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 $\pm 1/16$, 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 $8$ and fifty-six spin-$\frac{1}{2}$ states in the $56={\Lambda }^{3}8$ of the $SU(8)$ R-symmetry, cancel 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 and the Rarita-Schwinger ghost-subtraction weight 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.

4 pages + 1 page supplemental material, 1 figure, 1 table; v2 - 1 reference and corresponding discussion added