Abelian and Non–Abelian Dualities in String Backgrounds

We present a brief discussion of recent work on duality symmetries in non–trivial string backgrounds. Duality is obtained from a gauged non–linear σ–model with vanishing gauge field strength. Standard results are reproduced for abelian gauge groups, whereas a new type of duality is identified for non–abelian gauge groups. Examples of duals of WZW models and 4–d black holes are given. String Theory and Physics are at the moment two unrelated subjects which are expected to have an interesting connection. The only available approach towards this connection is to study the structure and physical properties of the semiclassical string vacua given by classes of conformal field theories (CFT). For applications to Particle Physics, it is enough to assume a flat four-dimensional spacetime and parametrize the different models by an internal CFT restricted by some consistency conditions such as worldsheet modular invariance. In this way many classes of models have been studied and some, close to the Standard Model of Particle Physics give hope that strings may actually be related to the real world. Furthermore, duality and mirror symmetries have been identified in these vacua and play an important role in determining the low energy properties of the theory. To ask questions about gravitation in the context of String Theory, we have to relax the flat spacetime assumption and substitute it by a general noncompact CFT. These questions are of fundamental importance since the main motivation for studying string theories is to provide a consistent way of quantizing gravity. In particular it is of prime importance to study singularities of cosmology and black hole-type of geometries in the context of String Theory, since it is in those regimes that the standard field theory methods of General Rel-ativity fail to apply. Duality, being a property of all string vacua with abelian isometries [1], has also been found in these backgrounds and could provide a way to understand how string theory probes those singular geometries. A generalization of this duality symmetry to backgrounds with non–abelian isometries was recently discovered [2]. In this talk we will briefly discuss these developments. 2. Non–Compact String Backgrounds We know several ways to construct non–compact string vacua. The straightforward approach is to look for solutions of the string background equations. Except for a few exceptions [3], these are solutions of only the lowest order in α ′ equations [4] :