Labeling Schemes for Tetrahedron Equations and Dualities between Them

TL;DR

This paper explores different labeling schemes for Zamolodchikov's tetrahedron equations, derives their relations, and establishes dualities between these schemes, including a novel nonlocal string labeling approach.

Contribution

It provides a detailed derivation of three labeling schemes for tetrahedron equations and introduces dualities between them, including a new nonlocal string labeling scheme.

Findings

Derived three labeling schemes for tetrahedron equations.

Established dualities between each pair of labeling schemes.

Identified cases with simultaneous dualities among all three labelings.

Abstract

Zamolodchikov's tetrahedron equations, which were derived by considering the scattering of straight strings, can be written in three different labeling schemes: one can use as labels the states of the vacua between the strings, the states of the string segments, or the states of the particles at the intersections of the strings. We give a detailed derivation of the three corresponding tetrahedron equations and show also how the Frenkel-Moore equations fits in as a {\em nonlocal} string labeling. We discuss then how an analog of the Wu-Kadanoff duality can be defined between each pair of the above three labeling schemes. It turns out that there are two cases, for which one can simultaneously construct a duality between {\em all} three pairs of labelings.