Supersymmetry. Theory, Experiment, and Cosmology

Compactification 271

nontrivial topology of the compact dimension. This is measured by the gauge-invariant

In such a configuration, the canonical momentum p25 becomes p25 + qA25 = p25 − qθ/(2πR) and the quantification condition now reads:

Thus a 26-dimensional field of charge q and mass m0 (pM pM = pµpµ − p25p25 = m20) appears as a tower of Kaluza–Klein states with a mass spectrum

[If we now consider an open string with U (N ) Chan–Paton degrees of freedom at both ends, the constant U (N ) gauge potential A25 may be diagonalized as

The open string spectrum shows that the Wilson line breaks the U (N ) gauge symmetry

#r

to U (n1) × · · · × U (nr), i=1 ni = N , if the θ’s appear in successive sets of ni equal eigenvalues10.]

Topological gauge symmetry breaking

The nontrivial nature of the topology of the Aharonov–Bohm set up or of the circle of compactification can be described using the notion of fundamental group. In both cases this group is nontrivial (Π1 = Z): space is multiply connected. In the following, we will often consider multiply-connected compact spaces K, most of the time of the form K = K0/G with K0 simply connected

and G a finite group of order n, in which case Π (K) G. For example, we can

1 =

distinguish three classes of curves in K = K0/Z3; (a) [1]: curves closed in K0;

(b) [g]: curves closed up to a g Z3 transformation, (c) [g2]: curves closed up

to a g2 transformation. These classes are the homotopy classes and they form

the fundamental group of the manifold K : Π1(K) = [1], [g], [g2] ≡ Z3.

10Let us consider a string in Chan–Paton state |i, j . Then, the quantification condition (10.56) yields the following mass spectrum:

Consider as above the massless vector fields (n = 0, N = 1). If all the θ’s are di erent, out of the N 2 vectors only N are massless and the Wilson line has broken the U (N ) symmetry down to U (1)N . If n of the θ’s are equal, there correspond n2 massless vector fields and thus a residual gauge symmetry U (n).

272 An overview of string theory and string models

If we consider a multiply connected compact space, then a nonvanishing Wilson line can develop along which Ak ≡ AakT a = 0 whereas Fkl = 0. In other words, along a noncontractible loop γ (γ / [1]), this solution corres-

ponds to

U (γ) = P exp i Ak dyk = 1 (10.60)

γ

where P is a path ordering operator. Such a matrix has the following properties:

(i)U (γ1) U (γ2) = U (γ1γ2).

(ii)U (γ) is identical for all γ in the same homotopy class. Take for example γ1, γ2 [gp]: γ1 and γ2 are curves between y and gpy (g G) closed in K0/G. Using Stokes’ theorem, we have

Ak dyk − Ak dyk = Ak dyk = Fij dsij = 0. (10.61)

γ1 γ2 γ=γ1−γ2

conditions. Take for example a curve γ [g] that goes from y to gy. It is possible to perform a gauge transformation (exp −i γ Akdyk) at the point gy such that it sets U (γ) to 1. In other words, as we go along the curve γ (closed

the gauge

in K0/G) we perform a gauge rotation that undoes the nonzero gauge field circulation Akdyk. Of course, this gauge transformation does not act only on fields and we have to gauge transform all the fields. Take for example

a scalar field φ(y). Since K = K0/G is obtained by identifying y and gy, φ must obey the boundary condition φ(y) = φ(gy) in order not to be multivalued. In the gauge rotated picture this becomes

To account for that, Ug−1 is also called a twist in the boundary condition.

10.2.4Orbifold compactification

Torus compactification is straightforward but it is often insu cient because it yields too many supersymmetries. Indeed, if we start with a 10-dimensional supersymmetric theory, the supersymmetric charge transforms as a Majorana spinor; hence it falls into a spinor representation 16 of the Lorentz group SO(10) (see Section B.2.2 of Appendix B). When we compactify six dimensions, this yields eight four-dimensional Majorana spinors (two degrees of freedom each). Thus, one supersymmetry charge in 10 dimensions yields N = 8 supersymmetry charges in four dimensions. We have seen that only N = 1 supersymmetry is compatible with the chiral nature of the Standard Model. One must therefore look for compactifications which leave less supersymmetry charges intact.

