Orbifold Basics: Does Torus Lattice Admit Only Z_N?

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In summary, an orbifold is a generalization of a manifold that includes spaces with singularities. A torus lattice is a periodic lattice formed by the intersection of two or more tori in Euclidean space. When a torus lattice admits only Z_N, it means it has a rotational symmetry of order N. However, torus lattices can also admit other types of symmetries such as translations and reflections. Orbifolds and torus lattices have various applications in fields such as physics, chemistry, and materials science, where they can be used to study crystal structures and particle behavior in high-dimensional spaces.
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Does a given lattice of a torus (either twisted or untwisted) admit only certain orbifolds - ie. only specific [tex]N[/tex] of [tex]Z_N[/tex] ?

For example, consider the twisted torus lattice (in complex plane) in page 121 of Green, Schwarz & Witten 's Vol-2 book. It is said (in page 122) that the torus is special, in that, it has a [tex]Z_3[/tex] symmetry. But why cannot it have a [tex]Z_2[/tex] symmetry? Here, under a [tex]Z_2[/tex] orbifold, the fixed points change to [tex]z=\{0, 0.5\}[/tex] - fixed under [tex]e^{i\pi}[/tex] - right?
 
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I would like to first clarify that a torus lattice refers to the arrangement of points on a torus, while an orbifold refers to a specific type of symmetry that can be applied to a space. So, the question of whether a given lattice of a torus admits only certain orbifolds is a bit misleading. Instead, we should ask whether a given lattice of a torus has certain symmetries or properties that allow for specific orbifolds to be applied.

In the case of the twisted torus lattice mentioned in the forum post, it is true that it has a Z_3 symmetry. This means that when we apply a rotation of 120 degrees around the origin, the lattice remains unchanged. This is because the points on the lattice are arranged in a way that is symmetric under this rotation. However, this does not mean that the twisted torus lattice cannot admit other orbifolds, such as a Z_2 symmetry.

In fact, a Z_2 symmetry can also be applied to the twisted torus lattice, but it would not be a full symmetry of the lattice. This is because a Z_2 symmetry would only fix two points on the lattice, as mentioned in the post. These two points, z=\{0, 0.5\}, would remain fixed under a rotation of 180 degrees, which is the defining property of a Z_2 symmetry. However, the rest of the lattice would not remain unchanged under this rotation.

In summary, the twisted torus lattice does have a Z_3 symmetry, but it can also admit other orbifolds, including a Z_2 symmetry. The reason why it is often referred to as having a Z_3 symmetry is because this is the most complete symmetry that the lattice possesses. It is important to note that the symmetries and properties of a lattice can vary depending on the specific arrangement of points, so not all torus lattices will have the same symmetries or admit the same orbifolds.
 
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The answer to this question lies in the concept of orbifolds and their symmetries. First, let's define what an orbifold is. An orbifold is a generalization of a manifold, which allows for certain types of singularities. These singularities are called fixed points and are points on the manifold that are fixed under the action of a group. In the case of a torus, the fixed points are the points where the lattice intersects itself.

Now, to address the question, we need to understand the symmetries of an orbifold. A symmetry of an orbifold is a transformation that leaves the orbifold invariant. In the case of a torus lattice, the symmetries are rotations and reflections. These symmetries correspond to the group Z_N, where N is the number of fixed points.

In the example given, the twisted torus lattice has a Z_3 symmetry because there are three fixed points on the lattice. This means that there are three different ways to rotate or reflect the lattice and still have it look the same. In the case of a Z_2 symmetry, there would only be two fixed points, and therefore, only two possible symmetries.

Therefore, the lattice of a torus does not admit only certain orbifolds, but rather, the orbifold is determined by the number of fixed points and the corresponding symmetry group. In the case of the twisted torus lattice, it admits a Z_3 orbifold because of its three fixed points. It is not possible for it to have a Z_2 orbifold because it does not have the necessary two fixed points.

In conclusion, the number of fixed points and their corresponding symmetries determine the orbifold of a torus lattice. The Z_N orbifold is not exclusive to the torus lattice, but rather, it is determined by the lattice's properties.
 

What is an orbifold?

An orbifold is a mathematical object that generalizes the concept of a manifold to include spaces with singularities.

What is a torus lattice?

A torus lattice is a type of periodic lattice that is formed by the intersection of two or more tori in Euclidean space.

What does it mean for a torus lattice to admit only Z_N?

When a torus lattice admits only Z_N, it means that the lattice is only compatible with rotations by multiples of 2π/N. This is often referred to as a rotational symmetry of order N.

Can a torus lattice admit other types of symmetries?

Yes, a torus lattice can admit other types of symmetries besides Z_N. These can include translations, reflections, and glide reflections.

What are some applications of orbifolds and torus lattices?

Orbifolds and torus lattices have applications in various fields such as physics, chemistry, and materials science. They can be used to study the structure and properties of crystals, as well as to understand the behavior of particles in high-dimensional spaces.

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