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 [math]\displaystyle{ \mathcal{A} }[/math] of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra [math]\displaystyle{ \mathcal{A} }[/math] 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 [math]\displaystyle{ \theta }[/math]-summable spectral triple (also known as a [math]\displaystyle{ \theta }[/math]-summable Fredholm module).
[math]\displaystyle{ \theta }[/math]-summable spectral triples
A [math]\displaystyle{ \theta }[/math]-summable spectral triple consists of the following data:
(a) A Hilbert space [math]\displaystyle{ \mathcal{H} }[/math] such that [math]\displaystyle{ \mathcal{A} }[/math] acts on it as an algebra of bounded operators.
(b) A [math]\displaystyle{ \mathbb{Z}_2 }[/math]-grading [math]\displaystyle{ \gamma }[/math] on [math]\displaystyle{ \mathcal{H} }[/math], [math]\displaystyle{ \mathcal{H}=\mathcal{H}_0\oplus\mathcal{H}_1 }[/math]. We assume that the algebra [math]\displaystyle{ \mathcal{A} }[/math] is even under the [math]\displaystyle{ \mathbb{Z}_2 }[/math]-grading, i.e. [math]\displaystyle{ a\gamma=\gamma a }[/math], for all [math]\displaystyle{ a\in\mathcal{A} }[/math].
(c) A self-adjoint (unbounded) operator [math]\displaystyle{ D }[/math], called the Dirac operator such that
- (i) [math]\displaystyle{ D }[/math] is odd under [math]\displaystyle{ \gamma }[/math], i.e. [math]\displaystyle{ D\gamma=-\gamma D }[/math].
- (ii) Each [math]\displaystyle{ a\in\mathcal{A} }[/math] maps the domain of [math]\displaystyle{ D }[/math], [math]\displaystyle{ \mathrm{Dom}\left(D\right) }[/math] into itself, and the operator [math]\displaystyle{ \left[D,a\right]:\mathrm{Dom}\left(D\right)\to\mathcal{H} }[/math] is bounded.
- (iii) [math]\displaystyle{ \mathrm{tr}\left(e^{-tD^2}\right)\lt \infty }[/math], for all [math]\displaystyle{ t\gt 0 }[/math].
A classic example of a [math]\displaystyle{ \theta }[/math]-summable spectral triple arises as follows. Let [math]\displaystyle{ M }[/math] be a compact spin manifold, [math]\displaystyle{ \mathcal{A}=C^\infty\left(M\right) }[/math], the algebra of smooth functions on [math]\displaystyle{ M }[/math], [math]\displaystyle{ \mathcal{H} }[/math] the Hilbert space of square integrable forms on [math]\displaystyle{ M }[/math], and [math]\displaystyle{ D }[/math] the standard Dirac operator.
The cocycle
The JLO cocycle [math]\displaystyle{ \Phi_t\left(D\right) }[/math] is a sequence
- [math]\displaystyle{ \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) }[/math]
of functionals on the algebra [math]\displaystyle{ \mathcal{A} }[/math], where
- [math]\displaystyle{ \Phi_t^0\left(D\right)\left(a_0\right)=\mathrm{tr}\left(\gamma a_0 e^{-tD^2}\right), }[/math]
- [math]\displaystyle{ \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, }[/math]
for [math]\displaystyle{ n=2,4,\dots }[/math]. The cohomology class defined by [math]\displaystyle{ \Phi_t\left(D\right) }[/math] is independent of the value of [math]\displaystyle{ t }[/math].
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