Topological Strings with Scaling Violation and Toda Lattice Hierarchy

TL;DR

This paper demonstrates a class of topological string theories with integrable structures described by the Toda lattice hierarchy, extending the topological $CP^1$ string theory to include scaling violation and complex monodromy groups.

Contribution

It introduces new topological string models with Toda lattice integrability and explores their monodromy groups and higher genus expansions, generalizing existing theories.

Findings

Integrable structure described by Toda lattice hierarchy.

Monodromy group extends affine Weyl group $ ilde{W}(A_N^{(1)})$.

Higher genus expansion derived from matrix form of Lax operator.

Abstract

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.