Hamiltonian Reduction and Topological Conformal Algebra in c≤1c\leq 1 Non-critical Strings

We study the hamiltonian reduction of affine Lie superalgebra $sl(2|1)^{(1)}$. Based on a scalar Lax operator formalism, we derive the free field realization of the classical topological topological algebra which appears in the $c\leq1$ non-critical strings. In the quantum case, we analyze the BRST cohomology to get the quantum free field expression of the algebra.Comment: 13 pages Latex, UTHEP-26

Lie Superalgebra and Extended Topological Conformal Symmetry in Non-critical W3W_{3} Strings

We obtain a new free field realization of $N=2$ super $W_{3}$ algebra using the technique of quantum hamiltonian reduction. The construction is based on a particular choice of the simple root system of the affine Lie superalgebra $sl(3|2)^{(1)}$ associated with a non-standard $sl(2)$ embedding. After twisting and a similarity transformation, this $W$ algebra can be identified as the extended topological conformal algebra of non-critical $W_{3}$ string theory.Comment: 14pages, UTHEP-27

Topological Strings with Scaling Violation and Toda Lattice Hierarchy

We show that there is a series of topological string theories whose integrable structure is described by the Toda lattice hierarchy. The monodromy group of the Frobenius manifold for the matter sector is an extension of the affine Weyl group $\widetilde W (A_N^{(1)})$ introduced by Dubrovin. These models are generalizations of the topological $CP^1$ string theory with scaling violation. The logarithmic Hamiltonians generate flows for the puncture operator and its descendants. We derive the string equation from the constraints on the Lax and the Orlov operators. The constraints are of different type from those for the $c=1$ string theory. Higher genus expansion is obtained by considering the Lax operator in matrix form.Comment: 27 pages, latex, no figure