Intertwiners and \ade Lattice Models

TL;DR

This paper develops a comprehensive theory of intertwiners for extbackslash ade lattice models, revealing their role in spectral analysis and linking eigenvalues, central charges, and scaling dimensions.

Contribution

It introduces a unified framework for intertwiners at multiple levels and a graphical representation, enhancing understanding of extbackslash ade models' spectra.

Findings

Intertwiners imply shared eigenvalues across models.

Existence of intertwiners links to central charges and scaling dimensions.

Graphical representation simplifies the analysis of intertwiners.

Abstract

Intertwiners between \ade lattice models are presented and the general theory developed. The intertwiners are discussed at three levels: at the level of the adjacency matrices, at the level of the cell calculus intertwining the face algebras and at the level of the row transfer matrices. A convenient graphical representation of the intertwining cells is introduced. The utility of the intertwining relations in studying the spectra of the \ade models is emphasized. In particular, it is shown that the existence of an intertwiner implies that many eigenvalues of the \ade row transfer matrices are exactly in common for a finite system and, consequently, that the corresponding central charges and scaling dimensions can be identified.