# Does WRT invariant detect hyperelliptic involution on the genus 2 surface?

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? $$Z(M_U) \stackrel{?}{=} Z(M_{-U})$$

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 $$a \not\cong a^*$$.

I’m not sure about the invariant due to the framing ambiguities. But for Witten-Reshetikhin-Turaev TQFT representations, I think it cannot detect the hyperelliptic involution in genus 1 and 2, since the basis for the TQFT vector space have the symmetry in the sense flipping by hyperelliptic involution gives the same basis vector back, hence in these two cases hyperelliptic involution acts trivially (projective). Which means also the invariant cannot detect it up to U(1).