As is well-known, the model (4.1) is asymptotically free [20]. The coupling

constant de¬ned in (4.2) runs according to the law

M2

1 1 1

= 2’ ln uv , (4.16)

g 2 (µ) µ2

g0 4π

2

where Muv is the ultraviolet cut-o¬ and g0 is the coupling constant at this cut-

o¬. At small momenta the theory becomes strongly coupled. The scale parameter

of the model is

4π

Λ2 = Muv exp ’ 2 .

2

(4.17)

g0

Our task is to study solitons in a pedagogical setting, which means, by de-

fault, that the theory must be weakly coupled. One can make the CP(1) model

(4.1) weakly coupled, still preserving N = 2 supersymmetry, by introducing the

so-called twisted mass [21].

4.1 Twisted Mass

I will explain here neither genesis of twisted masses nor the origin of the name.

Crucial is the fact that the target space of the CP(1) model has isometries. It was

noted by Alvarez-Gaum´ and Freedman that one can exploit these isometries to

e

introduce supersymmetric mass terms, namely,

¯

¯ 1 ’ φφ mΨL ΨR + mΨR ΨL

∆m LCP(1) = G ’|m|2 φφ ’ ¯ ¯¯ . (4.18)

χ

Here m is a complex parameter. Certainly, one can always eliminate the phase

of m by a chiral rotation of the fermion ¬elds. Due to the chiral anomaly, this

will lead to a shift of the vacuum angle θ. In fact, it is the combination θe¬ =

θ + 2arg m on which physics depends.

With the mass term included, the symmetry of the model is reduced to a

global U(1) symmetry,

φ ’ e’i± φ ,

¯ ¯

φ ’ ei± φ ,

Ψ ’ e’i± Ψ .

Ψ ’ ei± Ψ , ¯ ¯ (4.19)

Needless to say that in order to get the conserved supercurrent, one must modify

(4.7) appropriately,

J + = G (‚R φ)ΨR , J ’ = ’iG mφΨL ;

¯ ¯¯

R R

’ ¯ ¯

+

JL = G (‚L φ)ΨL , JL = iG mφΨR . (4.20)

Supersymmetric Solitons and Topology 267

The only change twisted mass terms introduce in the superalgebra is that (4.14)

and (4.15) are to be replaced by

{QL , QR } = mqU(1) ’ im

¯ dz ‚z h + anom. ,

{QR , QL } = mqU(1) + im

¯ ¯ ¯ dz ‚z h + anom. , (4.21)

where qU(1) is the conserved U(1) charge,

qU(1) ≡ 0

dzJU(1) ,

¯

”µ φφ ¯ µ

µ ¯

JU(1) = G φ i ‚ φ + Ψ γ µ Ψ ’ 2

¯ Ψγ Ψ , (4.22)

χ

and

21

h=’ . (4.23)

g2 χ

(Remember, χ is de¬ned after (4.2).) As already mentioned, in what follows, the

anomaly in (4.2) will be neglected. Equation (4.21) clearly demonstrates that the

very possibility of introducing twisted mass terms is due to the U(1) symmetry.

Most important for our purposes is the fact that the model at hand is weakly

Λ. Indeed, in this case the running of g 2 (µ) is frozen

coupled provided that m

at µ = m. Consequently, the solitons emerging in this model can be treated

quasiclassically.

4.2 BPS Solitons at the Classical Level

As already mentioned, the target space of the CP(1) model is S2 . The U(1)

invariant scalar potential term

¯

V = |m|2 G φφ (4.24)

lifts the vacuum degeneracy leaving us with two discrete vacua: at the south and

north poles of the sphere (Fig. 2) i.e. φ = 0 and φ = ∞.

The kink solutions interpolate between these two vacua. Let us focus, for

de¬niteness, on the kink with the boundary conditions

φ’0 at z ’ ’∞ , φ’∞ at z ’ ∞ . (4.25)

Consider the following linear combinations of supercharges

q = QR ’ i e’iβ QL , ¯¯ ¯

q = QR + i eiβ QL , (4.26)

where β is the argument of the mass parameter,

m = |m| eiβ . (4.27)

268 M. Shifman

Fig. 2. Meridian slice of the target space sphere (thick solid line). The arrows present

the scalar potential (4.24), their length being the strength of the potential. The two

vacua of the model are denoted by the closed circles at the north and south pole.

Then

{q, q } = 2H ’ 2|m| {q, q} = {¯, q } = 0 .

¯ dz ‚z h , q¯ (4.28)

Now, let us require q and q to vanish on the classical solution. Since for static

¯

¬eld con¬gurations

ΨR + ie’iβ ΨL ,

¯ ¯

q = ’ ‚z φ ’ |m|φ

the vanishing of these two supercharges implies

¯ ¯

‚z φ = |m|φ or ‚z φ = |m|φ . (4.29)

This is the BPS equation in the sigma model with twisted mass.

