JLO cocycle

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 $mathcal\{A\}$ of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra $mathcal\{A\}$ 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 $heta$-summable "Fredholm module".

$heta$-summable Fredholm Modules

A $heta$-summable Fredholm module (also known as a $heta$-summable "spectral triple") consists of the following data:

(a) A Hilbert space $mathcal\{H\}$ such that $mathcal\{A\}$ acts on it as an algebra of bounded operators.

(b) A $mathbb\{Z\}\_2$-grading $gamma$ on $mathcal\{H\}$ , $mathcal\{H\}=mathcal\{H\}\_0oplusmathcal\{H\}\_1$. We assume that the algebra $mathcal\{A\}$ is even under the $mathbb\{Z\}\_2$-grading, i.e. $agamma=gamma\; a$, for all $ainmathcal\{A\}$.

(c) A self-adjoint (unbounded) operator $D$, called the "Dirac operator" such that

:(i) $D$ is odd under $gamma$, i.e. $Dgamma=-gamma\; D$.

:(ii) Each $ainmathcal\{A\}$ maps the domain of $D$, $mathrm\{Dom\}left(D\; ight)$ into itself, and the operator $left\; [D,a\; ight]\; :mathrm\{Dom\}left(D\; ight)\; omathcal\{H\}$ is bounded.

:(iii)

A classic example of a $heta$-summable Fredholm module arises as follows. Let $M$ be a compact spin manifold, $mathcal\{A\}=C^inftyleft(M\; ight)$, the algebra of smooth functions on $M$, $mathcal\{H\}$ the Hilbert space of square integrable forms on $M$, and $D$ the standard Dirac operator.

The Cocycle

The JLO cocycle $Phi\_tleft(D\; ight)$ is a sequence

:$Phi\_tleft(D\; ight)=left(Phi\_t^0left(D\; ight),Phi\_t^2left(D\; ight),Phi\_t^4left(D\; ight),ldots\; ight)$

of functionals on the algebra $mathcal\{A\}$, where

:$Phi\_t^0left(D\; ight)left(a\_0\; ight)=mathrm\{tr\}left(gamma\; a\_0\; e^\{-tD^2\}\; ight),$:$Phi\_t^nleft(D\; ight)left(a\_0,a\_1,ldots,a\_n\; ight)=int\_\{0leq\; s\_1leqldots\; s\_nleq\; t\}mathrm\{tr\}left(gamma\; a\_0\; e^\{-s\_1\; D^2\}left\; [D,a\_1\; ight]\; e^\{-left(s\_2-s\_1\; ight)D^2\}ldotsleft\; [D,a\_n\; ight]\; e^\{-left(t-s\_n\; ight)D^2\}\; ight)ds\_1ldots\; ds\_n,$

for $n=2,4,dots$. The cohomology class defined by $Phi\_tleft(D\; ight)$ is independent of the value of $t$.

External links

* [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104161905] - The original paper introducing the JLO cocycle.
* [http://www.math.psu.edu/higson/Slides/trieste4.pdf] - A nice set of lectures.

* [ftp://ftp.alainconnes.org/book94bigpdf.pdf] - A comprehensive account of noncommutative geometry by its creator.