a pair, so really there are four supersymmetries. This is important because it implies there is a pair of operators acting on the Hilbert space of states, but only defined up to a sign. This ambiguity is what originally suggested to physicists there should exist a pair of Calabi-Yau manifolds which have dual string theories, one's that exchange this ambiguity between one another. The space has a complex structure, which is an integrable almost-complex structure , and because it is a Kähler manifold it necessarily has a symplectic structure called the Kähler form which can be complexified to a complexified Kähler form which is a closed -form, hence its cohomology class is in The main idea behind the Mirror Symmetry conjectures is to study the deformations, or moduli, of the complex structure and the complexified symplectic structure in a way that makes these two dual to each other. In particular, from a physics perspective, the super conformal field theory of a Calabi-Yau manifold should be equivalent to the dual super conformal field theory of the mirror manifold . Here conformal means conformal equivalence which is the same as and equivalence class of complex structures on the curve . There are two variants of the non-linear sigma models called the A-model and the B-model which consider the pairs and and their moduli. A-model Correlation functions from String theory Given a Calabi-Yau manifold with complexified Kähler class the nonlinear sigma model of the string theory should contain the three generations of particles, plus the electromagnetic, weak, and strong forces. In order to understand how these forces interact, a three-point function called the Yukawa coupling is introduced which acts as the correlation function for states in . Note this space is the eigenspace of an operator on the Hilbert space of states for the string theory. This three point function is "computed" as using Feynman path-integral techniques where the are the naive number of rational curves with homology class , and . Defining these instanton numbers is the subject matter of Gromov–Witten theory. Note that in the definition of this correlation function, it only depends on the Kahler class. This inspired some mathematicians to study hypothetical moduli spaces of Kahler structures on a manifold. Mathematical interpretation of A-model correlation functions In the A-model the corresponding moduli space are the moduli of pseudoholomorphic curves or the Kontsevich moduli spaces These moduli spaces can be