2

‚ ‚W

’i·

e W= , (3.34)

‚z ‚φ

which immediately entails (3.33).

If we deal with more than one ¬eld φ, the above “integral of motion” is of

limited help. However, for a single ¬eld φ it solves the problem: our boundary

conditions ¬x e’i· W to be real along the wall trajectory, which allows one to

¬nd the trajectory immediately. In this way we arrive at (3.18).

The constraint

Im e’i· W = const (3.35)

can be interpreted as follows: in the complex W plane the domain wall trajectory

is a straight line.

3.5 Does the BPS Equation Follow from the Second Order Equation

of Motion?

As we already know, every solution of the BPS equations is automatically a

solution of the second-order equations of motion. The inverse is certainly not

262 M. Shifman

true in the general case. However, in the minimal Wess“Zumino model under

consideration, given the boundary conditions appropriate for the domain walls,

this is true, much in the same way as in the minimal two-dimensional model

with which we began. Namely, every solution of the equations of motion with

the appropriate boundary conditions is simultaneously the solution of the BPS

equation (3.20).

The proof of this statement is rather straightforward [14]. Indeed, we start

from the equations of motion

2¯

‚z φ = W W , ‚z φ = W W ,

¯ ¯

2

(3.36)

where the prime denotes di¬erentiation with respect to the corresponding argu-

ment, and use them to show that

2

‚ ‚ ‚φ

2

|W | = . (3.37)

‚z ‚z ‚z

This implies, in turn, that

2

‚φ

2

|W | ’ = z independent const. (3.38)

‚z

From the domain wall boundary conditions, one immediately concludes that this

constant must vanish, so that in fact

2

‚φ

2

|W | ’ = 0. (3.39)

‚z

If z is interpreted as “time” this equation is nothing but “energy” conservation

along the wall trajectory.

Now, let us introduce the ratio

‚φ

’1

R≡ W

¯ . (3.40)

‚z

Please, observe that its absolute value is unity “ this is an immediate consequence

of (3.39). Our task is to show that the phase of R is z independent. To this end

we perform di¬erentiation (again exploiting (3.20)) to arrive at

2

‚R ‚φ

’2 2

=W W |W | ’

¯ ¯ = 0. (3.41)

‚z ‚z

The statement that R reduces to a z independent phase factor is equivalent to

the BPS equation (3.20), quod erat demonstrandum.

3.6 Living on a Wall

This section could have been entitled “The fate of two broken supercharges.”

As we already know, two out of four supercharges annihilate the wall “ these

Supersymmetric Solitons and Topology 263

supersymmetries are preserved in the given wall background. The two other su-

percharges are broken: being applied to the wall solution, they create two fermion

zero modes. these zero modes correspond to a (2+1)-dimensional (massless) Ma-

jorana spinor ¬eld ψ(t, x, y) localized on the wall.

To elucidate the above assertion it is convenient to turn ¬rst to the fate of

another symmetry of the original theory, which is spontaneously broken for each

given wall, namely, translational invariance in the z direction.

Indeed, each wall solution, e.g. (3.18), breaks this invariance. This means

that in fact we must deal with a family of solutions: if φ(z) is a solution, so is

φ(z ’ z0 ). The parameter z0 is a collective coordinate “ the wall center. People

also refer to it as a modulus (in plural, moduli). For the static wall, z0 is a ¬xed

constant.

Assume, however, that the wall is slightly bent. The bending should be negli-

gible compared to the wall thickness (which is of the order of m’1 ). The bending

can be described as an adiabatically slow dependence of the wall center z0 on t,

x, and y. We will write this slightly bent wall ¬eld con¬guration as

φ(t, x, y, z) = φw (z ’ ζ(t, x, y)) . (3.42)

Substituting this ¬eld in the original action, we arrive at the following e¬ective

(2+1)-dimensional action for the ¬eld ζ(t, x, y):

Tw

ζ

d3 x (‚ m ζ) (‚m ζ) ,

S2+1 = m = 0, 1, 2 . (3.43)

2

It is clear that ζ(t, x, y) can be viewed as a massless scalar ¬eld (called the

translational modulus) which lives on the wall. It is nothing but a Goldstone

¬eld corresponding to the spontaneous breaking of the translational invariance.

Returning to the two broken supercharges, they generate a Majorana (2+1)-

dimensional Goldstino ¬eld ψ± (t, x, y), (± = 1, 2) localized on the wall. The total

(2+1)-dimensional e¬ective action on the wall world volume takes the form

Tw ¯

d3 x (‚ m ζ) (‚m ζ) + ψi‚m γ m ψ

S2+1 = (3.44)

2

where γ m are three-dimensional gamma matrices in the Majorana representa-

tion, e.g.

γ0 = σ2 , γ1 = iσ3 , γ2 = iσ1 ,

with the Pauli matrices σ1,2,3 .

The e¬ective theory of the moduli ¬elds on the wall world volume is super-

symmetric, with two conserved supercharges. This is the minimal supersymmetry

in 2+1 dimensions. It corresponds to the fact that two out of four supercharges

are conserved.

