visual basic print barcode label Mathematical Aside: The Euler Characteristic in Java

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Mathematical Aside: The Euler Characteristic
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The Euler characteristic is a number which describes the shape of a topological space. Consider a polyhedron, and let V be the number of vertices, E be the number of edges, and F be the number of faces. Then the Euler characteristic is
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=V E+F
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(2.28)
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In string theory, we often want to know whether or not two geometric shapes or topologies are similar to one another in a speci c way. In particular, we want to know if we can continuously deform one shape into another (imagine working with clay and deforming one shape into another without breaking the clay apart, or introducing or losing any holes). Formally, a homeomorphism is a deformation of a geometric object into a new shape by stretching or compressing and being it, without tearing or breaking it. For instance, the quintessential example is a donut and a coffee cup (conveniently paired for police of cers). You could use a continuous deformation to transform one into the other or vice versa. So we say that a coffee cup and a donut are homeomorphic. On the other hand, a sphere and a donut are not
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CHAPTER 2 Equations of Motion
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homomorphic the donut has a hole but a sphere does not. The bottom line is there is no way to transform the donut into the sphere. If a geometric shape is homeomorphic to a sphere, then the Euler characteristic is (2.29) Many shapes have an Euler characteristic which vanishes. Some examples of this include a torus, a m bius strip, and a Klein bottle. Another example is a cylinder, which also has = 0 (see Fig. 2.2). Why is this interesting for us If the worldsheet of a string has a vanishing Euler characteristic, then it is possible to write the auxiliary eld h as a two-dimensional at space metric. That is, we take [using the choice of coordinates for the worldsheet as ( , )] 1 0 h = 0 1 (2.30)
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=V E+F = 2
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Mobius strip
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Klein bottle
Cylinder
Figure 2.2 Some surfaces with a vanishing Euler characteristic. When the Euler characteristic vanishes, we can de ne the auxiliary eld such that it has a representation of the at space Minkowski metric.
String Theory Demysti ed
Now notice that with this choice, h = det h = 1. We also have h X X = X X + X X = X 2 + X 2 In this case, we are able to write the Polyakov action in the remarkably simple form SP = T 2 d ( X 2 X 2 ) 2 (2.31)
EXAMPLE 2.3 Find the equations of motion using the Polyakov action as written in Eq. (2.27) when the auxiliary eld takes the form of the at space metric. SOLUTION In this case we have SP = = = T 2 d h h X X 2 T 2 d ( X X + X X ) 2 T 2 d ( X X + X X ) 2
So, we can write the lagrangian as L = X X + X X = X X + X X Therefore, L = X X + X X = X = X X X
L = X X + X X = X = X X X
CHAPTER 2 Equations of Motion
The Euler-Lagrange equations are L L + =0 X X Hence, we nd that the equations of motion for the relativistic string are 2 X 2 2 X 2 =0
(2.32)
(2.33)
Light-Cone Coordinates
It will be convenient to call upon light-cone coordinates in string theory. First, let s look at how light-cone coordinates can be de ned in Minkowski space-time in general and then consider having them in the context of the worldsheet and the equations of motion of the string. As we will see, this will simplify the way we write the action and the resulting equations of motion. For simplicity, let s take ordinary (3 + 1) dimensional space-time. The contravariant coordinates are x = ( x 0 , x1 , x 2 , x 3 ) where x 0 = ct and x 1 = x , x 2 = y, x 3 = z say. We form light-cone coordinates by choosing one spatial direction, which in this case we take to be x1, and forming linear combinations of it with x0 as follows: x+ = x 0 + x1 2 x = x 0 x1 2 (2.34)
These are two null or lightlike coordinates, but you can think of x+ as a timelike coordinate and x as a spacelike coordinate. Hence when we use indices and summations, we will treat + as a 0 index and as a 1 index. The other coordinates x2 and x3 are left alone. It is easy to derive the inverse relationship using Eq. (2.34). We have x0 = x+ + x 2 x1 = x+ x 2
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