### String Triality

String triality was considered in 1995 by M. J. Duff et al. See the hexagon on page 3 and the cube on page 9. From the abstract:

In six spacetime dimensions, the heterotic string is dual to a Type IIA string. On further toroidal compactification to four spacetime dimensions, the heterotic string acquires an SL(2,Z) strong/weak coupling duality and an SL(2,Z)×SL(2,Z) target space T,U duality ... 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 ... 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 the [S dual] SL(2,Z) 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)×O(6,22;Z). ... In three dimensions all three strings are related by O(8,24;Z) transformations.

That last comment is quite intriguing in light of the more recent appearance of Leech lattices, moonshine math and octonionic triality. Note also the analogy with Sparling's three copies of twistor space, and the prominence of the modular group.

Shortly after this active period for string theory, Shenker suggested the existence of a third length scale, shorter than the string scale, and presumably associated to a triality. From our point of view it is quite natural to consider three scales. In terms of mass, there is the Riofrio mass of the universe, the lightest masses (of the neutrinos) and an intermediate Planck scale mass related to transitions in the spectra from particles to black holes. It is of no concern here that one is effectively invoking a scale smaller than the string scale. A zero length scale is reached at $\hbar = 0$, corresponding to the Riofrio horizon. Since particles appear to be composed of three components, and are all composed of each other, it follows that the triality probably extends to cosmic observables.

Turning this around, it makes sense to think of mass spectra from the point of view of cubic numbers. That is, non-associative cubic arrays instead of associative matrices. Unfortunately, we might have to invent these first. In the meantime, recall that numbers usually come from vector spaces (perhaps as dimensions) and even matrices usually belong to vector spaces (perhaps as algebras) so the simplest thing to do is to just continue working with operad maps into an End(V) operad, only we would like to work with a 3-dimensional analogue, which means first defining a notion of 3-End(V). But then we want to throw away linearity, so we need the higher category theory to guide us towards the right structure, because without it we are blind.

In six spacetime dimensions, the heterotic string is dual to a Type IIA string. On further toroidal compactification to four spacetime dimensions, the heterotic string acquires an SL(2,Z) strong/weak coupling duality and an SL(2,Z)×SL(2,Z) target space T,U duality ... 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 ... 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 the [S dual] SL(2,Z) 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)×O(6,22;Z). ... In three dimensions all three strings are related by O(8,24;Z) transformations.

That last comment is quite intriguing in light of the more recent appearance of Leech lattices, moonshine math and octonionic triality. Note also the analogy with Sparling's three copies of twistor space, and the prominence of the modular group.

Shortly after this active period for string theory, Shenker suggested the existence of a third length scale, shorter than the string scale, and presumably associated to a triality. From our point of view it is quite natural to consider three scales. In terms of mass, there is the Riofrio mass of the universe, the lightest masses (of the neutrinos) and an intermediate Planck scale mass related to transitions in the spectra from particles to black holes. It is of no concern here that one is effectively invoking a scale smaller than the string scale. A zero length scale is reached at $\hbar = 0$, corresponding to the Riofrio horizon. Since particles appear to be composed of three components, and are all composed of each other, it follows that the triality probably extends to cosmic observables.

Turning this around, it makes sense to think of mass spectra from the point of view of cubic numbers. That is, non-associative cubic arrays instead of associative matrices. Unfortunately, we might have to invent these first. In the meantime, recall that numbers usually come from vector spaces (perhaps as dimensions) and even matrices usually belong to vector spaces (perhaps as algebras) so the simplest thing to do is to just continue working with operad maps into an End(V) operad, only we would like to work with a 3-dimensional analogue, which means first defining a notion of 3-End(V). But then we want to throw away linearity, so we need the higher category theory to guide us towards the right structure, because without it we are blind.

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