On the Inverse Mapping of the Formal Symplectic Groupoid of a Deformation Quantization

Abstract

Abstract. To each natural star product on a Poisson manifold M we associate an antisymplectic involutive automorphism of the formal neighborhood of the zero section of the cotangent bundle of M . If M is symplectic, this mapping is shown to be the inverse mapping of the formal symplectic groupoid of the star product. The construction of the inverse mapping involves modular automorphisms of the star product.

Formal Symplectic Groupoid of a Deformation Quantization

“…Now, if f = ab = a * b we see that L f = L a * b = L a L b = aL b and R f = R a * b = R b R a = bR a are natural differential operators. Using the same arguments as in Proposition 1 of [15] we can prove the following theorem. Theorem 4 was proved in [3] and [19] in the Kähler case.…”

“…We will prove that the mapping χ : C ∞ (Σ, Λ) → C is actually a bijection. Each element C ∈ C is completely determined by the family of polydifferential operators {C n }, n ≥ 0, on M, where C n is the n-differential operator such that (15) C n (f 1 , . .…”

“…, f n )}, n ≥ 0, of polydifferential operators on M coherent if it has Properties A and B. The correspondence C → {C n } given by formula (15) is a bijection between the space C and the set of all coherent families.…”

“…Symplectic groupoids are semiclassical geometric objects whose heuristic quantum counterparts are associative algebras treated as quantum objects. In [4], [14], and [15] evidence was given that the star algebra of a deformation quantization gives rise to a formal analogue of a symplectic groupoid. In this paper we give a global definition of a formal symplectic groupoid and show that to each natural deformation quantization (in the sense of Gutt and Rawnsley, [10]) there corresponds a canonical formal symplectic groupoid.…”

Symplectic microgeometry III: monoids

“…Moreover, it gives an appropriate categorical framework for Poisson manifold functorial quantization by symplectic (micro)groupoid methods. This framework may also be relevant for functorial aspects of star-product dequantization as in [14,15,16,17].…”