Richard Schoen)mentioning

confidence: 99%

“…Notice that it now suffices to find the eigenvalues of M h on Λ [0] h,h −1 (V * ) and Λ [1] h,h −1 (V * ). In Cao-Zhou [4], this is done by induction on the dimension of V .…”

Section: Furthermore If We Regard Multiplications By H and H −1 As Omentioning

confidence: 99%

On quantum de Rham cohomology theory

Cao

Zhou

1999

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. We define the quantum exterior product ∧ h and quantum exterior differential d h on Poisson manifolds. The quantum de Rham cohomology, which is a deformation quantization of the de Rham cohomology, is defined as the cohomology of d h . We also define the quantum Dolbeault cohomology. A version of quantum integral on symplectic manifolds is considered and the corresponding quantum Stokes theorem is stated. We also derive the quantum hard Lefschetz theorem. By replacing d by d h and ∧ by ∧ h in the usual definitions, we define many quantum analogues of important objects in differential geometry, e.g. quantum curvature. The quantum characteristic classes are then studied along the lines of the classical Chern-Weil theory. The quantum equivariant de Rham cohomology is defined in the similar fashion.In this note we announce a construction of a deformation of the de Rham complex for any Poisson manifold. In the case of a closed symplectic manifold, its cohomology provides a deformation of the ring structure on the de Rham cohomology. More precisely, on any Poisson manifold, we define a quantum exterior product ∧ h of exterior forms, and quantum exterior differential d h , such that d 2 h = 0, and d h is a derivation for ∧ h . Here h is an indeterminate. We define the quantum de Rham cohomology as the cohomology of d h . Since d h is a derivation with respect to ∧ h , there is an induced quantum multiplication on the quantum de Rham cohomology.This work grew out of our attempt to find a new way to define quantum cohomology, which has recently attracted much attention (see Tian [13] for a survey on this topic, and the introduction of Li-Tian [11] for more recent development). Intuitively, quantum cohomology provides a deformation of the ring structure on the vector space which underlies the de Rham cohomology by counting pseudoholomorphic curves in symplectic manifolds or stable curves in algebraic manifolds. Unlike many cohomology theories in algebraic topology, it is not defined as the cohomology of a graded differential algebra. For the sake of being consistent with other cohomology theories, it would be desirable to be able to do so, even though the applications of quantum cohomology do not really require this property. This is the first motivation for our construction of quantum de Rham cohomology. On the other hand, from the point of view of finding deformations of the ring structure on the de Rham cohomology per se, it is also interesting to see whether deformations of the de Rham complex can provide new deformations of the de Rham cohomology. This is the second motivation to our construction.

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Deformation theory and quantization. I. Deformations of symplectic structures

Cited by 1,332 publications

References 16 publications

Brane partons and singleton strings

Brane partons and singleton strings

Deformation quantization and Nambu Mechanics

On quantum de Rham cohomology theory

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