JLO cocycle

In noncommutative geometry, the Jaffe- Lesniewski-Osterwalder (JLO) cocycle (named after Arthur Jaffe, Andrzej Lesniewski, and Konrad Osterwalder) is a cocycle in an entire cyclic cohomology group. 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 spectral triple (also known as a $$\theta$$-summable Fredholm module). It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.

$$\theta$$-summable spectral triples
The input to the JLO construction is a $$\theta$$-summable spectral triple. These triples 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 $$a\in\mathcal{A}$$.

(c) A self-adjoint (unbounded) operator $$D$$, called the Dirac operator such that


 * (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 spectral triple 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.

The cocycle
Given a $$\theta$$-summable spectral triple, the JLO cocycle $$\Phi_t\left(D\right)$$ associated to the triple is a sequence


 * $$\Phi_t\left(D\right)=\left(\Phi_t^0\left(D\right),\Phi_t^2\left(D\right),\Phi_t^4\left(D\right),\ldots\right)$$

of functionals on the algebra $$\mathcal{A}$$, where


 * $$\Phi_t^0\left(D\right)\left(a_0\right)=\mathrm{tr}\left(\gamma a_0 e^{-tD^2}\right),$$
 * $$\Phi_t^n\left(D\right)\left(a_0,a_1,\ldots,a_n\right)=\int_{0\leq s_1\leq\ldots s_n\leq t}\mathrm{tr}\left(\gamma a_0 e^{-s_1 D^2}\left[D,a_1\right]e^{-\left(s_2-s_1\right)D^2}\ldots\left[D,a_n\right]e^{-\left(t-s_n\right)D^2}\right)ds_1\ldots ds_n,$$

for $$n=2,4,\dots$$. The cohomology class defined by $$\Phi_t\left(D\right)$$ is independent of the value of $$t$$