barcode in vb.net 2008 Copyright 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use. in Java

Generation QR in Java Copyright 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use.

Copyright 2009 by The McGraw-Hill Companies, Inc. Click here for terms of use.
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In bosonic string theory, there are 26 space-time dimensions. So, a D25-brane would be a space-filling brane. In superstring theory, there are 10 space-time dimensions. So, a space lling brane has 9 spatial dimensions and is called a D9-brane. Chances are if you re reading this book you ve completed calculus so you re familiar with the notion of a hyperplane. When rst getting started, the best way to think about a D-brane is It is a hyperplane-like object. The endpoints of open strings are attached to it. This is illustrated in Fig. 13.1. Note, however, that not all D-branes are hyperplanes, but this is a good way to visualize them. The spatial dimensions not associated with the brane are called the bulk. The volume of the brane is called the world-volume. Note that time ows everywhere, in the bulk and on the D-brane as well. A model of our universe has been proposed where we live in a D3-brane and the bulk consists of the remaining extra spatial dimensions. Perhaps the most fundamental physical insight that has resulted from the study of D-branes is that The interactions of the standard model (electromagnetism, strong, and weak forces) are constrained to the brane. Gravity can escape from the brane. Gravitational forces are distributed in the brane and also throughout the higher dimensions. Hence, the strength of gravity is diluted by the higher dimensions. This explains why its strength is so different from that of the other known forces. For simplicity, we will discuss branes within the context of bosonic string theory.
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D-brane
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Open string with ends attached to D-brane
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Space-time outside D-brane is the bulk
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Figure 13.1 A D-brane is a hyperplane-like object to which open strings attach.
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CHAPTER 13 D-Branes
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The Space-Time Arena
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The easiest way to describe a Dp-brane mathematically is to use the light-cone gauge. To specify the D-brane, we need to choose which coordinates will satisfy Neumann boundary conditions and which coordinates satisfy Dirichlet boundary conditions. To use the light-cone gauge, we also need to define lightcone coordinates that will satisfy Neumann boundary conditions, these will include: Time One spatial coordinate, which we choose to be X 1 ( , ) For a Dp-brane, we let i = 2, , p in the light-cone gauge. Then as usual we de ne: X ( , ) = X 0 X1 2 (13.1)
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Neumann boundary conditions can be written as X =0 (13.2)
= 0 ,
So, the coordinates chosen to satisfy Neumann boundary conditions are X + ( , ) X ( , ) X i ( , ) i = 2, , p (13.3)
Let us suppose that the D-brane is located at x a . That is, letting a = p + 1, , d : xa = x a (13.4)
The remaining spatial coordinates will satisfy Dirichlet boundary conditions. We use a = p + 1, , d to denote these coordinates. In bosonic string theory we take d = 25 while in superstring theory d = 9. So the Dirichlet boundary conditions will be applied to X a ( , ) a = p + 1, , d (13.5)
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Given x a = x a , the Dirichlet boundary condition can be written as X a ( 0, ) = X a ( , ) = x a a = p + 1, , d (13.6)
Notice that we can also specify the Dirichlet boundary conditions by de ning:
X a = X a ( , ) X a ( 0, )
a = p + 1, , d
(13.7)
Then we could write the Dirichlet boundary condition as
Xa = 0
(13.8)
The coordinates are divided into two groups and given labels depending on boundary conditions that are applied: The coordinates with indices = , i = 2, ..., p are called NN coordinates since they satisfy Neumann boundary conditions at both ends. The coordinates with indices a = p + 1, ..., d are called DD coordinates since they satisfy Dirichlet boundary conditions at both ends. A simpli ed illustration of the boundary conditions is shown in Fig. 13.2. To summarize, a Dp-brane is located at x a and has extension along the x i directions.
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