Abstract
Since the discovery of D-branes as solitons of string theory by Polchinski in 1995, the application research has been explosively developed. This word “explosively” is not an exaggeration. D-branes have given tremendous influences not only to the framework of string theory but also to various physics around, by their flexibility. They have brought us not a slight influence but a huge revolution such as a creation of new subjects and supply of new paradigms. In this chapter, I will pick up four topics which have been considerably developed intrinsically owing to the emergence of the D-branes, and will have a brief explanation for each. As you can read these four sections independently, you might start with whichever section you are interested in. Since they are leading-edge research results, I am forced to omit details a little bit. However, I hope you may read the great influences given by D-branes, the possibility of their future, and the excitement of researchers engaged in the study of D-branes.
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Notes
- 1.
Here, “EW” stands for electro-weak, that is, the “electro-weak theory.” This electro-weak theory is the theory unifying the electromagnetism and the “weak interaction,” and it is a main part of the Standard Model. In the electro-weak theory the Higgs mechanism is also used.
- 2.
It is considered that the scale at which the symmetry of this grand unified theory appears provides an upper limit of the energy for computing Feynman graphs of the Standard model (called “cut-off”). Since this cutoff is large as (6.4), an extremely large quantity comes out when one calculates for example a Feynman graph with loops of Higgs particles. The infinity which I mentioned previously means this cutoff in a concrete sense. Although this is made finite by the “renormalization,” the value after being finite must be the energy scale of the Standard model. The gauge hierarchy problems refer to that this procedure of making things finite is not natural. This problem is also called as “fine tuning problem” or “naturalness problem.”
- 3.
Even in string theory the word “brane” is used often. This word is for all extended objects including fundamental strings, D-branes and such. Since the D-branes have given influence on a number of subjects, the word “brane” has various definitions.
- 4.
Since the gravity is present in the 4 + n dimensions, this is a higher-dimensional black hole.
- 5.
As for the physical meaning of the metric (gravitational field), see the footnote including (4.5). In the case of the present metric (6.10), the distance between two points sharing the same value of the coordinate x 5 (that is, on the same brane) is multiplied by a factor \({\mathrm{e}}^{-k{x}^{5} }\). Even if the two points share the same \({x}^{\mu }\), once their coordinates x 5 are different, the distance gets different by constants.
- 6.
The Dirac’s delta function δ(x 5 − c) is the function that is infinite only at u = 0 while vanishes for the other values of u. The delta function satisfies the following integral equation
$$\begin{array}{rcl} \int \nolimits \nolimits \mathrm{d}uf(u)\delta (u - c) = f(c)& & \end{array}$$(6.11)for an arbitrary function f(u), that is, it can extract by the integration the information of the location of the delta function.
- 7.
In the model with large extra dimensions as we saw previously, the KK gravitons are extremely light, since the radius of the extra dimension is large. So you might wonder if a lot of the KK gravitons will be created at the accelerator. But it is not the case. These KK gravitons with the large extra dimensions have an extremely small coupling constant (given by 1 ∕ M Pl) since the interaction with particles of the Standard model is only the gravitational interaction in the 3 + 1 dimensions, so they are hardly created. On the other hand, in the case of the Randall–Sundrum model, it is known that this coupling constant is large (as about 1 ∕ M EW), because of the curved internal space. Once these KK gravitons at a particle accelerator are created and observed as the missing energy, information on the extra dimensional space should be revealed through the masses and the coupling constants.
- 8.
Precisely speaking, suppose A is the area of the triangle, then the magnitude of the elementary particle interaction is \({\mathrm{e}}^{-A/(2\pi {l}_{\mathrm{s}}^{2})}\). This is the probability that the worldsheet of a string with the tension 1 ∕ (2πl s 2) extends to the area A.
- 9.
