  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 E$n$-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 anon-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 may be an
unstable vacuum for the type $II^*$ theories, but they have supersymmetric
cosmological solutions.
