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CHAPTER 13 D-Branes
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a set of D-branes with spatial dimensions p, q, r ,... in various orientations. However, here we will stick to the simplest case, which is to consider two Dp-branes that are parallel but located at different coordinates x1a and x 2a. We will describe this case in a moment and see how the energy from stretching a string between the branes changes the mass spectrum. However, before doing that we take a brief aside to introduce Chan-Paton factors. Chan-Paton factors were introduced into string theory because Yang-Mills theories are necessary to describe the particle interactions of the standard model of particle physics. Before D-branes were known about, the technique used was to attach non-abelian degrees of freedom to the endpoints of open strings. These degrees of freedom were denoted quark and antiquark, respectively. These names came about by historical accident, string theory was originally proposed as a description of the strong interaction, but it was later displaced from that role by quantum chromodynamics(QCD). There are i = 1,..., N possible states of a string endpoint. Since an open string has two endpoints, it has two Chan-Paton indices ij. An open string state can be written as:
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a p; a = p; ij ij i , j =1 N
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a The ij are matrices that are called Chan-Paton factors. It turns out that amplitudes obtained when including Chan-Paton factors are invariant under U (N) transformations, which can be transformed into a local U (N) gauge symmetry in spacetime. This is exactly what is required for Yang-Mills theories, so it provides a basis for including the standard model in string theory. After D-branes were discovered, the Chan-Paton indices were reinterpreted. Now we suppose that there are multiple D-branes with integer labels, and string endpoints can be located at D-brane i and j for example. It turns out that multiple D-branes are what give rise to the standard model of particle physics in string theory. In particular, coincident D-branes give rise to massless gauge elds in the following way:
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If there are N coincident Dp-branes, there are N 2 massless gauge elds. This characterizes a U ( N ) Yang-Mills theory on the world-volume of the N coincident D-branes. We have already seen that a single Dp-brane has a photon state. This is consistent with the outline we are developing here. We have a single D-brane, and the gauge group of the electromagnetic eld is U(1). If we add more D-branes in the right way, we can get the number of gauge elds that we want.
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String Theory Demysti ed
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As we will see in a moment, strings with endpoints on different branes acquire mass from stretching of the string. Separating coincident D-branes provides a mechanism through which the gauge elds can acquire mass. Now, the gauge group of the electroweak theory is SU ( 2 ). There are four gauge elds with quanta: The photon The W + and W The Z 0 When we have two coincident D-branes, we have N = 2 and so there are N 2 = 4 gauge elds that transform under U ( 2 ) . This sounds like the right con guration we need to describe electroweak theory (and you might imagine more branes to include the strong interaction). However, the W + and W and Z 0 are massive. In quantum eld theory, we give them mass using the Higgs mechanism (see Chap. 9 and 10 in Quantum Field Theory Demysti ed for a description). In string theory, we separate the two coincident D-branes which will give mass to two of the string states, the states with ends attached to each of the branes. This isn t quite enough since we need one more massive state (and so will need a more complicated D-brane con guration to actually do it right). But you see how the process works. Now let s quantify the discussion. We consider bosonic string theory again with two D-branes that are parallel. The coordinate locations of the D-branes are given by x1a and x2a. There are four possibilities for open strings: A string has both endpoints attached to D-brane 1. A string has both endpoints attached to D-brane 2. A string starts on D-brane 1 and ends on D-brane 2. A string starts on D-brane 2 and ends on D-brane 1. Denoting the Chan-Paton indices by (i, j ) these possibilities correspond to: (1, 1) (2, 2) (1, 2) (2, 1) We already know how the (1, 1) and (2, 2) cases work out these are open strings with their endpoints attached to the same D-brane. So the spectrum will be unchanged. It includes a tachyon, the photon, and the Nambu-Goldstone boson. The cases (1, 2) and (2, 1) are string states stretched between the two branes. The descriptions of both cases are the same, so we focus on the (1, 2) case. First, we start with the boundary conditions, which are modi ed so that the string starts on
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