Supersymmetry. Theory, Experiment, and Cosmology

Representations of the supersymmetry algebra 59

λ−1/2 0 1/2

All the preceding supermultiplets meet the following CPT theorem requirement: for every state of helicity λ, one should have a parity reflected state of helicity −λ. They are CPT self-conjugate.

• N = 2 λ0 = −1

λ−1 −1/2 0

This supermultiplets does not satisfy the requirement above. One must add to it a supermultiplet built from |Ω0 = |E, λ0 = 0

In total, one obtains a N = 2 vector supermultiplet

Hence a non-CPT self-conjugate supermultiplet has 2N +1 helicity states.

• N = 1, λ0 = − 12

Similarly, one obtains by including the CPT conjugate (λ0 = 0)

λ−1/2 0 1/2

which describes the vector supermultiplet.

We should note that all the degrees of freedom discussed here are on-shell degrees of freedom.

60 The supersymmetry algebra and its representations

4.3.2Massive states, in the absence of central charges

In this case, one chooses a rest frame where Pµ = (m, 0, 0, 0) and W 2 = −m2J2, where the Lorentz generators (Ji) = (M23, M31, M12) form a SO(3) SU (2) algebra.

The only nontrivial anticommutator relation is now

In a way similar to the preceding case, one may therefore construct 22N states out of a state |Ω0 of given spin j0.

Let us take the case N = 1 for simplicity and consider a state |Ω0 of total spin j0 annihilated by the Qα. One may therefore form

˙

In the last line, we have used the fact that the indices β and γ˙ are necessarily distinct and that a†α˙ a†α˙ is a scalar entity (spin 0) made out of two spin 12 objects.

We have for example constructed:

Representations of the supersymmetry algebra 61

Let us note that the latter field content is compatible with the supersymmetric version of the Higgs mechanism. With a massless Higgs chiral superfield (two scalar and two spin 1/2 degrees of freedom) and a massless vector superfield (two spin 1/2 and two spin 1 degrees of freedom), we can make a massive vector superfield where one scalar component provides the missing longitudinal degree of freedom of the vector particle.

In the case of general N , starting again from a state |Ω0 of total spin j0, we

construct a state of maximal spin j = j0 + N/2 in the form of a1† · · · aN †|Ω0 . Thus,

˙ ˙

2 2

in the absence of central charges, massive multiplets of N > 1 supersymmetry always contain fields of spin larger or equal to 1.

4.3.3Massive states in the presence of central charges

In this case, the supersymmetry charges cannot be straightforwardly interpreted as creation and annihilation operators. We consider here only the case N even. Through a unitary transformation [388], one first puts the antisymmetric matrix Zij into the form

zN/2

The index i of the supersymmetry charges is then substituted to (ar) where a = 1, 2 and r = 1, . . . , N/2. The supersymmetry algebra (4.2, 4.7) reads

where [εab] = [εab] = iσ2.

One easily checks that massless representations are only compatible with zr = 0 (see Exercise 6).

For massive representations, one works again in the frame where Pµ = (m, 0, 0, 0).

Hence necessarily zr ≤ 2m.

•When zr < 2m, the multiplicities are the same as in the case of no central charge, i.e. 22N (2j + 1). One describes them as long multiplets.

62The supersymmetry algebra and its representations

•When zr = 2m for some or all zr, the multiplicity is decreased. For example, if

there are n0 central charges that saturate the bound, i.e. such that zr = 2m, the multiplicity is decreased to 22(N −n0)(2j + 1) and one speaks of short multiplets. Such states are called BPS-saturated states because of their connection with BPS monopoles, as we will see in the Section 4.5.

As an example, we may take for N = 2 supersymmetry n0 = 1, j = 0 which corresponds to the hypermultiplet constructed in the next section.

4.4Multiplets of N = 2 supersymmetry

Extended supersymmetries play an important rˆole in the understanding of the nonperturbative aspects of supersymmetric theories, as well in supersymmetric models arising from higher dimensional theories, such as string theories. In this section, we construct the basic N = 2 supermultiplets [147, 327] by combining the N = 1 supermultiplets that we are now acquainted with.

