Topological Strings and Integrable Hierarchies

TL;DR

This paper explores how topological string theory on Calabi-Yau manifolds connects to integrable hierarchies, matrix models, and non-critical strings through W-algebra symmetries and fermionic formulations.

Contribution

It introduces a W-algebra symmetry framework for solving topological B-model amplitudes and unifies various matrix models and string theories via Calabi-Yau geometries.

Findings

W-algebra symmetries encode topological string amplitudes.

Fermionic formulation provides a free fermion description.

Connections established between matrix models, non-critical strings, and Calabi-Yau geometries.

Abstract

We consider the topological B-model on local Calabi-Yau geometries. We show how one can solve for the amplitudes by using W-algebra symmetries which encodes the symmetries of holomorphic diffeomorphisms of the Calabi-Yau. In the highly effective fermionic/brane formulation this leads to a free fermion description of the amplitudes. Furthermore we argue that topological strings on Calabi-Yau geometries provide a unifying picture connecting non-critical (super)strings, integrable hierarchies, and various matrix models. In particular we show how the ordinary matrix model, the double scaling limit of matrix models, and Kontsevich-like matrix model are all related and arise from studying branes in specific local Calabi-Yau three-folds. We also show how A-model topological string on P^1 and local toric threefolds (and in particular the topological vertex) can be realized and solved as B-model topological string amplitudes on a Calabi-Yau manifold.