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 ${\mathcal {H}}$, ${\displaystyle {\mathcal {H}}={\mathcal {H}}_{0}\oplus {\mathcal {H}}_{1}}$. We assume that the algebra ${\mathcal {A}}$ is even under the $\mathbb {Z} _{2}$-grading, i.e. ${\displaystyle a\gamma =\gamma a}$, 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. ${\displaystyle D\gamma =-\gamma D}$.

(ii) Each $a\in\mathcal{A}$ maps the domain of $D$, ${\displaystyle \mathrm {Dom} \left(D\right)}$ into itself, and the operator ${\displaystyle \left[D,a\right]:\mathrm {Dom} \left(D\right)\to {\mathcal {H}}}$ is bounded.

(iii) ${\displaystyle \mathrm {tr} \left(e^{-tD^{2}}\right)<\infty }$, for all $t>0$.

A classic example of a $\theta$-summable Fredholm module arises as follows. Let $M$ be a compact spin manifold, ${\displaystyle {\mathcal {A}}=C^{\infty }\left(M\right)}$, 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 ${\displaystyle \Phi _{t}\left(D\right)}$ is a sequence

${\displaystyle \Phi _{t}\left(D\right)=\left(\Phi _{t}^{0}\left(D\right),\Phi _{t}^{2}\left(D\right),\Phi _{t}^{4}\left(D\right),\ldots \right)}$

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

${\displaystyle \Phi _{t}^{0}\left(D\right)\left(a_{0}\right)=\mathrm {tr} \left(\gamma a_{0}e^{-tD^{2}}\right),}$

${\displaystyle \Phi _{t}^{n}\left(D\right)\left(a_{0},a_{1},\ldots ,a_{n}\right)=\int _{0\leq s_{1}\leq \ldots s_{n}\leq t}\mathrm {tr} \left(\gamma a_{0}e^{-s_{1}D^{2}}\left[D,a_{1}\right]e^{-\left(s_{2}-s_{1}\right)D^{2}}\ldots \left[D,a_{n}\right]e^{-\left(t-s_{n}\right)D^{2}}\right)ds_{1}\ldots ds_{n},}$

for ${\displaystyle n=2,4,\dots }$. The cohomology class defined by ${\displaystyle \Phi _{t}\left(D\right)}$ 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.