This paper considers $A_\infty$-algebras satisfying an analytic bound with respect to a fixed norm. We define a notion of right Calabi–Yau (CY) structures on such $A_\infty$-algebras and show that these give rise to cyclic minimal models satisfying the same analytic bound. This strengthens a theorem of Kontsevich and Soibelman [Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, Preprint (2008), arXiv: 0811.2435], and yields a flexible method for obtaining the analytic potentials of Hua and Keller [Quivers with analytic potentials, Preprint (2019), arXiv: 1909.13517]. We apply these results to the endomorphism differential graded algebra (DGA) of polystable sheaves considered by Toda [Moduli stacks of semistable sheaves and representations of Ext-quivers, Geom. Topol. 22 (2018), 3083–3144], for which we construct a family of such right CY structures obtained from analytic germs of holomorphic volume forms. As a result, we find a canonical cyclic analytic $A_\infty$-structure on the Ext-algebra of a polystable sheaf, which depends only on the analytic-local geometry of its support. This yields an extension of Toda’s result [Geom. Topol. 22 (2018), 3083–3144] to the quasi-projective setting, and a new method for comparing cyclic $A_\infty$-structures of sheaves on different Calabi–Yau varieties.