The Zwegers shadow as $U(1)$-projected intrinsic torsion\\ on twelve-manifolds

Abstract

On a closed almost-quaternionic $\text{Spin}^c$ orbifold of real dimension $4n \geq 12$, with non-trivial determinant line bundle $L$ and a compatible involution $\sigma$, two obstruction classes from independent traditions coexist: Swann's intrinsic torsion, which measures the failure of quaternion-Kahler holonomy, and the Bruinier--Funke shadow, which measures the failure of modularity of the associated partition function. We show that the $U(1)$-projection of the intrinsic torsion under the $Sp(1) \to U(1)$ reduction induced by the $\text{Spin}^c$ structure and the shadow of the equivariant elliptic genus are cohomologically equivalent classes in the $\sigma$-odd component of rational cohomology, normalised by $c_{1}(L)$; the proof uses Swann's absorption of almost-quaternionic torsion into hyperkahler holonomy, its orbifold extension, Leray injectivity, and the Bruinier--Funke characterisation of shadows. A functorial refinement --- a spectral push-forward into $S_{3/2}(\Gamma_{0}(36), \chi)$, together with a compatible map of short exact sequences --- holds at the level of sheaves unconditionally, and pointwise under one explicitly stated structural hypothesis. On the twelve-dimensional orbifold $K_8 = [(\mathbb{CP}^2 \times S^2) \times_{w} (T^2/\mathbb{Z}_2)]^{\text{Spin}^c}$ the determinant line is $\mathcal{O}(3)$ by the Euler sequence, and the integer $3$ coincides with the coefficient $3$ in the completion $E_2^* = E_2 - \frac{3}{\pi \tau_2}$, the two anchorings independent. The oddness of $c_{1}(L)$ forces the involution to act freely on the shifted momentum lattice; constant configurations are consequently a symmetry-protected critical stratum of the spectral-rigidity functional on $\sigma$-even conformal deformations of the pillowcase factor, the kernel decouples, and the critical structure assembles multiplicatively to $6 \times 2 = 12 = \chi(K_8)$. The lowest eigenvalue of the $\text{Spin}^c$ Dirac operator on the toroidal factor is exactly $\pi$, independently of the modulus.