Supersymmetry. Theory, Experiment, and Cosmology

70 The supersymmetry algebra and its representations

S(1)

φ

Qm

C

S(2)

Fig. 4.1 Closed surface around a magnetic monopole

Under the gauge transformation, the wave function corresponding to a field of electric charge Qe transforms as

Requiring such wave functions to be single-valued (as φ → φ + 2π) implies that

which is the famous Dirac quantization condition [118, 381]: the presence of a single monopole imposes that all electric charges are multiples of a given unit of charge (2π /Qm). This gives a rationale for the problem of the quantization of charge mentioned in Chapter 1. We will encounter a seemingly di erent reason when we discuss grand unified theories in Chapter 9 but we will see that, at a deeper level, the two solutions coincide.

Until now, we have only considered electric charges and magnetic monopoles. One may envisage the presence of dyons [241], i.e. objects with both electric charge Qe and magnetic charge Qm. If we have two dyons, the Dirac quantization condition becomes [332, 390]:

We deduce that, if the theory has electrons of charges (−e, 0), then the quantization reads, for any dyon field (Qe, Qm): Qm = 2nπ /e. Let us consider for example the dyon (Qe, 2π /e) and assume that we have CP invariance. Under CP, the electric charge is odd whereas the magnetic charge is even. We thus also have a dyon (−Qe, 2π /e). Applying the quantization condition (4.72) to this pair, we obtain Qe = ne/2: such dyons must have integer or half-integer charge.

Moreover, applying again the quantization condition to (Qe, 2π /e) and (Qe, 2π /e), we obtain Qe − Qe = ne. Hence all of them must have integer charges, or have half-integer charges.

However, if there is a source of CP violation, we may expect that there will be a departure from these quantized values. Witten [370] has in particular stressed the importance of adding the CP violating term

BPS states 71

which gives a contribution of order θ to the dyon mass. We will return in more details to this issue in Section 4.5.7.

4.5.3Monopoles in the Georgi–Glashow model

In order to be able to consider the magnetic charge as a topological charge, we need to unify the electromagnetic U (1) symmetry as part of a larger nonabelian symmetry. The simplest case is the model of [180] (see also Exercise 9 of Appendix Appendix A) with a SU (2) gauge symmetry. We note that, in such cases, the monopole is a solution of the field equations and therefore must exist, in the context of these theories. From now on, we set = 1.

The Lagrangian of the Georgi–Glashow model reads

θ term does not contribute to the equation of motion thanks to the Bianchi identity

with standard definitions for the electric and magnetic fields (see the preceding section):

massless.

We are interested in finite energy configurations. Obviously, for the energy to remain finite, each term in (4.80) must fall o faster than 1/r2 at infinity. On the surface at infinity, the two-sphere S∞2 , we must recover the vacuum. Hence a finite

72 The supersymmetry algebra and its representations

energy configuration defines a mapping φa: S∞2 → Svac2 . Such a map is characterized by an integer, the winding number n, which counts the number of times one wraps one sphere around the other. This can be expressed as

ˆa We may express this in a more mathematical way. We first note that the vector φ

which describes the vacuum parametrizes the coset group G/H = SU (2)/U (1) in the breaking of G = SU (2) to H = U (1) (see Section A.2.3 of Appendix Appendix A). We have thus defined a mapping from S∞2 to SU (2)/U (1). The group of inequivalent mappings (i.e. mappings which cannot be deformed continuously into one another) from the k-sphere Sk to a manifold M (such as G/H) is called its kth homotopy group and noted πk(M). We have just established that π2(SU (2)/U (1)) is isomorphic to Z.

We must have in particular

where Ai is a SU (2) invariant potential. We then deduce from (4.82) that, at large distances, the field strength is aligned along the scalar field11, that is along the direction of unbroken U (1) symmetry:

The magnetic charge corresponding to this field configuration is expressed in terms of the winding number n given in (4.81) as:

(4.86) which corresponds to the Dirac quantization condition (4.71) since, in this theory, fields in the fundamental of SU (2) (not introduced here) would have an electric charge Qe = g/2. We conclude that the finite energy configurations of the Georgi–Glashow model are the magnetic monopoles.

