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.
of "functions" on the putative noncommutative space.
The cyclic cohomology of the algebra
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
It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.
[3] The input to the JLO construction is a
These triples consists of the following data: (a) A Hilbert space
acts on it as an algebra of bounded operators.
, called the Dirac operator such that A classic example of a
-summable spectral triple arises as follows.
be a compact spin manifold,
, the algebra of smooth functions on
the Hilbert space of square integrable forms on
the standard Dirac operator.
-summable spectral triple, the JLO cocycle
associated to the triple is a sequence of functionals on the algebra
The cohomology class defined by