Stability conditions on cyclic categories I: basic definitions and examples

A triangulated category $\mathcal{C}$ with a canonical Bott’s isomorphism $[2]\stackrel{\sim }{\to }id$ is called a cyclic category in this paper. We give a new notion of stability conditions on a $k$-linear Krull-Schmidt cyclic category. Given such a stability condition $\sigma$, we can assign a Maslov index to each basic loop in such a category. If all Maslov indexes vanish, we get ${\mathcal{C}}^{\mathrm{\prime }},{\sigma }^{\mathrm{\prime }}$ as the $\mathbb{Z}$-lifts of $\mathcal{C},\sigma$ respectively such that ${\mathcal{C}}^{\mathrm{\prime }}$ is a $\mathbb{Z}$-graded triangulated category and ${\sigma }^{\mathrm{\prime }}$ is a Bridgeland stability condition on ${\mathcal{C}}^{\mathrm{\prime }}$. Moreover, we showed that there is an isomorphism $$Sta{b}^{0,e}(\mathcal{C})\stackrel{\simeq }{\to }BStab({\mathcal{C}}^{\mathrm{\prime }})$$ where $Sta{b}^{0,e}(\mathcal{C})$ denotes the equivalence classes of stability conditions which are deformation equivalent to $\sigma$, and $BStab({\mathcal{C}}^{\mathrm{\prime }})$ denotes the space of Bridgeland stability conditions on ${\mathcal{C}}^{\mathrm{\prime }}$. We provide examples of stability conditions on a simple cyclic category. We also discuss some interesting phenomena in these examples, such as the chirality symmetry breaking phenomenon and nontrivial monodromy. The chirality symmetry breaking phenomenon involves stability conditions which can not be lifted to Bridgeland stability conditions.