Once cosmic strings are observed, how do we distinguish the vortex solitons of a field theory in Sect. 2.3 from the D-branes? This is an interesting question. Concretely speaking, it has been argued that there may be a difference in the number of the cosmic strings observed, for the field theory vortex solitons and for the D-branes. Figure 3.2 shows how the cosmic strings collide and reconnect with each other, and then form closed loops which contract to vanish. In this way, as the reconnection is a reason for the number of cosmic strings to decrease, what kinds of situation the reconnection occurs relates to how many cosmic strings remain in the end. It is known that the reconnection is classical in the case of the vortex solitons of the field theory, while it is probabilistic in the case of the D-branes, in fact. This difference should result in the number of the cosmic-string-like objects which we observe, at the end. By the way, we know that the D1-branes are vortex solitons coming from the tachyons of the D3-brane and the anti-D3-brane, then what is the origin of this difference? The reason is that in the annihilation of the D-branes there are related other stringy modes (infinite massive modes) which we omit, in addition to the tachyon. From the string connecting a D3-brane and an anti-D3-brane, there appear many oscillation modes as well as the tachyon, and these infinite number of modes condense too at the same time as the tachyon condensation. This is the reason why the D1-branes are different from just the vortex solitons. In the discussion of the annihilation of D-branes in Sect. 5.3, I described this points briefly. To prove the Sen’s conjecture was difficult (however finally it was proved), because of the difficulty to handle these infinite number of modes. The theory to handle the infinite number of modes simultaneously is the string field theory which appeared in Sect. 4.1.
- 10.
For instance, since black holes accelerate objects around them and absorb all of stuffs, there appears a disk of matters rounding at a high speed around the black hole, which is called an accretion disk. The acceleration energy is emitted as strong X-rays. We observe these X rays and guess the mass and the volume of the stars, the would-be black holes.
- 11.
In this section, basic knowledge of thermodynamics is supposed. The entropy S is a quantity characterizing “disorder” of the whole system where many microscopic elements interact with each other. When the microscopic state number of the system is d, the entropy is written as S = k Blogd (here, k B is the Boltzmann constant).
- 12.
They were awarded a Nobel prize for their discovery of the asymptotic freedom in QCD.
- 13.
These three do not mean the species of quarks called for example up quark and down quark. Each species of quarks such as up quark has three kinds. These three kinds are called “color.” The reason why the theory of quarks and gluons is called QCD, quantum chromodynamics, is the mechanism of this “color” of the SU(3).
- 14.
The index ⟨a, b⟩ of these gluons resembles the labels of the strings connecting various D-branes which appeared in Sect. 5.2, and as we will see later, indeed these are physically the same.
- 15.
In the solution, the Ramond-Ramond field C MNPQ is nonzero, but it is not written here. The important is that the dilaton field ϕ is constant in this solution. As seen in (4.13), if dilaton is constant, the coupling constant g s of string theory is constant regardless of the distance from the branes. As you will see next, as we need to see a region close to r = 0 in order to find a correspondence to the non-Abelian gauge theory, it helps a lot that g s is constant and does not diverge to infinity even there.
- 16.
For readers familiar with group theory, here I will precisely write what the symmetry is. Combining the conformal symmetry and Lorentz symmetry of the 3 + 1 dimensional spacetime, it turns out to be a symmetry called SO(2,4). This is called a conformal group. SO(2,4) corresponds to a symmetry which does not change the metric of the five-dimensional AdS spacetime (called an isometry). And the transformation of the phase of the scalar fields described here is understood as a group called SO(6) acting on the six scalar fields. This turns out to be an isometry of the metric of the five-dimensional sphere on the gravity side.
- 17.
For example, in non-Abelian gauge theories, it is known that there are states called glueballs which are bound states of two or more gluons. The mass spectrum of these glueballs calculated by using the gravity theory coincides almost with results of numerical calculations of the spectrum on computers.
- 18.
The “string field theory” which was mentioned in Sect. 4.1 might become the “field theory action” supposed here. However, in the string field theories, there are various problems: for example, it is difficult to include supersymmetries, and to perform a procedure called “second quantization” which is necessary for any field theory to have a particle picture.
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Hashimoto, K. (2012). Application of D-Brane Physics. In: D-Brane. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23574-0_6
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DOI: https://doi.org/10.1007/978-3-642-23574-0_6
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