Towards Integrability of Topological Strings

Usually the most interesting questions in Physics are connected with the dynamics. How the system evolves in time, what are the equilibrium states, how the known structures are dynamically generated. There are a lot of interesting questions about the dynamics. But underneath there are more fundamental kinematic questions what variables should we use to describe the system in the most natural way. Obviously the choice of the kinematic structure precedes the discussion of the dynamics. And the search for the appropriate structure is obviously not an easy task in general. There are indications that we do not know the right variables in the string theory. Thus the most important problem is to uncover the corresponding underlying kinematic structure. From this point of view the string theories supposedly describing our world are very complicated and it might be extremely difficult to decipher what are the right variables in this case. The problem is that the theories are highly non-local from the point of view of the space-time. In the string geometry the points are considered to be close if there is a curve passing through them. So in a sense all points coincide and it is not clear what is the right way to describe this situation. Obviously this is not a disadvantage but actually an advantage of the theory it clearly indicates that string theory is not about smooth manifolds or anything like standard geometry. Unfortunately it is not easy to say something meaningful in this situation. Thus one should look for more manageable yet non-trivial examples of string theories. It seems that the appropriate set of such examples is given by the topological string theories. The main advantage is the huge reduction of the degrees of freedom. Most commonly discussed topological string theories are of two types Type A and Type B. In mathematical terms the first one is connected with the symplectic geometry and Gromov-Witten invariants and the second one is described in terms of the variations of the Hodge structure in complex geometry. There is some redundancy in this description because Type A theory on a manifold M can be equivalent to Type B theory on another manifold M̃ (if we substitute the notion of the manifold by an appropriate generalization we get the equivalence of Type A and Type B topological strings). The Type A topological strings are intrinsically non-local. The points on M are considered to be close to each other if there is a holomorphic curve passing through them. Due to the rigidity of the holomorphic maps this case is intermediate between local theories and ”fully” non-local string theories. In particular it is difficult to get the explicit formula for GromovWitten invariants for a generic manifold. Type B topological string theory is the most simple case from this point of view it is a local quantum field theory. So one might suspect that it could give us nothing to help to understand the fundamental degrees of freedom behind the strings. Fortunately it seems not the case. The formulation of these theories in appropriate variables leads to a drastic simplification of the theory. It becomes a “free” theory with quadratic action functionals. This phenomena is well