Boundary from bulk integrability in three dimensions: 3D reflection maps from tetrahedron maps

Abstract

We established a method for obtaining set-theoretical solutions to the 3D reflection equation by using known ones to the Zamolodchikov tetrahedron equation, where the former equation was proposed by Isaev and Kulish as a boundary analog of the latter. By applying our method to Sergeev's electrical solution and a two-component solution associated with the discrete modified KP equation, we obtain new solutions to the 3D reflection equation. Our approach is closely related to a relation between the transition maps of Lusztig's parametrizations of the totally positive part of $S{L}_3$ and $S{O}_5$, which is obtained via folding the Dynkin diagram of $A_3$ into one of $B_2$.

14 pages, 8 figures, v2: updated abstract and minor corrections. For the interactive Figure 6, see the Mathematica notebook obtained from https://www.wolframcloud.com/obj/yoneyama/Published/R20_identity.nb