Compactification 273

The simplest possibility is to require further invariance under discrete symmetries of the compact manifold. For example, if we start with the circle described by the identification (10.27), we may impose a Z2 symmetry, often referred to as a twist,

This then describes the segment [0, πR[. On such a compact space, strings may be closed up to the identification (10.27) as in torus compactification (X(σ + π, τ ) = X(σ, τ ) + 2πmR), or closed up to the identification (10.63) (X(σ + π, τ ) = −X(σ, τ )). The latter strings are called twisted.

Such spaces obtained from manifolds by identifications based on discrete symmetries are called orbifolds. They are not as smooth as manifolds because the identifications lead to singularities of curvature. Let us illustrate this on the example of a tetrahedron which turns out ot be a two-dimensional orbifold.

We start with a torus obtained by identifying the opposite sides of a lozenge of angle π/3 (see Fig. 10.8a: AC OB, AO CB). This torus may thus be identified to a plane with periodic identifications of the type (10.27) in the directions corresponding to the two sides of the lozenge. Such a set-up is invariant under a rotation of π around the origin O (see Fig. 10.8b). Thanks to the periodic identifications, the points E, F , G are fixed points under this rotation.

We use this discrete symmetry to construct an orbifold out of this torus with only scissors and glue. Take the flat triangle AOB in Fig. 10.8c. The symmetry around O allows the following identifications: EA EO, GO GB, F A F B. By folding along the dashed lines, one obtains a tetrahedron with vertices E, F , G, A B O. Now a tetrahedron is flat everywhere except at each apex where there is a conical singularity of deficit angle π (the sum of angles is 3 × π/3 = π instead of 2π). In other words, curvature is zero everywhere except in E, F , G, O where it is infinite. For this reason, it is not a manifold and it falls into the class of orbifolds.

Obviously we have several classes of closed strings on such a space: closed strings which can shrink to a point by continuous deformations and closed strings which loop around one apex of the tetrahedron, i.e. one fixed point of the discrete symmetry which allowed us to construct the orbifold. The former are standard closed strings and

FE

Fig. 10.8 Constructing an orbifold (a) and manifold: torus (b) same torus represented as a plane with periodic identifications (c) orbifold: tetrahedron.

274 An overview of string theory and string models

are called untwisted in this context. The latter are closed only up to the identification associated with the discrete symmetry. They are the twisted strings. There is one twisted sector for each fixed point of the discrete symmetry.

Because of the curvature singularities associated with the fixed points, orbifolds were first considered only as toy models on which conjectures could be tested. It turns out that the presence of isolated singularities does not prevent from computing physical quantities in a string context, and orbifolds are now thought to be viable candidates for string compactification. Indeed, these singularities play an important rˆole because they allow us to combine the simplicity of a flat metric with the richness of a curved background (curvature is located at these singularities). Moreover, it is also possible to blow up these singularities by replacing the conical singularity by some smooth surface. One obtains in this way a manifold.

Twisted sectors

Let us consider first the example of the segment [0, πR[. A closed string twisted around the origin satisfies

Comparison with the untwisted case (10.42) shows the absence of terms relative to the center of mass: it is fixed at the fixed point considered herea i.e. x = 0. The oscillators satisfy the commutation relations:

The lowest levels are tachyonicb: the vacuum state |0 0 of mass squared −15/(4α ) and the first excited state α−1/2α˜−1/2|0 0 of mass squared −7/(4α ). The same procedure can be followed for the tetrahedron of Fig. 10.8 where the discrete symmetry used for orbifold identification is Z2. We may note that the torus of Fig. 10.8b is also invariant under rotations of angle 2π/3 around the origin. The corresponding identifications lead to a Z3 orbifold. In this case, the oscillators of any given twisted sector are of the form αn±1/3 (see Exercise 3).

aA closed string twisted around the other fixed point would have the same expansion with an extra additive term πR (the coordinate of the fixed point).

bThe indices 0 refer to the fixed point considered. We have similar states with index πR for the other fixed point.

String dualities and branes 275

10.3String dualities and branes

The ultimate goal of the unification program is to find eventually a single string theory. There are actually five distinct superstring theories in 10 dimensions. But very detailed relations – known as duality relations – exist between these theories, which may be compatible with having finally a single mother theory, the elusive M-theory.

Let us start by identifying the five superstring theories. We have already stressed that, for the closed string, the oscillation modes in one direction along the string decouple from the modes in the other direction. These are respectively the left-movers and the right-movers. Once one introduces fermions, the left-moving fermions may have an opposite or the same chirality as the right-moving fermions. A closed superstring theory on which left and right-moving fermions have opposite (resp. the same) chirality, is called a type IIA (resp. IIB) superstring. Type IIA or IIB strings have N = 2 supersymmetry as can be checked on the massless mode spectrum: it contains two gravitinos.

