ψ ≡ ψγ 0 , θ ≡ θγ 0 ,

and F is the auxiliary component (without kinetic term in the action). A con-

venient choice of gamma matrices is the following:

γ 5 = γ 0 γ 1 = ’σ1 ,

γ 0 = σ2 , γ 1 = iσ3 , σi are Pauli matrices. (A.2)

9

To set the scale properly, so that the U(1) charge of the vacuum state vanishes, one

must antisymmetrize the fermion current, Ψ γ µ Ψ ’ (1/2) Ψ γ µ Ψ ’ Ψ c γ µ Ψ c where

¯ ¯ ¯

the superscript c denotes C conjugation.

276 M. Shifman

In terms of the super¬eld N a the action of the original O(3) sigma model can

be written as follows:

1 ¯

d2 x d2 θ(D± N a )(D± N a )

S= (A.3)

2g 2

with the constraint

N a (x, θ)N a (x, θ) = 1 . (A.4)

Here g 2 is the coupling constant, integration over the Grassmann parameters is

normalized as

i¯

d2 θ θθ = 1 ,

2

while the spinorial derivatives are

‚ ‚ ¯

¯± ’ i(γ θ)± ‚µ , D± = ’

¯

µ

+ i(θγ µ )± ‚µ .

D± = (A.5)

‚θ±

‚θ

The mass deformation of (A.3) that preserves N = 2 but breaks O(3) down

to U(1) is

1 ¯

d2 x d2 θ (D± N a )(D± N a ) + 4imN 3

S= 2 (A.6)

2g

where m is a mass parameter. Note that N = 2 is preserved only because

the added term is very special “ linear in the third (a = 3) component of the

super¬eld N .

In components the Lagrangian in (A.6) has the form

1 2 ¯

(‚µ σ a ) + ψ a i ‚ψ a + F 2 + 2m F 3

L=

2g 2

1 1¯ 2

2 ¯

(‚µ σ a ) + ψ a i ‚ψ a +

= ψψ

2g 2 4

2 2

¯

+ mσ 3 ψψ ’ m2 σ1 + σ2

iθ µν abc a

µ µ σ ‚µ σ b ‚ν σ c .

+ (A.7)

8π

I added the θ term in the last line. The constraint (A.4) is equivalent to

1¯

σ2 = 1 , σψ = 0 , σF = (ψψ) (A.8)

2

while the auxiliary F term was eliminated through the equation of motion

1¯

(ψψ + 2mσ 3 )σ a ’ mδ 3a .

Fa = (A.9)

2

Supersymmetric Solitons and Topology 277

The equations of motion for σ and ψ have the form

¯

’δ ab + σ a σ b ‚ 2 σ b ’ σ b ψ a i ‚ψ b

1 1 3a ¯

2

¯

’ mσ 3 (ψψ) + m2 σ 3 σa + mδ (ψψ) + m2 σ 3 δ 3a = 0,

2 2

1¯

δ ab ’ σ a σ b i ‚ψ b + ψψ ψ a + mσ 3 ψ a = 0 . (A.10)

2

The ¬rst conserved supercurrent is

1

µ

(‚» σ a ) γ » γ µ ψ a + im γ µ ψ 3 .

S(1) = (A.11)

2

g

The second conserved supercurrent (remember that we deal with N = 2) is

1

µ

µabc σ a ‚» σ b γ » γ µ ψ c ’ im µ3ab σ a γ µ ψ b .

S(2) = (A.12)

2

g

In this form the model is usually called O(3) sigma model. The conversion to

the complex representation used in Sect. 4, in which form the model is usually

referred to as CP(1) sigma model, can be carried out by virtue of the well-known

formulae given, for example, in (67) and (69) of [24].

Appendix B.

Getting Started (Supersymmetry for Beginners)

To visualize conventional (non-supersymmetric) ¬eld theory one usually thinks of

a space ¬lled with a large number of coupled anharmonic oscillators. For instance,

in the case of 1+1 dimensional ¬eld theory, with a single spatial dimension, one

can imagine an in¬nite chain of penduli connected by springs (Fig. 5). Each

pendulum represents an anharmonic oscillator. One can think of it as of a massive

ball in a gravitational ¬eld. Each spring works in the harmonic regime, i.e. the

corresponding force grows linearly with the displacement between the penduli.

Letting the density of penduli per unit length tend to in¬nity, we return to ¬eld

theory.

If a pendulum is pushed aside, it starts oscillating and initiates a wave which

propagates along the chain. After quantization one interprets this wave as a

scalar particle.

Can one present a fermion in this picture? The answer is yes. Imagine that

each pendulum acquires a spin degree of freedom (i.e. each ball can rotate, see

Fig. 6). Spins are coupled to their neighbors. Now, in addition to the wave that

propagates in Fig. 5, one can imagine a spin wave propagating in Fig. 6. If one

perturbs a single spin, this perturbation will propagate along the chain.

Our world is 1+3 dimensional, one time and three space coordinates. In this

world bosons manifest themselves as particles with integer spins. For instance,

278 M. Shifman

Fig. 5. A mechanical analogy for the scalar ¬eld theory.

Fig. 6. A mechanical analogy for the spinor ¬eld theory.

the scalar (spin-0) particle from which we started is a boson. The photon (spin-1

particle) is a boson too. On the other hand, particles with semi-integer spins “

electrons, protons, etc. “ are fermions.

