Local index formula in noncommutative geometry

Statement

Let $(\mathcal A,H,D)$ be a finitely summable regular spectral triple with discrete dimension spectrum and the meromorphic-continuation hypotheses of Connes and Moscovici. The local index formula states that the periodic cyclic Chern character of the bounded transform of $D$ is represented by a finite $(b,B)$ -cocycle whose components are linear combinations of residues at specified poles of

\operatorname{Tr}_{s}\!\left( a_0[D,a_1]^{(k_1)}\cdots[D,a_n]^{(k_n)} |D|^{-n-2|k|-z}\right).

Here $T^{(k)}$ denotes the $k$ -fold commutator with $D^2$ , and $\operatorname{Tr}_{s}$ is the parity-appropriate trace.

Hypotheses and notation

Finite summability controls which cochain degrees and multi-indices can contribute. Regularity supplies the abstract pseudodifferential calculus needed to expand products and resolvents. The dimension-spectrum hypothesis provides meromorphic continuation of the weighted zeta functions whose residues occur in the formula. None of these three hypotheses alone implies the other two.

The displayed expression suppresses universal coefficients involving factorials and gamma functions. Their exact form depends on whether one uses the even or odd cocycle, how $D$ 's kernel is removed, and the normalization of the $(b,B)$ -complex. The complete formulas appear in Connes–Moscovici, Theorems II.1 and II.2.

Meaning of locality

The bounded-transform character is global: it involves the phase of $D$ and ordinary operator traces. The residue cocycle instead uses finitely many iterated commutators and coefficients extracted from spectral asymptotics. For classical Dirac operators these residues are integrals of local symbolic expressions, recovering the local character of the Atiyah–Singer index density.

In the abstract setting, “local” means residue-local relative to the pseudodifferential calculus of the triple. It does not assert that an underlying point-set space or coordinate neighborhood exists.

Consequences and use

Because the residue cocycle represents the Chern character of the Fredholm module, pairing it with $K$ -theory computes the same Fredholm index. The theorem therefore turns an index defined by a compressed operator into a finite sum of residues that can often be calculated from heat-kernel or symbol asymptotics. A detailed derivation of this cohomological replacement is given in Higson, §§5–7.

Conventions and scope

The theorem is an equality of cyclic-cohomology classes, not generally an identity between one chosen residue cochain and one chosen bounded character cochain. They can differ by a $(b,B)$ -coboundary.

References

A. Connes and H. Moscovici, “The Local Index Formula in Noncommutative Geometry,” Geometric and Functional Analysis 5 (1995), 174–243. DOI record. Relevant: §§II–III, especially Theorems II.1 and II.2 and the residue cocycle.
N. Higson, “The Local Index Formula in Noncommutative Geometry,” in Contemporary Developments in Algebraic K-Theory, ICTP Lecture Notes 15, 2004. Author-hosted manuscript. Relevant: §§5–7 on the residue formula and its identification with the Chern character.