Small Instantons in $C{P}^1$ and $C{P}^2$ Sigma Models

Abstract

The anomalous scaling behavior of the topological susceptibility ${\chi }_t$ in two-dimensional $C{P}^{N-1}$ sigma models for $N\le 3$ is studied using the overlap Dirac operator construction of the lattice topological charge density. The divergence of ${\chi }_t$ in these models is traced to the presence of small instantons with a radius of order $a$ (= lattice spacing), which are directly observed on the lattice. The observation of these small instantons provides detailed confirmation of Lüscher's argument that such short-distance excitations, with quantized topological charge, should be the dominant topological fluctuations in $C{P}^1$ and $C{P}^2$, leading to a divergent topological susceptibility in the continuum limit. For the $\CP$ models with $N>3$ the topological susceptibility is observed to scale properly with the mass gap. These larger $N$ models are not dominated by instantons, but rather by coherent, one-dimensional regions of topological charge which can be interpreted as domain wall or Wilson line excitations and are analogous to D-brane or ``Wilson bag'' excitations in QCD. In Lorentz gauge, the small instantons and Wilson line excitations can be described, respectively, in terms of poles and cuts of an analytic gauge potential.

33 pages, 12 figures

References in corpus (13)

• Large N Field Theories, String Theory and Gravity
• Instantons in QCD
• The index theorem in QCD with a finite cut-off
• Theta Dependence In The Large N Limit Of Four-Dimensional Gauge Theories
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• Some more remarks on the Witten-Veneziano formula for the ${\eta }^{\prime }$ mass
• The Negativity of the Overlap-Based Topological Charge Density Correlator in Pure-Glue QCD and the Non-Integrable Nature of its Contact Part
• Inherently Global Nature of Topological Charge Fluctuations in QCD
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• Coherent Topological Charge Structure in $C{P}^{N-1}$ Models and QCD
• The Euclidean two-point correlation function of the topological charge density
• Nonabelian Strings and Axion

Cited by in corpus (11)

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• Topological properties of $C{P}^{N-1}$ models in the large-$N$ limit
• Is $N=2$ Large?
• Theta-vacuum and large N limit in CP^{N-1} sigma models
• Fractionally charged Wilson loops as a probe of $\theta$-dependence in $C{P}^{N-1}$ sigma models: Instantons vs. large N
• Fractional topological charge in $SU(N)$ gauge theories without dynamical quarks
• Topological susceptibility of $2d{\mathrm{CP}}^1$ or $\mathrm{O}(3)$ non-linear $\sigma$-model: is it divergent or not?
• $\theta$-dependence in the small-$N$ limit of $2d$ $C{P}^{N-1}$ models
• Image-processing the topological charge density in the CP(N-1) model
• Tachyonic crystals and the laminar instability of the perturbative vacuum in asymptotically free gauge theories