Anomaly Matching between Topological Superconductors and $\mathcal{N}=8$ Supergravity

Abstract

The boundary of a $3{+}1$-dimensional topological superconductor carries a $\mathbb{Z}_{16}$-valued global (Dai-Freed) anomaly, famously matched by the $16$ fermions of a Standard Model generation once a certain $\mathbb{Z}_4$ symmetry is gauged. In a supergravity completion, though, a gravitino contributes $-7/16$ rather than $\pm1/16$, obstructing this $\mathbb{Z}_4$ in the Minimal Supersymmetric Standard Model. We observe that four-dimensional $\mathcal{N}=8$ supergravity evades this fate. Its fermions, eight gravitini in the $\mathbf 8$ and fifty-six spin-$\tfrac12$ states in the $\mathbf{56}=Λ^3\mathbf 8$ of the $SU(8)$ R-symmetry, saturate the anomaly. The cancellation holds for every admissible $\mathbb{Z}_4$ structure, is consistent with the vanishing of the continuous $SU(8)$ anomaly, and follows from a Smith homomorphism identifying the supergravity anomaly with that of the topological superconductor. In maximal supergravity this $\mathbb{Z}_4$ stays anomaly-free and gaugeable despite its eight gravitini. All numbers are reproduced by an open-source library described in the Supplemental Material.