Higher-supersymmetry refined generating-function modularity conjecture

Let $\mathfrak Y$ be a compactification with higher supersymmetry, let $\gamma=(p^0,p^A,q_A,q_0)$ be an electromagnetic charge vector, and let

$$F^{\rm cl}(X)=-\frac{\kappa_{ABC}X^AX^BX^C}{6X^0},\qquad \kappa_{AB}=\kappa_{ABC}p^C.$$

After restricting to the sublattice orthogonal to any null eigenvectors of $\kappa_{AB}$, let $\Lambda_p$ be the resulting charge lattice with quadratic form $\kappa_{AB}$, and define the refined generating functions $h^{\rm ref}_{p,\mu}(\tau,z)$ from the refined BPS indices. Higher-supersymmetry mock-Jacobi conjecture. The refined generating functions $h^{\rm ref}_{p,\mu}(\tau,z)$ are higher-depth mock Jacobi forms of the weight and index specified by the corresponding lattice data, with modular completions satisfying the refined analogues of the stated completion and derivative formulas. The functions entering these formulas are constructed using the same lattice and have a zero of order $n-1$ at $z=0$. This extends the mock-modular description of refined BPS generating functions to compactifications with $\mathcal N=4$ and $\mathcal N=8$ supersymmetry, including cases where the quadratic form has higher-signature or degenerate directions; the source assumes the required modification of the generalized error functions in problematic cases, and no explicit construction is supplied there.