Topological strings with scaling violation and Toda lattice hierarchy

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 ane Weyl group f W(A 1 N ) i n troduced by Dubrovin. These models are generalizations of the topological CP 1 string theory with scaling violation. The logarithmic Hamiltonians generate ows 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 dierent t ype from those for the c = 1 string theory. Higher genus expansion is obtained by considering the Lax operator in matrix form.

Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy

“…This model has been recently studied in [Du2,KO]. Here we show that the flat solutions of this model coincide with those of the W 0,2 -model just discussed (the equivalent pair discussed in Section 7).…”

Toda lattice hierarchy and generalized string equations

“…In the last few years, the Toda lattice hierarchy has come to be studied from renewed points of view, such as c = 1 strings [7,23,27,28], twodimensional topological strings [14,16,10,34,5], the topological CP 1 sigma model and its variations related to affine Coxeter groups [11,18,9]. As opposed to the (p, q) models in the KP hierarchy, these are related to string theories with a true continuous target space.…”

“…The generalized Kontsevich models, too, may be considered as a solution of the Toda lattice hierarchy obeying string equations of the twomatrix model type [19]. Meanwhile, string equations of the topological CP 1 model and its variants [11,18,9] are of the one-matrix type. Furthermore, the deformed c = 1 theory in the presence of black hole backgrounds [28] are known to obey more involved string equations, though this case, too, is essentially of the two-matrix model type.…”

A note on the extended dispersionless Toda hierarchy

“…Thus the extended dToda hierarchy becomes the master equation of the genus zero GW invariants whose generating function is characterized by the free energy of the extended dToda hierarchy. Based on the twistor theoretical method [20,11] the extended dToda hierarchy can be constructed by adding logarithmic-flow to the one-dimensional dToda hierarchy.The corresponding Orlov-Schulman operator is conjugated with the Lax operator under the Poisson bracket which imposes an extra condition (the so-called string equation) on the free energy of the extended dToda hierarchy. We will show that the full hierarchy flows can be expressed in terms of second derivatives of its associated free energy F and thus can be viewed as the corresponding dispersionless Hirota(dHirota) equations.…”

“…Thus the extended dToda hierarchy becomes the master equation of the genus zero GW invariants whose generating function is characterized by the free energy of the extended dToda hierarchy. Based on the twistor theoretical method [20,11] the extended dToda hierarchy can be constructed by adding logarithmic-flow to the one-dimensional dToda hierarchy.…”