equation is of ¬rst order, it can always be integrated in quadratures. Examples

of the solution for two popular choices of the superpotential are given in (2.7)

and (2.9).

The one-¬eld model is the simplest but certainly not the only model with

interesting applications. The generic multi-¬eld N = 1 SUSY model of the

Landau“Ginzburg type has a Lagrangian of the form

¯a γ µ ‚µ ψ a ’ ‚W ‚W ’ ‚ W ψ a ψ b

2

1 ¯

L= a µa

‚µ φ ‚ φ + iψ , (2.34)

‚φa ‚φa ‚φa ‚φb

2

where the superpotential W now depends on n variables, W = W(φa ); in what

follows a, b will be referred to as “¬‚avor” indices, a, b = 1, ..., n. The sum over

both a and b is implied in (2.34). The vacua (critical points) of the generic model

are determined by a set of equations

‚W

= 0, a = 1, ..., n . (2.35)

‚φa

If one views W(φa ) as a “mountain pro¬le,” the critical points are the extremal

points of this pro¬le “ minima, maxima, and saddle points. At the critical points

the potential energy

2

1 ‚W

a

U (φ ) = (2.36)

2 ‚φa

is minimal “ U (φa ) vanishes. The kink solution is a trajectory φa (z) interpolating

—

between a selected pair of critical points.

The BPS equations take the form

‚φa ‚W

=± a, a = 1, ..., n . (2.37)

‚z ‚φ

For n > 1 not all solutions of the equations of motion are the solutions of the

BPS equations, generally speaking. In this case the critical kinks represent a sub-

class of all possible kinks. Needless to say, as a general rule the set of equations

(2.37) cannot be analytically integrated.

Supersymmetric Solitons and Topology 245

A mechanical analogy exists which allows one to use the rich intuition one

has with mechanical motion in order to answer the question whether or not

a solution interpolating between two given critical points exist. Indeed, let us

again interpret z as a “time.” Then (2.37) can be read as follows: the velocity

vector is equal to the force (the gradient of the superpotential pro¬le). This is the

equation describing the ¬‚ow of a very viscous ¬‚uid, such as honey. One places a

droplet of honey at a given extremum of the pro¬le W and then one asks oneself

whether or not this droplet will ¬‚ow into another given extremum of this pro¬le.

If there is no obstruction in the form of an abyss or an intermediate extremum,

the answer is yes. Otherwise it is no.

Mathematicians developed an advanced theory regarding gradient ¬‚ows. It is

called Morse theory. Here I will not go into further details referring the interested

reader to Milnor™s well-known textbook [5].

2.4 Quantization. Zero Modes: Bosonic and Fermionic

So far we were discussing classical kink solutions. Now we will proceed to quan-

tization, which will be carried out in the quasiclassical approximation (i.e. at

weak coupling).

The quasiclassical quantization procedure is quite straightforward. With the

classical solution denoted by φ0 , one represents the ¬eld φ as a sum of the

classical solution plus small deviations,

φ = φ0 + χ . (2.38)

One then expands χ, and the fermion ¬eld ψ, in modes of appropriately chosen

di¬erential operators, in such a way as to diagonalize the Hamiltonian. The

coe¬cients in the mode expansion are the canonical coordinates to be quantized.

The zero modes in the mode expansion “ they are associated with the collective

coordinates of the kink “ must be treated separately. As we will see, for critical

solitons all nonzero modes cancel (this is a manifestation of the Bose“Fermi

cancelation inherent to supersymmetric theories). In this sense, the quantization

of supersymmetric solitons is simpler than the one of their non-supersymmetric

brethren. We have to deal exclusively with the zero modes. The cancelation of

the nonzero modes will be discussed in the next section.

