Four-dimensional string/string/string triality

In six space-time dimensions, the heterotic string is dual to a Type IIA string. On further toroidal compactification to four space-time dimensions, the heterotic string acquires an SL(2,Z)S strong/weak coupling duality and an SL(2,Z)T x SL(2,Z)U target space duality acting on the dilaton/axion, complex Kaehler form and the complex structure fields S, T, U respectively. Strong/weak duality in D=6 interchanges the roles of S and T in D=4 yielding a Type IIA string with fields T, S, U. This suggests the existence of a third string (whose six-dimensional interpretation is more obscure) that interchanges the roles of S and U. It corresponds in fact to a Type IIB string with fields U, T, S leading to a four-dimensional string/string/string triality. Since SL(2,Z)S is perturbative for the Type IIB string, this D=4 triality implies S-duality for the heterotic string and thus fills a gap left by D=6 duality. For all three strings the total symmetry is SL(2,Z)S x O(6,22;Z)TU. The O(6,22;Z) is perturbative for the heterotic string but contains the conjectured non-perturbative SL(2,Z)X, where X is the complex scalar of the D=10 Type IIB string. Thus four-dimensional triality also provides a (post-compactification) justification for this conjecture. We interpret the N=4 Bogomol'nyi spectrum from all three points of view. In particular we generalize the Sen-Schwarz formula for short multiplets to include intermediate multiplets also and discuss the corresponding black hole spectrum both for the N=4 theory and for a truncated S-T-U symmetric N=2 theory. Just as the first two strings are described by the four-dimensional elementary and dual solitonic solutions, so the third string is described by the stringy cosmic string solution. In three dimensions all three strings are related by O(8,24;Z) transformations. (orig.)