ODE/IM Correspondence in Toda Field Theories and Fermionic Basis in sin(h)-Gordon Model

Understand

The first part of this work consists of a study of the ODE/IM correspondence for simply-laced affine Toda field theories.

• It is a first step towards a full generalisation of the results of Lukyanov and Zamolodchikov on $\hat{\mathfrak a}_1$ to a general affine Lie-Ka\v{c}-Moody algebra $\hat{\mathfrak g}$.
• In order to achieve our goal, we investigate the structure of evaluation representations of $\hat{\mathfrak g}$ and show how their tensor products are related by what we call projected isomorphisms.
• These isomorphisms are used to construct a set of quadratic functional relations, called $\psi$-system, for the solutions to complex differential equations associated to $\hat{\mathfrak g}$.