6.9 About Solutions

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Chapter 7

The Branes: Three Viewpoints

Tu se’ certo il cantor del trino regno,

Tu lo spirto magnanimo e sovrano

Cui, quasi cervo a puro fonte, io vegno.

Giovanni Marchetti

7.1 Introduction and Conceptual Outline

Supergravity developed originally as the supersymmetric generalization of Einstein Gravity and, for several years, the construction of its various formulations in diverse dimensions, with diverse number of supercharges, went on independently from the theory of superstrings, whose origin was instead within the framework of the dual models of hadronic scattering amplitudes, namely within tentative theories of strong interactions. In the course of time, however, and as a result of the two string revolutions,1 the subjects of supergravity and of superstring theory merged completely, as soon as it became clear that the D = 10 theories described in the previous chapter are just effective low energy Lagrangians that encode the interactions of the massless modes of the corresponding perturbative string models.

Since the mid nineties the relation between supergravity and superstrings underwent a further substantial upgrading which is the essence of the second string revolution.

On one hand it became clear that each superstring theory, besides the elementary string states, includes also additional non-perturbative excitations, similar to

1By first string revolution it is meant the discovery by Green and Schwarz of the mechanism of anomaly cancellation which singled out five perturbatively consistent superstring models, namely:

1.Type II A

2.Type II B

3.Type I with SO(32) gauge group

4.Heterotic E8 × E8

5.Heterotic SO(32).

By second string revolution it is meant the series of discoveries around 1995–1996 that demonstrated that all the perturbatively consistent string models are related to each other by nonperturbative dualities pointing out to the fact that there is just one non-perturbative superstring theory.

the solitons of non-linear field theories, that can be associated with the propagation of extended objects of higher dimensions, the p-branes. Among them, particularly relevant are the Dp-branes that can be alternatively regarded as the loci where open strings have their end-points, or as space-time boundaries that can absorb or emit closed strings. On the other hand the p-branes could be identified with classical solutions of the relevant low energy supergravity and it was discovered that the symmetries of supergravity realize those non-perturbative duality transformations that can map string states into solitonic ones and vice-versa, demonstrating that all string theories are just different perturbative limits of a single theory, usually named M-theory.

From these considerations a new more profound understanding of (super-)gravity and (super-)branes emerged that is the goal of the present chapter to outline, emphasizing that superstrings are just a particularly relevant instance in a broader landscape.

Our starting point is the action of a charged particle in the background of an electromagnetic field. Naming xμ(τ ) the coordinates of the charged particle at proper time τ , we can write the following action:

where Aμ is the electromagnetic field, gμν the metric of the ambient space-time manifold, q the electric charge of the particle and x˙μ = dτd xμ. Varying the action (7.1.1) with respect to the trajectory function δxμ(τ ) we obtain the equation of motion of the charged particle subject to the Lorentz force and to the gravitational field encoded in the metric. On the other hand we can add the action Apart, which is a one-dimensional integral, to the action

which is a 4-dimensional integral and we can vary AMax + Apart in the electromagnetic field δAμ. What we obtain are Maxwell equations with a source term provided by the electric current localized on the world-line swept by the charged particle:

Similarly we can vary the action Apart with respect to the metric δgμν and this yields a stress-energy tensor, also localized on the particle world-line, that provides a source for the gravitational field in Einstein equation.

This short discussion puts into evidence the following two facts:

(A)If a field theory contains a gauge field that is a d-form A[d] then, setting p = d − 1, we can introduce a p-dimensional object which, by evolving through the

Fig. 7.1 In the above two pictures we present an intuitive image of the world-volume traced by p-branes in the ambient space-time. Since we cannot draw in higher dimensions we show the world volumes traced by 1-branes, namely strings. In the string case these world-volumes are actually world-sheets, namely 2-dimensional surfaces. Furthermore the string can be open, namely can admit end-points, or close. In the first case the world-sheet is of the type depicted on the left. In the second case the world-sheet is a sort of tube like that depicted on the right

ambient D-dimensional space-time MD , traces in this latter a d-dimensional world-volume (see Fig. 7.1):

The dynamics of such an extended object, which we name a p-brane, is described by an action given by a d-dimensional integral localized on the worldvolume Wd . Such a p-brane action is typically made of two terms

the first term being the area of the world-volume or generalization thereof, the second, often named the Wess-Zumino term, being the integral of the form A[d] on the world-volume.

(B)The extended objects described above provide sources for the bulk field-theory. These sources are localized on singular boundaries of space-time.

