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
.
