Ginsparg-Wilson Relation, Topological Invariants and Finite Noncommutative Geometry

Details Ginsparg-Wilson Relation, Topological Invariants and Finite Noncommutative Geometry Ginsparg-Wilson Relation, Topological Invariants and Finite Noncommutative Geometry

2002-09-26

arxiv.org/abs/hep-th/0209223v3

Phys.Rev. D67 (2003) 085005

Physics

High Energy Physics

High Energy Physics - Theory

Revtex4 file, 5 pages, references added, typo corrected, the final version to appear in Phys.Rev.D

Scientific paper

10.1103/PhysRevD.67.085005

We show that the Ginsparg-Wilson (GW) relation can play an important role to define chiral structures in {\it finite} noncommutative geometries. Employing GW relation, we can prove the index theorem and construct topological invariants even if the system has only finite degrees of freedom. As an example, we consider a gauge theory on a fuzzy two-sphere and give an explicit construction of a noncommutative analog of the GW relation, chirality operator and the index theorem. The topological invariant is shown to coincide with the 1st Chern class in the commutative limit.

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