‚ W(φ)

¯

V (φ, φ) = . (3.12)

‚φ

Supersymmetric Solitons and Topology 257

In what follows we will often denote the chiral super¬eld and its lowest (bosonic)

component by one and the same letter, making no distinction between capital

and small φ. Usually it is clear from the context what is meant in each particular

case.

If one limits oneself to renormalizable theories, the superpotential W must

be a polynomial function of ¦ of power not higher than three. In the model at

hand, with one chiral super¬eld, the generic superpotential then can be always

reduced to the following “standard” form

m2 »

W(¦) = ¦ ’ ¦3 . (3.13)

» 3

The quadratic term can be always eliminated by a rede¬nition of the ¬eld ¦.

Moreover, by using the symmetries of the model, one can always choose the

phases of the constants m and » at will. (Note that generically the parameters

m and » are complex.)

Let us study the set of classical vacua of the theory, the vacuum manifold.

In the simplest case of the vanishing superpotential, W = 0, any coordinate-

independent ¬eld ¦vac = φ0 can serve as a vacuum. The vacuum manifold is then

the one-dimensional (complex) manifold C 1 = {φ0 }. The continuous degeneracy

is due to the absence of the potential energy, while the kinetic energy vanishes

for any constant φ0 .

This continuous degeneracy is lifted in the case of a non-vanishing superpo-

tential. In particular, the superpotential (3.13) implies two degenerate classical

vacua,

m

φvac = ± . (3.14)

»

Thus, the continuous manifold of vacua C 1 reduces to two points. Both vacua are

physically equivalent. This equivalence could be explained by the spontaneous

breaking of the Z2 symmetry, ¦ ’ ’¦, present in the action (3.9) with the

superpotential (3.13).

The determination of the conserved supercharges in this model is a straight-

forward procedure. We have

¯™ ¯0

d3 xJ± ,

0

d3 xJ± ,

Q± = Q± = (3.15)

™

µ

where J± is the conserved supercurrent,

1 µ ββ™

µ

J± = (¯ ) J±β β ,

σ ™

2

√

¯™

J±β β = 2 2 (‚±β φ+ )ψβ ’ i F ψβ . (3.16)

™ ™ β±

The Golfand“Likhtman superalgebra in the spinorial notation takes the form

{Q± , Q± } = 2P±± ,

¯™ (3.17)

™

where P is the energy-momentum operator.

258 M. Shifman

3.3 Critical Domain Walls

The minimal Wess“Zumino model has two degenerate vacua (3.14). Field con-

¬gurations interpolating between two degenerate vacua are called domain walls.

They have the following properties: (i) the corresponding solutions are static and

depend only on one spatial coordinate; (ii) they are topologically stable and in-

destructible “ once a wall is created it cannot disappear. Assume for de¬niteness

that the wall lies in the xy plane. This is the geometry we will always keep in

mind. Then the wall solution φw will depend only on z. Since the wall extends

inde¬nitely in the xy plane, its energy Ew is in¬nite. However, the wall tension

Tw (the energy per unit area Tw = Ew /A) is ¬nite, in principle measurable, and

has a clear-cut physical meaning.

The wall solution of the classical equations of motion super¬cially looks very

similar to the kink solution in the SPM discussed in Sect. 2,

m

φw = tanh(|m|z) . (3.18)

»

Note, however, that the parameters m and » are not assumed to be real; the ¬eld

φ is complex in the Wess“Zumino model. A remarkable feature of this solution

is that it preserves one half of supersymmetry, much in the same way as the

critical kinks in Sect. 2. The di¬erence is that 1/2 in the two-dimensional model

meant one supercharge, now it means two supercharges.

Let us now show the preservation of 1/2 of SUSY explicitly. The SUSY

transformations (3.1) generate the following transformation of ¬elds,

√ √

δψ ± = 2 µ± F + i ‚µ φ (σ µ )±± µ± .

™

δφ = 2µψ , ¯™ (3.19)

The domain wall we consider is purely bosonic, ψ = 0. Moreover, let us impose

the following condition on the domain wall solution (the BPS equation):

F |φ=φ— = ’e’i· ‚z φw (z) , (3.20)

¯

w

where m3

· = arg , (3.21)

»2

¯¯

and, I remind, F = ’‚ W/‚ φ , see (3.11). This is a ¬rst-order di¬erential equa-

tion. The solution quoted above satis¬es this condition. The reason for the

occurrence of the phase factor exp(’i·) on the right-hand side of (3.20) will

become clear shortly. Note that no analog of this phase factor exists in the two-

dimensional N = 1 problem on which we dwelled in Sect. 2. There was only a

sign ambiguity: two possible choices of signs corresponded respectively to kink

and anti-kink.

The ¬rst-order BPS equations are, generally speaking, a stronger constraint

than the classical equations of motion.6 If the BPS equation is satis¬ed, then

the second supertransformation in (3.19) reduces to

δψ± ∝ µ± + i ei· (σ z )±± µ± .

™

™¯ (3.22)

6

I hasten to add that, in the particular problem under consideration, the BPS equation

follows from the equation of motion; this is explained in Sect. 3.5.

Supersymmetric Solitons and Topology 259

The right-hand side vanishes provided that

µ± = ’i ei· (σ z )±± µ± .

™

™¯ (3.23)

This picks up two supertransformations (out of four) which do not act on the

domain wall (alternatively people often say that they act trivially). Quod erat

demonstrandum.

