Construction of Hodge structures on the $\mathrm{SO}(3)$ modular functors

We prove that $\mathrm{SO}(3)$ modular functors in genus $0$ have geometric origin and support integral variations of Hodge structures for any odd level $r$ and $r$-th root of unity $\zeta_r\in\mathbb{C}$. We identify the TQFT intersection forms and integral structures with the geometric ones. Moreover, the gluing property of the modular functors is recovered geometrically as a K\"unneth formula. The construction is based on the homological models of Felder-Wieczerkowski and Martel.