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Wick type deformation quantization of Fedosov manifolds | | V.A. Dolgushev
; S.L. Lyakhovich
; A.A. Sharapov
; | | Date: |
5 Dec 2000 | | Journal: | Nucl.Phys. B606 (2001) 647-672 | | Subject: | High Energy Physics - Theory; Quantum Algebra | hep-th math.QA | | Affiliation: | Tomsk State Univ. | | Abstract: | A coordinate-free definition for Wick-type symbols is given for symplectic manifolds by means of the Fedosov procedure. The main ingredient of this approach is a bilinear symmetric form defined on the complexified tangent bundle of the symplectic manifold and subject to some set of algebraic and differential conditions. It is precisely the structure which describes a deviation of the Wick-type star-product from the Weyl one in the first order in the deformation parameter. The geometry of the symplectic manifolds equipped by such a bilinear form is explored and a certain analogue of the Newlander-Nirenberg theorem is presented. The 2-form is explicitly identified which cohomological class coincides with the Fedosov class of the Wick-type star-product. For the particular case of Kähler manifold this class is shown to be proportional to the Chern class of a complex manifold. We also show that the symbol construction admits canonical superextension, which can be thought of as the Wick-type deformation of the exterior algebra of differential forms on the base (even) manifold. Possible applications of the deformed superalgebra to the noncommutative field theory and strings are discussed. | | Source: | arXiv, hep-th/0101032 | | Services: | Forum | Review | PDF | Favorites | |
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