The transition from the ¬rst to the second line is approximate, valid to the lead-

ing order in the coupling constant. The quantization procedure and derivation

of (4.34) presented in Sect. 4.3 are also valid to the leading order in the coupling

constant. At the same time, the expressions in the second and last lines in (4.41)

are valid to all orders and, in this sense, are more general. They will be derived

below from the consideration of the relevant central charge.

The important circumstance to be stressed is that the kink mass depends on

a special combination of the coupling constant and θ, namely,

1 θ

„= +i (4.42)

g2 4π

In other words, it is the complexi¬ed coupling constant that enters.

It is instructive to make a pause here to examine the issue of the kink mass

from a slightly di¬erent angle. Equation (4.21) tells us that there is a central

charge ZLR in the anticommutator {QL cQR },

¯

¯

ZLR = ’i m dz ‚z h + i qU(1) , (4.43)

¯

where the anomalous term is omitted, as previously, which is fully justi¬ed at

weak coupling. If the soliton under consideration is critical “ and it is “ its

mass must be equal to the absolute value |ZLR |. This leads us directly to (4.41).

¯

However, one can say more.

Indeed, g 2 in (4.41) is the bare coupling constant. It is quite clear that the

kink mass, being a physical parameter, should contain the renormalized constant

g 2 (m), after taking account of radiative corrections. In other words, switching on

radiative corrections in ZLR one must replace the bare 1/g 2 by the renormalized

¯

2

1/g (m). We will see now how it comes out, verifying en route a very important

assertion “ the dependence of ZLR on all relevant parameters, „ and m, being

¯

holomorphic.

I will perform the one-loop calculation in two steps. First, I will rotate the

mass parameter m in such a way as to make it real, m ’ |m|. Simultaneously,

272 M. Shifman

Fig. 4. h renormalization.

the θ angle will be replaced by an e¬ective θ,

θ ’ θe¬ = θ + 2β , (4.44)

where the phase β is de¬ned in (4.26). Next, I decompose the ¬eld φ into a

classical and a quantum part,

φ ’ φ + δφ .

Then the h part of the central charge ZLR becomes

¯

¯

2 1 ’ φφ ¯

h’h+ 2 δ φ δφ . (4.45)

g 1 ’ φφ 3

¯

¯

Contracting δ φ δφ into a loop (Fig. 4) and calculating this loop “ quite a trivial

exercise “ we ¬nd with ease that

2

21 12 Muv

h’h+’ 2 + ln . (4.46)

|m|2

g χ χ 4π

Combining this result with (4.40) and (4.42), we arrive at

2

1 Muv k

= 2im „ ’ ln 2 ’ i

ZLR (4.47)

¯

4π m 2

(remember, the kink mass M = |ZLR |). A salient feature of this formula, to

¯

be noted, is the holomorphic dependence of ZLR on m and „ . Such a holomor-

¯

phic dependence would be impossible if two and more loops contributed to h

renormalization. Thus, h renormalization beyond one loop must cancel, and it

does.8 Note also that the bare coupling in (4.47) conspires with the logarithm in

such a way as to replace the bare coupling by that renormalized at |m|, as was

expected.

8

Fermions are important for this cancelation.

Supersymmetric Solitons and Topology 273

The analysis carried out above is quasiclassical. It tells us nothing about the

possible occurrence of non-perturbative terms in ZLR . In fact, all terms of the

¯

type

2

Muv

exp (’4π„ ) , = integer

m2

are fully compatible with holomorphy; they can and do emerge from instantons.

An indirect calculation of non-perturbative terms was performed in [22]. I will

skip it altogether referring the interested reader to the above publication.

4.5 Switching On Fermions

Fermion non-zero modes are irrelevant for our consideration since, being com-

bined with the boson non-zero modes, they cancel for critical solitons, a usual

story. Thus, for our purposes it is su¬cient to focus on the (static) zero modes in

the kink background (4.30). The coe¬cients in front of the fermion zero modes

will become (time-dependent) fermion moduli, for which we are going to build

the corresponding quantum mechanics. There are two such moduli, · and ·.¯

The equations for the fermion zero modes are

¯

1 ’ φφ

2¯

‚z ΨL ’ φ‚z φ ΨL ’ i |m|eiβ ΨR = 0 ,

χ χ

¯

1 ’ φφ

2¯

|m|e’iβ ΨL = 0

‚z ΨR ’ φ‚z φ ΨR + i (4.48)

χ χ

¯

(plus similar equations for Ψ ; since our operator is Hermitean we do not need to

consider them separately.)

It is not di¬cult to ¬nd solutions to these equations, either directly or by us-

ing supersymmetry. Indeed, if we know the bosonic solution (4.30), its fermionic

superpartner “ and the fermion zero modes are such superpartners “ is obtained

¯

from the bosonic one by those two supertransformations which act on φ , φ

nontrivially. In this way we conclude that the functional form of the fermion

zero mode must coincide with the functional form of the boson solution (4.30).