The BPS equation (4.29) has a number of peculiarities compared to those in

more familiar Landau“Ginzburg N = 2 models. The most important feature is

its complexi¬cation, i.e. the fact that (4.29) is holomorphic in φ. The solution

of this equation is, of course, trivial and can be written as

φ(z) = e|m|(z’z0 )’i± . (4.30)

Here z0 is the kink center while ± is an arbitrary phase. In fact, these two

parameters enter only in the combination |m|z0 + i±. We see that the notion of

the kink center also gets complexi¬ed.

The physical meaning of the modulus ± is obvious: there is a continuous

family of solitons interpolating between the north and south poles of the target

space sphere. This is due to the U(1) symmetry. The soliton trajectory can follow

Supersymmetric Solitons and Topology 269

Fig. 3. The soliton solution family. The collective coordinate ± in (4.30) spans the

interval 0 ¤ ± ¤ 2π. For given ± the soliton trajectory on the target space sphere

follows a meridian, so that when ± varies from 0 to 2π all meridians are covered.

any meridian (Fig. 3). It is instructive to derive the BPS equation directly from

the (bosonic part of the) Lagrangian, performing the Bogomol™nyi completion,

¯ ¯

d2 x L = d2 x G ‚µ φ‚ µ φ ’ |m|2 φφ

¯ ¯

’’ dz G ‚z φ ’ |m|φ (‚z φ ’ |m|φ)

+ |m| dz ‚z h , (4.31)

where I assumed φ to be time-independent and the following identity has been

used

¯¯

‚z h ≡ G(φ‚z φ + φ‚z φ) .

Equation (4.29) ensues immediately. In addition, (4.31) implies that (classically)

the kink mass is

2|m|

M0 = |m| (h(∞) ’ h(0)) = 2 . (4.32)

g

The subscript 0 emphasizes that this result is obtained at the classical level.

Quantum corrections will be considered below.

4.3 Quantization of the Bosonic Moduli

To carry out conventional quasiclassical quantization we, as usual, assume the

moduli z0 and ± in (4.30) to be (weakly) time-dependent, substitute (4.30) in

270 M. Shifman

the bosonic Lagrangian (4.31), integrate over z and thus derive a quantum-

mechanical Lagrangian describing moduli dynamics. In this way we obtain

M0 2 1 θ

LQM = ’M0 + ±2 ’

z+

™ ™ ±.

™ (4.33)

g 2 |m|

20 2π

The ¬rst term is the classical kink mass, the second describes the free motion of

the kink along the z axis. The term in the braces is most interesting (I included

the θ term which originates from the last line in (4.1)).

Remember that the variable ± is compact. Its very existence is related to

the exact U(1) symmetry of the model. The energy spectrum corresponding to

± dynamics is quantized. It is not di¬cult to see that

g 2 |m| 2

E[±] = qU(1) , (4.34)

4

where qU(1) is the U(1) charge of the soliton,

θ

qU(1) = k + , k = an integer . (4.35)

2π

This is the same e¬ect as the occurrence of an irrational electric charge θ/(2π)

on the magnetic monopole, a phenomenon ¬rst noted by Witten [15]. Objects

which carry both magnetic and electric charges are called dyons. The standard

four-dimensional magnetic monopole becomes a dyon in the presence of the θ

term if θ = 0. The qU(1) = 0 kinks in the CP(1) model are sometimes referred

to as Q-kinks.

A brief comment regarding (4.34) and (4.35) is in order here. The dynamics

of the compact modulus ± is described by the Hamiltonian

1

±2

HQM = ™ (4.36)

g 2 |m|

while the canonic momentum conjugated to ± is

δLQM 2 θ

±’

p[±] = =2 ™ . (4.37)

g |m|

δ±

™ 2π

In terms of the canonic momentum the Hamiltonian takes the form

2

g 2 |m| θ

HQM = p[±] + (4.38)

4 2π

The eigenfunctions obviously are

Ψk (±) = eik± , k = an integer , (4.39)

which immediately leads to E[±] = (g 2 |m|/4)(k + θ(2π)’1 )2 .

Let us now calculate the U(1) charge of the k-th state. Starting from (4.22)

we arrive at

2 θ θ

’k+

qU(1) = 2 ± = p[±] +

™ , (4.40)

g |m| 2π 2π

quod erat demonstrandum, cf. (4.35).

Supersymmetric Solitons and Topology 271

4.4 The Soliton Mass and Holomorphy

Taking account of E[±] “ the energy of an “internal motion” “ the kink mass can

be written as

2

2|m| g 2 |m| θ

M= 2 + k+

g 4 2π

1/2

2

g4

2|m| θ

=2 1+ k+

g 4 2π

1 θ + 2πk

= 2|m| +i . (4.41)