BPS states 73

More generally, if the symmetry group of a theory is G and the unbroken subgroup H, then monopoles exist if the second homotopy group π2(G/H) is nontrivial, i.e. not reduced to the identity. If G is simply connected (i.e. any loop can be continuously deformed into a point: π1(G) is trivial) such as SU (n) or any simple compact connected Lie group besides SO(n), then

For example, if H = U (1), π1(U (1)) = Z, then monopoles exist and are characterized by a integer n. This is what happens with the ’t Hooft–Polyakov model just discussed or the breaking of the grand unified group SU (5) (see Chapter 9). If H = SO(n) (n ≥ 3), π1(SO(n)) = Z2, there is a single type of monopole corresponding to the nontrivial element −1 of Z2. If G is not simply connected, one may find useful the following theorem: π2(G/H) is isomorphic to the kernel of the homomorphism of π1(H) into π1(G) (a loop in H is naturally a loop in G).

4.5.4BPS monopoles

We note that, according to (4.84), we may write the electric and magnetic charges with respect to U (1) as

Following [47], we may derive a lower bound for the energy of the stationary configurations that we have just identified. For simplicity, we work in the temporal gauge12 Aa0 = 0. We obtain for the energy density (4.80)

where we have separated the contribution which is quadratic in time derivatives

(“kinetic energy” T ) from the one which does not involve time derivatives (“potential”

V). Since, according to (4.88), aQm = d3x BiaDiφa, one obtains (choosing the upper, respectively lower, sign for positive, respectively negative, magnetic charge):

12Accordingly, we must impose the Gauss law constraint, which corresponds to ensuring the A0

in the limit of vanishing potential V (φ). This limit is known as the Prasad–Sommerfield limit [315] and the states which saturate the bound are thus known as BPS states. The upper (lower) sign corresponds to the monopole (antimonopole) solution.

As we will emphasize later, the Prasad–Sommerfield condition of vanishing potential is unstable under quantum fluctuations unless we are in a supersymmetric context.

The Bogomol’nyi equations can be solved using the following static and spherically symmetric ansatz:

where r = |x| and H and K are functions of the rescaled radial variable ξ = agr. Plugging this ansatz into the Bogomol’nyi equation (4.93) for the (anti)monopole yields

the boundary conditions being H → 0, K → 1 as ξ → 0 and H → 1, K → 0 as ξ → ∞. The solution corresponding to the monopole (upper sign in the preceding equation) reads:

4.5.5Moduli space

Before quantizing the monopole, we need to identify its collective coordinates. The monopole solution which has been constructed in the previous subsection sits at the origin but the translation invariance of the equations implies that a solution with the same energy can be constructed at any point in space. Thus the spatial coordinates of the monopole center of mass provide three collective coordinates.

Since the inclusion of these degrees of freedom does not change the (potential) energy of the monopole, they are zero modes, or moduli, of the system and their quantization requires a special treatment. Just as for the kink, the standard way [316] is to start with the classical solution (4.94) which we denote φacl(x), Aai cl(x) and to

BPS states 77

We thus consider a N = 2 vector supermultiplet, whose action is written in (4.36) (we will set P r = 0 and thus consider the on-shell action). We note that the condition of vanishing potential requires [A, B] = 0, which may be satisfied for nonzero vacuum expectation values of A and B. Supersymmetry ensures that this holds at the quantum level.

Choosing the gauge symmetry group SU (2), as in the Georgi–Glashow model, we may set the vacuum expectation value for B to zero. A nonzero value for A (sayA3 = a = 0) breaks SU (2) to U (1). The supersymmetry current for the model considered reads (see (3.37) in Chapter 3 and (C.73) in Appendix C):

One may then deduce the supersymmetry charge anticommutation relations, with special attention to the boundary terms. Introducing

We see that, in (4.108), U and V play the rˆole of central charges. We thus expect mass bounds similar to the ones derived in subsection 4.3.3. Indeed, if we work in the frame where Pµ = (M, 0, 0, 0), then (4.109) reads

Obviously states which saturate the bound are annihilated by at least one of the supersymmetry generators.

We note that all states in the theory saturate the bound: for example, the photon Aµ is massless whereas Wµ± has mass ga; and dyon masses are given by (4.106). This explains what could have been seen as a puzzle otherwise. Indeed, we started with the eight states of the (non-CPT self-conjugate) N = 2 vector supermultiplet: Aµ, λi, A, B. Symmetry breaking gives mass to some of these states but does not provide the