The semisimplification principle for Rozansky–Witten invariants

Let $X$ be a non-compact space whose Rozansky–Witten invariant $Z_{\mathrm{RW}[X]}(M_3)$ computes invariants associated with a non-semisimple modular tensor category, denoted $\operatorname{MTC}[X]$. In the equivariant localization formula

$$Z_{\mathrm{RW}[X]}(S^1\times\Sigma_g)=\sum_{\lambda}(S_{0\lambda})^{2-2g},$$

the labels $\lambda$, equivalently the Bethe vacua in the corresponding expression, are considered. The semisimplification principle. The sum runs over simple modules in the semisimplification of $\operatorname{MTC}[X]$. This proposal is motivated by examples in which non-semisimple modular categories produce Rozansky–Witten invariants for non-compact targets; its general validity is presented as a suggestion and remains open.