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 equation which 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 on a generic quintic threefold in . B-model Correlation functions from String theory For the same Calabi-Yau manifold in the A-model subsection, there is a dual superconformal field theory which has states in the eigenspace of the operator . Its three-point correlation function is defined as where is a holomorphic 3-form on and for an infinitesimal deformation (since is the tangent space of the moduli space of Calabi-Yau manifolds containing , by the Kodaira–Spencer map and the Bogomolev-Tian-Todorov theorem) there is the Gauss-Manin connection taking a class to a class, hence can be integrated on . Note that this correlation function only depends on the complex structure of . Another formulation of Gauss-Manin connection The action of the cohomology classes on the can also be understood as a cohomological variant of the interior product. Locally, the class corresponds to a Cech cocycle for some nice enough cover giving a section . Then, the insertion product gives an element which can be glued back into an element of . This is because on the overlaps giving hence it defines a 1-cocycle. Repeating this process gives a 3-cocycle which is equal to . This is because locally the Gauss-Manin connection acts as the interior product. Mathematical interpretation of B-model correlation functions Mathematically, the B-model is a variation of hodge structures which was originally given by the construction from the Dwork family. Mirror conjecture Relating these two models of string theory by resolving the ambiguity of sign for the operators led physicists to the following conjecture: for a Calabi-Yau manifold there should exist a mirror Calabi-Yau manifold such that there exists a mirror isomorphism giving the compatibility of the associated A-model and B-model. This means given and such that under the mirror map, there is the equality of correlation functions This is significant because it relates the number of degree genus curves on a quintic threefold in (so ) to integrals in a variation of Hodge structures. Moreover, these