Calabi-yau manifold

Compactification in string theory: the Calabi-yau manifold

Compactification is when the extra dimensions are small and closed up on themselves. Compactification in string theory borrows from Kaluza-Klein theory. Theodor Kaluza in 1919, proposed a mathematical theory that could incorporate electromagnetism into general relativity. His idea was that we live in a 5-dimensional spacetime, instead of 4. In Kaluza's theory, electromagnetic waves were ripples in the 5th dimension. However, this theory lacked substantial evidence. Extra dimensions also made a comeback in the 1970s in theories of supergravity, which also had problems. The leading candidate today for a theory of everything or a theory of quantum gravity is string theory. Superstring theory suggests that we live in a 10-dimensional universe. But where are the extra dimensions?

In string theory, the extra dimensions, in this setting, are wrapped up on themselves: taking on the geometry of the Calabi-yau manifold. The Calabi-yau manifold is a shape or space into which the extra dimensions required by string theory are curled up into. These geometries are consistent with the equations of string theory. The extra dimensions of string theory cannot be crumpled up in any arbitrary way. The equations that emerge from string theory, severely restrict their geometry to a particular manifold. These are hidden and curled up dimensions. The Calabi-yau manifold, is the geometry that extra dimensions in string theory must take on, in order to provide a realistic model of particle physics. Indeed, the extra dimensions of spacetime in string theory are conjectured to take on the geometry of the Calabi-yau manifold. They satisfy the stringent requirements demanded by the theory. The Calabi-yau manifold in named after the mathematician Eugenio Calabi from the university of Pennsylvania and Shing-Tung Yau from Harvard, who proved his conjecture. The Calabi-yau manifold is typically taken to be 6-dimensional, as it serves it's function in superstring theory. These dimensions are curled up so small that they cannot be observed in even our most high energy experiments. We can also picture, that if superstring theory is correct, than at every point of 3-dimensional extended space, there are also 6 compactified dimensions, taking on the Calabi-yau geometry that strings can vibrate in. These dimensions would exist everywhere as a component of the spatial fabric. However, they are so tiny that we are completely unaware of their existence. 

There are tens of thousands of shapes that meet these requirements: of being a Calabi-yau manifold. I know, this sounds like a lot, however, compared to infinity, this collection is, indeed, quite exclusive. That being said, the Calabi-yau geometry is a rare one. Ironically, the hope was that string theory would give us a single picture of reality. However, many possible models of the universe arise from string theory and physicists don't know how to choose the right manifold! A particular Calabi-yau compactification can determine the number of generations of elementary particles. Of course there are 3 generations of matter, each successive generation having greater mass than the preceding generation. The first generation (up quark, down quark and electron) particles are the most common and the most stable, while the more massive particles will decay into these lower generation states. We are looking for the unique Calabi-yau manifold candidate for the Standard Model! String theorists have yet to single out a single Calabi-yau compactification that describes our physical world. In fact, just when Andrew Strominger thought that he had made a unique discovery of a Calabi-yau compactification, Gary Horowitz found several more. Shing-Tung Yau admitted that there are thousands of Calabi-yau manifold candidates! But which one (if any) is correct? Getting string theory to match the real world has proven to be a real adventure, and is still incomplete!

The Calabi conjecture is a mathematical hypothesis put forward in the 1950s by the geometer Eugenio Calabi. This conjecture states that a space that satisfies certain topological requirements can also satisfy a stringent geometric requirement for curvature. This requirement is known as Ricci flatness. Calabi’s conjecture also covered cases where the Ricci curvature was not necessarily zero. Proving the Calabi conjecture meant proving the existence of a Ricci flat metric.

Calabi-yau manifolds are a broad class of geometric shapes with zero Ricci curvature. They were shown to be physically possible by the proof of the Calabi conjecture. These shapes are complex, meaning, they have an even number of real dimensions. 

1. Compact Kahler manifold: A Kahler manifold is a complex manifold named for the geometer Erich Kahler.  Kahler manifolds are endowed with a special kind of holonomy. This holonomy preserves the manifold’s complex structure under the operation of parallel transport. 

2. Vanishing first Chern class: Chern classes are a set of fixed properties that characterize complex manifolds. The number of Chern classes for a complex manifold equals the number of complex dimensions. The last Chern class is equal to the Euler characteristic. Chern classes were introduced in the 1940s by S. S. Chern. 

3. Ricci flat metric: Ricci curvature is a kind of curvature related to the flow of matter in spacetime. This is according to the equations of Albert Einstein’s general theory of relativity. 

In 1984-85, Edward Witten, Andrew Strominger, Gary Horowitz and Philip Candelas, realized that the extra 6 dimensions of superstring theory had to be compactified on the geometry of the Calabi-yau manifold. These theorists showed the significance of this particular method of compactifying these extra dimensions. They showed that if you wrap up 6 of the 10 dimensions in superstring theory, while still preserving the supersymmetry from the other 4 dimensions, the 6 dimensions could be represented by a Calabi-yau manifold! Compactification on a Calabi-yau manifold preserves just the right amount of supersymmetry for the theory to reproduce the Standard Model. If there was too much supersymmetry, you can not have left handed particles that have different interactions from right handed ones. 

Thus, the Calabi-yau manifold became relevant in string theory, in order to obtain or derive the Standard Model. After a few choices of Calabi-yau manifolds, these theorists showed that you could break down the symmetry of string theory to resemble the Standard Model much more closely. This particular class of 6-dimensional shapes can meet these conditions. These are the Calabi-yau shapes, now relevant in string theory!

In 1984, Andrew Strominger and Philip Candelas, were seeking the 6-dimensional geometry that could account for the compact hidden extra dimensions of string theory. The internal shape of that space had to be compact. The curvature also had to satisfy the requirements of general relativity and the symmetries of string theory. This led, Gary Horowitz and Edward Witten to the spaces that Shing-Tung Yau had proven in the Calabi conjecture in the late 1970s. This bridge between mathematics and physics is how the Calabi-yau manifold is made applicable in string theory. These four theorists finished their work in 1984 and their formal paper came about in 1985. This paper coined the term “Calabi-yau space”. 

Mirror symmetry is a correspondence between two topologically distinct Calabi-yau manifolds that can give rise to the exact same physical theory.

1. Homological mirror symmetry

2. SYZ conjecture