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 ${\displaystyle {\mathcal {A}}}$ of "functions" on the putative noncommutative space. The cyclic cohomology of the algebra ${\displaystyle {\mathcal {A}}}$ 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.^[1][2]

The JLO cocycle is associated with a metric structure of non-commutative differential geometry known as a ${\displaystyle \theta }$-summable spectral triple (also known as a ${\displaystyle \theta }$-summable Fredholm module). It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.^[3]

${\displaystyle \theta }$-summable spectral triples

The input to the JLO construction is a ${\displaystyle \theta }$-summable spectral triple. These triples consists of the following data:

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

(b) A ${\displaystyle \mathbb {Z} _{2}}$-grading ${\displaystyle \gamma }$ on ${\displaystyle {\mathcal {H}}}$, ${\displaystyle {\mathcal {H}}={\mathcal {H}}_{0}\oplus {\mathcal {H}}_{1}}$. We assume that the algebra ${\displaystyle {\mathcal {A}}}$ is even under the ${\displaystyle \mathbb {Z} _{2}}$-grading, i.e. ${\displaystyle a\gamma =\gamma a}$, for all ${\displaystyle a\in {\mathcal {A}}}$.

(c) A self-adjoint (unbounded) operator ${\displaystyle D}$, called the Dirac operator such that

(i) ${\displaystyle D}$ is odd under ${\displaystyle \gamma }$, i.e. ${\displaystyle D\gamma =-\gamma D}$.

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

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

A classic example of a ${\displaystyle \theta }$-summable spectral triple arises as follows. Let ${\displaystyle M}$ be a compact spin manifold, ${\displaystyle {\mathcal {A}}=C^{\infty }\left(M\right)}$, the algebra of smooth functions on ${\displaystyle M}$, ${\displaystyle {\mathcal {H}}}$ the Hilbert space of square integrable forms on ${\displaystyle M}$, and ${\displaystyle D}$ the standard Dirac operator.

The cocycle

Given a ${\displaystyle \theta }$-summable spectral triple, the JLO cocycle ${\displaystyle \Phi _{t}\left(D\right)}$ associated to the triple is a sequence

${\displaystyle \Phi _{t}\left(D\right)=\left(\Phi _{t}^{0}\left(D\right),\Phi _{t}^{2}\left(D\right),\Phi _{t}^{4}\left(D\right),\ldots \right)}$

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

${\displaystyle \Phi _{t}^{0}\left(D\right)\left(a_{0}\right)=\mathrm {tr} \left(\gamma a_{0}e^{-tD^{2}}\right),}$

${\displaystyle \Phi _{t}^{n}\left(D\right)\left(a_{0},a_{1},\ldots ,a_{n}\right)=\int _{0\leq s_{1}\leq \ldots s_{n}\leq t}\mathrm {tr} \left(\gamma a_{0}e^{-s_{1}D^{2}}\left[D,a_{1}\right]e^{-\left(s_{2}-s_{1}\right)D^{2}}\ldots \left[D,a_{n}\right]e^{-\left(t-s_{n}\right)D^{2}}\right)ds_{1}\ldots ds_{n},}$

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

See also

• Cyclic homology
• Chern class
• Arthur Jaffe

References