Construction of Hodge structures on the SO(3) modular functors

We prove that 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 ζᵣ. 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.