4.4.1Vector supermultiplet

We have encountered the N = 2 vector supermultiplet in Section 4.3.1 above. It is easy to check from the field content that it can be decomposed into a N = 1 vector and N = 1 chiral supermultiplets:

We may thus try to construct an N = 2 supersymmetric action out of a N = 1 vector supermultiplet (Aaµ, λa, Da) and a N = 1 chiral supermultiplet (φa, ψa, F a) in the adjoint representation. We use equations (C.81) and (C.84) of Appendix C to write

64 The supersymmetry algebra and its representations

This action is invariant under the following N = 2 supersymmetry transformations:

The transformation with parameter ηα1 , η¯1α˙ is simply the N = 1 supersymmetry transformation that acts between the components of the chiral and vector supermultiplets (see Exercise 7).

4.4.2Hypermultiplet

Matter may be described by hypermultiplets which involve on-shell two complex scalar

ψ1

fields φi, i = 1, 2, forming a doublet of U (2), and one Dirac fermion Ψ ≡ ¯ α .

ψ2α˙

O -shell, this is complemented by two complex auxiliary fields F i, i = 1, 2, and the supersymmetry transformation reads

S

where the transformation parameter ξi, i = 1, 2, is again the symplectic Majorana spinor of (4.38). Under the N = 1 supersymmetry transformation with parameter

BPS states 65

We see that the supersymmetry algebra closes on shell only in the case of massless fields (F i = 0, γµ∂µΨ = 0, φi = 0). In the case of nonvanishing mass, one must take into account the central charge. We note that

is the operator equivalent of the condition z2 = 4m2, or z = 2m found at the end of the previous section to obtain short multiplets such as the hypermultiplet.

The U (2) symmetry is not compatible with any self-interactions of the hypermultiplet. The only allowed interactions are gauge interactions. If we couple the hypermultiplet with the vector supermultiplet introduced in the previous subsection, the Lagrangian (4.43) becomes, for m = 0,

L µ i† i i† i ¯ µ

= D φ Dµφ + F F + iΨγ DµΨ

The supersymmetry and central charge transformations (4.39) and (4.41) of the hypermultiplet then include extra terms involving the gauge supermultiplet.

4.5BPS states

Solitons, which can be defined as a (partially) localized and nondispersive concentration of energy will play an increasingly important rˆole in the remainder of this book. Since they provide a beautiful illustration of the uses of short supermultiplets we will take the opportunity to present some key notions using as examples kinks and monopoles. This will allow us to illustrate important new notions such as BPS states, moduli space, zero modes, duality, and their connection with supersymmetry.

4.5.1Introduction: The kink solution and domain walls

We start by illustrating some of the basic methods used in this section by considering the simpler case of domain walls. Domain walls appear in models where a discrete symmetry is broken: two regions of space which correspond to two di erent ground states are separated by a sheet-like structure where energy is concentrated (corresponding to the energy barrier between the two ground states). We take as an example a real scalar field whose potential energy V (φ), being invariant under the reflection φ ↔ −φ, has minima at φ = ± φ0 (φ0 > 0): when we need to be more explicit, we choose V (φ) = 12 λ(φ2 − φ20)2. We will assume that the domain wall is flat and parallel to the y–z plane. The problem becomes 1 + 1 dimensional, i.e. depends only on time t and on the spatial coordinate x and we can write the scalar field action

66 The supersymmetry algebra and its representations

The corresponding energy per unit area reads:

Finiteness requires |φ| → |φ0| for x → ±∞. We will be focussing our attention on the kink (resp. antikink) solution where φ → ±φ0 (resp. φ → φ0) as x → ±∞. This twisted boundary condition stabilizes the solution: changing for example from −φ0 to +φ0 all the way to x → −∞ in a finite time would require an infinite amount of energy. In other words

may be interpreted as a topological charge: being the integral of a total divergence, it vanishes for topologically trivial backgrounds8.

