dimensional multiplets was discarded in the literature for years. It is for a rea-

son: while the fermion parity (’1)F is granted in any local ¬eld theory based on

fermionic and bosonic ¬elds, it is not de¬ned in the one-dimensional irreducible

representation. Indeed, if it were de¬ned, it would be ’1 for Q1 , which is in-

compatible with any of the equations (2.74). The only way to recover (’1)F is

to have a reducible representation containing both | sol+ and | sol ’ . Then,

√

(’1)F = σ1 ,

Q1 = σ3 2Z , (2.75)

where σ1,2,3 stand for the Pauli matrices.

Does this mean that the one-state supermultiplet is not a possibility in the

local ¬eld theory? As I argued above, in the simplest two-dimensional super-

symmetric model (2.1) the BPS solitons do exist and do realize such supershort

multiplets defying (’1)F . These BPS solitons are neither bosons nor fermions.

Further details can be found in [7], in which a dedicated research of this partic-

ular global anomaly is presented. The important point is that short multiplets

of BPS states are protected against becoming non-BPS under small perturba-

tions. Although the overall sign of Q1 on the irreducible representation is not

observable, the relative sign is. For instance, there are two types of reducible rep-

resentations of dimension two: one is {+, ’} (see (2.75)), and the other {+, +}

(equivalent to {’, ’}). In the ¬rst case, two states can pair up and leave the

BPS bound as soon as appropriate perturbations are introduced. In the second

case, the BPS relation M = Z is “bullet-proof.”

254 M. Shifman

To reiterate, the discrete Z2 symmetry G = (’1)F discussed above is nothing

but the change of sign of all fermion ¬elds, ψ ’ ’ψ. This symmetry is seemingly

present in any theory with fermions. How on earth can this symmetry be lost in

the soliton sector?

Technically the loss of G = (’1)F is due to the fact that there is only one

(real) fermion zero mode on the soliton in the model at hand. Normally, the

¯

fermion degrees of freedom enter in holomorphic pairs, {ψ, ψ}. In our case of a

single fermion zero mode we have “one half” of such a pair. The second fermion

zero mode, which would produce the missing half, turns out to be delocalized.

More exactly, it is not localized on the soliton, but, rather, on the boundary

of the “large box” one introduces for quantization (see Sect. 2.5). For physical

measurements made far away from the auxiliary box boundary, the fermion

parity G is lost, and the supermultiplet consisting of a single state becomes

a physical reality. In a sense, the phenomenon is akin to that of the charge

fractionalization, or the Jackiw“Rebbi phenomenon [8]. The essence of this well-

known phenomenon is as follows: in models with complex fermions, where the

fermion number is de¬ned, it takes integer values only provided one includes in

the measurement the box boundaries. Local measurements on the kink will yield

a fractional charge.

3 Domain Walls in (3+1)-Dimensional Theories

Kinks are topological defects in (1+1)-dimensional theories. Topological defects

of a similar nature in 1+3 dimensions are domain walls. The corresponding

geometry is depicted in Fig. 1. Just like kinks, domain walls interpolate (in the

transverse direction, to be denoted as z) between distinct degenerate vacua of

the theory. Unlike kinks, domain walls are not localized objects “ they extend

into the longitudinal directions (x and y in Fig. 1). Therefore, the mass (energy)

of the domain wall is in¬nite and the relevant parameter is the wall tension “ the

mass per unit area. In (1+3)-dimensional theories the wall tension has dimension

m3 .

In this section I will discuss supersymmetric critical (BPS-saturated) do-

main walls. Before I will be able to proceed, I have to describe the simplest

(1+3)-dimensional supersymmetric theory in which such walls exist. Unlike in

two dimensions, where ¬eld theories with minimal supersymmetry possess two

supercharges, in four dimensions the minimal set contains four supercharges,

{Q± , Q± } ,

¯™ ±, ± = 1, 2 .

™

¯™

Q± and Q± are spinors with respect to the Lorentz group.

3.1 Superspace and Super¬elds

The four-dimensional space xµ (with Lorentz vectorial indices µ = 0, ..., 3) can

¯™

be promoted to superspace by adding four Grassmann coordinates θ± and θ± ,

Supersymmetric Solitons and Topology 255

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Fig. 1. Domain wall geometry.

