Fractionally charged Wilson loops as a probe of theta dependence in CPN−1 sigma models

We present a method for exploring θ -dependence in two-dimensional U(1) gauge theories which is based on the calculation of fractionally charged Wilson loops in the θ = 0 theory. Including a Wilson loop with charge q = θ/2π in the path integral of a U(1) gauge theory in two spacetime dimensions is equivalent to including a θ -term in the two-volume enclosed by the loop. We study the θ -dependence of the free energy density ε(θ) for the CP1, CP5 and CP9 sigma models by extracting ε(θ) from the area law of a fractionally charged Wilson loop. For CP1, ε(θ) is smooth in the region θ ≈ π and well-described by a dilute instanton gas approximation throughout the range 0 < θ < 2π . For CP5 and CP9 the energy density exhibits a clear cusp and evidence for discrete degenerate vacua at θ = π , as expected from large N arguments. For CP9 the θ dependence is in good quantitative agreement with the leading order large N prediction ε(θ) = 1 2 χtθ 2 throughout the range 0 < θ < π .