v

Here m, », and v are real (positive) parameters. The ¬rst model is referred

to as superpolynomial (SPM), the second as super“sine“Gordon (SSG). The

classical vacua in the SPM are at φ = ±m(2»)’1 ≡ φ± . I will assume that

—

»/m 1 to ensure the applicability of the quasiclassical treatment. This is

the weak coupling regime for the SPM. A kink solution interpolates between

φ’ = ’m/2» at z ’ ’∞ and φ+ = m/2» at z ’ ∞, an anti-kink solution

— —

between φ— = m/2» at z ’ ’∞ and φ’ = ’m/2» at z ’ ∞. The classical

+

—

kink solution has the form

m mz

φ0 = tanh . (2.7)

2» 2

The weak coupling regime in the SSG case is attained at v 1. In the

super“sine“Gordon model there are in¬nitely many vacua; they lie at

π

k

φ— = v + kπ , (2.8)

2

where k is an integer, either positive or negative. Correspondingly, there exist

solitons connecting any pair of vacua. In this case we will limit ourselves to

240 M. Shifman

consideration of the “elementary” solitons, which connect adjacent vacua, e.g.

φ—0,’1 = ±πv/2,

φ 0 = v arcsin [ tanh(mz)] . (2.9)

In D = 1+1 the real scalar ¬eld represents one degree of freedom (bosonic),

and so does the two-component Majorana spinor (fermionic). Thus, the number

of bosonic and fermionic degrees of freedom is identical, which is a necessary

condition for supersymmetry. One can show in many di¬erent ways that the La-

grangian (2.1) does actually possess supersymmetry. For instance, let us consider

the supercurrent,

‚W µ

J µ = ( ‚φ)γ µ ψ + i γ ψ. (2.10)

‚φ

This object is linear in the fermion ¬eld; therefore, it is obviously fermionic. On

the other hand, it is conserved. Indeed,

‚2W ‚W

‚µ J µ = (‚ 2 φ)ψ + ( ‚φ)( ‚ψ) + i ( ‚φ)ψ + i ‚ψ . (2.11)

‚φ2 ‚φ

The ¬rst, second, and third terms can be expressed by virtue of the equations

of motion, which immediately results in various cancelations. After these cance-

lations only one term is left in the divergence of the supercurrent,

1 ‚3W ¯

‚µ J = ’

µ

(ψψ) ψ . (2.12)

2 ‚φ3

If one takes into account (i) the fact that the spinor ψ is real and two-component,

and (ii) the Grassmannian nature of ψ1,2 , one immediately concludes that the

right-hand side in (2.12) vanishes.

The supercurrent conservation implies the existence of two conserved charges,2

‚W

0

γ0ψ

Q± = dz J± = dz ‚φ + i , ± = 1, 2 . (2.13)

‚φ ±

These supercharges form a doublet with respect to the Lorentz group in D = 1+1.

They generate supertransformations of the ¬elds, for instance,

‚W

¯

[Q± , φ] = ’iψ± , {Q± , ψβ } = ( ‚)±β φ + i δ±β , (2.14)

‚φ

and so on. In deriving (2.14) I used the canonical commutation relations

™ ¯

ψ± (t, z), ψβ (t, z ) = γ 0

φ(t, z), φ(t, z ) = iδ(z’z ) , δ(z’z ) . (2.15)

±β

Note that by acting with Q on the bosonic ¬eld we get a fermionic one and vice

versa. This fact demonstrates, once again, that the supercharges are symmetry

generators of fermionic nature.

2

Two-dimensional theories with two conserved supercharges are referred to as N = 1.

Supersymmetric Solitons and Topology 241

Given the expression for the supercharges (2.13) and the canonical commu-

tation relations (2.15) it is not di¬cult to ¬nd the superalgebra,

{Q± , Qβ } = 2 (γ µ )±β Pµ + 2i (γ 5 )±β Z .

¯ (2.16)

Here Pµ is the operator of the total energy and momentum,

Pµ = dzT µ 0 , (2.17)

where T µν is the energy-momentum tensor,

1¯ 1 2

T µν = ‚ µφ ‚ νφ + ψγ µ i‚ νψ ’ g µν ‚γ φ ‚ γφ ’ (W ) , (2.18)

2 2

and Z is the central charge,

Z= dz ‚z W(φ) = W[φ(z = ∞)] ’ W[φ(z = ’∞)] . (2.19)

The local form of the superalgebra (2.16) is

µ¯

J± , Qβ = 2 (γν )±β T µν + 2i (γ 5 )±β ζ µ , (2.20)

where ζ µ is the conserved topological current,

‚ν W .

ζµ = µν

(2.21)

Symmetrization (antisymmetrization) over the bosonic (fermionic) operators in

the products is implied in the above expressions.

I pause here to make a few comments. Equation (2.16) can be viewed as

a general de¬nition of supersymmetry. Without the second term on the right-

hand side, i.e. in the form {Q± , Qβ } = 2 (γ µ )±β Pµ , it was obtained by two of

¯

the founding fathers of supersymmetry, Golfand and Likhtman, in 1971 [1]. The

Z term in (2.16) is referred to as the central extension. At a naive level of con-

sideration one might be tempted to say that this term vanishes since it is the

integral of a full derivative. Actually, it does not vanish in problems in which one

deals with topological solitons. We will see this shortly. The occurrence of the

central charge Z is in one-to-one correspondence with the topological charges

“ this fact was noted by Witten and Olive [2]. Even before the work of Witten

and Olive, the possibility of central extensions of the de¬ning superalgebra was

observed, within a purely algebraic consideration, by Haag, Lopuszanski, and

Sohnius [3]. The theories with centrally extended superalgebras are special: they

admit critical solitons. Since the central charge is the integral of the full deriva-

tive, it is independent of details of the soliton solution and is determined only by

the boundary conditions. To ensure that Z = 0 the ¬eld φ must tend to distinct

limits at z ’ ±∞.

