JLO cocycle

In noncommutative geometry, the JLO cocycle is a cocycle (and thus defines a cohomology class) in entire cyclic cohomology. It is a non-commutative version of the classic Chern character of the conventional differential geometry. In noncommutative geometry, the concept of a manifold is replaced by a noncommutative algebra $\mathcal{A}$ of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra $\mathcal{A}$ contains the information about the topology of that noncommutative space, very much as the de Rham cohomology contains the information about the topology of a conventional manifold.

The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a $\theta$ -summable Fredholm module (also known as a $\theta$ -summable spectral triple).

$\theta$ -summable Fredholm Modules

A $\theta$ -summable Fredholm module consists of the following data:

(a) A Hilbert space $\mathcal{H}$ such that $\mathcal{A}$ acts on it as an algebra of bounded operators.

(b) A $\mathbb{Z}_2$ -grading $\gamma$ on $\mathcal{H}$ , $\mathcal{H}=\mathcal{H}_0\oplus\mathcal{H}_1$ . We assume that the algebra $\mathcal{A}$ is even under the $\mathbb{Z}_2$ -grading, i.e. $a\gamma=\gamma a$ , for all $a\in\mathcal{A}$ .

(i)

D

is odd under

\gamma

, i.e.

D\gamma=-\gamma D

(ii) Each

a\in\mathcal{A}

maps the domain of

D

\mathrm{Dom}\left(D\right)

into itself, and the operator

\left[D,a\right]:\mathrm{Dom}\left(D\right)\to\mathcal{H}

is bounded.

(iii)

\mathrm{tr}\left(e^{-tD^2}\right)<\infty

, for all

t>0

A classic example of a $\theta$ -summable Fredholm module arises as follows. Let $M$ be a compact spin manifold, $\mathcal{A}=C^\infty\left(M\right)$ , the algebra of smooth functions on $M$ , $\mathcal{H}$ the Hilbert space of square integrable forms on $M$ , and $D$ the standard Dirac operator.