2022/11/30 by Yucheng Liu, Liu, Yucheng
Mathematics · #14B05 #14F08 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph)
paper · pdf · doi:10.48550/arxiv.2211.16939
openalex publication_date 2022/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A triangulated category C with a canonical Bott's isomorphism [2]\xrightarrow∼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 σ, we can assign a Maslov index to each basic loop in such a category. If all Maslov indexes vanish, we get C',σ' as the ℤ-lifts of C,σ respectively such that C' is a ℤ-graded triangulated category and σ' is a Bridgeland stability condition on C'. Moreover, we showed that there is an isomorphism Stab0,e(C)\xrightarrow≃ BStab(C') where Stab0,e(C) denotes the equivalence classes of stability conditions which are deformation equivalent to σ, and BStab(C') denotes the space of Bridgeland stability conditions on C'. 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.