L2 χn (z) = ωn χn (z) . (2.53)

Supersymmetric Solitons and Topology 249

Introduce

1

χn (z) =

˜ P χn (z) . (2.54)

ωn

˜

Then, χn (z) is a (normalized) eigenfunction of L2 with the same eigenvalue,

˜

1 1

L2 χn (z) = P P †

˜˜ 2 2

P χn (z) = P ωn χn (z) = ωn χn (z) .

˜ (2.55)

ωn ωn

In turn,

1†

χn (z) = P χn (z) .

˜ (2.56)

ωn

The quantization of the nonzero modes is quite standard. Let us denote the

Hamiltonian density by H,

dz H .

H=

Then in the quadratic in the quantum ¬elds χ approximation the Hamiltonian

density takes the following form:

12

H ’ ‚z W = χ + [(‚ z ’ W )χ]2

™

2

+ i ψ2 (‚ z + W )ψ1 + iψ1 (‚ z ’ W )ψ2 } , (2.57)

where W is evaluated at φ = φ0 . We recall that the prime denotes di¬erentiation

over φ,

‚2W

W= .

‚φ2

The expansion in eigenmodes has the form,

χ(x) = bn (t) χn (z) , ψ2 (x) = ·n (t) χn (z) ,

n=0 n=0

ψ1 (x) = ξn (t) χn (z) .

˜ (2.58)

n=0

Note that the summation does not include the zero mode χ0 (z). This mode is

not present in ψ1 at all. As for the expansions of χ and ψ2 , the inclusion of the

zero mode would correspond to a shift in the collective coordinates z0 and ·.

Their quantization has been already considered in the previous section. Here we

set z0 = 0.

The coe¬cients bn , ·n and ξn are time-dependent operators. Their equal time

commutation relations are determined by the canonical commutators (2.15),

™ {·m , ·n } = δmn , {ξm , ξn } = δmn .

[bm , bn ] = iδmn , (2.59)

Thus, the mode decomposition reduces the dynamics of the system under con-

sideration to quantum mechanics of an in¬nite set of supersymmetric harmonic

oscillators (in higher orders the oscillators become anharmonic). The ground

250 M. Shifman

state of the quantum kink corresponds to setting each oscillator in the set to the

ground state.

Constructing the creation and annihilation operators in the standard way, we

¬nd the following nonvanishing expectations values of the bilinears built from

the operators bn , ·n , and ξn in the ground state:

ωn 1 i

™

b2 b2

= , = , ·n ξn = . (2.60)

sol sol sol

n n

2 2ωn 2

The expectation values of other bilinears obviously vanish. Combining (2.57),

(2.58), and (2.60) we get

sol |H(z) ’ ‚z W| sol

1 ωn 2 1 ωn 2

[(‚ z ’ W )χn ]2 ’

= χn + χ

2n

2 2 2ωn

n=0

1

’ [(‚ z ’ W )χn ]2 ≡ 0. (2.61)

2ωn

In other words, for the critical kink (in the ground state) the Hamiltonian den-

sity is locally equal to ‚z W “ this statement is valid at the level of quantum

corrections!

The four terms in the braces in (2.61) are in one-to-one correspondence with

those in (2.57). Note that in proving the vanishing of the right-hand side we did

not perform integrations by parts. The vanishing of the right-hand side of (2.57)

demonstrates explicitly the residual supersymmetry “ i.e. the conservation of Q2

and the fact that M = Z. Equation (2.61) must be considered as a local version

of BPS saturation (i.e. conservation of a residual supersymmetry).

The multiplet shortening guarantees that the equality M = Z is not corrected

in higher orders. For critical solitons, quantum corrections cancel altogether;

M = Z is exact.

What lessons can one draw from the considerations of this section? In the

case of the polynomial model the target space is noncompact, while the one in

the sine“Gordon case can be viewed as a compact target manifold S 1 . In these

both cases we get one and the same result: a short (one-dimensional) soliton

multiplet defying the fermion parity (further details will be given in Sect. 2.7).

2.6 Anomaly I

We have explicitly demonstrated that the equality between the kink mass M

and the central charge Z survives at the quantum level. The classical expression

for the central charge is given in (2.19). If one takes proper care of ultraviolet

regularization one can show [6] that quantum corrections do modify (2.19). Here

we will present a simple argument demonstrating the emergence of an anomalous

term in the central charge. We also discuss its physical meaning.

Supersymmetric Solitons and Topology 251

To begin with, let us consider γ µ Jµ where Jµ is the supercurrent de¬ned in

(2.10). This quantity is related to the superconformal properties of the model

under consideration. At the classical level

(γ µ Jµ )class = 2i W ψ . (2.62)

Note that the ¬rst term in the supercurrent (2.10) gives no contribution in (2.62)

due to the fact that in two dimensions γµ γ ν γ µ = 0.

The local form of the superalgebra is given in (2.20). Multiplying (2.20) by

γµ from the left, we get the supertransformation of γµ J µ ,

1

γ 5 = γ 0 γ 1 = ’σ1 .

¯

γ µ Jµ , Q = Tµ + iγµ γ 5 ζ µ ,

µ

(2.63)

2

This equation establishes a supersymmetric relation between γ µ Jµ , Tµ , and ζ µ

µ

and, as was mentioned above, remains valid with quantum corrections included.