242 M. Shifman

2.1 Critical (BPS) Kinks

A kink in D = 1+1 is a particle. Any given soliton solution obviously breaks

translational invariance. Since {Q, Q} ∝ P , typically both supercharges are bro-

¯

ken on the soliton solutions,

Q± |sol = 0 , ± = 1, 2 . (2.22)

However, for certain special kinks, one can preserve 1/2 of supersymmetry, i.e.

Q1 |sol = 0 and Q2 |sol = 0 , (2.23)

or vice versa. Such kinks are called critical, or BPS-saturated.3

The critical kink must satisfy a ¬rst order di¬erential equation “ this fact,

as well as the particular form of the equation, follows from the inspection of

(2.13) or the second equation in (2.14). Indeed, for static ¬elds φ = φ(z), the

supercharges Q± are proportional to

‚z φ + W 0

Q± ∝ . (2.24)

’‚z φ + W

0

One of the supercharges vanishes provided that

‚φ(z) ‚W(φ)

=± , (2.25)

‚z ‚φ

or, for short,

‚z φ = ±W . (2.26)

The plus and minus signs correspond to kink and anti-kink, respectively. Gener-

ically, the equations that express the conditions for the vanishing of certain

supercharges are called BPS equations.

The ¬rst order BPS equation (2.26) implies that the kink automatically sat-

is¬es the general second order equation of motion. Indeed, let us di¬erentiate

both sides of (2.26) with respect to z. Then one gets

‚z φ = ±‚z W = ±W ‚z φ

2

‚U

=W W = . (2.27)

‚φ

3

More exactly, in the case at hand we deal with 1/2 BPS-saturated kinks. As I have

already mentioned, BPS stands for Bogomol™nyi, Prasad, and Sommer¬eld [4]. In

fact, these authors considered solitons in a non-supersymmetric setting. They found,

however, that under certain conditions they can be described by ¬rst order di¬erential

equations, rather than second order equations of motion. Moreover, under these

conditions the soliton mass was shown to be proportional to the topological charge.

We understand now that the limiting models considered in [4] are bosonic sectors of

supersymmetric models.

Supersymmetric Solitons and Topology 243

The latter presents the equation of motion for static (time independent) ¬eld

con¬gurations. This is a general feature of supersymmetric theories: compliance

with the BPS equations entails compliance with the equations of motion.

The inverse statement is generally speaking wrong “ not all solitons which

are static solutions of the second order equations of motion satisfy the BPS

equations. However, in the model at hand, with a single scalar ¬eld, the inverse

statement is true. In this model any static solution of the equation of motion

satis¬es the BPS equation. This is due to the fact that there exists an “integral

of motion.” Indeed, let us reinterpret z as a “time,”.. for a short while. Then

the equation ‚z φ ’ U = 0 can be reinterpreted as φ ’U = 0, i.e. the one-

2

dimensional motion of a particle of mass 1 in the potential ’U (φ). The conserved

™

“energy” is (1/2) φ2 ’ U . At ’∞ both the “kinetic” and “potential” terms tend

to zero. This boundary condition emerges because the kink solution interpolates

between two critical points, the vacua of the model, while supersymmetry ensures

™

that U (φ— ) = 0. Thus, on the kink con¬guration (1/2) φ2 = U implying that

™

φ = ±W .

We have already learned that the BPS saturation in the supersymmetric

setting means the preservation of a part of supersymmetry. Now, let us ask

ourselves why this feature is so precious.

To answer this question let us have a closer look at the superalgebra (2.16).

In the kink rest frame it reduces to

2 2

(Q1 ) = M + Z , (Q2 ) = M ’ Z

{Q1 , Q2 } = 0 , (2.28)

where M is the kink mass. Since Q2 vanishes on the critical kink, we see that

M =Z. (2.29)

Thus, the kink mass is equal to the central charge, a nondynamical quantity

which is determined only by the boundary conditions on the ¬eld φ (more exactly,

by the values of the superpotential in the vacua between which the kink under

consideration interpolates).

2.2 The Kink Mass (Classical)

The classical expression for the central charge is given in (2.19). (Anticipating a

turn of events I hasten to add that a quantum anomaly will modify this classical

expression; see Sect. 2.6.) Now we will discuss the critical kink mass.

In the SPM

m3

m ±

φ± W[φ± ]

=± , W0 ≡ =± (2.30)

— —

12»2

2»

and, hence,

m3

MSPM = 2. (2.31)

6»

244 M. Shifman

In the SSG model

π ±

φ± = ±v W0 ≡ W[φ± ] = ±mv 2 .

, (2.32)

— —

2

Therefore,

MSSG = 2mv 2 . (2.33)

Applicability of the quasiclassical approximation demands m/» 1 and

v 1, respectively.

2.3 Interpretation of the BPS Equations. Morse Theory