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In the model described above we deal with a single scalar ¬eld. Since the BPS
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 ,

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