tions of comparable magnitude had to wait for experimental con¬rmation that

long. For example, the neutrino had a time lag of 27 years. A natural question

arises: why do we believe that this concept is so fundamental?

Supersymmetry may help us to solve two of the the deepest mysteries of

nature “ the cosmological term problem and the hierarchy problem.

Supersymmetric Solitons and Topology 281

B.2 Cosmological Term

An additional term in the Einstein action of the form

√

d4 x g Λ

∆S = (B.2)

goes under the name of the cosmological term. It is compatible with general

covariance and, therefore, can be added freely; this fact was known to Einstein.

Empirically Λ is very small, see below. In classical theory there is no problem

with ¬ne-tuning Λ to any value.

The problem arises at the quantum level. In conventional (non-supersym-

metric) quantum ¬eld theory it is practically inevitable that

Λ ∼ MPl ,

4

(B.3)

where MPl is the Planck scale, MPl ∼ 1019 GeV. This is to be confronted with

the experimental value of the cosmological term,

Λexp ∼ (10’12 GeV)4 . (B.4)

The divergence between theoretical expectations and experiment is 124 orders

of magnitude! This is probably the largest discrepancy in the history of physics.

Why may supersymmetry help? In supersymmetric theories Λ is strictly for-

bidden by supersymmetry, Λ ≡ 0. Of course, supersymmetry, even if it is there,

must be broken in nature. People hope that the breaking occurs in a way en-

suring splittings between the superpartners™ masses in the ball-park of 100 GeV,

with the cosmological term in the ball-park of the experimental value (B.4).

B.3 Hierarchy Problem

The masses of the spinor particles (electrons, quarks) are protected against large

quantum corrections by chirality (“handedness”). For scalar particles the only

natural mass scale is MPl . Even if originally you choose this mass in the “human”

range of, say, 100 GeV, quantum loops will inevitably drag it to MPl . A crucial

element of the Standard Model of electroweak interactions is the Higgs boson

(not yet discovered). Its mass has to be in the ball-park of 100 GeV. If you

let its mass to be ∼ MPl , this will drag, in turn, the masses of the W bosons.

Thus, you would expect (MW )theor ∼ 1019 GeV while (MW )exp ∼ 102 GeV. The

discrepancy is 17 orders of magnitude.

Again, supersymmetry comes to rescue. In supersymmetry the notion of chi-

rality extends to bosons, through their fermion superpartners. There are no

quadratic divergences in the boson masses, at most they are logarithmic, just

like in the fermion case. Thus, the Higgs boson mass gets protected against large

quantum corrections.

Having explained that supersymmetry may help to solve two of the most

challenging problems in high-energy physics, I hasten to add that it does a

lot of other good things already right now. It proved to be a remarkable tool in

282 M. Shifman

Fig. 9. SUSY time arrow.

dealing with previously “uncrackable” issues in gauge theories at strong coupling.

Let me give a brief list of achievements: (i) ¬rst ¬nite four-dimensional ¬eld

theories; (ii) ¬rst exact results in four-dimensional gauge theories [26]; (iii) ¬rst

fully dynamical (albeit toy) theory of con¬nement [27]; (iv) dualities in gauge

theories [28]. The latter ¬nding was almost immediately generalized to strings

which gave rise to the breakthrough discovery of string dualities.

To conclude my mini-introduction, I present an arrow of time in supersym-

metry (Fig. 9).

References

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2. E. Witten and D. I. Olive, Phys. Lett. B 78, 97 (1978).

3. R. Haag, J. T. Lopuszanski, and M. Sohnius, Nucl. Phys. B 88, 257 (1975).

Supersymmetric Solitons and Topology 283

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Particles, Eds. C. Rebbi and G. Soliani (World Scienti¬c, Singapore, 1984) p. 389];

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[hep-th/9810068].

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ume, Eds. M. Olshanetsky and A. Vainshtein (World Scienti¬c, Singapore, 2002),

p. 585“625.

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c

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16. G. ™t Hooft, Nucl. Phys. B 79, 276 (1974).

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Lett. 20, 194 (1974), reprinted in Solitons and Particles, Eds. C. Rebbi and G.

Soliani, (World Scienti¬c, Singapore, 1984), p. 522].

18. B. Zumino, Phys. Lett. B 87, 203 (1979).

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Supersymmetry, Proc. NATO Advanced Study Institute on Supersymmetry, Bonn,

Germany, August 1984, Eds. K. Dietz, R. Flume, G. von Gehlen, and V. Rittenberg

(Plenum Press, New York 1985) pp. 45-87, and in Supergravities in Diverse Di-

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284 M. Shifman

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[hep-th/9506077].

