
JLO cocycle

In noncommutative geometry, the JLO cocycle is a cocycle (and thus defines a cohomology class) in entire cyclic cohomology. 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 formula_1 of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra formula_1 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 formula_3-summable spectral triple (also known as a formula_3-summable Fredholm module).
formula_3-summable spectral triples.
A formula_3-summable spectral triple consists of the following data:
(a) A Hilbert space formula_7 such that formula_1 acts on it as an algebra of bounded operators.
(b) A formula_9-grading formula_10 on formula_7, formula_12. We assume that the algebra formula_1 is even under the formula_9-grading, i.e. formula_15, for all formula_16.
(c) A self-adjoint (unbounded) operator formula_17, called the "Dirac operator" such that
A classic example of a formula_3-summable spectral triple arises as follows. Let formula_28 be a compact spin manifold, formula_29, the algebra of smooth functions on formula_28, formula_7 the Hilbert space of square integrable forms on formula_28, and formula_17 the standard Dirac operator.
The cocycle.
The JLO cocycle formula_34 is a sequence
of functionals on the algebra formula_1, where
for formula_39. The cohomology class defined by formula_34 is independent of the value of formula_41.