Supersymmetry. Theory, Experiment, and Cosmology

280 An overview of string theory and string models

fields belong to 10-dimensional supermultiplets that propagate on the boundaries of

spacetime, that is on the 10-dimensional surfaces at 0 and πR(11) . The e ective field theory is a version of 11-dimensional supergravity on a manifold with boundaries,

known as Hoˇrava–Witten supergravity. Eleven-dimensional supergravity also contains D2-branes and D5-branes.

Open strings, T -duality, and branes

Branes were encountered [311] by studying the behavior of open strings under T -duality. Let us see this in more details.

There does not seem to be any winding mode associated with an open string since, in principle, an open string can always unwind itself. On the other hand, under a duality transformation (10.47), the Neumann boundary condition is turned into a Dirichlet condition: since z = e2i(τ +σ), z¯ = e2i(τ −σ) we have

Hence in the dual theory, the end of the open string is fixed (in the associated compact dimension). One can go one step further and show that all endpoints lie on the same hyperplane. For example we have

In other words, the corresponding open string winds n times around the dual dimension. This corresponds to a stable configuration because the ends of the string are attached to the hyperplane (the open string cannot unwind itself).

It is reasonable to expect that this hyperplane is a dynamical object which can fluctuate in shape and position. Indeed, one finds among the massless string states a scalar field which describes the position of this plane. This dynamical object is called a Dirichlet brane or D-brane.

We have seen that open strings allow us to introduce a gauge symmetry through the Chan–Paton degrees of freedom. Let us consider a Wilson line with a constant gauge potential (10.58). Following the discussion of Section 10.2.3, in particular (10.56), we see that a string state |ij has momentum p25 = (2πn + qθi − qθj )/(2πR). Thus (10.73) reads in this case

Thus the end points of the open string |ij lie at branes located at θiR and θj R respectively (up to a constant). If n values of θ are equal, that is if n branes are coincident, we have seen in Section 10.2.3 that there is an enhanced gauge symmetry U (n).

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We have discussed compactification from the point of view of duality relations between string theories. Of course, we need to compactify six out of the 10 dimensions of the perturbative superstring (or seven out of 11 in the strongly interacting case). This is unfortunately where we lose the uniqueness of the string picture (if indeed M-theory provides a unique framework): the vast number of compactification schemes leads to an equally vast number of low energy models. It is only a complete understanding of the nonperturbative dynamics of compactification which would determine which four-dimensional model corresponds to the true ground state. Meanwhile, however, the string picture gets richer.

We have already encountered orbifolds: they provide standard examples of six or seven-dimensional spaces on which to compactify string theories. For instance, one may take the product of three two-dimensional orbifolds such as the ones constructed in Section 10.2.4 (a classic example uses the Z3 orbifold studied in Exercise 3). Another type of compact space which is widely used are Calabi–Yau spaces. They are devised in such a way that compactification on these six-dimensional spaces yields N = 1 supersymmetry in four dimensions, as well as scalar fields charged under the gauge symmetry.

New gauge symmetries find their origin in the symmetries of the six or sevendimensional compact manifold. As stressed before, symmetries, including supersymmetry, can also be broken by the boundary conditions that one chooses to impose on the fields for each of the compact coordinates. In the case of brane models, we have just seen that nonabelian gauge symmetries arise at the place where several branes coincide.

Some low energy observables which remain unexplained by the Standard Model find a new interpretation in the context of compactified string models. For example, in the context of compactification of the heterotic string model on a Calabi–Yau manifold, the number of families is interpreted as a topological number which can be computed for each Calabi–Yau manifold. Models with three families have been found, which resemble closely the Standard Model at low energy. Similar determination of the number of families exists also in the context of brane models.