One may try to solve the static equation of motion

which is second order. [47] has introduced a trick which allows us to identify a first order equation and simplifies the task of finding an explicit solution in more complicated situations. One may write (4.47) for the static solution φ(x)

where the sign is chosen in order to obtain the absolute value in (4.51) and depends on the asymptotic values of φ: the choice φ(±∞) = ±φ0 yields a plus sign.

We have thus obtained a bound on the energy which depends only on the asymptotic values of the field. Moreover, any solution of (4.52) minimizes E and thus the

8More precisely, we may define a current jµ = εµν ∂ν φ where εµν is the antisymmetric tensor with

two indices (µ = 0 for t and 1 for x). This current is conserved by construction: ∂ jµ = εµν ∂ ∂ φ = 0.

µ µ ν

Thus the topological charge T is conserved: T = dx j0.

BPS states 67

action S in the static case: it is therefore a solution of the equation of motion (4.49) as can be checked directly. In the specific case considered, one finds

√

where we have introduced the mass of the φ excitation: m = 2λφ0. The energy (rest mass) per unit area is obtained directly from the boundary value (4.51):

Note that, for a given value of the elementary excitation mass, the soliton rest mass is inversely proportional to the coupling λ: the smaller the coupling, the larger the mass. This is to be contrasted with the standard situation of the Higgs mechanism where masses are proportional to couplings.

The kink is thus characterized by the unique collective coordinate x0 which fixes the position of the domain wall (i.e. where energy is localized). One calls such a collective coordinate a modulus and the space of solutions of minimal energy (for a given topological charge) is referred to as the moduli space: in this case, since x0 R, it is isomorphic to R.

A boost in the x direction may move this position to any arbitrary value. Thus, one may turn the static solution into a dynamical one by letting x0 be time-dependent: x0(t). This motion in moduli space corresponds to an adiabatic evolution through the space of static solutions. Of course, other time dependent configurations of the system may be considered but this motion dominates at long wavelength over all other excitations: it corresponds to a massless fluctuation around the kink solution, what we will call later a zero mode. If we want to quantize around the kink background, we must first factor out this type of mode [316].

Replacing φ(x, t) by φ(x) in (4.53) with x0 = x0(t), one obtains for the action (4.46)

which is, up to an additive constant, the action of a nonrelativistic particle.

The analysis has been conducted until now at the classical level: for example, the estimate of the rest mass of the soliton has been purely classical. We will now see that one can go further in the context of supersymmetry. Indeed, if we turn supersymmetric, we must replace the action (4.46) by a supersymmetric two-dimensional action. For simplicity, in what follows, we simply ignore the two extra spatial dimensions

where µ = 0, 1 correspond to the variables x0 ≡ t and x1 ≡ x, respectively, and ψ is a Majorana fermion in two dimensions. The scalar potential is, as usual, written in

terms of the superpotential V (φ) = W 2(φ)/2. We recover the previous potential for

√

W (φ) = λ(φφ20 − φ3/3).

68 The supersymmetry algebra and its representations

One can then define [368] (see Problem 1) a supersymmetry charge with components Q+, Q− satisfying

Note that Z is of a topological nature, much in the way of the charge T introduced

in (4.48). It vanishes for topologically trivial backgrounds but is nonzero for the kink:

√ √

for W (φ) = λ(φφ20 − φ3/3), we have Z = 8 λφ30/3.

The kink solution satisfies this equation with a − sign. The orthogonal combination is broken by the kink solution: indeed, the domain wall obviously breaks translational invariance in the x direction.

We now turn to truly four-dimensional examples. In this case, the rˆole of the topological charge is played by the electric or magnetic charge. Indeed, the electric charge can be written as a surface integral: using Gauss’ law, one may write for the electric charge

which can be compared with the two-dimensional case just discussed, equation (4.58). A discussion of the magnetic charge requires some background on magnetic monopoles which we now present.

4.5.2Monopoles and the Dirac quantization condition

Magnetic monopoles appear naturally when one tries to introduce sources that respect the electric–magnetic duality observed in Maxwell equations in the vacuum. We start by reviewing the argument: this notion of duality will play an important rˆole in subsequent chapters (see Chapter 8).