- Source: JLO cocycle
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. In noncommutative geometry, the concept of a manifold is replaced by a noncommutative algebra
A
{\displaystyle {\mathcal {A}}}
of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra
A
{\displaystyle {\mathcal {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
{\displaystyle \theta }
-summable spectral triple (also known as a
θ
{\displaystyle \theta }
-summable Fredholm module). It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.
θ
{\displaystyle \theta }
-summable spectral triples
The input to the JLO construction is a
θ
{\displaystyle \theta }
-summable spectral triple. These triples consists of the following data:
(a) A Hilbert space
H
{\displaystyle {\mathcal {H}}}
such that
A
{\displaystyle {\mathcal {A}}}
acts on it as an algebra of bounded operators.
(b) A
Z
2
{\displaystyle \mathbb {Z} _{2}}
-grading
γ
{\displaystyle \gamma }
on
H
{\displaystyle {\mathcal {H}}}
,
H
=
H
0
⊕
H
1
{\displaystyle {\mathcal {H}}={\mathcal {H}}_{0}\oplus {\mathcal {H}}_{1}}
. We assume that the algebra
A
{\displaystyle {\mathcal {A}}}
is even under the
Z
2
{\displaystyle \mathbb {Z} _{2}}
-grading, i.e.
a
γ
=
γ
a
{\displaystyle a\gamma =\gamma a}
, for all
a
∈
A
{\displaystyle a\in {\mathcal {A}}}
.
(c) A self-adjoint (unbounded) operator
D
{\displaystyle D}
, called the Dirac operator such that
(i)
D
{\displaystyle D}
is odd under
γ
{\displaystyle \gamma }
, i.e.
D
γ
=
−
γ
D
{\displaystyle D\gamma =-\gamma D}
.
(ii) Each
a
∈
A
{\displaystyle a\in {\mathcal {A}}}
maps the domain of
D
{\displaystyle D}
,
D
o
m
(
D
)
{\displaystyle \mathrm {Dom} \left(D\right)}
into itself, and the operator
[
D
,
a
]
:
D
o
m
(
D
)
→
H
{\displaystyle \left[D,a\right]:\mathrm {Dom} \left(D\right)\to {\mathcal {H}}}
is bounded.
(iii)
t
r
(
e
−
t
D
2
)
<
∞
{\displaystyle \mathrm {tr} \left(e^{-tD^{2}}\right)<\infty }
, for all
t
>
0
{\displaystyle t>0}
.
A classic example of a
θ
{\displaystyle \theta }
-summable spectral triple arises as follows. Let
M
{\displaystyle M}
be a compact spin manifold,
A
=
C
∞
(
M
)
{\displaystyle {\mathcal {A}}=C^{\infty }\left(M\right)}
, the algebra of smooth functions on
M
{\displaystyle M}
,
H
{\displaystyle {\mathcal {H}}}
the Hilbert space of square integrable forms on
M
{\displaystyle M}
, and
D
{\displaystyle D}
the standard Dirac operator.
The cocycle
Given a
θ
{\displaystyle \theta }
-summable spectral triple, the JLO cocycle
Φ
t
(
D
)
{\displaystyle \Phi _{t}\left(D\right)}
associated to the triple is a sequence
Φ
t
(
D
)
=
(
Φ
t
0
(
D
)
,
Φ
t
2
(
D
)
,
Φ
t
4
(
D
)
,
…
)
{\displaystyle \Phi _{t}\left(D\right)=\left(\Phi _{t}^{0}\left(D\right),\Phi _{t}^{2}\left(D\right),\Phi _{t}^{4}\left(D\right),\ldots \right)}
of functionals on the algebra
A
{\displaystyle {\mathcal {A}}}
, where
Φ
t
0
(
D
)
(
a
0
)
=
t
r
(
γ
a
0
e
−
t
D
2
)
,
{\displaystyle \Phi _{t}^{0}\left(D\right)\left(a_{0}\right)=\mathrm {tr} \left(\gamma a_{0}e^{-tD^{2}}\right),}
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
,
{\displaystyle \Phi _{t}^{n}\left(D\right)\left(a_{0},a_{1},\ldots ,a_{n}\right)=\int _{0\leq s_{1}\leq \ldots s_{n}\leq t}\mathrm {tr} \left(\gamma a_{0}e^{-s_{1}D^{2}}\left[D,a_{1}\right]e^{-\left(s_{2}-s_{1}\right)D^{2}}\ldots \left[D,a_{n}\right]e^{-\left(t-s_{n}\right)D^{2}}\right)ds_{1}\ldots ds_{n},}
for
n
=
2
,
4
,
…
{\displaystyle n=2,4,\dots }
. The cohomology class defined by
Φ
t
(
D
)
{\displaystyle \Phi _{t}\left(D\right)}
is independent of the value of
t
{\displaystyle t}
See also
Cyclic homology
Chern class
Arthur Jaffe