MUB

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 deRham 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 $\mathcal{\mathscr{H}}$ such that $\mathcal{𝒜}$ acts on it as an algebra of bounded operators.

(b) A ${\mathbb{\mathbb{Z}}}_2$-grading $\gamma$ on $\mathcal{\mathscr{H}}$, $\mathcal{\mathscr{H}}={\mathcal{\mathscr{H}}}_0\oplus {\mathcal{\mathscr{H}}}_1$. We assume that the algebra $\mathcal{𝒜}$ is even under the ${\mathbb{\mathbb{Z}}}_2$-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 $D$, $\mathrm{D}\mathrm{o}\mathrm{m}\left(D\right)$ into itself, and the operator $[D,a]:\mathrm{D}\mathrm{o}\mathrm{m}\left(D\right)\to \mathcal{\mathscr{H}}$ is bounded.

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

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

The Cocycle

The JLO cocycle ${\mathrm{\Phi }}_t\left(D\right)$ is a sequence

${\mathrm{\Phi }}_t\left(D\right)=({\mathrm{\Phi }}_t^0\left(D\right),{\mathrm{\Phi }}_t^2\left(D\right),{\mathrm{\Phi }}_t^4\left(D\right),\dots )$

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

${\mathrm{\Phi }}_t^0\left(D\right)\left(a_0\right)=\mathrm{t}\mathrm{r}\left(\gamma a_0{e}^{-t{D}^2}\right),$

${\mathrm{\Phi }}_t^n\left(D\right)(a_0,a_1,\dots ,a_n)=\underset{0\le s_1\le \dots s_n\le t}{\int }\mathrm{t}\mathrm{r}(\gamma a_0{e}^{-s_1{D}^2}[D,a_1]e^{-(s_2-s_1)D^2}\dots [D,a_n]e^{-(t-s_n)D^2})d{s}_1\dots d{s}_n,$

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

External links

• [1] - The original paper introducing the JLO cocycle.
• [2] - A nice set of lectures.

• [3] - A comprehensive account of noncommutative geometry by its creator.