Conventional symmetries, such as isotopic invariance, do not mix bosons with

fermions. Isosymmetry tells us that the proton and neutron masses are the same.

It also tells us that the masses of π 0 and π + are the same. However, no prediction

for the ratio of the pion to proton masses emerges.

Supersymmetry is a very unusual symmetry. It connects masses and other

properties of bosons with those of fermions. Thus, each known particle acquires

a superpartner: the superpartner of the photon (spin 1) is the photino (spin

1/2), the superpartner of the electron (spin 1/2) is the selectron (spin 0). Since

spin is involved, which is related to geometry of space-time, it is clear that

supersymmetry has a deep geometric nature. Unfortunately, I have no time to

dwell on further explanations. Instead, I would like to present here a quotation

from Witten which nicely summarizes the importance of this concept for modern

physics. Witten writes [25]:

“... One of the biggest adventures of all is the search for supersymmetry.

Supersymmetry is the framework in which theoretical physicists have

sought to answer some of the questions left open by the Standard Model

of particle physics.

Supersymmetry, if it holds in nature, is part of the quantum structure of

space and time. In everyday life, we measure space and time by numbers,

“It is now three o™clock, the elevation is two hundred meters above sea

Supersymmetric Solitons and Topology 279

y

x

θ

Fig. 7. Superspace.

level,” and so on. Numbers are classical concepts, known to humans since

long before Quantum Mechanics was developed in the early twentieth

century. The discovery of Quantum Mechanics changed our understand-

ing of almost everything in physics, but our basic way of thinking about

space and time has not yet been a¬ected.

Showing that nature is supersymmetric would change that, by revealing

a quantum dimension of space and time, not measurable by ordinary

numbers. .... Discovery of supersymmetry would be one of the real mile-

stones in physics.”

I have tried to depict “a quantum dimension of space and time” in Fig. 7.

Two coordinates, x and y represent the conventional space-time. I should have

drawn four coordinates, x, y, z and t, but this is impossible “ we should try to

imagine them.

The axis depicted by a dashed line (going in the perpendicular direction)

is labeled by θ (again, one should try to imagine four distinct θ™s rather than

one). The dimensions along these directions cannot be measured in meters, the

coordinates along these directions are very unusual, they anticommute,

θ1 θ2 = ’θ2 θ1 , (B.1)

and, as a result, θ2 = 0. This is in sharp contrast with ordinary coordinates

for which 5 meters — 3 meters is, certainly, the same as 3 meters — 5 meters.

In mathematics the θ™s are known as Grassmann numbers, the square of every

given Grassmann number vanishes. These extra θ directions are pure quantum

structures. In our world they would manifest themselves through the fact that

every integer spin particle has a half-integer spin superpartner.

A necessary condition for any theory to be supersymmetric is the balance

between the number of the bosonic and fermionic degrees of freedom, having the

same mass and the same “external” quantum numbers, e.g. electric charge. To

give you an idea of supersymmetric ¬eld theories, let us turn to the most fa-

miliar and simplest gauge theory, quantum electrodynamics (QED). This theory

describes electrons and positrons (one Dirac spinor with four degrees of freedom)

280 M. Shifman

˜

e e

e

»

c)

a) b)

˜ ˜

e e

e

Fig. 8. Interaction vertices in QED and its supergeneralization, SQED. (a) eeγ ver-

¯

tex; (b) selectron coupling to photon; (c) electron“selectron“photino vertex. All ver-

tices have the same coupling constant. The quartic self-interaction of selectrons is also

present but not shown.

interacting with photons (an Abelian gauge ¬eld with two physical degrees of

freedom). Correspondingly, in its supersymmetric version, SQED, one has to add

one massless Majorana spinor, the photino (two degrees of freedom), and two

complex scalar ¬elds, the selectrons (four degrees of freedom).

Balancing the number of degrees of freedom is a necessary but not su¬cient

condition for supersymmetry in dynamically nontrivial theories, of course. All

interaction vertices must be supersymmetric too. This means that each line

in every vertex can be replaced by that of a superpartner. Say, we start from

the electron“electron“photon coupling (Fig. 8a). Now, as we already know, in

SQED the electron is accompanied by two selectrons. Thus, supersymmetry

requires the selectron“selectron“photon vertices (Fig. 8b) with the same coupling

constant. Moreover, the photon can be replaced by its superpartner, photino,

which generates the electron“selectron“photino vertex (Fig. 8c) with the same

coupling.

With the above set of vertices one can show that the theory is supersymmetric

at the level of trilinear interactions, provided that the electrons and selectrons

are degenerate in mass, while the photon and photino ¬elds are both massless.

To make it fully supersymmetric, one should also add some quartic terms, which

describe the self-interactions of the selectron ¬elds. Historically, SQED was the

¬rst supersymmetric theory discovered in four dimensions [1].

B.1 Promises of Supersymmetry

Supersymmetry has yet to be discovered experimentally. In spite of the ab-

sence of direct experimental evidence, immense theoretical e¬ort was invested

in this subject in the last thirty years; over 30,000 papers are published. The

so-called Minimal Supersymmetric Standard Model (MSSM) became a gener-

ally accepted paradigm in high-energy physics. In this respect the phenomenon

is rather unprecedented in the history of physics. Einstein™s general relativity,

the closest possible analogy one can give, was experimentally con¬rmed within