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Andrew Strominger
Shing-Tung Yau
Eric Zaslow
SYZ is an acronym for those who published the 1996 paper: Andrew Strominger, Shing-Tung Yau and Eric Zaslow from Northwestern University.
SYZ 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.
2-dimensions
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.
4-dimensions
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.
6-dimensions
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.
The simplest way to visualize the SYZ conjecture of mirror symmetry is with a 2-dimensional torus. This torus is composed of circles and it parametrized by a 1-dimensional loop. In 4-dimensions, we have a K3 surface, composed of 2-dimensional tori, where B is an ordinary sphere. In 6-dimensions, we have a Calabi-yau, composed of 3-dimensional tori and parametrized by a 3-sphere.
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