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 A of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra 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.
The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a θ -summable Fredholm module (also known as a θ -summable spectral triple).
A θ -summable Fredholm module consists of the following data:
(a) A Hilbert space H such that A acts on it as an algebra of bounded operators.
(b) A Z 2 -grading γ on H , H = H 0 ⊕ H 1 . We assume that the algebra A is even under the Z 2 -grading, i.e. a γ = γ a , for all a ∈ A .
(c) A self-adjoint (unbounded) operator D , called the Dirac operator such that
(i)
D is odd under
γ , i.e.
D γ = − γ D .(ii) Each
a ∈ A maps the domain of
D ,
D o m ( D ) into itself, and the operator
[ D , a ] : D o m ( D ) → H is bounded.(iii)
t r ( e − t D 2 ) < ∞ , for all
t > 0 .
A classic example of a θ -summable Fredholm module arises as follows. Let M be a compact spin manifold, A = C ∞ ( M ) , the algebra of smooth functions on M , H the Hilbert space of square integrable forms on M , and D the standard Dirac operator.
The JLO cocycle Φ t ( D ) is a sequence
Φ t ( D ) = ( Φ t 0 ( D ) , Φ t 2 ( D ) , Φ t 4 ( D ) , … ) of functionals on the algebra A , where
Φ t 0 ( D ) ( a 0 ) = t r ( γ a 0 e − t D 2 ) , Φ t n ( D ) ( a 0 , a 1 , … , a n ) = ∫ 0 ≤ s 1 ≤ … s n ≤ t t r ( γ a 0 e − s 1 D 2 [ D , a 1 ] e − ( s 2 − s 1 ) D 2 … [ D , a n ] e − ( t − s n ) D 2 ) d s 1 … d s n , for n = 2 , 4 , … . The cohomology class defined by Φ t ( D ) is independent of the value of t .
