Stability conditions on \(\mathrm{CY}_N\) categories associated to \(A_n\)-quivers and period maps
Stability conditions on triangulated categories were introduced by \textit{T. Bridgeland} [Ann. Math. (2) 166, No. 2, 317--345 (2007; Zbl 1137.18008)], motivated by \textit{M. R. Douglas}' work on \(\pi\)-stability [in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. III: Invited lectures. Beijing: Higher Education Press. 395--408 (2002; Zbl 1008.81074)]. Their existences on the derived category \(D^{b}(X)\) of an algebraic variety \(X\) and the geometric, topological structures of their moduli space \(Stab(D^{b}(X))\) are very interesting questions. Note that a stability condition on \(D^{b}(X)\) involves the choice of a heart \(\mathcal{A}\) of a t-structure, which is an abelian category, like \(Coh(X)\subseteq D^{b}(X)\). However, even we start with an algebraic variety, the heart \(\mathcal{A}\) could parametrize something `non-commutative', e.g. representations of some quiver, etc. So it is also natural to study \(Stab(D^{b}(Q,I))\) for the derived category of representations of a quiver \(Q\) with relation \(I\). In this paper, the author considers the Calabi-Yau \(N\)-category \(\mathcal{D}_{n}^{N}\) associated with an \(A_{n}\)-quiver, where objects are finite dimensional dg-modules over the Ginzburg \(CY_{N}\) algebra associated with an \(A_{n}\)-quiver. He shows that the distinguished connected component of \(Stab(\mathcal{D}_{n}^{N})\) can be described as the universal cover of the space of certain polynomials with simple zeros. As \(\mathcal{D}_{n}^{N}\) can be viewed as the derived Fukaya category of Lagrangian \(N\)-spheres in the universal deformation space of \(A_{n}\)-singularity, his result is expected to be compared with corresponding structures in the B-side, under the philosophy of mirror symmetry. The author discusses Frobenius structure on \(Stab(\mathcal{D}_{n}^{N})\) and its connection with \textit{K. Saito}'s theory of primitive forms in the B-side [Publ. Res. Inst. Math. Sci. 19, 1231--1264 (1983; Zbl 0539.58003)]. The proof is based on the recent work of \textit{T. Bridgeland} and \textit{I. Smith} [Publ. Math., Inst. Hautes Étud. Sci. 121, 155--278 (2015; Zbl 1328.14025)] on describing stability conditions of certain \(CY_3\) category (associated with quivers with superpotential) by the space of quadratic differentials with simple zeros on Riemann surfaces.
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