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

Abstract

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 $ζ_r\in\mathbb{C}$. 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 Künneth formula. The construction is based on the homological models of Felder-Wieczerkowski and Martel.

Figures (26)

• Figure 1.1: The gluing map $q:{\newline}^1\overline{\mathcal{M}}_{0,{n_1}}(r)\times{\newline}^1\overline{\mathcal{M}}_{0,{n_2+1}}(r)\rightarrow {\newline}^1\overline{\mathcal{M}}_{0,{n}}(r)$ represented at the level of generic Riemann surfaces. The curve on the right is nodal.
• Figure 2.1: An example of a cut of a disk $\mathcal{D}=(D,\nu,\underline{\lambda})$ with $D=D^4$ into $\mathcal{D}_{\mu}'=(D^2,\mu,(\lambda_1,\lambda_3))$ and $\mathcal{D}_{\mu}"=(D^3,\nu,(\mu,\lambda_2,\lambda_4))$.
• Figure 4.1: Skein relations.
• Figure 4.2: The marked cube $([0,1]^3,P)$ for the Temperley Lieb algebra $T_4$ and a tangle in it. Only the boundary of the tangle is represented.
• Figure 4.3: The element $e_k$ of $T_n$.

Theorems & Definitions (167)

• Definition 1.1
• Definition 1.2
• Theorem 1.3: \ref{['theoremgeometricconstruction']}
• Corollary 1.4
• Theorem 1.5: \ref{['theoremintersectionform']}, \ref{['intersectionformpolarizes']} and \ref{['theoremintegralstructure']}
• Theorem 1.6: \ref{['gluingishodge']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4