Related papers: The (n,1)-Reduced DKP Hierarchy, the String Equati…

The $(n,m)^{\th}$ KdV hierarchy is a restriction of the KP hierarchy to a submanifold of pseudo-differential operators in a radio form. Explicit formula of the restricted Hamiltonian structure of KP is given which provides a new, more…

Recently much attention has been paid to the restriction of KP to the submanifold of operators which can be represented as a ratio of two purely differential operators L=AB^{-1}. Whereas most of the aspects concerning this reduced…

In this paper, we extend the matrix-resolvent method to the study of the Dubrovin--Zhang type tau-functions for the constrained KP hierarchy and the bigraded Toda hierarchy of $(M,1)$-type. We show that the Dubrovin--Zhang type tau-function…

Warped compactifications with significant warping provide one of the few known mechanisms for naturally generating large hierarchies of physical scales. We demonstrate that this mechanism is realizable in string theory, and give examples…

We solve Virasoro constraints on the KP hierarchy in terms of minimal conformal models. The constraints we start with are implemented by the Virasoro generators depending on a background charge $Q$. Then the solutions to the constraints are…

We use the representation theory of the infinite matrix group to show that (in the polynomial case) the $n$--vector $k$--constrained KP hierarchy has a natural geometrical interpretation on Sato's infinite Grassmannian. This description…

We apply the new method for constructing integrable Hamiltonian hierarchies of Lax type equations developed in our previous paper, to show that all W-algebras W(gl_N,f) carry such a hierarchy. As an application, we show that all vector…

In this thesis, we obtain the formula for the Kac determinant of the algebra arising from the level $N$ representation of the Ding-Iohara-Miki algebra. This formula can be proved by decomposing the level $N$ representation into the deformed…

We formulate the constrained KP hierarchy (denoted by \cKP$_{K+1,M}$) as an affine ${\widehat {sl}} (M+K+1)$ matrix integrable hierarchy generalizing the Drinfeld-Sokolov hierarchy. Using an algebraic approach, including the graded…

We study the representations of the W-algebra W(g) associated to an arbitrary finite-dimensional simple Lie algebra g via the quantized Drinfeld-Sokolov reductions. The characters of irreducible representations with non-degenerate highest…

The BRST quantisation of the Drinfeld - Sokolov reduction applied to the case of $A^{(1)}_2\,$ is explored to construct in an unified and systematic way the general singular vectors in ${\cal W}_3$ and ${\cal W}_3^{(2)}$ Verma modules. The…

The Toda lattice hierarchy is discussed in connection with the topological description of the $c=1$ string theory compactified at the self-dual radius. It is shown that when special constraints are imposed on the Toda hierarchy, it…

Under the Neumann constraints, each equation of the KdV hierarchy is decomposed into two finite dimensional systems, including the well-known Neumann model. Like in the case of the Bargmann constraint, the explicit Lax representations are…

We give a matrix formulation of the Hamiltonian structures of constrained KP hierarchy. First, we derive from the matrix formulation the Hamiltonian structure of the one-constraint KP hierarchy, which was originally obtained by Oevel and…

In this paper we give a geometric description in terms of the Grassmann manifold of Segal and Wilson, of the reduction of the KP hierarchy known as the vector $k$-constrained KP hierarchy. We also show in a geometric way that these…

We prove that the total descendant potential functions of the theory of Fan-Jarvis-Ruan-Witten for D_4 with symmetry group <J> and D_4^T with symmetry group G_{max}, respectively, are both tau-functions of the D_4…

We discuss a differential integrable hierarchy, which we call the (N, M)$--th KdV hierarchy, whose Lax operator is obtained by properly adding $M$ pseudo--differential terms to the Lax operator of the N--th KdV hierarchy. This new hierarchy…

We describe a geometric theory of Virasoro constraints in generalized Drinfeld-Sokolov hierarchies. Solutions of Drinfeld-Sokolov hierarchies are succinctly described by giving a principal bundle on a complex curve together with the data of…

For each r = (r_1, r_2,...,r_N) we construct a highest weight module M_r of the Lie algebra W_{1+infty}. The highest weight vectors are specific tau-functions of the N-th Gelfand--Dickey hierarchy. We show that these modules are quasifinite…

We construct generalized additional symmetries of a two-component BKP hierarchy defined by two pseudo-differential Lax operators. These additional symmetry flows form a Block type algebra with some modified(or additional) terms because of a…