Special Lagrangian geometry as slightly deformed algebraic geometry (geometric quantization and mirror symmetry)

The special geometry of calibrated cycles, which is closely related to the mirror symmetry among Calabi–Yau 3-manifolds, is in fact only a specialization of a more general geometry, which may naturally be called slightly deformed algebraic geometry or phase geometry. On the other hand, both of these geometries are parallel to classical gauge theory and its complexification. This article explains this parallelism. Hence the appearance of new invariants in complexified gauge theory (see [9] and [24]) is accompanied by the appearance of analogous invariants in the theory of special Lagrangian cycles, whose development is at present much more modest. Algebraic geometry is transformed into special Lagrangian geometry by the geometric Fourier transform (GFT). Roughly speaking, this construction coincides with the well-known “spectral curve” constructions (see [3], [11] and elsewhere) plus phase geometry.