In the context of the Kaluza–Klein compactification, the size of the compact dimensions is microscopic: the nonobservation of Kaluza–Klein modes at high energy colliders implies a bound of approximately (1 TeV)−1. The constraint is di erent in the context of brane models. There, if the standard matter and gauge interactions are localized on the brane (or coincident branes), one must distinguish compact dimensions along the brane and orthogonal to the brane. The previous limit holds for dimensions along the brane. Compact dimensions orthogonal to the brane only support excitations with gravitational-type forces. The experimental limits then only arise from Cavendish-type experiments and give a bound of a few fractions of a millimeter for R. We will see in what follows that such macroscopic values for the radius of compactification allow us to decrease dramatically the string scale MS , or the Planck scale of the higher-dimensional theory, down to a few TeV. This opens the remarkable possibility that the next high energy colliders probe directly the quantum gravity regime.

282 An overview of string theory and string models

Calabi–Yau manifolds

Our starting point is the heterotic string theory which has been discussed above. Our discussion here is purely field theoretical and for this purpose we consider that, to a first approximation, the heterotic string theory yields a 10-dimensional supersymmetric Yang–Mills theory with gauge group E8 × E8 coupled to N = 1 supergravity.

More precisely, this means that, in the Yang–Mills sector, the massless fields are the gauge fields and their supersymmetric partners, the gauginos, in the adjoint representation of E8 × E8, (1, 248) + (248, 1). These are the only massless fields with gauge degrees of freedom. The algebra of N = 1 supersymmetry in 10 dimensions reads:

where A, B are SO(1, 9) spinor indices. Q is a Majorana–Weyl spinor. Indeed, it is possible to impose both conditions in 2n = 2 mod 8 dimensions, hence in 10 dimensions (see Section B.2.2 of Appendix B). As such, Q represents 25−2 = 8 degrees of freedom.

We now need to compactify the theory on M4 × K where K is a compact manifold. We narrow down the possible choices by making the following requirements:

(i)N = 1 supergravity in four dimensions;

(ii)the low energy four-dimensional theory includes some scalar fields with nonzero gauge quantum numbers (typically Higgs fields).

We now follow the general strategy of Candelas, Horowitz, Strominger and Witten [66].We want to find a vacuum solution |Ω which is the solution of the equations of motion and which respects one four-dimensional supersymmetry charge Q. Clearly, the vacuum must be Lorentz invariant which implies, for any fermion F ,

To determine Ω|B|Ω for the boson fields, we use the supersymmetry transformations in the following way: denoting by δF the transformation of F under supersymmetry, it follows from the fact that the vacuum is supersymmetric (Q|Ω = 0) that

Explicitly, δF can be expressed as a function of the boson fields B. Hence the system of constraints (10.77) can be used to determine the bosonic expectation values Ω|B|Ω [66].

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We can now write the supersymmetry transformations of the fermion fieldsa:

1

δψM = κ10 DM η

where φ is the dilaton field, η is the 10-dimensional supersymmetry transformation parameter (a Majorana–Weyl spinor), DM η its covariant derivative and κ10 is the 10-dimensional gravitational constant. We have made in (10.78) the simplifying assumptionsb that φ is constant and the antisymmetric tensor field bM N has vanishing field strength hM N P . The latter hypothesis imposes (dh = tr R R− tr F F )

where k, l = 4, . . . , 9 (Fµνa = 0 in the vacuum |Ω to ensure Lorentz invariance). First look at (10.80) which expresses the fact that there exists a covariantly constant spinor. One can show that this implies that the manifold K is a complex K¨ahler manifold (cf. equation (C.55) of Appendix C). We recall in Section D.1 of Appendix D the notion of holonomy group and mention that for a general six-dimensional complex manifold, the holonomy group H is SO(6). But (10.80) expresses the fact that there is one direction in tangent space which remains untouched by the action of H

The spinor η is Majorana–Weyl and is therefore in the 4 of SO(6). The holon-

omy group H is the subgroup of SO(6) SU (4) that leaves invariant a 4;

=

hence H = SU (3). This is where Calabi–Yau manifolds enter the game.

