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).
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 Cocycle
The JLO cocycle
is a sequence
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/739bda567416aa39e9ffd88b08c6fa3fe5325f92)
of functionals on the algebra
, where
![{\displaystyle \Phi _{t}^{0}\left(D\right)\left(a_{0}\right)=\mathrm {tr} \left(\gamma a_{0}e^{-tD^{2}}\right),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8edcf87ce430384e7a9e549692630fd984702547)
![{\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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd1faa7874ce3542de63eff6922749f2cb46a5e)
for
. The cohomology class defined by
is independent of the value of
.
External links
- [1] - The original paper introducing the JLO cocycle.
- [2] - A nice set of lectures.
- [3] - A comprehensive account of noncommutative geometry by its creator.