Toda equations associated with loop groups of complex classical Lie groups

A Toda equation is specified by a choice of a Lie group and a $\mathbb Z$-gradation of its Lie algebra. The Toda equations associated with loop groups of complex classical Lie groups, whose Lie algebras are endowed with integrable $\mathbb Z$-gradations with finite dimensional grading subspaces, are described in an explicit form.Comment: 39 page

Maximally nonabelian Toda systems

A detailed consideration of the maximally nonabelian Toda systems based on the classical semisimple Lie groups is given. The explicit expressions for the general solution of the corresponding equations are obtained.Comment: 28 pages, LaTeX file. A few references and appendix B are added; this version will appear in Nuclear Physics

On the supersymmetric vacua of the Veneziano-Wosiek model

We study the supersymmetric vacua of the Veneziano-Wosiek model in sectors with fermion number F=2, 4 at finite 't Hooft coupling lambda. We prove that for F=2 there are two zero energy vacua for lambda > lambda_c = 1 and none otherwise. We give the analytical expressions of both vacua. One of them was previously known, the second one is obtained by solving the cohomology of the supersymmetric charges. At F=4 we compute the would-be supersymmetric vacua at high order in the the strong coupling expansion and provide strong support to the conclusion that lambda = 1 is a critical point in this sector too. It separates a strong coupling phase with two symmetric vacua from a weak coupling phase with positive spectrum.Comment: 17 pages, 2 eps figure

Highest ℓ\ell-Weight Representations and Functional Relations

We discuss highest $\ell$-weight representations of quantum loop algebras and the corresponding functional relations between integrability objects. In particular, we compare the prefundamental and $q$-oscillator representations of the positive Borel subalgebras of the quantum group $\mathrm{U}_q(\mathcal L(\mathfrak{sl}_{l+1}))$ for arbitrary values of $l$. Our article has partially the nature of a short review, but it also contains new results. These are the expressions for the $L$-operators, and the exact relationship between different representations, as a byproduct resulting in certain conclusions about functional relations