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

Abstract: 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 $\hat{\mathfrak g}$. Finally, from the $\psi$-system for the algebras $\mathfrak a$ and $\mathfrak d$, we derive a set of Bethe Ansatz equations satisfied by the eigenvalues of some particular boundary problem for the above mentioned differential equations. The second part of this work deals with the study of one-point functions in sine- and sinh-Gordon models. The approach to the computation of these quantities follows a powerful method, which we call fermionic basis, developed by Boos et al. for the XXZ, quantum Liouville and quantum sine-Gordon models. We show how the determinant formula for one-point functions obtained there can be generalised to the sinh-Gordon model. In doing so we give an interpretation of the fermionic basis in terms of certain symmetries of the system. This new perspective will also allow us to solve trivially the reflection relations introduced by Fateev et al. We then provide analytical and numerical results supporting our finding.