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Mathematics > Algebraic Geometry

arXiv:1304.0937 (math)
[Submitted on 3 Apr 2013 (v1), last revised 18 Aug 2013 (this version, v2)]

Title:Quintic periods and stability conditions via homological mirror symmetry

Authors:So Okada
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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].
Comments: 20 pages, 5 figures, a substantial revision, Comments are welcome
Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Classical Analysis and ODEs (math.CA); Category Theory (math.CT); Symplectic Geometry (math.SG)
Cite as: arXiv:1304.0937 [math.AG]
  (or arXiv:1304.0937v2 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1304.0937

Submission history

From: So Okada [view email]
[v1] Wed, 3 Apr 2013 12:40:39 UTC (14 KB)
[v2] Sun, 18 Aug 2013 11:12:21 UTC (34 KB)
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