Four-dimensional string-string-string triality

Abstract 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 × SL(2, Z)u target space duality acting on the dilaton/axion, complex Kahler 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 × 0(6,22; Z)tu. The 0(6,22; Z) is perturbativefor 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 0(8,24; Z) transformations.