Type-IIB supersymmetric theories have an SL(2,$Z$) invariance, known as $U$ duality, which controls the nonperturbative behavior of the theory. Under SL(2,$Z$) the supercharges are doublets, implying that the bosonic charges would be singlets or triplets. However, among the bosonic charges there are doublet strings and doublet five-branes which are in conflict with the doublet property of the supercharges. It is shown that the conflict is resolved by structure constants that depend on moduli, such as the tau parameter, which transform under the same SL(2,$Z$). The resulting superalgebra encodes the nonperturbative duality properties of the theory and is valid for any value of the string coupling constant. The usefulness of the formalism is illustrated by applying it to purely algebraic computations of the tension of $\mathrm{}(p,q)\mathrm{}$ strings, and the mass and entropy of extremal black holes constructed from $D$-1-branes and $D$-5-branes. In the latter case the nonperturbative coupling dependence of the BPS mass and renormalization is computed for the first time in this paper. It is further argued that the moduli dependence of the superalgebra provides hints for four more dimensions beyond ten, such that the superalgebra is embedded in a fundamental theory which would be covariant under SO(11,3). An outline is given for a matrix theory in 14 dimensions that would be consistent with $M$(atrix) theory as well as with the above observations.

• Received 27 June 1997

DOI:https://doi.org/10.1103/PhysRevD.56.7954