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Meet Homological Mirror Symmetry

In this paper, we introduce the interested reader to homological mirror symmetry. After recalling a little background knowledge, we tackle the simplest cases of homological mirror symmetry: curves of genus zero and one. We close by outlining the current state of the field and mentioning what homological mirror symmetry has to say about other aspects of mirror symmetry.

Planar Homological Mirror Symmetry

In this article, we formulate a planar limited version of the B-side in homological mirror symmetry that formularizes Chern-Simons-type topological open string field theory using homotopy associative algebra (A∞ algebra). This formulation is based on the works by Dijkgraaf and Vafa. We show that our formularization includes gravity/gauge theory correspondence which originates in the AdS/CFT dua...

Equivariant Localization and Mirror Symmetry

Planar Homological Mirror Symmetry Ii

In the planar limit of B-model open string field theory in the Dijkgraaf-Vafa context, an enlarged cyclic symmetry is formulated by introducing the idea of homotopy Lie categories (L∞ categories) on a set of sequences of holomorphic functions (prepotentials with Whitham deformations) following the work of Itoyama and Kanno.

Homological Mirror Symmetry for the Quartic Surface

Warning. This preprint is in finished form, and should be readable by itself. However, the argument uses certain general properties of Fukaya categories, the proofs of which will appear in [61] (under preparation). I apologize for this unsatisfactory state of affairs.