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Geometric structures on moduli spaces Lagrangian submanifolds. (English) Zbl 1220.53064

Introduction: Let \(M, N\) be a pair of Calabi-Yau 3-folds: we say that \(N\) is a mirror partner of \(M\) if the \(A\)-model string theory of \(M\) is isomorphic to the \(B\)-model string theory of \(N\) and so, in particular, conformal field theories associated to mirror partners are equivalent; this is a general pattern, up to now lacking precise mathematical formulation, one aspect of which is, roughly speaking, that “counting the numbers of rational curves of various degree in \(M\)” is the same as “calculating the period integrals of holomorphic forms in \(N\)” (and, consequently, we have the famous relations between the Hodge numbers of the mirror partners).
Few years ago, A. Strominger, S.-T. Yau and E. Zaslow [AMS/IP Stud. Adv. Math. 23, 333–347 (2001; Zbl 0998.81091), Nucl. Phys., B 479, No. 1–2, 243–259 (1996; Zbl 0896.14024), see also hep-th/9606040] argued that every Calabi-Yau 3-fold \(M\) with mirror partner \(N\) admits a family of “supersymmetric toroidal 3-cycles” (i.e. special Lagrangian tori equipped with flat \(U(1)\)-bundles) and \(N\) arises precisely as the compactification of the moduli space of such cycles, the latter being nothing but the complexification of the moduli space of special Lagrangian tori.
Immediately after, N. J. Hitchin [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 25, No. 3-4, 503–515 (1997; Zbl 1015.32022)] provided an excellent description of SYZ construction for arbitrary dimension.
Pushing a bit forward Hitchin’s result, in the present paper we show how the non-compactified moduli space of special Lagrangian submanifolds and its complexification can be endowed with a very tamed geometric structure; in particular, they allow atlases of local coordinates whose transition functions are restrictions of elements of a finite presentation subgroup of the special linear group.
This fact seems to be of general interest: in fact, on the one hand, compactification at the present time appears out of reach, on the other hand, the geometric structure allows us to treat the moduli space of special Lagrangian submanifolds and its complexification largely as compact manifolds. We intend to pursue to a larger extent this aspect of the theory in a forthcoming paper.
In the last section of the paper, some further features and generalizations in the special cases \(n=2, 3\) are also considered.

MSC:

53C38 Calibrations and calibrated geometries
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
32Q25 Calabi-Yau theory (complex-analytic aspects)
53D12 Lagrangian submanifolds; Maslov index
58D27 Moduli problems for differential geometric structures
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References:

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