Chern character of a Fredholm module - Imported Knowlpedia Content

Definition

Let $(H,\pi,F)$ be a $p$ -summable Fredholm module over a complex algebra $\mathcal A$ . Its Chern character is the class

\operatorname{Ch}(H,\pi,F)\in HP^\varepsilon(\mathcal A)

in periodic cyclic cohomology, where $\varepsilon$ is the module's parity. For every integer $n>p-1$ of parity $\varepsilon$ , the class has a representative obtained, up to the standard degree-dependent normalization, by tracing

\pi(a_0)[F,\pi(a_1)]\cdots[F,\pi(a_n)],

with the grading operator inserted in the even case. Summability makes this product trace class. Representatives in successive admissible degrees correspond under cyclic periodicity.

Why the construction is cohomological

The trace property and $F^2=1$ imply the cyclic symmetry and cocycle identity for the displayed cochain. If one starts with an unnormalized Fredholm module, compact perturbation and normalization produce the same periodic class. Operator homotopies, unitary equivalence, and addition of degenerate modules also leave the class unchanged. Consequently the construction factors through analytic K-homology rather than depending on a chosen cycle representative Connes, Chapter IV, §1, Propositions 1–2.

The constants in the trace formula are chosen so that representatives in degrees $n,n+2,\ldots$ match under Connes's periodicity operator. Omitting them can preserve the cocycle equation but changes the normalization of the index pairing.

Index pairing

Pairing the Chern character with a $K$ -theory class reproduces the Fredholm-module index pairing. In even parity, a projection is paired with the index of the compressed off-diagonal part of $F$ ; in odd parity, a unitary is compressed by the positive spectral projection of $F$ . Thus the cohomological expression is integral on $K$ -theory classes Connes, Chapter IV, §1, Proposition 2.

This equality is the bridge used by local index formulas: a trace cocycle defined from the bounded phase $F$ may be replaced, in the same cyclic cohomology class, by a residue cocycle built from an unbounded operator $D$ .

Conventions and scope

Formulas vary by powers of $2$ , signs, factors of $i$ , and whether a modified trace is used in low degree. These choices are harmless only when the cyclic periodicity and $K$ -theory pairing conventions are changed consistently.

References

A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted text. Relevant: Chapter IV, §1, especially the character formula and Propositions 1–2.
A. Connes, “Non-Commutative Differential Geometry,” Publications Mathématiques de l'IHÉS 62 (1985), 41–144. DOI record. Relevant: §§II.1–II.3 on cyclic cohomology and the Chern character of summable Fredholm modules.