Nonlinear O(3)O(3)O(3) sigma model in discrete complex analysis (original) (raw)

arXiv: High Energy Physics - Theory, 2016

Abstract

We examine a discrete version of the two-dimensional nonlinear O(3)O(3)O(3) sigma model derived from discrete complex analysis. We adopt two lattices, one rectangular, the other polar. We define a discrete energy E(f)rmdisc.E({f})^{\rm disc.}E(f)rmdisc. and a discrete area calA(f)rmdisc.{\cal{A}}({f})^{\rm disc.}calA(f)rmdisc., where the function fff is related to a stereographic projection governed by a unit vector of the model. The discrete energy and area satisfy the inequality E(f)rmdisc.ge∣calA(f)rmdisc.∣E({f})^{\rm disc.} \ge |{\cal{A}}({f})^{\rm disc.}|E(f)rmdisc.ge∣calA(f)rmdisc.∣, which is saturated if and only if the function fff is discrete (anti-)holomorphic. We show for the rectangular lattice that, except for a factor 2, the discrete energy and the area tend to the usual continuous energy E(f)E({f})E(f) and the area calA(f)=4piN,,,Ninpi_2(S2){\cal{A}}({f})=4 \pi N, \,\,N\in \pi_2(S^2)calA(f)=4piN,,,Ninpi_2(S2) as the lattice spacings tend to zero. In the polar lattice, we section the plane by 2M2M2M lines passing through the origin into 2M2M2M equal sectors and place vertices radially in a geometric progression with a common ratio $q...

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