Indices, Triality, and Ultraviolet Divergences for Supersymmetric Theories

Transverse rotation-group representation indices are shown to appear in one-loop radiative corrections. Some conjectures are made on how such index structure generalizes to higher orders. An unusual correlation between the absence of UV divergences in supersymmetric theories and the equality of such indices for fermions and bosons is described. This correlation is argued to be a fundamental indicator of higher-order UV behavior for $D=10\mathrm{}$ ($N=4\mathrm{}$) supersymmetric Yang-Mills theory and $D=11\mathrm{}$ ($N=8\mathrm{}$) supergravity. Within this context, the significance of O(8) triality is discussed.