The conjecture allows one to relate the number of rational curves on a Calabi-Yau manifold (encoded as Gromov–Witten invariants) to integrals from a family of varieties (encoded as period integrals on a variation of Hodge structures).
These relations were original discovered by Candelas, de la Ossa, Green, and Parkes[1] in a paper studying a generic quintic threefold in
Shortly after, Sheldon Katz wrote a summary paper[3] outlining part of their construction and conjectures what the rigorous mathematical interpretation could be.
Originally, the construction of mirror manifolds was discovered through an ad-hoc procedure.
The string theory in the A-model only depended upon the Kahler or symplectic structure on
This polynomial is equivalently described as a global section of the line bundle
[1][5] Notice the vector space of global sections has dimension
[6] (non-zero scalers of the base field) given equivalent spaces.
corresponds to the equivalence classes of polynomials which define smooth Calabi-Yau quintic threefolds in
[7] Now, using Serre duality and the fact each Calabi-Yau manifold has trivial canonical bundle
Using the Lefschetz hyperplane theorem the only non-trivial cohomology group is
Because of the Bogomolev-Tian-Todorov theorem, all such deformations are unobstructed, so the smooth space
Now, notice there is only a single dimension of complex deformations of this family, coming from
are called world-sheets and represent the birth and death of a particle as a closed string.
For simplicity, only genus 0 curves were considered originally, and many of the results popularized in mathematics focused only on this case.
acting on the Hilbert space of states, but only defined up to a sign.
The main idea behind the Mirror Symmetry conjectures is to study the deformations, or moduli, of the complex structure
In particular, from a physics perspective,[8]: 1–2 the super conformal field theory of a Calabi-Yau manifold
should be equivalent to the dual super conformal field theory of the mirror manifold
There are two variants of the non-linear sigma models called the A-model and the B-model which consider the pairs
the nonlinear sigma model of the string theory should contain the three generations of particles, plus the electromagnetic, weak, and strong forces.
are the naive number of rational curves with homology class
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.
These moduli spaces can be equipped with a virtual fundamental class
With the original construction, the A-model considered was on a generic quintic threefold in
in the A-model subsection, there is a dual superconformal field theory which has states in the eigenspace
This is because locally the Gauss-Manin connection acts as the interior product.
Mathematically, the B-model is a variation of hodge structures which was originally given by the construction from the Dwork family.
Relating these two models of string theory by resolving the ambiguity of sign for the operators