By “one-point function,” we will always mean the torus one-point function on nonbounding, aka Ramond–Ramond, tori.

The spectator fermions furthermore allow the sign ambiguity to be handled by demanding that $\operatorname{Fer}(n) \otimes {\mathcal F}$ have *trivialized* anomaly, including a choice of parity for its Ramond-sector ground state.[^10] More precisely, the sign ambiguity can be swapped for an ambiguity in the choice of generators of $\operatorname{Fer}(n)$. We will largely ignore the sign ambiguity in this paper, since we will not try to add different SQFTs together (and so we will never risk thinking we have found a cancellation when there was not one).

Another phenomenon in gravitationally-anomalous SQFTs becomes transparent when working with spectator fermions. Consider the Ramond-sector Hilbert space ${\mathcal H}_R$ for the spectated theory $\operatorname{Fer}(n) \otimes {\mathcal F}$. The decoupled $\operatorname{Fer}(n)$ subalgebra of the full observable algebra provides operators on this Hilbert space: specifically, ${\mathcal H}_R$ is naturally a module for the fermion zero modes,[^11] which form a copy of the $n$th Clifford algebra $\operatorname{Cliff}(n)$. Moreover, the SQFT $\operatorname{Fer}(n) \otimes {\mathcal F}$ compactified on $S^1$ automatically possesses a time-reversal structure — showing this is an interesting exercise, solved in [@GPPV §3.2.2] — and hence ${\mathcal H}_R$ possesses a real 