TITLE: Does WRT invariant detect hyperelliptic involution on the genus 2 surface?
QUESTION [5 upvotes]: The Witten-Reshetikhin-Turaev invariant cannot detect the hyperelliptic involution on the genus 1 surface, and that if $M_U$ is the mapping torus for a mapping class group element $U\in \mathrm{Mod}(\Sigma_1)$, then 
$$Z(M_U) = Z(M_{-U})$$
where $-I\in \mathrm{Mod}(\Sigma_1)$ is the hyperelliptic involution. 
What about the genus 2 case? If $-I\in \mathrm{Mod}(\Sigma_2)$ is the hyperelliptic involution on the genus 2 surface, then is there any $U\in \mathrm{Mod}(\Sigma_2)$, for which the following does not hold?
$$Z(M_U) \stackrel{?}{=} Z(M_{-U})$$

REPLY [2 votes]: Unless I'm misunderstanding what's meant by hyperelliptic involution, WRT invariants do, in fact, detect the hyperelliptic involution in genus 1.  Let $C$ be the modular tensor category (MTC) corresponding to the WRT theory.  The Hilbert space of a torus has a basis indexed by the simple objects $\{a\}$ of $C$, and the hyperelliptic involution sends $a$ to $a^*$.  It is not true that $a \cong a^*$ for all MTCs C and simple objects $a$, so WRT invariants detect the hyperelliptic involution in genus 1.  (Note that this argument works regardless of how we lift the hyperelliptic involution to a morphism of framed manifolds.)
A similar argument works in higher genus.  One can choose a "spine" basis for the Hilbert space such that the basis vectors are permuted by the involution, and it's easy to check that this permutation is non-trivial for an MTC which contains a simple object $a$ such that $a \not\cong a^*$.