Topological strings and integrable hierarchies. (English) Zbl 1095.81049
The simplest string theories are topological and come in two basic varieties known as the A- and B-model. They are of interest because they provide access to many deep phenomena of string theory such as large N transitions that encode the connection between gauge theory and geometry. The paper studies the topological B-model on a special class of non-compact Calabi-Yau threefolds and develops various techniques to solve it completely. In particular it shows how to solve for the amplitudes by using \(\mathcal{W}\)-algebra symmetries which encode the symmetries of holomorphic diffeomorphisms of the Calabi-Yau space. In the highly effective fermionic/brane formulation this leads to a free fermion description of the amplitudes. By these techniques a unified picture connecting non-critical (super)strings, integrable manifolds and various matrix models emerges. In particular the ordinary matrix model, the double scaling limit of matrix models, and the Kontsevich-like matrix model are all shown to be related and to arise from studying branes in specific Calabi-Yau manifolds. Furthermore an A-model topological string on \(\mathbb {P}^1\) and local toric threefolds (and in particular the topological vertex) are realized and solved as B-model string amplitudes on a Calabi-Yau threefold.
Reviewer: Helmut Rumpf (Wien)
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81T10 | Model quantum field theories |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
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