Special Langrangian geometry and slightly deformed algebraic geometry (spLag and sdAG)

TL;DR

This paper explores the relationship between special Lagrangian geometry, mirror symmetry, and a new field called slightly deformed algebraic geometry, highlighting parallels with complexified gauge theories and invariants.

Contribution

It introduces the concept of slightly deformed algebraic geometry and connects it to special Lagrangian geometry and mirror symmetry, expanding the understanding of invariants in these fields.

Findings

Parallelism between special Lagrangian geometry and gauge theories.

Introduction of slightly deformed algebraic geometry as a new subject.

Discussion of geometric invariants related to mirror symmetry.

Abstract

The special geometry of calibrated cycles, closely related to mirror symmetry among Calabi--Yau 3-folds, is itself a real form of a new subject, which we call slightly deformed algebraic geometry. On the other hand, both of these geometries are parallel to classical gauge theories and their complexifications. This article explains this parallelism, so that the appearance of invariants of new type in complexified gauge theory (see Donaldson--Thomas [D-T] and Thomas [T]) can be accompanied by analogous invariants in the theory of special Lagrangian cycles, for which the development is at present much more modest than in gauge theory. We discuss related geometric constructions, arising from mirror symmetry and symplectic geometry.