When p > 0, namely when the object tracing the world-volume is really extended and not simply a point-particle, on top of moving it can deform its own shape, vibrate, split and join with other similar entities. In other words there are dynamical processes occurring on the world-volume and besides the bulk field theory, that we name macroscopic, we have also a world-volume field theory that we name microscopic. The microscopic field theory can be quantized for its own sake and its elementary excitations are paired in a precise way to the classical fields of the macroscopic theory. This is what we do in the case of superstrings. In this case the microscopic field theory lives in two-dimensions and has distinctive miraculous properties: it is typically conformal, which means invariant with respect to a very specific infinite Lie group, its spectrum can be derived by means of algebraic techniques and its Green functions can be calculated exactly in a large variety of cases.

Fig. 7.2 The three intertwined aspects of brane theory

We do not dwell on the microscopic aspects of string theory that form the topics of large specialized text books. We just emphasize that the relation of supergravity to strings is the same as the relation of the former with other p-branes allowed by the existence of suitable (p + 1)-forms. The difference is that the quantum microscopic theory of p > 1 branes usually cannot be solved exactly: the spectrum is mostly unknown and the Green functions are out of reach of exact calculations.

It must be stressed that introducing p-brane boundary actions gives rise to solutions of the bulk field theory that are determined by such sources and have singularities on the world-volume of the source. This is just a generalization of the electric and magnetic fields generated by point-like charged particles. It follows that p-branes can also be identified with appropriate classical solutions of supergravity. Hence the new theory of strings and branes that emerged from the second string revolution has a challenging triadic structure which we have tried to summarize in Fig. 7.2. On one hand the superstring massless modes perfectly match the field content of that supergravity which is necessary to write the considered superstring microscopic action. On the other hand the field spectrum of supergravity determines which additional p > 1 branes can be coupled to it. The microscopic action of such p-branes can be constructed according to a procedure which we outline in the following section issuing a generalized gauge theory living on the world-volume. Finally for each allowed p-brane we have a corresponding classical solution of supergravity. Therefore there are three complementary aspects and as many complementary approaches to the study of p-branes that can be alternatively viewed as:

(a)classical solutions of the low energy supergravity field equations in the bulk,

(b)world-volume gauge theories described by suitable world-volume actions characterized by κ-supersymmetry,

(c)boundary states in the superconformal field theory description (SCFT) of superstring vacua.

As we explained the three descriptions are intertwined. The relation between (a) and (b) was already illustrated. To the constructive principle of κ-supersymmetry we dedicate the next sections. The viewpoint (c) is explained as follows. When we adopt the abstract language of superconformal field theories (SCFT), classical string backgrounds are identified with a specific SCFT and the brane is identified with a suitable composite state constructed in the framework of the same. The worldvolume action encodes the interactions within the chosen boundary conformal field theory.

The above discussions can be summarized in the following list of statements:

1.There is just one non-perturbative ten dimensional string theory that can also be identified as the mysterious M-theory having D = 11 supergravity as its low energy limit.

2.All p-branes, whether electric or magnetic, whether coupled to Neveu Schwarz or to Ramond (p + 1)-forms encode noteworthy aspects of the unique M-theory.

3.Microscopically the p-brane degrees of freedom are described by a suitable gauge theory G T p+1 living on the p + 1 dimensional world volume W V p+1 that can be either conformal or not.

4.Macroscopically each p-brane is a generalized soliton in the following sense. It is a classical solution of D = 10 or D = 11 supergravity interpolating between

two asymptotic geometries that, with some abuse of language, we respectively name the the geometry at infinity geo∞ and the the near horizon geometry geoH . The latter which only occasionally corresponds to a true event horizon is instead

universally characterized by the following property. It can be interpreted as a solution of some suitable p + 2 dimensional supergravity S G p+2 times an appropriate internal space ΩD−p−2.

5.Because of the statement above, all space-time dimensions 11 ≥ D ≥ 3 are relevant and supergravities in these diverse dimensions describe various perturbative and non-perturbative aspects of superstring theory. In particular we have a

most intriguing gauge/gravity correspondence implying that classical supergravity S G p+2 expanded around the vacuum solution geoH is dual to the quantum gauge theory G T p+1 in one lower dimension.

In line with our previous choices, we do not address the superconformal aspects of p-branes which relate with the microscopic theory of superstrings. We just focus on the following two aspects that complete the landscape of far reaching consequences of General Relativity when enlarged by supersymmetry:

(A)Construction of the world-volume actions with κ-supersymmetry,

(B)p-brane solutions of bulk supergravity as classical solitons.