NONCOMMUTATIVE GEOMETRY AND DUALITY OF ORIENTIFOLD FIELD THEORIES

An important phenomenon in modern physics is that, often, very different field theories give rise to identical physical predictions. This is what physicists usually call “duality”, though it does not always agree with the mathematical meaning of the term. In string theories, important invariants are given by “brane charges”, which take their values in K-groups of the space-time manifold. Thus a duality of string theories should give rise to isomorphisms of K-groups. A number of years ago, V. Mathai and I studied this phenomenon in the context of T-duality of type II string theories and complex K-theory, and we showed that the desired K-theory isomorphisms arise from applications of Connes’ “Thom isomorphism” for crossed products by R. In this talk, I will discuss the more complicated situation of orientifold string theories, where one needs Atiyah’s KR-theory (slightly modified). We will see that in many cases, the desired KR-theory isomorphisms arise from explicit KK-equivalences for real C*-algebras, coming out of standard constructions in noncommutative geometry.