Differential forms and the noncommutative residue

For a pseudo-differential operator S of order 0 acting on sections of a vector bundle B on a compact manifold M without boundary, we associate a differential form of order dimension of M acting on C∞(M)×C∞(M). This differential form Ωn,S is given in terms of the noncommutative 1-density res([S,f][S,h]). In the particular case of an even-dimensional, compact, conformal manifold without boundary, we study this differential form for the case (B,S)=(H,F), that is, the Fredholm module associated by Connes [A. Connes, Quantized calculus and applications, in: XIth International Congress of Mathematical Physics (Paris, 1994), Internatl. Press, Cambridge, MA, 1995, pp. 15-36] to the manifold M. We give its explicit expression in the flat case and we address a possible approach to the computations for the general case.