to define “the” elliptic index.

Our convention, standard in algebraic topology, will be to trade nontrivial multipliers for nontrivial weights of modular forms, by including a normalization factor of $\eta(\tau)^n$ whenever necessary, where $\eta(\tau) = q^{1/24} \prod_{j=1}^\infty (1-q^j)$ is Dedekind’s eta function. The combination $\eta(\tau)^n Z_{RR}(\tau,\bar\tau)$ is sometimes called the [*Witten genus*]{} of a gravitationally-anomalous SQFT, and we will use that term. It is automatically modular without multiplier, of weight $(\frac n 2, 0)$.

When $n>0$, the Witten genus can be interpreted as follows. Consider the nonanomalous SQFT $\operatorname{Fer}(n) \otimes {\mathcal F}$, where $\operatorname{Fer}(n) = \operatorname{Fer}(1)^{\otimes n}$ means the holomorphic CFT of $n$ chiral fermions $\psi_1,\dots,\psi_n$, acted upon trivially by the right-moving supersymmetry. Deformations of ${\mathcal F}$ correspond to deformations of $\operatorname{Fer}(n) \otimes {\mathcal F}$ which preserve the $\operatorname{Fer}(n)$-subsector. The $n$ free chiral fermions in $\operatorname{Fer}(n)$ are called [*spectators*]{}. Because of the zero modes of the chiral fermions, the plain elliptic genus of $\operatorname{Fer}(n) \otimes {\mathcal F}$ vanishes. But because we have a distinguished $\operatorname{Fer}(n) \subset \operatorname{Fer}(n) \otimes {\mathcal F}$, we find a distinguished observable, namely ${:}\psi_1\cdots\psi_n{:}$. The Witten genus of the gravitationally-anomalous SQFT ${\mathcal F}$ is precisely the one-point function of $(-1)^{\frac{n}{4}} {:}\psi_1\cdots\psi_n{:}$ in $\operatorname{Fer}(n) \otimes {\mathcal F}$. [^9] 