JLO cocycle

{{short description|Cocycle in an entire cyclic cohomology group}}

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 $\mathcal{A}$ 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 de Rham cohomology contains the information about the topology of a conventional manifold.{{cite arXiv |last=Jaffe |first=Arthur |title=Quantum Harmonic Analysis and Geometric Invariants |date=1997-09-08 |eprint=physics/9709011 }}{{Cite book |last=Higson |first=Nigel |url=http://www.math.psu.edu/higson/Slides/trieste4.pdf |title=K-Theory and Noncommutative Geometry |date=2002 |publisher=Penn State University |pages=Lecture 4|archive-url=https://web.archive.org/web/20100624222022/http://www.math.psu.edu/higson/Slides/trieste4.pdf |archive-date=2010-06-24 }}

The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a $\theta$ -summable spectral triple (also known as a $\theta$ -summable Fredholm module). It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.{{Cite journal |last1=Jaffe |first1=Arthur |last2=Lesniewski |first2=Andrzej |last3=Osterwalder |first3=Konrad |date=1988 |title=Quantum $K$-theory. I. The Chern character |url=https://projecteuclid.org/journals/communications-in-mathematical-physics/volume-118/issue-1/Quantum-K-theory-I-The-Chern-character/cmp/1104161905.full |journal=Communications in Mathematical Physics |volume=118 |issue=1 |pages=1–14 |doi=10.1007/BF01218474 |bibcode=1988CMaPh.118....1J |issn=0010-3616|url-access=subscription }}

<math>\theta</math>-summable spectral triples

The input to the JLO construction is a $\theta$ -summable spectral triple. These triples 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 $\mathcal{H}$ , $\mathcal{H}=\mathcal{H}_0\oplus\mathcal{H}_1$ . We assume that the algebra $\mathcal{A}$ is even under the $\mathbb{Z}_2$ -grading, i.e. $a\gamma=\gamma a$ , for all $a\in\mathcal{A}$ .

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

:(ii) Each $a\in\mathcal{A}$ maps the domain of $D$ , $\mathrm{Dom}\left(D\right)$ into itself, and the operator $\left[D,a\right]:\mathrm{Dom}\left(D\right)\to\mathcal{H}$ is bounded.

:(iii) $\mathrm{tr}\left(e^{-tD^2}\right)<\infty$ , for all $t>0$ .

A classic example of a $\theta$ -summable spectral triple arises as follows. Let $M$ be a compact spin manifold, $\mathcal{A}=C^\infty\left(M\right)$ , 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

Given a $\theta$ -summable spectral triple, the JLO cocycle $\Phi_t\left(D\right)$ associated to the triple is a sequence

: $\Phi_t\left(D\right)=\left(\Phi_t^0\left(D\right),\Phi_t^2\left(D\right),\Phi_t^4\left(D\right),\ldots\right)$

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

: $\Phi_t^0\left(D\right)\left(a_0\right)=\mathrm{tr}\left(\gamma a_0 e^{-tD^2}\right),$

: $\Phi_t^n\left(D\right)\left(a_0,a_1,\ldots,a_n\right)=\int_{0\leq s_1\leq\ldots s_n\leq t}\mathrm{tr}\left(\gamma a_0 e^{-s_1 D^2}\left[D,a_1\right]e^{-\left(s_2-s_1\right)D^2}\ldots\left[D,a_n\right]e^{-\left(t-s_n\right)D^2}\right)ds_1\ldots ds_n,$

for $n=2,4,\dots$ . The cohomology class defined by $\Phi_t\left(D\right)$ is independent of the value of $t$

References

Category:Noncommutative geometry

JLO cocycle

<math>\theta</math>-summable spectral triples

The cocycle

See also

References