On the phase form of a deformation quantization with separation of variables

Abstract

Given a star product with separation of variables on a pseudo-Kähler manifold, we obtain a new formal (1,1)-form from its classifying form and call it the phase form of the star product. The cohomology class of a star product with separation of variables equals the class of its phase form. We show that the phase forms can be arbitrary and they bijectively parametrize the star products with separation of variables. We also describe the action of a change of the formal parameter on a star product with separation of variables, its formal Berezin transform, classifying form, phase form, and canonical trace density.

Deformation quantization with separation of variables on a super-Kähler manifold

“…If ⋆ is a star product with separation of variables on a pseudo-Kähler manifold (M, ω −1 ) with a classifying form ω, one can construct a trace density of the product ⋆ on a contractible coordinate chart U ⊂ M as follows (see [12] and [14]). Let Φ be a formal potential of ω on U.…”

“…Since δ I and δJ are (left) differential operators, it follows from ( 13) and ( 15) that the product (14)…”

Formal oscillatory integrals and deformation quantization

“…Then Λ 0 = αδ x 0 for some α = 0. We have that div ρ ∂ ∂x i = ∂u ∂x i and ( 4) is equivalent to the condition (11) Λ…”

“…We see that the leading term of Λ is Λ0 = Λ 0 = αδ x 0 . It remains to show that Λ satisfies (11). The operators…”

“…is a (not normalized) trace density of the product ⋆ on U. It can be normalized up to a multiplicative constant from C as follows, as shown in [10] (see also [11]). There exists a potential Ψ of the dual form ω that satisfies (25), (26), and the equation ( 28)…”