A Calabi–Yau manifold is a complex K¨ahler manifold with a metric of SU (3) holonomy (we will denote the holonomy group by SU (3)H ).

We may now write the decomposition of QA under the subgroup SO(2) × SU (3)H of the transverse Lorentz group SO(2) × SO(6) (the quantum number associated with the four-dimensional SO(2) is the helicity):

QA = (h = +1/2, 1) + (h = −1/2, 1) + (h = +1/2, 3) + (h = −1/2, 3). (10.83)

aSince we only consider values of the fields in the vacuum |Ω , we will write from now on F or B instead of Ω|F |Ω and Ω|B|Ω and we use (10.76) to set F to zero.

bThe dilatino variation δλ is identically zero under these assumptions.

284 An overview of string theory and string models

The residual charges at low energy are those which leave invariant the background fields, which are in SU (3)H . As expected, the only supersymmetry charge that survives is the singlet under SU (3)H which gives two degrees of freedom, i.e. N = 1 supergravity in four dimensionsa.

We also want to fulfill condition (ii) and must therefore take a closer look at the only nonsinglet bosons available, the gauge fields. Their decomposition under SO(2) × [SO(6) ≡ SU (4)] is given by

Since 6 ≡ 3 + 3 under SU (3)H , the decomposition of a 10-dimensional gauge field under SO(2) × SU (3)H reads:

Hence the only fields invariant under SU (3)H are gauge fields (h = ±1) and there does not seem to be any four-dimensional scalar with gauge interactions. However proper care must be taken of the condition (10.79). The easiest way to satisfy this condition is to identify the spin connection (Christo el symbol, see (D.5) of Appendix D) with some of the Yang–Mills fields (gauge connection).We therefore choose one of the two E8 (refer to the other one as E8), decompose it into E6 × SU (3)YM and set

(remember that AM stands for Ω|AM |Ω ). For M = 4, . . . , 9, the (AM )ij are (four-dimensional) scalar fields in the adjoint of SU (3)YM, with a nonzero vacuum expectation value through (10.86) (if K is not flat, the spin connection ΓiM j cannot be set to zero globally). They therefore break the E8 gauge symmetry to E6.

A Yang–Mills field AaM a = 1, . . . ,dim E8 = 248 transforms under SO(8) and E8 as (8v , 248), with the following decomposition:

SU (3)H

aOne may alternatively count the number of (Weyl–Majorana) supersymmetry charges on the two-dimensional world-sheet. Just by counting the number of degrees of freedom, one concludes that two charges are needed in the world-sheet to make N = 1 spacetime supersymmetry. In the context of the heterotic string, they come from the right-moving sector and the minimal model has (0, 2) supersymmetry, i.e. 0 left-moving charges and two rightmoving ones. In the case of Calabi–Yau supersymmetry, the identifications made enhance the system of two-dimensional supersymmetry charges to (2, 2) world-sheet supersymmetry.

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The arrow ! connects representations for which a SU (3)H rotation can be undone by a SU (3)YM rotation. Hence, the fields which commute with the nontrivial background are in the representations:

where the second entry is the E6 representation content. We find gauge fields

(h = ±1) in the adjoint of E6 (78) and scalar fields in 27 and 27. We also have gauge fields in the adjoint representation 248 of E8.

For completeness we give the four-dimensional decomposition of the 10-dim- ensional Yang–Mills supermultiplet in Table 10.3 and of the 10-dimensional supergravity multiplet in Table 10.4. The numbers hp,q are topological numbers called Hodge numbers. They count the number of (p, q)-forms (that is di erential forms with p holomorphic indices and q antiholomorphic indices; remember that a Calabi–Yau manifold is a complex manifold) that are annihilated by the Laplacian. For a manifold of complex dimension 3 (hence p, q ≤ 3), the property of SU (3) holonomy implies h0,0 = h3,3 = h0,3 = 1 and h0,1 = h0,2 = h1,3 = h2,3 = 0. Since hp,q = hq,p, the only Hodge numbers to determine for a specific Calabi–Yau manifold are h1,1 and h2,1. We note that the Euler characteristics (see Section 10.4.1) of the compact manifold is given in terms of the Hodge numbers by the relation

