The p-modular TQFT structure conjecture

Let $p\geq 5$ be a prime and let

$$q_p=\frac{p-3}{2}.$$

Let $\Gamma_g$ denote the mapping class group of a genus-$g$ surface, let $\dov{\cal U}^{(k)}_p$ be the $p$-modular TQFTs extending the $\Gamma_g$-representations arising from the short exact sequences in the Johnson–Morita extension, let ${\cal V}_p^{[\leq 0]}$ be the reduced TQFT over $\mathbb F_p$, and let ${\cal V}_{\zeta_p}^I$ be the integral Reshetikhin–Turaev TQFT over $\mathbb Z[\zeta_p]$. A TQFT is half-projective with parameter ${\sf x}$ if its composition law is modified by a factor ${\sf x}^{\mu(C_2,C_1)}$, where $\mu(C_2,C_1)$ is the rank of the connecting map in the relevant Mayer–Vietoris sequence.

The p-modular TQFT structure conjecture.

A) There are TQFTs $\dov{\cal U}^{(k)}_p$ which extend the $\Gamma_g$-representations from the Johnson–Morita extension.

B) The TQFT ${\cal V}_p^{[\leq 0]}$ over $\mathbb F_p$ is a quotient of sub-TQFTs of the $q_p$-fold symmetric product $\operatorname{S}^{q_p}\dov{\cal U}^{(1)}_p$.

C) The TQFT ${\cal V}_{\zeta_p}^I$ is half-projective with parameter ${\sf x}=(\zeta_p-1)^{q_p}$ and consequently has a block-structured $\mathbb Z[\zeta_p]$-basis.

The three assertions are related: B depends on A, while the block structure in C is intended to imply half-projectivity. The conjecture describes the structure of the $p$-modular and integral TQFTs and their relation to mapping class group representations; its resolution is not supplied in the source.