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.

The cocycle

The JLO cocycle $\Phi _{t}\left(D\right)$ 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 $t$ .

External links

- The original paper introducing the JLO cocycle.
- A nice set of lectures.
- A comprehensive account of noncommutative geometry by its creator.

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