JLO cocycle

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 ${\mathcal {A}}$ 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 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 ${\mathcal {H}}$ 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 $a\in {\mathcal {A}}$.

(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 $D$, $\mathrm {Dom} \left(D\right)$ 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 spectral triple 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

Given a $\theta$-summable spectral triple, the JLO cocycle $\Phi _{t}\left(D\right)$ associated to the triple 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 $t$

See also

• Cyclic homology
• Chern class
• Arthur Jaffe