Symmetries of string, M- and F-theories

The d = 10 type II string theories, d = I I 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 of 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 11 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.