One may also consider open strings together with closed strings (for consistency and in order to obtain a graviton among the massless modes). In this case only one supersymmetry charge is allowed and the corresponding theory is referred to as type I. As we have seen, open strings may carry gauge charges at their ends, which allows them to describe gauge theories.

Orientifolds

One way to obtain N = 1 supersymmetry from type IIB theories is to correlate left and right movers by requiring invariance under a world-sheet symmetry. Let us introduce the world-sheet parity Ω : σ ↔ π − σ or z ↔ z¯, then we obtain from (10.6)

If we gauge this discrete symmetry, only states invariant under the symmetry remain in the spectrum of this by now unoriented string. This means for example that we must discard the antisymmetric tensor field among the massless modes of the closed string. This procedure is somewhat reminiscent of the orbifold twist except that we have used a Z2 world-sheet symmetry instead of a spacetime symmetry. The corresponding string theory is called a type IIB orientifold. As in the orbifold case, consistency of the theory requires the addition of twisted strings with respect to Ω. These are nothing else but the type I open strings.

Finally, since left-movers and right-movers may be quantized independently, it has been realized that one can describe simultaneously the right-movers by a superstring theory (and thus obtain N = 1 spacetime supersymmetry) in 10 dimensions and the left-movers by a standard bosonic string theory in 26 dimensions. Spacetime obviously has only the standard 10 dimensions. The extra 16 compact dimensions found in the right-movers are considered as internal: the corresponding momenta are quantized (because the dimensions are compact) and can thus be interpreted as quantum numbers. Indeed, it was shown that they can describe a nonabelian gauge symmetry with a gauge group of rank 16. These form the so-called heterotic string theories. The cancellation of quantum anomalies imposes that the gauge groups are either SO(32) or E8×E8.

276 An overview of string theory and string models

The heterotic string theory

We have seen in Section 10.2.2 that it is possible to generate gauge symmetry by compactifying some coordinates on the torus. More precisely, we found that, by requiring the coordinates X25 = XL25 + XR25 to lie on a circle of radius R = (α )1/2, one generates a gauge symmetry SU (2) × SU (2). Here we have separated the leftand right-moving coordinate because a closer look at (10.50) shows that each SU (2) is associated with one sector (α−M1 are the creation operators for the left-moving sector, and α˜−M1 for the right-moving sector). The rank of the group (the rank of SU (2) is one: t3 is the only diagonal generator) is equal to the number of compactified coordinates in each sector.

In the context of the heterotic string, one thus expects a gauge symmetry group of rank 16 (i.e. 16 gauge quantum numbers) from the available degrees of freedom in the left-moving sector. One finds in fact SO(32) or E8 × E8. We describe here briefly the E8 × E8 heterotic string construction of [212, 213].

We start with the right-moving sector. It is constructed along the lines of the superstring theory described in Section 10.1. We work in the light-cone

where a˜R = 0 in the R case, a˜NS = 1/2 in the NS case. The Ramond vacuum is a spinor which we write |A , where A = 1, . . . , 16 are indices in the spinor representation 8+ + 8− of the transverse Lorentz group SO(8). In the NS

We conclude that we have spacetime vectors |I · · · and spacetime spinors |A · · · , where · · · represents the left-moving state that we now proceed to determine.

The left-moving sector consists of 24 transverse coordinates, 16 of which are internal degrees of freedoma. For these 16 coordinates, we use the following property: in two dimensions, one boson is equivalent to two Majorana fermionsb. We therefore replace the 16 internal coordinates by 2 × 16 fermions λi. In the same way that the coordinates XM , M = 1, . . . , 2n, are in the vector representation of SO(2n), these fermions form a vector representation of a SO(16) × SO(16) group. They can have periodic or antiperiodic boundary conditions. However modular invariance imposes restrictions on the possible choices. If all 32 fermions have the same boundary conditions, one obtains

aThe remaining eight spacetime coordinates XI (z) provide the N = 1 supergravity multiplet which consists of |I αJ−1|0 (graviton [35 degrees of freedom], antisymmetric tensor

[28] and dilaton [1]) and |A αJ−1|0 (gravitino [56] and a spinor field called the dilatino [8]).

bSchematically, the correspondence reads at the level of currents: ∂z X ≡ ψ1ψ2.