Mirror symmetry

There are two main avenues of inquiry for mirror symmetry: the SYZ conjecture (which is a more geometric approach to mirror symmetry) and the homological mirror symmetry approach (which uses more algebraic methods).

It began with casual conversations between Shing-Tung Yau and Andrew Strominger in 1995. The topic of the conversation was D-branes, which had entered string theory in an important way. Strominger’s paper showed how D-branes fit into the Calabi-yau geometry. Strominger and Yau were curious about these submanifolds and their role in string theory. The three theorists got together to publish a paper in June of 1996.  Proving of the SYZ conjecture would mean the existence of these Calabi-yau substructures. What does the SYZ conjecture say? A Calabi-yau can be divided into two highly entangled 3-dimensional shapes. One of these shapes is a 3-dimensional torus. Inverting this torus, or changing its radius from r to 1/r, will give rise to the mirror manifold. To understand the SYZ conjecture, we must understand the submanifold that the Calabi-yau is composed of. The submanifolds of Calabi-yau that the SYZ conjecture is concerned with are wrapped by D-branes. These submanifold D-branes have half of the dimensions of the manifold of which they are embedded. The simplest way to visualize this process is with a donut or 2-dimensional torus. There will be a special Lagrangian submanifold. In this case, it is a 1-dimensional loop through the hole of the donut. We can see this overall geometry as being a union of circles or loops. We call this auxiliary space B. B parametrizes this union of circles. Every point on B corresponds to a different circle. B is called the moduli space and it is an index of every subspace of the bigger manifold. B shows how all of these subspaces are arranged. Going up a complex dimension, from 2 to 4, our Calabi-yau becomes a K3 surface. The submanifolds here are 2-dimensional tori. Many of these 2-dimensional tori fit into this K3 surface. B, in this case, is a 2-dimensional sphere. Each point on this 2-dimensional sphere B corresponds to a different 2-dimensional tori. That is, except for 24 bad points. These pinched donuts have singularities. Going up one more complex dimension from 4 to 6, we have a Calabi-yau 3-fold. B, in this case is 3-sphere. This is a sphere with a 3-dimensional surface. We cannot readily picture this. The subspaces are 3-dimensional tori. We can now consider mirror symmetry. We have a manifold X, composed of submanifolds, catalogued by the moduli space B. We take these submanifolds, which have a radius of r, and invert them to the radius 1/r. A feature of string theory (T-duality to be specific) is that a change like this does not change the physics at all. The way T-duality works is that momentum and winding number, which can be reduced to integer values are swapped moving from the initial to the inverted radius of tori. The mathematics and physics in these two seemingly distinct situation are actually identical under this T-duality transformation. T-duality, according to the SYZ conjecture can be extended from circles to tori. The T in T-duality, in fact, stands for tori. It thus follows that the name of SYZ’s paper was “Mirror symmetry is T-duality.” Mirror symmetry and T-duality can go hand in hand in this picture. Inverting the radii of the submanifolds gives you a manifold that has an overall smaller radius. We can invert the radii of the 3-tori submanifolds that make up the Calabi-yau, gives rise to a set of mirror manifolds that are indistinguishable from a physics perspective. 

There is also homological mirror symmetry. This 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.