Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2001-01-05
Nucl.Phys. B606 (2001) 647-672
Physics
High Energy Physics
High Energy Physics - Theory
20 pages, no figures
Scientific paper
10.1016/S0550-3213(01)00241-3
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 K\"ahler 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.
Dolgushev Vasily A.
Lyakhovich Simon L.
Sharapov Alexey A.
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