Concretely,

1/2

’ie’iβ

g 2 |m|

ΨR

e|m|(z’z0 )

=· (4.49)

ΨL 1

2

and

1/2

g 2 |m|

¯ ieiβ

ΨR

e|m|(z’z0 ) ,

=·

¯ (4.50)

¯

ΨL 1

2

where the numerical factor is introduced to ensure the proper normalization of

the quantum-mechanical Lagrangian. Another solution which asymptotically, at

large z, behaves as e3|m|(z’z0 ) must be discarded as non-normalizable.

Now, to perform the quasiclassical quantization we follow the standard route:

the moduli are assumed to be time-dependent, and we derive the quantum me-

chanics of the moduli starting from the original Lagrangian (4.1) with the twisted

274 M. Shifman

mass terms (4.18). Substituting the kink solution and the fermion zero modes

for Ψ , one gets

LQM = i · · .

¯™ (4.51)

In the Hamiltonian approach the only remnants of the fermion moduli are the

anticommutation relations

{¯, ·} = 1 , {¯, · } = 0 , {·, ·} = 0 ,

· ·¯ (4.52)

which tell us that the wave function is two-component (i.e. the kink supermul-

tiplet is two-dimensional). One can implement (4.52) by choosing, e.g., · = σ + ,

¯

’

· = σ , where σ m = (σ1 ± σ2 )/2.

p

The fact that there are two critical kink states in the supermultiplet is consis-

tent with the multiplet shortening in N = 2. Indeed, in two dimensions the full

N = 2 supermultiplet must consist of four states: two bosonic and two fermionic.

1/2 BPS multiplets are shortened “ they contain twice less states than the full

supermultiplets, one bosonic and one fermionic. This is to be contrasted with

the single-state kink supermultiplet in the minimal supersymmetric model of

Sect. 2. The notion of the fermion parity remains well-de¬ned in the kink sector

of the CP(1) model.

4.6 Combining Bosonic and Fermionic Moduli

Quantum dynamics of the kink at hand is summarized by the Hamiltonian

M0 ¯ ™

™

HQM = ζζ (4.53)

2

acting in the space of two-component wave functions. The variable ζ here is a

complexi¬ed kink center,

i

ζ = z0 + ±. (4.54)

|m|

For simplicity, I set the vacuum angle θ = 0 for the time being (it will be

reinstated later).

The original ¬eld theory we deal with has four conserved supercharges. Two

of them, q and q , see (4.26), act trivially in the critical kink sector. In moduli

¯

quantum mechanics they take the form

√ √ ™

™ ¯¯

q = M 0 ζ· , q = M 0 ζ· ;

¯ (4.55)

they do indeed vanish provided that the kink is at rest. The superalgebra de-

scribing kink quantum mechanics is {¯, q} = 2HQM . This is nothing but Wit-

q

ten™s N = 1 supersymmetric quantum mechanics [23] (two supercharges). The

realization we deal with is peculiar and distinct from that of Witten. Indeed,

the standard supersymmetric quantum mechanics of Witten includes one (real)

bosonic degree of freedom and two fermionic ones, while we have two bosonic

degrees of freedom, x0 and ±. Nevertheless, the superalgebra remains the same

due to the fact that the bosonic coordinate is complexi¬ed.

Supersymmetric Solitons and Topology 275

Finally, to conclude this section, let us calculate the U(1) charge of the kink

states. We start from (4.22), substitute the fermion zero modes and get 9

1

∆qU(1) = [¯·]

· (4.56)

2

(this is to be added to the bosonic part given in (4.40)). Given that · = σ + and

¯

’ 1

· = σ we arrive at ∆qU(1) = 2 σ3 . This means that the U(1) charges of two

kink states in the supermultiplet split from the value given in (4.40): one has

the U(1) charges

1 θ

k+ + ,

2 2π

and

1 θ

k’ + .

2 2π

5 Conclusions

Supersymmetric solitons is a vast topic, with a wide range of applications in ¬eld

and string theories. In spite of almost thirty years of development, the review

literature on this subject is scarce. Needless to say, I was unable to cover this

topic in an exhaustive manner. No attempt at such coverage was made. Instead,

I focused on basic notions and on pedagogical aspects in the hope of providing a

solid introduction, allowing the interested reader to navigate themselves in the

ocean of the original literature.

Appendix A.

CP(1) Model = O(3) Model (N = 1 Super¬elds N )

In this Appendix we follow the review paper [24]. One introduces a (real) super-

¬eld

1¯

¯

N a (x, θ) = σ a (x) + θψ a (x) + θθF a , a = 1, 2, 3, (A.1)

2

where σ is a scalar ¬eld, ψ is a Majorana two-component spinor,