The role of flat SL(2,C) connections in counting special Lagrangians

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There are numerous speculative proposals for constructing enumerative invariants of Calabi-Yau 3-folds by counting special Lagrangian submanifolds. One of the many difficulties in defining such invariants is that special Lagrangians may collapse to multiple covers as parameters vary. For two-fold covers specifically, work of Siqi He shows that the local deformation theory is governed by the existence of Z_2-harmonic 1-forms. On the other hand, Taubes originally defined Z_2-harmonic 1-forms to describe limits of diverging sequences of flat SL(2,C) connections. In this talk I will discuss the connection between these two theories, and discuss progress toward and obstructions to gluing results providing the obverse of He's result in the realm of flat SL(2,C) connections.

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