# Existence of twisted metaplectic categories

The paper Classification of metaplectic modular categories by Ardonne-Cheng-Rowell-Wang (2016) mentions (in Section 3) the Grothendieck ring for the metaplectic modular categories, i.e. $$SO(N)_2$$, $$N>1$$ odd (see below).

Now by exchanging $$1$$ and $$Z$$ in lines (2) and (3), we get a new family of fusion rings.

Question 1: Are there (unitary) fusion categories corresponding to this new family?
(which could be called twisted metaplectic)

Question 2: Can this procedure be extended to other fusion categories?

There is a way to make two new families: in the right hand-side of (2) and (3), put $$\oplus (X_1 \oplus X_2)^{\oplus \frac{N-1}{2}}$$ By this modification, you still have a fusion ring, but the objects $$X_1, X_2$$ are now of FPdim $$N$$. Again you have the usual and the twisted version. Note that after this modification, the usual case with $$N=3$$ corresponds to the Grothendieck ring of $$\mathrm{Rep}(S_4)$$.

Bonus question 1: Are there (unitary) fusion categories corresponding to these new families?

Bonus question 2: Can this procedure be extended to other fusion categories?

In this case, let $$\mathcal{C}$$ be an odd metaplectic category. In the notation above, let $$\lambda\in Z^2(\mathbb{Z}/2,Inv(\mathcal{C}))$$ be chosen with $$\lambda(1,1)=Z$$. Then a new tensor product defined by $$U\hat{\otimes} V\cong U\otimes V\otimes \lambda(d(U),d(V))$$ where $$d$$ is the grading degree function, utilizing the $$\mathbb{Z}/2$$ grading on $$\mathcal{C}$$. In particular $$X_1\hat{\otimes} X_2\cong X_1\otimes X_2\otimes Z\cong X_1\otimes X_1$$ etc. As $$Z$$ is a boson (we are using the braiding on $$\mathcal{C}$$ in our construction) Proposition 6.3 in loc. cit. says that we further twist the associativity by a $$3$$-cocycle.
One can proceed in a similar way for general $$N$$-metaplectic categories too (i.e. $$N$$ can be odd or even).