To properly de¬ne the mode expansion we have to discretize the spectrum,

i.e. introduce an infrared regularization. To this end we place the system in a

large spatial box, i.e., we impose the boundary conditions at z = ±L/2, where

L is a large auxiliary size (at the very end, L ’ ∞). The conditions we choose

are

[‚z φ ’ W (φ)]z=±L/2 = 0 , ψ1 |z=±L/2 = 0 ,

[‚z ’ W (φ)] ψ2 |z=±L/2 = 0 , (2.39)

where ψ1,2 denote the components of the spinor ψ± . The ¬rst line is nothing but

a supergeneralization of the BPS equation for the classical kink solution. The

246 M. Shifman

second line is a consequence of the Dirac equation of motion: if ψ satis¬es the

Dirac equation, there are essentially no boundary conditions for ψ2 . Therefore,

the second line is not an independent boundary condition “ it follows from

the ¬rst line. These boundary conditions fully determine the eigenvalues and

the eigenfunctions of the appropriate di¬erential operators of the second order;

see (2.40) below.

The above choice of the boundary conditions is de¬nitely not unique, but it is

particularly convenient because it is compatible with the residual supersymmetry

in the presence of the BPS soliton. The boundary conditions (2.39) are consistent

with the classical solutions, both for the spatially constant vacuum con¬gurations

and for the kink. In particular, the soliton solution φ 0 given in (2.7) (for the

SPM) or (2.9) (for the SSG model) satis¬es ‚z φ ’ W = 0 everywhere. Note that

the conditions (2.39) are not periodic.

Now, for the mode expansion we will use the second order Hermitean di¬er-

˜

ential operators L2 and L2 ,

L2 = P † P , L2 = P P † ,

˜ (2.40)

where

P † = ’‚z ’ W |φ=φ0 (z) .

P = ‚z ’ W |φ=φ0 (z) , (2.41)

The operator L2 de¬nes the modes of χ ≡ φ ’ φ0 , and those of the fermion ¬eld

˜

ψ2 , while L2 does this job for ψ1 . The boundary conditions for ψ1,2 are given in

(2.39), for χ they follow from the expansion of the ¬rst condition in (2.39),

[‚z ’ W (φ0 (z))] χ|z=±L/2 = 0 . (2.42)

It would be natural at this point if you would ask me why it is the di¬erential

˜

operators L2 and L2 that are chosen for the mode expansion. In principle, any

Hermitean operator has an orthonormal set of eigenfunctions. The choice above

is singled out because it ensures diagonalization. Indeed, the quadratic form

following from the Lagrangian (2.1) for small deviations from the classical kink

solution is

1

d2 x ’χL2 χ ’ iψ1 P ψ2 + iψ2 P † ψ1 .

S (2) ’ (2.43)

2

where I neglected time derivatives and used the fact that dφ0 /dz = W (φ0 ) for

the kink under consideration. If diagonalization is not yet transparent, wait for

an explanatory comment in the next section.

It is easy to verify that there is only one zero mode χ0 (z) for the operator

L2 . It has the form

±

1

cosh2 (mz/2) (SPM) .

dφ0

χ0 ∝ ∝ W |φ=φ0 (z) ∝ (2.44)

1

dz (SSG) .

cosh (mz)

It is quite obvious that this zero mode is due to translations. The corresponding

collective coordinate z0 can be introduced through the substitution z ’’ z ’ z0

Supersymmetric Solitons and Topology 247

in the classical kink solution. Then

‚φ0 (z ’ z0 )

χ0 ∝ . (2.45)

‚z0

The existence of the zero mode for the fermion component ψ2 , which is

proportional to the same function ‚φ0 /‚z0 as the zero mode in χ, (in fact, this

is the zero mode in P ), is due to supersymmetry. The translational bosonic

zero mode entails a fermionic one usually referred to as “supersymmetric (or

supertranslational) mode.”

˜

The operator L2 has no zero modes at all.

