Topological Strings and Integrable Hierarchies

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.Comment: 82 pages, harvmac, 1 figur

Matrix models, 4D black holes and topological strings on non-compact Calabi-Yau manifolds

“…Indeed, considering the commutation relations of the generators, it seems natural to expect some kind of non-commutative geometry to arise in this process. Probably the techniques of [4] will be very useful in this regard.…”

“…This will be discussed in section 2.2.3. But let us now return to (4) and see what it means in terms of the matrix model. From the point of view of the matrix model (4) looks rather intriguing since it is precisely this type of transformation which takes us from the grand canonical (fixed chemical potential µ) to the canonical (fixed fermion number N) free energy.…”

“…In principle, one could calculate the normalization by carefully comparing our normalization of Ω to the one in [5]. 4 Strictly speaking, for the deformed matrix model, there is no clear distinction between electric and magnetic charges. In this context, however, it is natural to call q magnetic charges.…”

“…In fact, it is now known that there are very general relations between topological string theories and matrix models, see for example [4]. In general, one can identify the partition function of a certain matrix model with the partition function of a corresponding topological string theory.…”

Tachyon Condensation, Open-Closed Duality, Resolvents, and Minimal Bosonic and Type 0 Strings

“…5 We suspect that there is a relation with the work of ref. [34], which recently discussed the organization of a number of topological string theory structures in terms of modes of a chiral boson φ. Interestingly, in that work, of which we do not understand much (due to no fault of the authors) an exponentiation of the boson also plays a natural role in the insertion of D-branes in the background, which reminds us of the observation of ref. [20], discussed above.…”