STRING COMPACTIFICATIONS — OLD AND NEW

Abstract

In this talk, after a lightning review of some old and some new string compactifications, I will mostly discuss recent work in compactifications with duality twists and their relation to the other compactifications such as orbifolds and flux compactifications. Much of the discussion here is based on work done in collaboration with Chris Hull [1]. A central program in string theory over the last twenty years concerns compactifications of the ten-dimensional string theory to four dimensions on some internal six-dimensional manifold. The six manifold must satisfy the string equations of motion which at low energies are the equations of ten-dimensional supergravity with thirty-two or sixteen supersymmetries depending on whether our starting point is the Type-II string or the heterotic string. I will restrict myself here to the lOd string theory, but many of the remarks can be generalized easily to eleven-dimensional supergravity. For phenomenological reasons, one wants at most N = 1 supersymmetry in fourdimensions so that the fermions can appear in chiral representations of the gauge group. For the heterotic string this means that the manifold of compactification must be a Calabi-Yau manifold, that is, a six dimensional manifold with 5(7(3) holonomy. For the Type-II string, compactifications on a Calabi-Yau manifold gives N = 2 supersymmetry in four dimensions. The N=2 compactifications, even though not useful phenomenologically, have been a rich source of data for exploring the highly nontrivial generalization of Riemannian geometry that string theory seems to provide. Mirror symmetry, which relates Calabi-Yau manifolds that are completely distinct at the level of conventional geometry but are equivalent at the stringy level, or geometric transitions within the perturbative or nonperturbative string theory which connect up completely distinct Calabi-Yau manifolds are just a few examples of the rich stringy phenomena that have been uncovered. The string equations of motion are equivalent to the requirement of superconformal invariance of the worldsheet and the low energy low energy supergravity equations typically receive higher order stringy corrections that become important when the compactifications manifold has large curvature. There is a subclass of string compactifications which can be studied exactly in conformal field theory. The simplest manifold for compactification is a torus that has no curvature and thus satisfies string equations of motion to all orders. Toroidal Compactifications