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Homological mirror symmetry was first unveiled in 1993 by Maxim Kontsevich. It has been developing ever since. We can understand homological mirror symmetry in terms of D-branes. This is despite the fact that D-branes were not discovered until 1995 by Joseph Polchinski. D-branes are the subsurfaces that open strings attach to at their endpoints. Homological mirror symmetry provides us with two kinds of D-branes: A-branes and B-branes. These are terms introduced by Edward Witten. If we consider two Calabi-yau manifolds, X and X’, then A-branes on X are equivalent to B-branes on X’. A-branes are objects of symplectic geometry while B-branes are the product of algebraic geometry. These were two branches of geometry that were thought to be completely separate. However, they are related by mirror symmetry.
The end points of open strings lie on a D-brane.
Maxim Kontsevich
Before we can understand homological mirror symmetry, we must first understand what a brane is in string theory. A brane is the notion of a point particle that has been raised or generalized to higher dimensions.
There are also objects in string theory known as D-branes. Open strings (which are shaped like a line segment, unlike closed loop shaped strings) have two end points, that are required to lie on a D-brane. "D" stands for Dirichlet, which is a boundary condition that they satisfy.
Some more definitions that are important for our conversation:
Objects: are mathematical structures. These could be sets, vector spaces or topological spaces.
Categories: a mathematical construction that consists of objects.
Morphisms: there is a set of morphisms between any pair of two objects. These are functions between the objects.
Now, we can consider a category where the objects are D-branes. There can be two branes, known as α and β. The morphisms will be open strings between α and β. It would thus follow that there was an A and B model of topological string theory.
In the homological mirror symmetry program, the derived category of coherent sheaves on one Calabi-yau is dual to the Fukaya category of it's mirror manifold. This program is an example of what mirror symmetry in topological string theory would look like. It is also a bridge between complex and symplectic geometry.
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