The translational and supertranslational zero modes discussed above imply

that the kink 4 is described by two collective coordinates: its center z0 and a

fermionic “center” ·, which is a Grassmann parameter,

φ = φ0 (z ’ z0 ) + nonzero modes , ψ2 = · χ0 + nonzero modes , (2.46)

where χ0 is the normalized mode obtained from (2.44) by normalization. The

nonzero modes in (2.46) are those of the operator L2 . As for the component ψ1

˜

of the fermion ¬eld, we decompose ψ1 in modes of the operator L2 ; thus, ψ1 is

˜

given by the sum over nonzero modes of this operator (L2 has no zero modes).

Now, we are ready to derive a Lagrangian describing the moduli dynamics.

To this end we substitute (2.46) in the original Lagrangian (2.1) ignoring the

nonzero modes and assuming that time dependence enters only through (an

adiabatically slow) time dependence of the moduli, z0 and ·,

2

12 dφ0 (z) i 2

LQM = ’M + z0

™ dz + ··

™ dz (χ0 (z))

2 dz 2

M2 i

= ’M + z + ·· ,

™ ™ (2.47)

20 2

where M is the kink mass and the subscript QM emphasizes the fact that the

original ¬eld theory is now reduced to quantum mechanics of the kink moduli.

The bosonic part of this Lagrangian is quite evident: it corresponds to a free

non-relativistic motion of a particle with mass M .

A priori one might expect the fermionic part of LQM to give rise to a Fermi“

Bose doubling. While generally speaking this is the case, in the simplest example

at hand there is no doubling, and the “fermion center” modulus does not manifest

itself.

Indeed, the (quasiclassical) quantization of the system amounts to imposing

the commutation (anticommutation) relations

1

[ p, z0 ] = ’i , ·2 = , (2.48)

2

where p = M z0 is the canonical momentum conjugated to z0 . It means that in

™

the quantum dynamics of the soliton moduli z0 and ·, the operators p and · can

4

Remember, in two dimensions the kink is a particle!

248 M. Shifman

be realized as

d 1

·=√ .

p = M z0 = ’i

™ , (2.49)

dz0 2

√

(It is clear that we could have chosen · = ’ 1/ 2 as well. The two choices are

physically equivalent.)

Thus, · reduces to a constant; the Hamiltonian of the system is

1 d2

=M’

HQM 2. (2.50)

2M dz0

The wave function on which this Hamiltonian acts is single-component.

One can obtain the same Hamiltonian by calculating supercharges. Substi-

tuting the mode expansion in the supercharges (2.13) we arrive at

√ √

Q1 = 2 Z · + ... , Z z0 · + ... ,

Q2 = ™ (2.51)

and Q2 = HQM ’ M . (Here the ellipses stand for the omitted nonzero modes.)

2

The supercharges depend only on the canonical momentum p,

√ p

Q2 = √

Q1 = 2Z , . (2.52)

2Z

In the rest frame in which we perform our consideration {Q1 , Q2 } = 0; the

only value of p consistent with it is p = 0. Thus, for the kink at rest, Q1 =

√

2Z and Q2 = 0, which is in full agreement with the general construction.

The representation (2.52) can be used at nonzero p as well. It reproduces the

superalgebra (2.16) in the non-relativistic limit, with p having the meaning of

the total spatial momentum P1 .

The conclusion that there is no Fermi“Bose doubling for the supersymmetric

kink rests on the fact that there is only one (real) fermion zero mode in the

kink background, and, consequently, a single fermionic modulus. This is totally

counterintuitive and is, in fact, a manifestation of an anomaly. We will discuss

this issue in more detail later (Sect. 2.7).

2.5 Cancelation of Nonzero Modes

Above we have omitted the nonzero modes altogether. Now I want to show that

for the kink in the ground state the impact of the bosonic nonzero modes is

canceled by that of the fermionic nonzero modes.

For each given nonzero eigenvalue, there is one bosonic eigenfunction (in the

operator L2 ), the same eigenfunction in ψ2 , and one eigenfunction in ψ1 (of the

˜ ˜

operator L2 ) with the same eigenvalue. The operators L2 and L2 have the same

spectrum (except for the zero modes) and their eigenfunctions are related.

Indeed, let χn be a (normalized) eigenfunction of L2 ,