Versality in mirror symmetry

Abstract

One of the attractions of homological mirror symmetry is that it not only implies the previous predictions of mirror symmetry (e.g., curve counts on the quintic), but it should in some sense be 'less of a coincidence' than they are and therefore easier to prove. In this survey we explain how Seidel's approach to mirror symmetry via versality at the large volume/large complex structure limit makes this idea precise.

Special representatives of complexified Kähler classes

“…Let X be a Kähler manifold of dimension n. We will always assume that X is compact. Following the conventions of [44], we define a complexified Kähler form on X as…”

“…The B-field enters crucially in the definition of Fuk(X, ω C ): for example, objects are Lagrangians L ⊂ X, with respect to ω, endowed with a unitary connection, with curvature id ⊗B| L . Similarly, morphisms are defined in terms of holomorphic discs and the monodromy of the unitary connections, twisted by the B-field (see [44,Section 4.2.4]).…”

“…where Recall that adding to B a closed, integral (1, 1)-form changes the Fukaya category by an equivalence of A ∞ categories (see [44,Remark 4.11]), so (1.7) is still trying to fix a special complexified Kähler form yielding the same Fukaya category, up to equivalence.…”

“…A crucial notion in mirror symmetry is that of large volume limit (see e.g. [44,Section 4]). For us, it means that we should analyse the formal behaviour of our equation under the scaling…”