Equivariant Quantisation of Hypertoric Varieties via Symplectic Cuts

We study the equivariant geometric quantisation of a non-compact hypertoric orbifold of complex dimension $2n$ by compact symplectic cuts. When the defining arrangement has a non-empty bounded base region and the fixed lift is integral, we prove that, at every positive regular integral level $d$, the space of sections on the cut is the sum of the fibre-rotation weight spaces of weight at most $d$ in the original quantisation. The higher cohomology of the cut pre-quantum line orbibundle vanishes, so its equivariant index is the character of its actual section space. Using the moment polyptychs introduced in a companion paper, we give an orbifold Kawasaki-Riemann-Roch localisation formula for this character. We also compute it as a weighted lattice-point sum: a residual weight $x$ occurs at level $d$ if and only if the arrangement slack function satisfies $\mathcal{S}(x) \leq d$, with multiplicity $\binom{d - \mathcal{S}(x) + n}{n}$. The examples compare these calculations in rank one, including a cut with an orbifold point, and in rank two, for smooth cuts that are not toric.