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 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 deRham 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 -summable Fredholm module (also known as a -summable spectral triple).
-summable Fredholm Modules
A -summable Fredholm module consists of the following data:
(a) A Hilbert space such that acts on it as an algebra of bounded operators.
(b) A -grading on , . We assume that the algebra is even under the -grading, i.e. , for all .
(c) A self-adjoint (unbounded) operator , called the Dirac operator such that
- (i) is odd under , i.e. .
- (ii) Each maps the domain of , into itself, and the operator is bounded.
- (iii) , for all .
A classic example of a -summable Fredholm module arises as follows. Let be a compact spin manifold, , the algebra of smooth functions on , the Hilbert space of square integrable forms on , and the standard Dirac operator.
The JLO cocycle is a sequence
of functionals on the algebra , where
for . The cohomology class defined by is independent of the value of .
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