Please help me on calculating orbifold euler characteristics

[dm368]

how could v 'calculate' the orbifold euler char. using String theorists' formula for the same ? For example, i know the Euler char. of S^1/Z_2 is 1, when we identify the x to -x points of S^1(not the antipodal points i.e. dimetrically opposite points, but putting a mirror along the horizontal line cutting the circle and identifying the mirror image points). But i don't know how to use the string theoreists' formula and don't get what summation over commutating group elements means(in S^1/Z_2 case, they are {1,-1} ).
S^2/Z_2 would be more elaborative perhaps ?
Thanks

 

[AndyW]

I am familiar with string theory and its application in understanding the properties of orbifolds. In order to calculate the orbifold Euler characteristic using the string theorists' formula, we first need to understand the concept of the Euler characteristic and its relevance in string theory.

The Euler characteristic is a topological invariant that describes the shape of a manifold, which is a mathematical object that is locally similar to Euclidean space. In string theory, the Euler characteristic is used to describe the number of degrees of freedom in a particular configuration of strings.

To calculate the orbifold Euler characteristic using the string theorists' formula, we need to first understand the concept of orbifolds. An orbifold is a generalization of a manifold, where some points have a symmetry group acting on them. In the case of S^1/Z_2, the orbifold is obtained by identifying points on a circle that are related by a reflection symmetry.

The string theorists' formula for calculating the orbifold Euler characteristic is given by:

χ = (1/|G|)Σχ(g),

where χ is the orbifold Euler characteristic, G is the symmetry group of the orbifold, and g is an element of the symmetry group. This formula takes into account the contributions from each element of the symmetry group to the overall Euler characteristic.

In the case of S^1/Z_2, the symmetry group is {1,-1}, and the summation over the group elements can be written as:

Σχ(g) = χ(1) + χ(-1),

where χ(1) and χ(-1) are the contributions to the Euler characteristic from the identity element and the reflection symmetry, respectively.

For S^1/Z_2, the Euler characteristic of the underlying manifold (in this case, a circle) is 1. Therefore, using the string theorists' formula, we can calculate the orbifold Euler characteristic as:

χ = (1/2)(1+1) = 1.

This result is consistent with the known Euler characteristic of S^1/Z_2, which is also 1.

For a more elaborate example, let's consider S^2/Z_2. In this case, the orbifold is obtained by identifying points on a sphere that are related by a reflection symmetry. The symmetry group is again {1,-1}, but the underlying manifold (in this case, a sphere) has an Euler characteristic of

 

[R0man]

The orbifold Euler characteristic is a mathematical concept that is used to describe the topological properties of a space or surface that has been modified by a symmetry group. In simple terms, it is a way to measure the number of holes or handles in a surface after it has been folded, twisted, or otherwise modified.

To calculate the orbifold Euler characteristic, you can use the following formula:

χ = (1/|G|) * Σχ(g)

Where χ is the orbifold Euler characteristic, G is the symmetry group, and χ(g) is the Euler characteristic of the individual group elements. The summation is taken over all the commutating group elements, which means that the group elements that do not change the shape of the orbifold.

In the case of S^1/Z_2, the symmetry group is {1, -1} and the individual Euler characteristics are 1 and -1. Plugging these values into the formula, we get:

χ = (1/2) * (1 + (-1)) = 0

This means that the orbifold Euler characteristic of S^1/Z_2 is 0, indicating that there are no holes or handles in the surface.

For S^2/Z_2, the calculation would be more complex as the symmetry group is larger and there are more individual Euler characteristics to consider. It would involve identifying the group elements and their corresponding Euler characteristics, and then applying the formula to get the final result.

In the context of string theory, the orbifold Euler characteristic is important for understanding the properties of Calabi-Yau spaces, which are used in superstring theory to compactify extra dimensions. The formula above is a simplified version of the full string theorists' formula, which takes into account additional factors such as the number of dimensions and the curvature of the space.

In summary, to calculate the orbifold Euler characteristic, you need to identify the symmetry group, determine the individual Euler characteristics of the group elements, and then apply the formula to get the final result. While it may seem complicated at first, with practice and understanding of the underlying concepts, you can easily use this formula for various orbifolds, including more complex ones like S^2/Z_2.