ar X iv : m at h - ph / 0 20 60 21 v 1 1 3 Ju n 20 02 Quantum Systems Exactly Habilitation

ly as 2-cochain: Fa = ρ ∗ ◦ 1 2 (θ − θ)∂i ∧ ∂k = ρ ◦ 1 2 (f )jlθ θ∂i ∧ ∂k (5.85) with θf = θf , or f ′ = 1 1 + fθ f, (5.86) 5.2. ORDINARY VERSUS NONCOMMUTATIVE GAUGE THEORY 99 which we recognize as the noncommutative field strength (with lower indices) for constant f , θ [128]. The general result for non-constant f , θ is thus simply obtained by the application of the covariantizing map ρ (after raising indices with θ’s). The Seiberg-Witten map in the semiclassical regime for constant θ has previously been discussed in [151, 153], where it was understood as a coordinate redefinition that eliminates fluctuations around a constant background. We will now use Kontsevich’s formality theorem to quantize everything. The goal is to obtain (5.1) in the form (5.68) of which (5.82) is the semiclassical limit. Kontsevich formality map Kontsevich’s formality map is a collection of skew-symmetric multilinear maps Un for n = 0 . . .∞ that map tensor products of n polyvector fields to differential operators. More precisely Un maps the tensor product of n ki-vector fields to an m-differential operator, where m is determined by the matching condition m = 2− 2n+ n ∑