JLO cocycle

Short description: Cocycle in an entire cyclic cohomology group

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 ${𝒜}$ of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra ${𝒜}$ 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 ${\mathscr{H}}$ such that ${𝒜}$ acts on it as an algebra of bounded operators.

(b) A ${\mathbb{\mathbb{Z}}}_2$-grading $\gamma$ on ${\mathscr{H}}$, ${\mathscr{H}}={{\mathscr{H}}}_0\oplus {{\mathscr{H}}}_1$. We assume that the algebra ${𝒜}$ is even under the ${\mathbb{\mathbb{Z}}}_2$-grading, i.e. $a\gamma =\gamma a$, for all $a\in {𝒜}$.

(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 $a\in {𝒜}$ maps the domain of $D$, $\mathrm{D}\mathrm{o}\mathrm{m}\left(D\right)$ into itself, and the operator $[D,a]:\mathrm{D}\mathrm{o}\mathrm{m}\left(D\right)\to {\mathscr{H}}$ 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 spectral triple arises as follows. Let $M$ be a compact spin manifold, ${𝒜}=C^{\mathrm{\infty }}\left(M\right)$, the algebra of smooth functions on $M$, ${\mathscr{H}}$ 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 ${\mathrm{\Phi }}_t\left(D\right)$ associated to the triple 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 ${𝒜}$, 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 $t$

See also

• Cyclic homology
• Chern class
• Arthur Jaffe