4 Extended Supersymmetry in Two Dimensions:

The Supersymmetric CP(1) Model

In this part I will return to kinks in two dimensions. The reason is three-fold.

First, I will get you acquainted with a very interesting supersymmetric model

264 M. Shifman

which is routinely used in a large variety of applications and as a theoretical lab-

oratory. It is called, rather awkwardly, O(3) sigma model. It also goes under the

name of CP(1) sigma model. Initial data for this model, which will be useful in

what follows, are collected in Appendix A. Second, supersymmetry of this model

is extended (it is more than minimal). It has four conserved supercharges rather

than two, as was the case in Sect. 2. Since the number of supercharges is twice as

large as in the minimal case, people call it N = 2 supersymmetry. So, we will get

familiar with extended supersymmetries. Finally, solitons in the N = 2 sigma

model present a showcase for a variety of intriguing dynamical phenomena. One

of them is charge “irrationalization:” in the presence of the θ term (topological

term) the U(1) charge of the soliton acquires an extra θ/(2π). This phenomenon

was ¬rst discovered by Witten [15] in the ™t Hooft“Polyakov monopoles [16,17].

The kinks in the CP(1) sigma model are subject to charge irrationalization too.

Since they are simpler than the ™t Hooft“Polyakov monopoles, it makes sense to

elucidate the rather unexpected addition of θ/(2π) in the CP(1) kink example.

The Lagrangian of the original CP(1) model is [18]

” ”

i

¯

LCP(1) = G ¯ ¯

‚µ φ‚ µ φ + ΨL ‚R ΨL + ΨR ‚L ΨR

2

” ”

i 2¯

¯ ¯

’ ’

¯ ¯ ¯

ΨL ΨL φ ‚R φ + ΨR ΨR φ ‚L φ ΨL ΨL ΨR ΨR

χ2

χ

iθ 1 µν ¯

+ µ ‚µ φ‚ν φ , (4.1)

2π χ2

where G is the metric on the target space,

2 1

G≡ , (4.2)

2

2 ¯

g 1 + φφ

and χ ≡ 1 + φφ. (It is useful to note that R = 2 χ’2 is the Ricci tensor.) The

¯

derivatives ‚R,L are de¬ned as

‚ ‚ ‚ ‚

’

‚R = , ‚L = + . (4.3)

‚t ‚z ‚t ‚z

The target space in the case at hand is the two-dimensional sphere S2 with

radius RS2 = g ’1 .

¯

As is well-known, one can introduce complex coordinates φ , φ on S2 . The choice

of coordinates in (4.1) corresponds to the stereographic projection of the sphere.

The term in the last line of (4.1) is the θ term. It can be represented as an

integral over a total derivative. Moreover, the fermion ¬eld is a two-component

Dirac spinor

ΨR

Ψ= . (4.4)

ΨL

Bars over φ and ΨL,R denote Hermitean conjugation.

Supersymmetric Solitons and Topology 265

This model has the extended N = 2 supersymmetry since the Lagrangian

(4.1) is invariant (up to total derivatives) under the following supertransforma-

tions (see e.g. the review paper [19])

δφ = ’i¯R ΨL + i¯L ΨR ,

µ µ

¯

φ

δΨR = ’i (‚R φ) µL ’ 2i (¯R ΨL ’ µL ΨR ) ΨR ,

µ ¯

χ

¯

φ

δΨL = i (‚L φ) µR ’ 2i (¯R ΨL ’ µL ΨR ) ΨL ,

µ ¯ (4.5)

χ

with complex parameters µR,L . The corresponding conserved supercurrent is

J µ = G (‚» •) γ » γ µ Ψ .

¯ (4.6)

Since the fermion sector is most conveniently formulated in terms of the chiral

components, it makes sense to rewrite the supercurrent (4.6) accordingly,

’

¯

+

JR = G (‚R φ)ΨR , JR = 0 ;

’ ¯ +

JL = G (‚L φ)ΨL , JL = 0 . (4.7)

where

10

J± =

J ± J1 .

2

The current conservation law takes the form

‚L J + + ‚R J ’ = 0 . (4.8)

The superalgebra induced by the four supercharges

dz J 0 (t, x)

Q= (4.9)

is as follows:

{QL , QL } = (H + P ) , {QR , QR } = (H ’ P ) ;

¯ ¯ (4.10)

{QL , QR } = 0 , {QR , QL } = 0 ; (4.11)

{QR , QR } = 0 , {QR , QR } = 0 ;

¯¯ (4.12)

{QL , QL } = 0 , {QL , QL } = 0 ;

¯¯ (4.13)

i

dz ‚z χ’2 ΨR ΨL ,

{QR , QL } =

¯ ¯ (4.14)

π

i

dz ‚z χ’2 ΨL ΨR .

{QL , QR } = ’

¯ ¯ (4.15)

π

266 M. Shifman

where (H, P ) is the energy-momentum operator,

dzθ0i ,

(H, P ) = i = 0, 1 ,

and θµν is the energy-momentum tensor. Equations (4.14) and (4.15) present a

quantum anomaly “ these anticommutators vanish at the classical level. These

anomalies will not be used in what follows. I quote them here only for the sake