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 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 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 -summable spectral triple (also known as a -summable Fredholm module). It was first introduced in a 1988 paper by Jaffe, Lesniewski, and Osterwalder.[3]

-summable spectral triples

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The input to the JLO construction is a  -summable spectral triple. These triples 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 spectral triple 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

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Given a  -summable spectral triple, the JLO cocycle   associated to the triple is a sequence

 

of functionals on the algebra  , where

 
 

for  . The cohomology class defined by   is independent of the value of  

See also

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References

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  1. ^ Jaffe, Arthur (1997-09-08). "Quantum Harmonic Analysis and Geometric Invariants". arXiv:physics/9709011.
  2. ^ Higson, Nigel (2002). K-Theory and Noncommutative Geometry (PDF). Penn State University. pp. Lecture 4. Archived from the original (PDF) on 2010-06-24.
  3. ^ Jaffe, Arthur; Lesniewski, Andrzej; Osterwalder, Konrad (1988). "Quantum $K$-theory. I. The Chern character". Communications in Mathematical Physics. 118 (1): 1–14. Bibcode:1988CMaPh.118....1J. doi:10.1007/BF01218474. ISSN 0010-3616.