Let $G$ be a finite group, $\tau$ a group automorphism of $G$ of period two and $m$ a natural number. Following [1, Definition 2.1], a complex fusion category $\mathcal{C}$ is called a quadratic category with $(G,\tau,m)$ if its Grothendieck ring has basis $\{X_g, Y_g \ | \ g \in G\}$ and fusion rules (let $e$ be the neutral element of $G$):
- $X_gX_h = X_{gh}$,
- $X_gY_e = Y_g = Y_eX_{g^{\tau}}$,
- $Y_e^2 = X_e + m\sum_{g \in G} Y_g$.
Note that quadratic just requires the action of $G$ on the non-group part of the basis to be transitive, as stated in the second line above, in particular, we can have $Y_{g_1}=Y_{g_2}$ with $g_1 \neq g_2$.
Let $\mathcal{C}$ be a spherical quadratic category with $(G,\tau,m)$. By [1, Theorem 2.2], if $G$ is an odd group and $m$ is an odd number, then $G$ is abelian and $g^{\tau} = g^{-1}$ for any $g \in G$.
The Haagerup category is a quadratic category with $(C_3, -1, 1)$.
Question: Is there a complex fusion category, quadratic with $(C_3, 1, 1)$?
According to above statements, such a fusion category would be a twisted Haagerup category without spherical structure, in fact without pivotal structure by applying [2, Corollary 2.14].
Recall that the existence of a pivotal structure on every fusion category is a well-known conjecture [3, Conjecture 2.8]. If this conjecture is true then above question has a negative answer, but if this conjecture is false then above category could be a first counter-example. Anyway, it should be possible to answer above question independently of this conjecture.
References
[1]: Izumi, Masaki The classification of $3^n$ subfactors and related fusion categories. Quantum Topol. 9 (2018), no. 3, 473–562.
[2]: Ostrik, Victor. Pivotal fusion categories of rank 3. Mosc. Math. J. 15 (2015), no. 2, 373-396, 405.
[3]: Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor. On fusion categories. Ann. of Math. (2) 162 (2005), no. 2, 581-642.
Best Answer
Quick answer: no.
Andrew Schopieray pointed out to me the PhD thesis of Josiah E. Thornton [2].
Let $\mathcal{F}$ be a fusion ring with basis $B=\{ b_1, \dots, b_r \}$. Let $G$ be the group of invertible elements $b_i$ of $B$ (i.e. $\textrm{FPdim}(b_i)=1$). The fusion ring $\mathcal{F}$ is called quadratic if the action of $G$ on $B \setminus G$ is transitive. If $G=B$ such a fusion ring is called pointed, and if $|B \setminus G|=1$ it is called near-group. A fusion category with a quadratic Grothendieck ring is called (in the literature) a quadratic category or a generalized near-group category.
Theorem. A categorification $\mathcal{C}$ of a quadratic fusion ring must admit a spherical (so pivotal) structure.
Proof. By [2, Theorem IV.3.6.], $\mathcal{C}$ must be $\varphi$-pseudo-unitary (i.e. pseudo-unitary up to Galois automorphism), and then spherical (so pivotal) by [1, Proposition 2.16].
References
[1] V. Drinfeld, S. Gelaki, D. Nikshych, V. Ostrik, On braided fusion categories. I., Selecta Math. (N.S.) 16 (2010), no. 1, 1--119.
[2] J.E. Thornton, Generalized near-group categories, Thesis (Ph.D.)–University of Oregon (2012) 72 pp.