Symmetries of string, M and F-theories

The d=10 type II string theories, d=11 M-theory and d=12 F-theory have the same symmetry group. It can be viewed either as a subgroup of a conformal group OSp(1|64) or as a contraction of OSp(1|32). The theories are related by different identifications of their symmetry operators as generators of OSp(1|32). T- and S-dualities are recognized as redefinitions of generators. Some (s,t) signatures of spacetime allow reality conditions on the generators. All those that allow a real structure are related again by redefinitions within the algebra, due to the fact that the algebra OSp(1|32) has only one real realization. The redefinitions include space/space, time/time and space/time dualities. A further distinction between the theories is made by the identification of the translation generator. This distinguishes various versions of type II string theories, in particular the so-called *-theories, characterized by the fact that the P_0 generator is not the (unique) positive-definite energy operator in the algebra.