Now, let us calculate the wall tension. To this end we rewrite the expression

for the energy functional as follows

+∞

¯

E= ¯

dz ‚z φ ‚z φ + F F

’∞

+∞

2

e’i· ‚z W + h.c. + ‚z φ + ei· F

≡ dz , (3.24)

’∞

where φ is assumed to depend only on z. In the literature this procedure is

called the Bogomol™nyi completion. The second term on the right-hand side is

non-negative “ its minimal value is zero. The ¬rst term, being full derivative,

depends only on the boundary conditions on φ at z = ±∞.

Equation (3.24) implies that E ≥ 2 Re e’i· ∆W . The Bogomol™nyi com-

pletion can be performed with any ·. However, the strongest bound is achieved

provided e’i· ∆W is real. This explains the emergence of the phase factor in the

BPS equations. In the model at hand, to make e’i· ∆W real, we have to choose

· according to (3.21).

When the energy functional is written in the form (3.24), it is perfectly

obvious that the absolute minimum is achieved provided the BPS equation (3.20)

is satis¬ed. In fact, the Bogomol™nyi completion provides us with an alternative

derivation of the BPS equations. Then, for the minimum of the energy functional

“ the wall tension Tw “ we get

Tw = |Z| . (3.25)

Here Z is the topological charge de¬ned as

8 m3

Z = 2 {W(φ(z = ∞)) ’ W(φ(z = ’∞))} = . (3.26)

3 »2

How come that we got a nonvanishing energy for the state which is anni-

hilated by two supercharges? This is because the original Golfand“Likhtman

superalgebra (3.17) gets supplemented by a central extension,

{Q± , Qβ } = ’4 Σ±β Z , = ’4 Σ±β Z ,

¯ ¯™ ¯™ ¯™

Q± , Qβ (3.27)

™

where

1

Σ±β = ’ dx[µ dxν] (σ µ )±± (¯ ν )±

™

™σ β (3.28)

2

is the wall area tensor. The particular form of the centrally extended algebra is

somewhat di¬erent from the one we have discussed in Sect. 2. The central charge

260 M. Shifman

is no longer a scalar. Now it is a tensor. However, the structural essence remains

the same.

As was mentioned, the general connection between the BPS saturation and

the central extension of the superalgebra was noted long ago by Olive and Wit-

ten [2] shortly after the advent of supersymmetry. In the context of supersym-

metric domain walls, the topic was revisited and extensively discussed in [10]

and [12] which I closely follow in my presentation.

Now let us consider representations of the centrally extended superalgebra

(with four supercharges). We will be interested not in a generic representation

but, rather, in a special one where one half of the supercharges annihilates all

states (the famous short representations). The existence of such supercharges

was demonstrated above at the classical level. The covariant expressions for the

˜

residual supercharges Q± are

2

Q± = ei·/2 Q± ’ e’i·/2 Σ±β nβ Q± ,

˜ ¯™ (3.29)

±

™

A

where A is the wall area (A ’ ∞) and

P±±™

n±± = (3.30)

™

Tw A

is the unit vector proportional to the wall four-momentum P±± ; it has only the

™

time component in the rest frame. The subalgebra of these residual supercharges

in the rest frame is

Q± , Qβ = 8 Σ±β {Tw ’ |Z|} .

˜˜ (3.31)

The existence of the subalgebra (3.31) immediately proves that the wall ten-

sion Tw is equal to the central charge Z. Indeed, Q|wall = 0 implies that

˜

Tw ’ |Z| = 0. This equality is valid both to any order in perturbation theory

and non-perturbatively.

From the non-renormalization theorem for the superpotential [13] we addi-

tionally infer that the central charge Z is not renormalized. This is in contradis-

tinction with the situation in the two-dimensional model 7 of Sect. 2. The fact

that there are more conserved supercharges in four dimensions than in two turns

out crucial. As a consequence, the result

8 m3

Tw = (3.32)

3 »2

for the wall tension is exact [12,10].

The wall tension Tw is a physical parameter and, as such, should be express-

ible in terms of the physical (renormalized) parameters mren and »ren . One can

easily verify that this is compatible with the statement of non-renormalization

of Tw . Indeed,

» = Z 3/2 »ren ,

m = Z mren ,

7

There one has to deal with the fact that Z is renormalized and, moreover, a quantum

anomaly was found in the central charge. See Sect. 2.6. What stays exact is the

relation M ’ Z = 0.

Supersymmetric Solitons and Topology 261

where Z is the Z factor coming from the kinetic term. Consequently,

m3 m3

ren

=2.

2

» »ren

Thus, the absence of the quantum corrections to (3.32), the renormalizability of

the theory, and the non-renormalization theorem for superpotentials “ all these

three elements are intertwined with each other. In fact, every two elements taken

separately imply the third one.

What lessons have we drawn from the example of the domain walls? In the

centrally extended SUSY algebras the exact relation Evac = 0 is replaced by the

exact relation Tw ’ |Z| = 0. Although this statement is valid both perturba-

tively and non-perturbatively, it is very instructive to visualize it as an explicit

cancelation between bosonic and fermionic modes in perturbation theory. The

non-renormalization of Z is a speci¬c feature of four dimensions. We have seen

previously that it does not take place in minimally supersymmetric models in

two dimensions.

3.4 Finding the Solution to the BPS Equation

In the two-dimensional theory the integration of the ¬rst-order BPS equation

(2.26) was trivial. Now the BPS equation (3.20) presents in fact two equations

“ one for the real part and one for the imaginary part. Nevertheless, it is still

trivial to ¬nd the solution. This is due to the existence of an “integral of motion,”

‚

Im e’i· W = 0 . (3.33)

‚z

The proof is straightforward and is valid in the generic Wess“Zumino model

with an arbitrary number of ¬elds. Indeed, di¬erentiating W and using the BPS