The level-dependent decomposition conjecture for quantum K-theory of gerbes

Let $\mathfrak{X}$ be a banded $\mathbb{Z}_k$ gerbe on the Fano GIT quotient $X$. Let the Chern–Simons level correspond to a Ruan–Zhang level $(E,\ell)$ on $\mathfrak{X}$, for some bundle $E\to\mathfrak{X}$.

Gerbe decomposition conjecture. For the Ruan–Zhang/Chern–Simons level corresponding to ordinary quantum $K$-theory,

$$QK(\mathfrak{X})=QK\left(\coprod_{k^2}X\right),$$

whereas for general choices of level,

$$QK^\ell(\mathfrak{X})=QK^\ell\left(\coprod_k X\right).$$

The predicted $k^2$-fold decomposition in the ordinary-level case follows from independent one-form and finite symmetries with no self-'t Hooft anomaly. For general levels, the anomaly is expected to reduce this to $k$ copies, although special anomaly-free subgroups may produce larger multiplicities.