JLO cocycle

This article includes a list of references, related reading, or external links, but its sources remain unclear because it lacks inline citations. Please help improve this article by introducing more precise citations. (November 2018) (Learn how and when to remove this message)

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 ${\displaystyle {\mathcal {A}}}$ 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 the de 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 ${\displaystyle \theta }$-summable spectral triple (also known as a ${\displaystyle \theta }$-summable Fredholm module). It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.^[3]