the anti-holomorphic component of the supercurrent. In a compact theory, the one-point function of any anti-commutator $[\bar{G}_0, O]$ would vanish. In a non-compact theory, with non-compact direction parameterized by the expectation value of the operator $\Phi$, we can imagine integrating by parts in the space of fields to obtain a term proportional to[^14] the torus one-point function of $O$ evaluated in the boundary theory ${\mathcal B}$, resulting in the following [*holomorphic anomaly equation*]{}. Including the spectator fermions to fix the normalizations, we propose: [^15] $$\begin{gathered}
\label{eqn.holomorphicanomaly}
\sqrt{-8\tau_2} \frac{\partial}{\partial\bar{\tau}} \bigl[\text{Witten genus of }{\mathcal F}\bigr] \\
=(\text{torus one-point function of } (-1)^{\frac{n}{4}}{:}\psi_1\cdots\psi_{n-1}\bar{G}{:}  \mbox{ in $\operatorname{Fer}(n-1) \times {\mathcal B}$})\end{gathered}$$ Here $\tau_2 = \frac1{2i}(\tau - \bar\tau)$ is the imaginary part of $\tau$. The sign of the square root $\sqrt{-8}$ is essentially arbitrary, and can be absorbed in the ambiguity in the sign of $\bar{G}$ or in the sign of the Ramond sector of ${\mathcal B}$. As a reality check, note that, in the IR limit, both sides are real-analytic modular of weight $(\frac{n-1}2, \frac32)$ with trivial multipliers.

Being a bit loose with the phase of the torus one-point function, we can write equation (\[eqn.holomorphicanomaly\]) as $$\label{eqn.holomorphicanomaly2}
\sqrt{-8\tau_2} \, \eta(\tau) \frac{\partial}{\partial\bar{\tau}} Z_{RR}({\mathcal F}) = (\text{torus 