Toda Lattice Hierarchy and Generalized String Equations

Abstract

String equations of the $p$-th generalized Kontsevich model and the compactified $c=1$ string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at $p=-1$ does not coincide with the $c=1$ string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of $c\le 1$ string models is discussed in this new framework.

35pages, LaTeX Errors are corrected in Eqs. (2.21), (2.36), (2.33), (3.3), (5.10), (6.1), sentences after (3.19) and theorem 5. A few references are updated

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Cited by in corpus (21)

• Time-dependent backgrounds of 2D string theory
• Toda hierarchies and their applications
• Multivariate hypergeometric functions as tau functions of Toda lattice and Kadomtsev-Petviashvili equation
• Scalar products of symmetric functions and matrix integrals
• The multicomponent 2D Toda hierarchy: Discrete flows and string equations
• String Equation for String Theory on a Circle
• The multicomponent 2D Toda hierarchy: dispersionless limit
• The "ghost" symmetry in the CKP hierarchy
• The Additional Symmetries for the BTL and CTL Hierarchies
• Generalized string equations for double Hurwitz numbers
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• KP integrability of triple Hodge integrals. II. Generalized Kontsevich matrix model
• Orbifold melting crystal models and reductions of Toda hierarchy
• Thermodynamic limit of random partitions and dispersionless Toda hierarchy
• On the squared eigenfunction symmetry of the Toda lattice hierarchy
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• Open/closed duality for FZZT branes in c=1
• Squared Eigenfunction Symmetries for the BTL and CTL Hierarchies
• An hbar-expansion of the Toda hierarchy: a recursive construction of solutions
• Notes about KP/BKP correspondence
• Integrable Deformations of $\hat{c}=1$ Strings in Flux Backgrounds