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.
Highlights
Topological strings on Calabi-Yau threefolds have served as a unifying theme of many aspects of string theory
Many deep phenomena in string theory have a simpler and better understood description in the context of topological strings, in particular large N transitions that encode the connections between gauge theory and geometry
For example it has been known that non-critical bosonic strings have two different matrix model descriptions: a double scaling limit of a matrix model, in which the string world-sheets emerge through the ’t Hooft ribbon diagrams as triangulations, as well as a finite N matrix model, introduced by Kontsevich, in which the matrix diagrams can be considered as open string field theory diagrams that triangulate moduli space
Summary
Topological strings on Calabi-Yau threefolds have served as a unifying theme of many aspects of string theory. In the large N limit, with suitable tuning of these parameters, one can end up with a function with the lower terms vanishing This gets identified with non-critical bosonic string with the background corresponding to the (r, s) minimal model. The open string field theory on these branes turns out to be a Kontsevich-like matrix model whose classical action can be read off from the Calabi-Yau geometry: W. The case (r, s) = (1, 2) gives the usual Kontsevich model with action From this point of view it is natural to compute the change in the closed string partition function, as a result of the back reaction to the presence of N branes, as a rank N matrix integral. In this sense we can think of these as being ‘brane/anti-brane’ pairs.
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