Abstract
As I mentioned a little in Chap. 1, D-branes are objects defined as a space on which end points of a string can be attached. Why is such a space important? For a preparation for answering this question, we have learned the solitons and their importance, and the construction of string theory and the compactification of higher-dimensional spacetime, in detail in Chaps. 2 and 3. In this chapter, based on those, I will show that the D-branes are solitons of string theory, and I will explain the importance of the D-branes. First, in Sect. 4.1, we consider what the solitons of string theory should be like, and after all we will find that they turn out to be “black holes” which are holes of spacetime. The black holes are objects whose existence is shown in Einstein’s general relativity. In the sense that even light cannot escape due to too large gravity force, they are holes in spacetime. I will explain also the black holes later, but in fact the black holes are solitons, because they are defined as solutions of equations of motion of gravity. Since string theory contains gravity, solitons of string theory turn out to be the black holes. However, since string theory has higher-dimensional spacetime, the black holes appearing there show various dimensions. Here, we will see how higher-dimensional black holes appear in string theory. And, in Sect. 4.2, finally I will introduce the D-brane as a space on which the end points of a string can be attached. As a result of the definition, surprisingly, it turns out that D-branes can be identified with the black holes, and thus be solitons of string theory. And furthermore, we will see that in string theory there may exist a “duality” which is a symmetry exchanging the black holes and the strings. That is, the D-branes can be exchangeable with the string, and might play a role as a fundamental constituent of string theory. What is the ultimate theory describing all the particles and interactions in this world? As a matter of fact, string theory, which is a candidate for that, may be constructed by the D-branes. I will explain the path to the ultimate theory in Chap. 7. In Sect. 4.2, first I will start with the explanation of what D-branes are, and then of the role played by the D-branes as solitons in string theory, and finally about the duality.
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- 1.
Actually, there are two kinds of the “equation of motion of string theory,” and only one of them is important for our solitons. To understand this situation, first you may think of a particle theory. For a description of particle(s), we have the one for a one-body problem and a many-body problem. The former is a worldline described by \({X}^{\mu }(\tau )\), and it is a method used for the case of a motion of one particle. The latter is, for example if it is for photons, described by a gauge field \({A}_{\mu }({x}^{\nu })\). Now if we look back what the soliton is in Sect. 2.3, we know that it is a solution of equations of motion of the very latter many body problem, namely, the solution of a field equation depending on x. You can easily understand this if you remember the viewpoint that solitons are a collective motion of elementary particles. Then, what is the equation of motion of string theory? There are two answers for this question, as in the same way, and the first one is equations of motion controlling \({X}^{\mu }(\sigma,\tau )\) of the one-body problem. However, this is not the answer we want for the soliton. The second, the action of the many-body problem, is important. It is the string field theory.
- 2.
The Kalb–Ramond field was named from two scientists, Kalb and Ramond, on the other hand in the case of the Ramond–Ramond field, it does not mean there are two Ramond’s. It was named from the fact that both the right- and left- movers of the fermion on the 1+1-dimensional worldsheet satisfy the boundary condition of the Ramond type.
- 3.
We can obtain the field action of particles corresponding to various string oscillation modes by calculating scattering amplitudes of string theory. (In the case of ordinary field theories of particles, first as a starting point we give an action of a field by hand, and then we draw Feynman graphs to compute scattering amplitudes. However, in fact in string theory, the process is opposite.) In string theory, by deforming the 2-dimensional worldsheet, we can find the scattering amplitudes of multi-strings. So as to reproduce the scattering amplitudes obtained in that manner, one can write an action of the corresponding particle fields (the correspondence can be seen as in Fig. 1.3.) In this way, in string theory, an action obeyed by the massless field like the gravity field having appeared above is written. The action obtained in this way is known to be equal to the action determined by the supersymmetry.
- 4.
The physical meaning of the gravity field (metric) is as follows. Given the gravity field \({g}_{\mu \nu }(x)\), by using a vector \((d{x}^{\mu })\) connecting two arbitrary points separated infinitesimally, the “proper length” ds between the two points is given as \(d{s}^{2} = {g}_{\mu \nu }(x)d{x}^{\mu }d{x}^{\nu }\). This proper length is a “physical” length which is invariant under any general coordinate transformation, namely, arbitrary relabeling of the coordinates, \({x\prime}^{\mu } = {x\prime}^{\mu }(x)\). This can be understood by the fact that the coordinate transformation for the gravity field is given by
$$\begin{array}{rcl}{ g}_{\mu \nu }\prime = \frac{d{x}^{\rho }d{x}^{\sigma }} {d{x\prime}^{\mu }d{x\prime}^{\nu }}{g}_{\rho \sigma }.& & \end{array}$$(4.5)The general relativity is a theory with the principle that physics is invariant under this relabeling of the coordinates, the gravity field is a basic field of that. You may understand the physical meaning of the gravity field from this proper length ds. For instance, to multiply the gravity field by a constant means that the physical distance between two points is multiplied by the constant.
- 5.
This claim that the black hole is a soliton may be correct in a strict sense if we exactly calculate it in string theory in fact. String theory becomes a supergravity theory at low energy. However, in the region near the singularity where fields vary quite rapidly, one cannot take the low energy approximation. If we treat this without the low energy approximation in a proper way in string theory, the singularity of this black hole may be actually “resolved” and the singularity might go away. There various massive fields, as well as the massless field such as the gravity, could take complicated configurations. The appearance of the singularity may be due to that we approximately focus on only the low energy of the string theory such as the gravity theory and ignore the massive fields. Therefore, the black hole could be exactly an soliton if we look at it in the whole string theory. For this, some evidence is known in string theory.
- 6.
The “BPS” means the capital letters of three scientists, E. Bogomol’nyi, M. Prasad and C. Sommerfield. The BPS method, which was originally invented as a convenient way to find a monopole solution of a certain gauge theory, turned out to be related with the supersymmetries in later days.
- 7.
This applies to C(x) too, and the object has a 0-dimensional worldvolume which is a point in the spacetime. This is called an “instanton.” Although the instanton is a very important physical concept, I will not deal with it in this book.
- 8.
J. Polchinski already found D-branes in 1989 with J. Dai and G. Leigh, the importance of the D-branes has been hidden until he identified the D-branes with the black branes in 1995.
- 9.
In string theory, one expects that the value of the coupling constant g s too is determined automatically in the framework of the theory. This is based on the relation (4.13) of this dilaton field. The reasoning is that once the “vacuum” of string theory is determined, the value of the dilaton field is fixed, and then the coupling constant of string theory is also determined.
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© 2012 Springer-Verlag Berlin Heidelberg
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Hashimoto, K. (2012). D-Branes. In: D-Brane. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23574-0_4
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DOI: https://doi.org/10.1007/978-3-642-23574-0_4
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