Quintic periods and stability conditions via homological mirror symmetry

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].

20 pages, 5 figures, a substantial revision, Comments are welcome

References in corpus (7)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary
• Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
• The Landau-Ginzburg to Calabi-Yau Dictionary for D-Branes
• Wall-crossing structures in Donaldson-Thomas invariants, integrable systems and Mirror Symmetry
• Gepner type stability conditions on graded matrix factorizations
• The Physical Mirror Equivalence for the Local P^2