The action functional in non-commutative geometry

We establish the equality between the restriction of the Adler-Manin-Wodzicki residue or non-commutative residue to pseudodifferential operators of order –n on ann-dimensional compact manifoldM, with the trace which J. Dixmier constructed on the Macaev ideal. We then use the latter trace to recover the Yang Mills interaction in the context of non-commutative differential geometry.     

Noncommutative Finite Dimensional Manifolds II: Moduli Space and Structure of Noncommutative 3-Spheres

This paper contains detailed proofs of our results on the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and a general theory which creates a bridge between noncommutative differential geometry and its purely algebraic counterpart. It allows to construct a morphism from an involutive quadratic algebra to a C*-algebra constructed from the characteristic variety and the hermitian line bundle associated to the central quadratic form. We apply the general theory in the case of noncommutative 3-spheres and show that the above morphism corresponds to a natural ramified covering by a noncommutative 3-dimensional nilmanifold. We then compute the Jacobian of the ramified covering and obtain the answer as the product of a period (of an elliptic integral) by a rational function. We describe the real and complex moduli spaces of noncommutative 3-spheres, relate the real one to root systems and the complex one to the orbits of a birational cubic automorphism of three dimensional projective space. We classify the algebras and establish duality relations between them.     

From Local Perturbation Theory to Hopf and Lie Algebras of Feynman Graphs

We review the algebraic structures imposed on the renormalization procedure in terms of Hopf and Lie algebras of Feynman graphs, and exhibit the connection to diffeomorphisms of physical observables.     

Quasi homomorphismes,cohomologie cyclique et positivité

We show that cyclic cohomology of an algebraA is obtained from traces with suitable domains on the algebraqA of the second author. WhenA is aC* algebra so isqA and the notion of positive trace makes sense. We hence get a notion of positivity for cyclic cocycles. We prove that a positive trace onqA defines a type I or II Fredholm module onA.     

Geometry from the spectral point of view

In this Letter, we develop geometry from a spectral point of view, the geometric data being encoded by a triple (A. H. D.) of an algebraA represented in a Hilbert spaceH with selfadjoint operatorD. This point of view is dictated by the general framework of noncommutative geometry and allows us to use geometric ideas in many situations beyond Riemannian geometry.

This paper is dedicated to the memory of J. Schwinger     

The local index formula in noncommutative geometry

In noncommutative geometry a geometric space is described from a spectral vantage point, as a tripleA, H, D consisting of a *-algebraA represented in a Hilbert spaceH together with an unbounded selfadjoint operatorD, with compact resolvent, which interacts with the algebra in a bounded fashion. This paper contributes to the advancement of this point of view in two significant ways: (1) by showing that any pseudogroup of transformations of a manifold gives rise to such a spectral triple of finite summability degree, and (2) by proving a general, in some sense universal, local index formula for arbitrary spectral triples of finite summability degree, in terms of the Dixmier trace and its residue-type extension.We dedicate this paper to Misha Gromov     

Fourier's three

Perfecting instruments is all in the day's work; introducing truly novel experiments remains a rarer achievement. Still, we have seen over the last thirty years the growth of three highly similar and yet independent Fourier spectroscopies. Who feels ready to stick the fourth feather in Fourier's cap?     

On the Chern character of θ summable Fredholm modules

We show that the entire cyclic cohomology class given by the Jaffe-Lesniewski-Osterwalder formula is the same as the class we had constructed earlier as the Chern character of -summable Fredholm modules.