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Innoncommutative geometry, theJaffe- Lesniewski-Osterwalder (JLO) cocycle (named afterArthur Jaffe, Andrzej Lesniewski, andKonrad Osterwalder) is acocycle in an entirecyclic cohomology group. It is a non-commutative version of the classicChern character of the conventionaldifferential geometry. In noncommutative geometry, the concept of a manifold is replaced by a noncommutative algebra${\displaystyle \mathcal{A}}$ of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra${\displaystyle \mathcal{A}}$ contains the information about the topology of that noncommutative space, very much as thede Rham cohomology contains the information about the topology of a conventional manifold.^[1][2]

The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a${\displaystyle \theta }$-summablespectral triple (also known as a${\displaystyle \theta }$-summable Fredholm module). It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.^[3]

The input to the JLO construction is a${\displaystyle \theta }$-summable spectral triple. These triples consists of the following data:

(a) AHilbert space${\displaystyle \mathcal{H}}$ such that${\displaystyle \mathcal{A}}$ acts on it as an algebra of bounded operators.

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

(c) A self-adjoint (unbounded) operator${\displaystyle D}$, called theDirac operator such that

(i)${\displaystyle D}$ is odd under${\displaystyle \gamma }$, i.e.${\displaystyle D\gamma =-\gamma D}$.

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

(iii), for all${\displaystyle t>0}$.

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

Given a${\displaystyle \theta }$-summable spectral triple, the JLO cocycle${\displaystyle {\mathrm{\Phi }}_t\left(D\right)}$ associated to the triple is a sequence

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

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

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

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

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

• Cyclic homology
• Chern class
• Arthur Jaffe

• Noncommutative geometry

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