(with spinorial indices ±, ± = 1, 2). The coordinate transformations

™

¯™ ¯™ ¯™

{xµ , θ± , θ± } : δx±± = ’2i θ± µ± ’ 2i θ± µ±

δθ± = µ± , δ θ ± = µ± ,

¯™ ¯™ (3.1)

™

add SUSY to the translational and Lorentz transformations.5

Here the Lorentz vectorial indices are transformed into spinorial ones accord-

ing to the standard rule

1 ™

Aβ β = Aµ (σ µ )β β , Aµ = A±β (¯ µ )β± ,

™σ (3.2)

™ ™

2

where

(σ µ )±β = {1, „ }±β , (¯ µ )β± = (σ µ )±β .

σ™ (3.3)

™ ™ ™

We use the notation „ for the Pauli matrices throughout these lecture notes.

The lowering and raising of the spinorial indices is performed by virtue of the

±β

symbol ( ±β = i(„2 )±β , 12 = 1). For instance,

™ ™™

(¯ µ )ργ = {1, ’„ }β± .

(¯ µ )β± = β ρ ±γ

σ σ™ (3.4)

™

5

My notation is close but not identical to that of Bagger and Wess [9]. The main

distinction is the conventional choice of the metric tensor gµν = diag(+ ’ ’’) as

opposed to the diag(’ + ++) version of Bagger and Wess. For further details see

Appendix in [10]. Both, the spinorial and vectorial indices will be denoted by Greek

letters. To di¬erentiate between them we will use the letters from the beginning of

the alphabet for the spinorial indices (e.g. ±, β etc.) reserving those from the end of

the alphabet (e.g. µ, ν, etc.) for the vectorial indices.

256 M. Shifman

Two invariant subspaces {xµ , θ± } and {xµ , θ± } are spanned on 1/2 of the

¯™

L R

Grassmann coordinates,

{xµ , θ± } : δ(xL )±± = ’4i θ± µ± ;

δθ± = µ± , ¯™

™

L

{xµ , θ± } :

¯™ ¯™ ¯™

δ(xR )±± = ’4i θ± µ± ,

δ θ ± = µ± ,

¯™ (3.5)

™

R

where

¯™

(xL,R )±± = x±± “ 2i θ± θ± . (3.6)

™ ™

The minimal supermultiplet of ¬elds includes one complex scalar ¬eld φ(x) (two

bosonic states) and one complex Weyl spinor ψ ± (x) , ± = 1, 2 (two fermionic

states). Both ¬elds are united in one chiral super¬eld,

√

¦(xL , θ) = φ(xL ) + 2θ± ψ± (xL ) + θ2 F (xL ) , (3.7)

where F is an auxiliary component, which appears in the Lagrangian without

the kinetic term.

The superderivatives are de¬ned as follows:

‚ ‚

¯™

’ i‚±± θ± , D± = ’ ¯± + iθ± ‚±± ,

¯™ ¯™

D± = D± , D± = 2i‚±± . (3.8)

™ ™ ™

‚θ± ‚θ ™

3.2 Wess“Zumino Models

The Wess“Zumino model describes interactions of an arbitrary number of chi-

ral super¬elds. We will consider the simplest original Wess“Zumino model [11]

(sometimes referred to as the minimal model).

The model contains one chiral super¬eld ¦(xL , θ) and its complex conjugate

¦(x ¯

¯ R , θ), which is anti-chiral. The action of the model is

1 ¯1 1 ¯¯ ¯

d4 x d2 θ W(¦) + d4 x d2 θ W(¦) .

d4 x d4 θ ¦¦ +

S= (3.9)

4 2 2

Note that the ¬rst term is the integral over the full superspace, while the second

and the third run over the chiral subspaces. The holomorphic function W(¦) is

called the superpotential. In components the Lagrangian has the form

1

¯ ¯™

L = (‚ µ φ)(‚µ φ) + ψ ± i‚±± ψ ± + F F + F W (φ) ’ W (φ)ψ 2 + h.c.

¯ . (3.10)

™

2

From (3.10) it is obvious that F can be eliminated by virtue of the classical

equation of motion,

‚ W(φ)

F =’

¯ , (3.11)

‚φ

so that the scalar potential describing the self-interaction of the ¬eld φ is