Topological String Theory - Admissible Spacetimes

Admissible Spacetimes

The fundamental strings of string theory are two-dimensional surfaces. A quantum field theory known as the N = (1,1) sigma model is defined on each surface. This theory consist of maps from the surface to a supermanifold. Physically the supermanifold is interpreted as spacetime and each map is interpreted as the embedding of the string in spacetime.

Only special spacetimes admit topological strings. Classically one must choose a spacetime such that the theory respects an additional pair of supersymmetries, and so is in fact an N = (2,2) sigma model. This will be the case for example if the spacetime is a Kähler manifold and the H-flux is identically equal to zero, although there are more general cases in which the target is a generalized Kähler manifold and the H-flux is nontrivial.

So far we have described ordinary strings on special backgrounds. These strings are never topological. To make these strings topological, one needs to modify the sigma model via a procedure called a topological twist which was invented by Edward Witten in 1988. The central observation is that these theories have two U(1) symmetries known as R-symmetries, and one may modify the Lorentz symmetry by mixing rotations and R-symmetries. One may use either of the two R-symmetries, leading to two different theories, called the A model and the B model. After this twist the action of the theory is BRST exact, and as a result the theory has no dynamics, instead all observables depend on the topology of a configuration. Such theories are known as topological theories.

While classically this procedure is always possible, quantum mechanically the U(1) symmetries may be anomalous. In this case the twisting is not possible. For example, in the Kahler case with H = 0 the twist leading to the A-model is always possible but that leading to the B-model is only possible when the first Chern class of the spacetime vanishes, implying that the spacetime is Calabi-Yau. More generally (2,2) theories have two complex structures and the B model exists when the first Chern classes of associated bundles sum to zero whereas the A model exists when the difference of the Chern classes is zero. In the Kahler case the two complex structures are the same and so the difference is always zero, which is why the A model always exists.

There is no restriction on the number of dimensions of spacetime, other than that it must be even because spacetime is generalized Kahler. However all correlation functions with worldsheets that are not spheres vanish unless the complex dimension of the spacetime is three, and so spacetimes with complex dimension three are the most interesting. This is fortunate for phenomenology, as phenomenological models often use a physical string theory compactified on a 3 complex-dimensional space. The topological string theory is not equivalent to the physical string theory, even on the same space, but certain supersymmetric quantities agree in the two theories.