Some Conformal Invariants from the Noncommutative Residue for Manifolds with Boundary

Abstract

We review previous work of Alain Connes, and its extension by the author, on some conformal invariants obtained from the noncommutative residue on even dimensional compact manifolds without boundary. Inspired by recent work of Yong Wang, we also address possible generalizations of these conformal invariants to the setting of compact manifolds with boundary.Furthermore, in the 4-dimensional case, Connes has also shown that the Paneitz operator [14] (critical GJMS for n = 4 [10]), can be derived from B 4 by the relationAiming to extend the work of Connes to even dimensional manifolds, in [18] we have proved the following two results:Theorem 1 of [18]. Let M be an n-dimensional compact conformal manifold without boundary. Let S be a pseudodifferential operator of order 0 acting on sections of a vector bundle over M such that S 2 f 1 = f 1 S 2 and the pseudodifferential operator P = [S, f 1 ][S, f 2

Spectral triples and manifolds with boundary

“…From a physics point of view, applications of noncommutative integrals on manifolds to classical gravity has begun with Connes' remark that D −2 coincides in dimension 4 with Einstein-Hilbert action, a fact recovered in [40,41]. Then, a generalization to manifolds with boundaries was proposed in [53,[55][56][57]. From the quantum side, a noncommutative approach of the unit disk is proposed in [10].…”