Dynkin TBA's
F. Ravanini, R. Tateo, A. Valleriani
TL;DR
The work establishes a key identity for ADE S-matrices that enables transforming the relative TBA into Zamolodchikov's universal ADE form and derives the associated Y-system. It then classifies magnonic TBA systems on ADE and tadpole diagrams, extending the framework to ADET magnonic TBA and systematically extracting UV data, perturbing dimensions, and Casimir-energy non-perturbative terms. The paper reveals rich RG structures, including previously known and novel massive and massless flows across $A_n$, $D_n$, $E_n$, and $T_n$ cases, and connects these to parafermionic cosets, tensor products of minimal models, and level-rank duality. It also outlines generalizations to higher-rank algebras and highlights intriguing dualities and open questions that motivate further study.
Abstract
We prove a useful identity valid for all $ADE$ minimal S-matrices, that clarifies the transformation of the relative thermodynamic Bethe Ansatz (TBA) from its standard form into the universal one proposed by Al.B.Zamolodchikov. By considering the graph encoding of the system of functional equations for the exponentials of the pseudoenergies, we show that any such system having the same form as those for the $ADE$ TBA's, can be encoded on $A,D,E,A/Z_2$ only. This includes, besides the known $ADE$ diagonal scattering, the set of all $SU(2)$ related {\em magnonic} TBA's. We explore this class sistematically and find some interesting new massive and massless RG flows. The generalization to classes related to higher rank algebras is briefly presented and an intriguing relation with level-rank duality is signalled.