Quintic periods and stability conditions via homological mirror symmetry

TL;DR

This paper establishes a connection between quintic periods, stability conditions, and homological mirror symmetry for the Fermat Calabi-Yau threefold, leading to new categorical bases, wall-crossings, and modular forms.

Contribution

It introduces Bridgeland stability conditions via homological mirror symmetry, linking quintic periods with categorical and modular structures in Calabi-Yau geometry.

Findings

Derived bases of quintic periods categorically

Wall-crossings explained by quintic periods

Identification of a quasimodular form related to quintic periods

Abstract

For the Fermat Calabi-Yau threefold and the theory of stability conditions [Bri07], there have been two mathematical aims given by physical reasoning. One is that we should define stability conditions by central charges of quintic periods [Hos04,Kon12,KonSoi13], which extend the Gamma class [KKP,Iri09,Iri11]. The other is that for well-motivated stability conditions on a derived Fukaya-type category, each stable object should be a Lagrangian [ThoYau]. We answer affirmatively to these aims with the simplest homological mirror symmetry (HMS for short) of the Fermat Calabi-Yau threefold [Oka09,FutUed] and stability conditions of Bridgeland type, which we introduce. With HMS, we naturally obtain stability conditions of Bridgeland type by the monodromy around the Gepner point. As consequences, we obtain bases of quintic periods and the mirror map [CdGP] categorically, wall-crossings by quintic periods, and a quasimodular form [KanZag] attached to quintic periods by motivic Donaldson-Thomas invariants [KonSoi08]. The quasimodular form is of the quantum dilogarithm and of a mock modular form [Zag06].