JLO cocycle

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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 \theta-summable Fredholm module (also known as a \theta-summable spectral triple).

\theta-summable Fredholm Modules[edit]

A \theta-summable Fredholm module 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}.

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

(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 Fredholm module 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[edit]

The JLO cocycle \Phi_t\left(D\right) 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.

External links[edit]

  • [1] - The original paper introducing the JLO cocycle.
  • [2] - A nice set of lectures.
  • [3] - A comprehensive account of noncommutative geometry by its creator.