Probability distribution of the index in gauge theory on 2d non-commutative geometry

Abstract

We investigate the effects of non-commutative geometry on the topological aspects of gauge theory using a non-perturbative formulation based on the twisted reduced model. The configuration space is decomposed into topological sectors labeled by the index nu of the overlap Dirac operator satisfying the Ginsparg-Wilson relation. We study the probability distribution of nu by Monte Carlo simulation of the U(1) gauge theory on 2d non-commutative space with periodic boundary conditions. In general the distribution is asymmetric under nu -> -nu, reflecting the parity violation due to non-commutative geometry. In the continuum and infinite-volume limits, however, the distribution turns out to be dominated by the topologically trivial sector. This conclusion is consistent with the instanton calculus in the continuum theory. However, it is in striking contrast to the known results in the commutative case obtained from lattice simulation, where the distribution is Gaussian in a finite volume, but the width diverges in the infinite-volume limit. We also calculate the average action in each topological sector, and provide deeper understanding of the observed phenomenon.

16 pages,10 figures, version appeared in JHEP

References in corpus (7)

• Dynamical generation of fuzzy extra dimensions, dimensional reduction and symmetry breaking
• A non-perturbative study of 4d U(1) non-commutative gauge theory -- the fate of one-loop instability
• Fermions on spontaneously generated spherical extra dimensions
• Higher Order Terms of Improved Mean Field Approximation for IIB Matrix Model and Emergence of Four-dimensional Space-time
• The index of the overlap Dirac operator on a discretized 2d non-commutative torus
• Stability of 4-dimensional Space-time from IIB Matrix Model via Improved Mean Field Approximation
• Noncommutative perturbative quantum field theory: Wilson loop in two-dimensional Yang-Mills, and unitarity from string theory

Cited by in corpus (10)

• Ginsparg-Wilson Dirac operator in the monopole backgrounds on the fuzzy 2-sphere
• Higher-Dimensional Unified Theories with Fuzzy Extra Dimensions
• Index theorem in spontaneously symmetry-broken gauge theories on a fuzzy 2-sphere
• Wilson Loops in 2D Noncommutative Euclidean Gauge Theory: 2. 1/θExpansion
• Finite-matrix formulation of gauge theories on a non-commutative torus with twisted boundary conditions
• Probability of the Standard Model Appearance from a Matrix Model
• Construction of a topological charge on fuzzy S^2 x S^2 via Ginsparg-Wilson relation
• Ginsparg-Wilson relation on a fuzzy 2-sphere for adjoint matter
• Dominance of a single topological sector in gauge theory on non-commutative geometry
• The instability of intersecting fuzzy spheres