Calabi-Yau manifold

It is said that there are 10 dimensions in String Theory, out of which 4 are our ordinary Length, Width, Height and Time. The other 6 dimensions are said to be closely packed together in a manifold called the Calabi-Yau manifold.

A 2D slice of the 6D Calabi-Yau quintic manifold.

A Calabi–Yau manifold, or 'Calabi–Yau space', is a special type of manifold. It is described in certain branches of mathematics such as algebraic geometry.

The Calabi–Yau manifold's properties, such as Ricci flatness(condition when the average curvature of every point on a space cancels out to be zero), have applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime might take the form of a 6-dimensional Calabi–Yau manifold. This led to the idea of string theory mirror symmetry.

The Calabi-Yau manifolds are complex manifolds are extensions or broader forms of K3 Surfaces in any number of complex dimensions (1 complex dimension = 2 real dimensions). K3 surfaces is a 4-dimensional (4D) complex surface which is smooth, compact, and Ricci Flat (look at the former paragraph).

There are different types of such manifolds, depending on the number of complex dimensions they have. For example, Calabi-Yau 2-fold has 2 complex dimensions, Calabi-Yau 3-fold has 3, and so on.

Use in String Theory

The various properties of the Calabi-Yau manifold, like ricci-flatness, supersymmetry make it very useful in string theory.

1. The Ricci Flatness property of the Calabi-Yau manifold makes it useful in String Theory. String Theory requires the background spacetime to be flat at large scales Compactifying extra dimensions on the Ricci Flat space ensures that the overall 4-D shape of the Universe remains constant.
2. Supersymmetry is a symmetry that relates bosons (particles with integer spin) and fermions (particles with half-integer spin). Supersymmetry is a key part of string theory because it helps explain various phenomena, such as particle masses and forces.Calabi-Yau manifolds preserve a special kind of supersymmetry, making them the ideal candidate for compactifying extra dimensions while maintaining the consistency of supersymmetric string theories.
3. The complex geometry of Calabi-Yau manifolds provides a lot of flexibility in how the extra dimensions are compactified, which impacts the types of particles and forces in the low-energy world we observe. The Hodge Numbers and other topographical properties of this manifolds influence things like the number of groups of particles, the types of interactions, and more.