Tau Functions Of Drinfeld-Sokolov Hierarchies And Some Relevant Problems

The Drinfeld-Sokolov hierarchies, introduced by Drinfeld and Sokolov in 1980's,are important integrable systems that play a significant role in the development of thesoliton theory as well as in its applications in mathematical physics. In this thesis westudy tau functions of Drinfeld-Sokolov hierarchies and some relevant problems.Tau functions act as a bridge between integrable systems and related branchesof mathematical physics such as quantum field theory, matrix models, representationtheory and algebraic geometry. In the literature there are several ways to define taufunctions of Drinfeld-Sokolov hierarchies that admit certain restrictions, but the re-lation between them are not clear yet. As our first main result, we define tau func-tions of Drinfeld-Sokolov hierarchies based on a class of tau-symmetric Hamiltoniandensities, and present explicitly the relation between tau functions defined variouslyfor Drinfeld-Sokolov hierarchies. Such tau functions include those constructed frompseudo-di?erential operator representations of the hierarchies, the ones constructed byHollowood and Miramontes for Drinfeld-Sokolov hierarchies corresponding to a?neLie algebras of A-D-E type, the ones constructed by Enriquez and Frenkel for hierar-chies of mKdV type, as well as those given by Miramontes for generalized Drinfeld-Sokolov hierarchies and for generating conserved densities of them.Our second main result is that we describe the Drinfeld-Sokolov hierarchies cor-responding to untwisted a?ne Lie algebras of type D and their tau functions in termsof pseudo-di?erential operators, then find the bilinear equations satisfied by such taufunctions. These bilinear equations coincide with the integrable hierarchies constructedby Date, Jimbo, Kashiwara, Miwa and by Kac, Wakimoto via the basic representationof untwisted a?ne Lie algebras of type D.Our third main result is the verification of the D-type simple singularities case ofGivental and Milanov's conjecture, which was proposed in 2004 to connect singularitytheory and integrable hierarchies. From our result it follows that the Givental-Milanov hierarchies for D-type simple singularities are equivalent to the Drinfeld-Sokolov hier-archies corresponding to untwisted a?ne Lie algebras of type D.We also generalize the notion of pseudo-di?erential operators, which is crucialto represent the two-component BKP hierarchy and its reductions to Drinfeld-Sokolovhierarchies of type D. Also with the method of R-matrix, we construct a bi-Hamiltonianstructure of the two-component BKP hierarchy, and clarify the relation between theHamiltonian densities and the tau function.