Non-commutative Differential Calculus and the Axial Anomaly in Abelian Lattice Gauge Theories

Details Non-commutative Differential Calculus and the Axial Anomaly in Abelian Lattice Gauge Theories Non-commutative Differential Calculus and the Axial Anomaly in Abelian Lattice Gauge Theories

1999-06-17

arxiv.org/abs/hep-lat/9906015v6

Nucl.Phys. B569 (2000) 643-660

Physics

High Energy Physics

High Energy Physics - Lattice

uses PHYZZX, 28 pages, the final version to appear in Nucl. Phys. B

Scientific paper

10.1016/S0550-3213(99)00706-3

The axial anomaly in lattice gauge theories has a topological nature when the Dirac operator satisfies the Ginsparg-Wilson relation. We study the axial anomaly in Abelian gauge theories on an infinite hypercubic lattice by utilizing cohomological arguments. The crucial tool in our approach is the non-commutative differential calculus~(NCDC) which makes the Leibniz rule of exterior derivatives valid on the lattice. The topological nature of the ``Chern character'' on the lattice becomes manifest in the context of NCDC. Our result provides an algebraic proof of L\"uscher's theorem for a four-dimensional lattice and its generalization to arbitrary dimensions.

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