String Theory

Abstract

String theory models physical phenomena by closed vibrating loops (“strings”) moving in space. As the string moves, it forms a surface, its world sheet ∑. The movement in space is described by a map ∑ → T to some target space T. (This is the starting point for the data of a nonlinear sigma model.) This space is usually required to be 10 = 4 + 6-dimensional and is often assumed to be of the form T = M ⁴ x X ⁶, where M ⁴ is a 4-manifold which, at least locally, may be thought of as the space–time of special relativity. The additional 6 dimensions are necessary because a string needs a sufficient number of directions in which it can vibrate. If this number is smaller than 6, then problems such as negative probabilities occur. The space X carries a Riemannian metric and is very small compared to M. Among other constraints, supersymmetry imposes conditions on the metric of X that imply that it has to be a Calabi–Yau space. A Calabi–Yaumanifold has a complex structure such that the first Chern class vanishes, and the metric is Kähler for this complex structure. (A large class of examples of Kähler manifolds are complex submanifolds of complex projective spaces.) Calabi conjectured that all Kähler manifolds with vanishing first Chern class admit a Ricci-flat metric, which was later proven by S. T. Yau.