Affine and super affine Toda field theories

Abstract

Classical integrability of affine Toda field theory has been known. But its quantum counterpart had not been well understood. Motivated form the work on the deformed conformal field theory, it was suggested that ATFTs are quantum integrable. Subsequently the exact S matrices were contructed for the based on the simply laced and some super Lie algebras, using the bootstrap and factorization property as well as the first principles of the quantum field theory. We found various consistent results between the proposed $\beta$ dependent S matrices and conventional perturbative field theory. We observed the complete agreement for all channels in $A_2$, $A_3$, $A_5$, $D_4$ and 23 channels in $D_5$ theory. We also proved that the simple pole residue at the arbitrary double pole positions vanish up to one loop order for all $A_n$ theories. Renormalization of mass and vertex function and Landau singularity analysis on the Leading and Subleading singularity are the main ingredient of this work. On the other hand. ATFTs based on the non simply laced Lie algebras was thought not to be quantum integrable from the study on the renormalization of the mass spectrum. But quite recently. Direct construction of the quantum conserved higher spin currents implies quantum integrability and a generalization of the standard bootstrap idea made it possible to construct the S matrices for the ATFTs based on the non simply laced Lie algebras.