Homological Mirror Symmetry - Hodge Diamond

The following diamond is called Hodge Diamond on where the dimension of (p,q)-differential forms hp,q are aligned as the coordinate (p,q) and form a diamond shape. In the case of p = 0, 1, 2, q = 0, 1, 2 i.e. 2-dimensional,

h2,2 h2,1 h1,2 h2,0 h1,1 h0,2 h1,0 h0,1 h0,0

In the case of an elliptic curve, which is viewed as a 1-dimensional Calabi–Yau manifold, the Hodge diamond is especially simple: it is the following figure.

1 1 1 1

In the case of a K3 surface, which is viewed as 2-dimensional Calabi–Yau manifold, since the Betti numbers are {1, 0, 22, 0, 1}, their Hodge diamond is the following figure.

1 0 0 1 20 1 0 0 1

In the 3-dimensional case a very interesting thing happens. There are sometimes mirror pairs, say M and W, that have symmetric Hodge diamonds each other along diagonal straight line.

M's diamond:

1 0 0 0 a 0 1 b b 1 0 a 0 0 0 1

W's diamond:

1 0 0 0 b 0 1 a a 1 0 b 0 0 0 1

M and W correspond to A- and B-model in string theory. Mirror symmetry does not only replace the homological dimensions but also symplectic structure and complex structure on the mirror pairs. That is the origin of homological mirror symmetry.