. 56
( 78 .)


‚ W(φ)
V (φ, φ) = . (3.12)
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
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
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± ,
d3 xJ± ,
Q± = Q± = (3.15)

where J± is the conserved supercurrent,
1 µ ββ™
J± = (¯ ) J±β β ,
σ ™

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,
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)

where m3
· = arg , (3.21)
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)
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
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

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)

Σ±β = ’ dx[µ dxν] (σ µ )±± (¯ ν )±

™σ β (3.28)
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
Q± = ei·/2 Q± ’ e’i·/2 Σ±β nβ Q± ,
˜ ¯™ (3.29)

where A is the wall area (A ’ ∞) and
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 ,
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
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)
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


. 56
( 78 .)