But the expressions for these operators can (and will) be changed. Classically

the trace of the energy-momentum tensor is

1 ¯

W ψψ ,

µ

= (W )2 +

Tµ (2.64)

class 2

as follows from (2.18). The zero component of the topological current ζ µ in the

second term in (2.63) classically coincides with the density of the central charge,

‚z W, see (2.21). It is seen that the trace of the energy-momentum tensor and

the density of the central charge appear in this relation together.

It is well-known that in renormalizable theories with ultraviolet logarith-

mic divergences, both the trace of the energy-momentum tensor and γ µ Jµ have

anomalies. We will use this fact, in conjunction with (2.63), to establish the

general form of the anomaly in the density of the central charge.

To get an idea of the anomaly, it is convenient to use dimensional regular-

ization. If we assume that the number of dimensions is D = 2 ’ µ rather than

D = 2, the ¬rst term in (2.10) does generate a nonvanishing contribution to

γ µ Jµ , proportional to (D ’ 2)(‚ν φ) γ ν ψ. At the quantum level this operator gets

an ultraviolet logarithm (i.e. (D ’ 2)’1 in dimensional regularization), so that

D ’ 2 cancels, and we are left with an anomalous term in γ µ Jµ .

To do the one-loop calculation, we apply here (as well as in some other

instances below) the background ¬eld technique: we substitute the ¬eld φ by its

background and quantum parts, φ and χ, respectively,

φ ’’ φ + χ . (2.65)

Speci¬cally, for the anomalous term in γ µ Jµ ,

(γ µ Jµ )anom = (D ’ 2) (‚ν φ) γ ν ψ = ’(D ’ 2) χγ ν ‚ν ψ

= i (D ’ 2) χ W (φ + χ) ψ , (2.66)

where an integration by parts has been carried out, and a total derivative term

is omitted (on dimensional grounds it vanishes in the limit D = 2). We also used

252 M. Shifman

the equation of motion for the ψ ¬eld. The quantum ¬eld χ then forms a loop

and we get for the anomaly,

(γ µ Jµ )anom = i (D ’ 2) 0|χ2 |0 W (φ) ψ

dD p 1

= ’(D ’ 2) W (φ) ψ

(2π)D p2 ’ m2

i

W (φ) ψ .

= (2.67)

2π

The supertransformation of the anomalous term in γ µ Jµ is

1 1 1

¯

W WW

¯

(γ µ Jµ )anom , Q = ψψ +

2 8π 4π

1

W

+iγµ γ 5 µν

‚ν . (2.68)

4π

The ¬rst term on the right-hand side is the anomaly in the trace of the energy-

momentum tensor, the second term represents the anomaly in the topological

current. The corrected current has the form

1

‚ν W + W

ζµ = µν

. (2.69)

4π

Consequently, at the quantum level, after the inclusion of the anomaly, the cen-

tral charge becomes

1 1

Z= W+ W ’ W+ W . (2.70)

4π 4π

z=+∞ z=’∞

2.7 Anomaly II (Shortening Supermultiplet Down to One State)

In the model under consideration, see (2.1), the fermion ¬eld is real which implies

that the fermion number is not de¬ned. What is de¬ned, however, is the fermion

parity G. Following a general tradition, G is sometimes denoted as (’1)F , in

spite of the fact that in the case at hand the fermion number F does not exist.

The tradition originates, of course, in models with complex fermions, where the

fermion number F does exist, but we will not dwell on this topic.

The action of G reduces to changing the sign for the fermion operators leaving

the boson operators intact, for instance,

G Q± G’1 = ’Q± , G Pµ G’1 = Pµ . (2.71)

The fermion parity G realizes Z2 symmetry associated with changing the sign of

the fermion ¬elds. This symmetry is obvious at the classical level (and, in fact,

in any ¬nite order of perturbation theory). This symmetry is very intuitive “ this

Supersymmetric Solitons and Topology 253

is the Z2 symmetry which distinguishes fermion states from the boson states in

the model at hand, with the Majorana fermions.

Here I will try to demonstrate (without delving too deep into technicalities)

that in the soliton sector the very classi¬cation of states as either bosonic or

fermionic is broken. The disappearance of the fermion parity in the BPS soliton

sector is a global anomaly [7].

Let us consider the algebra (2.28) in the special case M 2 = Z 2 . Assuming Z

to be positive, we consider the BPS soliton, M = Z, for which the supercharge

Q2 is trivial, Q2 = 0. Thus, we are left with a single supercharge Q1 realized

nontrivially. The algebra reduces to a single relation

(Q1 )2 = 2 Z . (2.72)

The irreducible representations of this algebra are one-dimensional. There are

two such representations, √

Q1 = ± 2Z , (2.73)

i.e., two types of solitons,

√ √

Q1 | sol+ = 2Z | sol+ , Q1 | sol ’ = ’ 2Z | sol ’ . (2.74)

It is clear that these two representations are unitary non-equivalent.

The one-dimensional irreducible representation of supersymmetry implies

multiplet shortening: the short BPS supermultiplet contains only one state while