Axial Anomalies of Maximally Supersymmetric Tensor Theories

We revisit anomalies of $(4,0)$ and $(3,1)$ maximally supersymmetric tensor theories in $d=6$. A $(4,0)$ on-shell tensor multiplet descends to that of the $d=5$ maximal supergravity upon a dimensional reduction, hypothesized to offer a strong-coupled UV completion of the latter in the same sense of $(2,0)$ theories as the UV completion of $d=5$ $N=2$ pure Yang-Mills. The gravitational anomalies, found to be nonvanishing, had been computed, although its relevance in the absence of the $d=6$ metric is not obvious. We perform a comprehensive anomaly computation for (4,0) and (3,1) tensor supermultiplets, respectively, for $Sp(4)$ and Sp(3)× Sp(1) R-symmetry anomalies and the mixed R-gravitational anomaly thereof, and find that anomalies involving R-symmetries cancel out identically. We close with questions on how to address the anomaly in this class of theories with no general covariance.