Forces from Connes™ Geometry

T. Sch¨cker

u

Centre de Physique Th´orique, CNRS “ Luminy, Case 907, 13288 Marseille Cedex 9,

e

France

Abstract. Einstein derived general relativity from Riemannian geometry. Connes ex-

tends this derivation to noncommutative geometry and obtains electro“magnetic, weak,

and strong forces. These are pseudo forces, that accompany the gravitational force just

as in Minkowskian geometry the magnetic force accompanies the electric force. The

main physical input of Connes™ derivation is parity violation. His main output is the

Higgs boson which breaks the gauge symmetry spontaneously and gives masses to

gauge and Higgs bosons.

1 Introduction

Still today one of the major summits in physics is the understanding of the

spectrum of the hydrogen atom. The phenomenological formula by Balmer and

Rydberg was a remarkable pre-summit on the way up. The true summit was

reached by deriving this formula from quantum mechanics. We would like to

compare the standard model of electro“magnetic, weak, and strong forces with

the Balmer“Rydberg formula [1] and review the present status of Connes™ deriva-

tion of this model from noncommutative geometry, see Table 1. This geometry

extends Riemannian geometry, and Connes™ derivation is a natural extension of

another major summit in physics: Einstein™s derivation of general relativity from

Riemannian geometry. Indeed, Connes™ derivation uni¬es gravity with the other

three forces.

Let us brie¬‚y recall four nested, analytic geometries and their impact on our

understanding of forces and time, see Table 2. Euclidean geometry is underly-

ing Newton™s mechanics as space of positions. Forces are described by vectors

living in the same space and the Euclidean scalar product is needed to de¬ne

work and potential energy. Time is not part of geometry, it is absolute. This

point of view is abandoned in special relativity unifying space and time into

Minkowskian geometry. This new point of view allows to derive the magnetic

¬eld from the electric ¬eld as a pseudo force associated to a Lorentz boost. Al-

though time has become relative, one can still imagine a grid of synchronized

clocks, i.e. a universal time. The next generalization is Riemannian geometry =

curved spacetime. Here gravity can be viewed as the pseudo force associated to

a uniformly accelerated coordinate transformation. At the same time, universal

time loses all meaning and we must content ourselves with proper time. With

today™s precision in time measurement, this complication of life becomes a bare

necessity, e.g. the global positioning system (GPS).

T. Sch¨ cker, Forces from Connes™ Geometry, Lect. Notes Phys. 659, 285“350 (2005)

u

http://www.springerlink.com/ c Springer-Verlag Berlin Heidelberg 2005

286 T. Sch¨cker

u

Table 1. An analogy

atoms particles and forces

Balmer“Rydberg formula standard model

quantum mechanics noncommutative geometry

Table 2. Four nested analytic geometries

geometry force time

F · dx

Euclidean E= absolute

1

E, ’ B, µ0 =

Minkowskian universal

0 2

0c

Coriolis ” gravity

Riemannian proper, „

12

∆„ ∼ 10’40 s

gravity ’ YMH, » =

noncommutative g

32

Our last generalization is to Connes™ noncommutative geometry = curved

space(time) with uncertainty. It allows to understand some Yang“Mills and some

Higgs forces as pseudo forces associated to transformations that extend the two

coordinate transformations above to the new geometry without points. Also,

proper time comes with an uncertainty. This uncertainty of some hundred Planck

times might be accessible to experiments through gravitational wave detectors

within the next ten years [2].

Prerequisites

On the physical side, the reader is supposed to be acquainted with general rel-

ativity, e.g. [3], Dirac spinors at the level of e.g. the ¬rst few chapters in [4]

and Yang“Mills theory with spontaneous symmetry break-down, for example

the standard model, e.g. [5]. I am not ashamed to adhere to the minimax prin-

ciple: a maximum of pleasure with a minimum of e¬ort. The e¬ort is to do

a calculation, the pleasure is when its result coincides with an experiment re-

sult. Consequently our mathematical treatment is as low-tech as possible. We

do need local di¬erential and Riemannian geometry at the level of e.g. the ¬rst

few chapters in [6]. Local means that our spaces or manifolds can be thought of

as open subsets of R4 . Nevertheless, we sometimes use compact spaces like the

torus: only to simplify some integrals. We do need some group theory, e.g. [7],

mostly matrix groups and their representations. We also need a few basic facts

on associative algebras. Most of them are recalled as we go along and can be

found for instance in [8]. For the reader™s convenience, a few simple de¬nitions

from groups and algebras are collected in the Appendix. And, of course, we need

some chapters of noncommutative geometry which are developped in the text.

For a more detailed presentation still with particular care for the physicist see

Refs. [9,10].

Forces from Connes™ Geometry 287

2 Gravity from Riemannian Geometry

In this section we brie¬‚y review Einstein™s derivation of general relativity from

Riemannian geometry. His derivation is in two strokes, kinematics and dynamics.

2.1 First Stroke: Kinematics

˜

Consider ¬‚at space(time) M in inertial or Cartesian coordinates x» . Take as

˜

matter a free, classical point particle. Its dynamics, Newton™s free equation,

˜

¬xes the trajectory x» (p):

˜

˜