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

Abstract

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 Lüscher's theorem for a four-dimensional lattice and its generalization to arbitrary dimensions.

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

References in corpus (2)

• Axial Anomaly in Lattice Abelian Gauge Theory in Arbitrary Dimensions
• Relation $Tr{\gamma }_5=0$ and the index theorem in lattice gauge theory

Cited by in corpus (22)

• Various Super Yang-Mills Theories with Exact Supersymmetry on the Lattice
• Lattice Chiral Gauge Theories
• Anomaly Cancellation Condition in Lattice Gauge Theory
• Gauge anomaly cancellations in SU(2)_L \times U(1)_Y Electroweak theory on the lattice
• Dirac-Kaehler Fermion from Clifford Product with Noncommutative Differential Form on a Lattice
• Noncompact chiral U(1) gauge theories on the lattice
• Domain wall fermion and chiral gauge theories on the lattice with exact gauge invariance
• Axial anomaly with the overlap-Dirac operator in arbitrary dimensions
• Global obstructions to gauge-invariance in chiral gauge theory on the lattice
• Gauge fixing, families index theory, and topological features of the space of lattice gauge fields
• Differential Geometry of Group Lattices
• Solving the local cohomology problem in U(1) chiral gauge theories within a finite lattice
• Chiral anomalies in the reduced model
• A simple construction of fermion measure term in U(1) chiral lattice gauge theories with exact gauge invariance
• More about the axial anomaly on the lattice
• A construction of the Glashow-Weinberg-Salam model on the lattice with exact gauge invariance
• Optimised Dirac Operators on the Lattice: Construction, Properties and Applications
• A numerical solution to the local cohomology problem in U(1) chiral gauge theories
• Anomalous gauge theories revisited
• Axial anomaly in the reduced model: Higher representations
• Spectrum of the Hermitian Wilson-Dirac Operator for a Uniform Magnetic Field in Two Dimensions
• Chern-Simons Type Characteristic Classes of Abelian Lattice Gauge Theory