Topological Field Theory

T-Duality on a timelike circle does not interchange IIA and IIB string theories, but takes the IIA theory to a type IIB * theory and the IIB theory to a type IIA * theory. The type II * theories admit E-branes, which are the images of the type II D-branes under timelike T-duality and correspond to imposing Dirichlet boundary conditions in time as well as some of the spatial directions. The effective action describing an En-brane is the n-dimensional Euclidean super-Yang-Mills theory obtained by dimensionally reducing 9+1 dimensional super-Yang-Mills on 9 − n spatial dimensions and one time dimension. The IIB * theory has a solution which is the product of 5-dimensional de Sitter space and a 5-hyperboloid, and the E4-brane corresponds to a non-singular complete solution which interpolates between this solution and flat space. This leads to a duality between the large N limit of the Euclidean 4-dimensional U(N) super-Yang-Mills theory and the IIB * string theory in de Sitter space, and both are invariant under the same de Sitter supergroup. This theory can be twisted to obtain a large N topological gauge theory and its topological string theory dual. Flat space-time is an unstable vaccum for the type II * theories, but they have supersymmetric cosmological solutions.