In mathematics, Mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold" which allows one to relate Gromov–Witten invariants to period integrals on a variation of Hodge structures. In short, this means there is a relation between the number of genus algebraic curves of degree on a Calabi-Yau variety and integrals on a dual variety. These relations were original discovered by Candelas, De la Ossa, Green, and Schwarts in a paper studying a generic quintic threefold in as the variety and a construction from the quintic Dwork family giving. Shortly after, Sheldon Katz wrote a summary paper outlining part of their construction and conjectures what the rigorous mathematical interpretation could be.
Recall that a generic quintic threefold in is defined by a homogeneous polynomial of degree. This polynomial is equivalently described as a global section of the line bundle. Notice the vector space of global sections has dimensionbut there are two equivalences of these polynomials. First, polynomials under scaling by given equivalent spaces. Second, projective equivalence is given by the automorphism group of, which is dimensional. This gives a dimensional parameter spacesince, which can be constructed using Geometric invariant theory. Now, using Serre duality and the fact each Calabi-Yau manifold has trivial canonical bundle, the space of deformations has an isomorphismwith the part of the Hodge structure on. Using the Lefschetz hyperplane theorem the only non-trivial cohomology group is since the others are isomorphic to. Using the Euler characteristic and the Euler class, which is the top Chern class, the dimension of this group is. This is becauseUsing the Hodge structure we can find the dimensions of each of the components. First, because is Calabi-Yau, sogiving the Hodge numbers, hencegiving the dimension of the moduli space of Calabi-Yau manifolds. Because of the Bogomolev-Tian-Todorov theorem, all such deformations are unobstructed, so the smooth space is in fact the moduli space of quintic threefolds. The whole point of this construction is to show how the complex parameters in this moduli space are converted into Kähler parameters of the mirror manifold.
Mirror manifold
There is a distinguished family of Calabi-Yau manifolds called the Dwork family. It is the projective familyover the complex plane. Now, notice there is only a single dimension of complex deformations of this family, coming from having varying values. This is important because the Hodge diamond of the mirror manifold has
.
Anyway, the family has symmetry groupacting byNotice the projectivity of is the reason for the conditionThe associated quotient variety has a crepant resolution given by blowing up the singularitiesgiving a new Calabi-Yau manifold with parameters in. This is the mirror manifold and has where each Hodge number is.
In string theory there is a class of models called non-linear sigma models which study families of maps where is a genus algebraic curve and is Calabi-yau. This space has a complex structure, which is an integrablealmost-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 complexifiedKähler formwhich is a closed -form, hence its cohomology class is inThe 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. 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 modulich 38 pg 729.
A-model
In the A-model the corresponding moduli space are the moduli of pseudoholomorphic curvespg 153or the Kontsevich moduli spacesThese moduli spaces can be equipped with a virtual fundamental class
or
which is represented as the vanishing locus of a section of a sheaf called the Obstruction sheaf over the moduli space. This section comes from the differential equationwhich can be viewed as a perturbation of the map. It can also be viewed as the Poincaré dual of the Euler class of if it is a Vector bundle. With the original construction, the A-model considered was a generic quintic threefold in.
B-model
Mathematically, the B-model is a variation of hodge structures which was originally given by the construction from the Dwork family.