p,q

A closer look at Table 10.3 shows that the combination h2,1 − h1,1 is the number of fermions in 27 of E6 minus the number of fermions in 27, that is the net number of families (as seen in Chapter 9, a representation 27 of E6 encompasses a full family of the Standard Model). Thus, in the context of Calabi–Yau models, the number of families is directly related to the Euler characteristics of the manifold. Manifolds with χ = −6 have been actively searched for.

Looking at Table 10.4, one notes the presence of real scalar fields. In order to fall into chiral supermultiplets, they should pair up to form complex scalar fields. Indeed, the spin 0 component of the chiral supermultiplets reads:

where a is the pseudoscalar field obtained by duality transformation from the antisymmetric tensor field bµν (see Section 10.4.2 below).

286An overview of string theory and string models

Table 10.3 Decomposition of the 10-dimensional Yang–Mills supermultiplet.

10.4Phenomenological aspects of superstring models

We discuss in this section the most salient features of string theory applied to the description of fundamental interactions. We have seen in preceding sections that string models generically have a gauge symmetry large enough to include the Standard Model interactions, and thus provide a scheme unifying all known interactions, including gravity. However, the gravitational sector is richer than in standard Einstein gravity: it involves moduli fields, the value of which determines the low energy physics. The determination of these values thus puts, once again, the issue of supersymmetry breaking center stage.

Before we address these questions, it is important to discuss the energy scales present in these models, and their possible range, since this determines the possible ways to put these issues to the experimental test.

10.4.1Scales

The di erent string theories that we have discussed in Section 10.3 involve various relations between the mass scales and couplings [378]. This is best seen by looking at some key terms in the e ective field theory action. The di erent regimes are partly due to the fact that, depending on the string theory considered, the relevant terms appear

288 An overview of string theory and string models

Fig. 10.10 (a) Topology of a tree-level closed string diagram such as in Fig. 10.5 (the crosses indicate vertices of particle emission corresponding to the incoming and outgoing strings of Fig. 10.5); (b) topology of a one-loop closed string diagram; (c) topology of a tree-level open string diagram such as in Fig. 10.4.

(Euler characteristics χ = 2), a one-loop diagram the topology of a torus (χ = 0), and so on (see Fig. 10.10). Thus a general diagram has an overall coupling factor λ−χ, hence an overall dilaton dependence e−χφ. This result extends to the open string case: a tree-level diagram has the topology of a disk (χ = 1) and thus a dependence e−φ.

Euler characteristics

The Euler characteristics is easily computed in the case of simple compact surfaces. A standard method uses the triangulation of the compact manifold. In the case of a two-dimensional surface, it consists in dividing the surface into triangles, making sure that any two distinct triangles either are disjoint, have a single vertex or an entire edge in common. Examples of triangulations are given in Fig. 10.11.

The Euler characteristic is then given by

where b0, b1, b2 are respectively the number of vertices, edges, and triangles. From looking at Fig. 10.11a,c, one obtains χ(S1) = 3 − 3 and χ(S2) = 5 − 9 + 6 = 2. This generalizes to χ(Sn) = 1 + (−1)n for a sphere Sn in (n + 1) dimensions. We also obtain for a disk (Fig. 10.11b), χ(D2) = 4 − 6 + 3 = 1. From this last result, one can infer another very convenient formula for the Euler characteristics of a two-dimensional surface:

where h is the number of handles and b the number of boundaries. Indeed, a surface with neither handles nor boundaries is the sphere S2. Adding a boundary amounts to puncturing a hole in this sphere, that is removing a disk (δχ = −1). Adding a handle amounts to puncturing two holes (δχ = −2) and gluing their boundaries. We deduce that χ = 0 for the torus: by deforming a torus, one easily obtains a sphere with one handle.