6.1 Historical Outline and Introduction

215

against transformations that rescale lengths and shapes preserving only angles. It was discovered that the quantum string is free from conformal anomalies only if it propagates in a space-time with 26 dimensions, precisely one time and twenty-five space directions.

The second problem met in the stringy interpretation of Veneziano amplitudes related with the absence of fermions. So far the harmonic oscillators associated with the string vibrational modes were just bosonic and no state corresponding to particles with half-integer spin could be constructed. Yet the hadronic spectrum contains both bosons and fermions. Where were the latter hidden? The answer to both questions came soon and it opened new broad horizons.

6.1.1 Fermionic Strings and the Birth of Supersymmetry

To the question where in the theory of tiny strings the fermions were hidden, Neveu and Schwarz on one side and Pierre Ramond on the other (see Fig. 6.6) gave two answers which, although different, are not alternative, rather complementary. The followed approach was algebraic in both cases.

In 1971 while traveling back from Europe to the US on board of a big ship, John Schwarz met André Neveu1 and during the Atlantic crossing they found a generalization of the infinite dimensional symmetry algebra of Nambu string that goes under the name of Virasoro algebra.2 In its original approach Virasoro found that the physical string states constructed with the harmonic oscillators of the Fubini Veneziano approach should be annihilated by the action of an infinite number of operators Ln, that are in one-to-one correspondence with the integer numbers: n Z:

n Z : Ln|phys.! = 0

(6.1.1)

and satisfy the following infinite dimensional Lie algebra:

[Lm, Ln] = (m − n)Lm+n +	c	m3	− m δm+n,0
	12			(6.1.2)

[c, Ln] = 0

The operator c commutes with all other generators of the algebra and for this reason is called the central charge. Therefore in every irreducible representation, c takes a fixed numerical value which, as Virasoro showed [3], equals the number of spacetime dimensions D in which the string is propagating. The cancellation of anomalies

1This episode is known to the author by private conversations with Prof. Schwarz who told him this story while visiting him at his place Torino in 1981.

2The name of the algebra refers to his discoverer, the brilliant Italo-Argentinian physicist Miguel Virasoro born in Buenos Aires in 1940, who is presently full professor of Theoretical Physics at La Sapienza of Rome and from 1995 to 2002 was Director of the International Centre of Theoretical Physics of Trieste, founded by Abdus Salam.

216	6 Supergravity: The Principles

Fig. 6.6 The three founders of fermionic strings: on the left André Neveu, in the middle John Schwarz, on the right Pierre Ramond. Born in Paris in 1946, Neveu studied at the Ecole normale superériure (ENS). In the early seventies he was for some time at Princeton where he collaborated with John Schwarz and David Gross. These collaborations resulted in two very important results: the Neveu-Schwarz algebra on one side and the Gross-Neveu toy model of quantum chromodynamics on the other. Later Neveu worked at the Laboratory of Theoretical Physics of ENS in Paris, at the CERN Theory Division in Geneve and from 1989 he has been director of the Laboratory of Theoretical Physics of the University of Montpellier II. Born in Massachusetts in 1941, John Schwarz studied as an undergraduate at Harvard and as a graduate at Berkeley University. Assistant professor in Princeton from 1966 to 1972 he moved next to the California Institute of Technology where he is currently the Harold Brown Professor of Theoretical Physics. For several years one of the very few believers in superstring theory, John Schwarz was responsible for the first string revolution in 1984 when, together with Michael Green he found the mechanism of anomaly cancellation establishing the set of five consistent perturbative string theories. John Schwarz was awarded the Dirac Medal in 1989 and the Dannie Heineman Prize for Mathematical Physics in 2002. Born in France in 1943, Pierre Ramond studied in the United States where he graduated in 1969 from Syracuse University. Assistant Professor at Yale University and at the California Institute of Technology, Ramond joined the University of Florida at Gainesville in 1980. There he is currently Distinguished Professor of Physics. His contribution to the development of superstring theory has been a fundamental one, the Ramond sector being an essential part of the superstring spectrum where all the fermionic particles are located

and the quantum consistency of the theory required a value c = 26 which explains the unexpected result quoted above.

During their boat trip André and John derived an extension of Virasoro algebra by means of another infinite set of operators Gm+ 12 , in one to one correspondence with

the half integer numbers (n + 12 Z + 12 ), that satisfy the following commutation, anti-commutation relations:

[Lm, Gn+ 12 ] =		2 m − n −		2 Gm+n+		21
		1		1
{Gm+ 21 , Gn+ 12	} =	2Ln+m+1 +		c	m2 + m δm+n+1,0		(6.1.3)
				3
[c, Gn+ 12	] =	0

6.1 Historical Outline and Introduction

217

The union of (6.1.2) with (6.1.3) constitutes the Neveu Schwarz algebra [7] which is a first example of a super Lie Algebra. The novelty that entitles it to the new qualifier “super” is the presence of a grading that splits the set of all generators in two classes, the even ones (in our case the Ln) and the odd ones (in our case the

G + 1 ). The algebra is specified by providing the commutators of the even operators

n 2

with the even ones that necessarily produces another even operator, of the evens with the odds that produces an odd and finally the anti-commutator of two odds that necessarily produces an even.

Chronologically the Neveu Schwarz superalgebra was not really the very first, since a few months before, also in 1971, Pierre Ramond had found another very similar extension of the Virasoro algebra adding to it a set of odd generators Gn that are in correspondence with the integer numbers. Ramond superalgebra [4] is obtained by adjoining to (6.1.2) the following commutation, anti-commutation relations:

[Lm, Gn] =	2 m − n Gm+n
	1
					c	+m2 −	1	,δm+n,0
{Gm, Gn} =	2Ln+m +								(6.1.4)
				3			4
[c, Gn] =	0

Which algebra was the right one for superstrings? Both were right since they have a common origin in an extension of Nambu string theory by means of a new fermionic field that lives on the world-sheet traced by the string while propagating through space-time. The evolution history of the string, for instance that drawn in Fig. 6.3 or in Fig. 6.4 is described by giving the coordinate Xμ of the ambient space as functions Xμ(σ, τ ) of the two Gaussian coordinates {σ, τ } that label the world-sheet points. So doing we can regard the world-sheet itself as a two-dimensional spacetime and the functions Xμ(σ, τ ) as a set of scalar fields living on it. Adopting this point of view why not consider, besides spin 0 fields also new fermionic fields of spin 12 living in the same space-time? Let us do it and let us introduce as many new fields of this type as there were scalar fields: let us name such newcomers Ψ μ(σ, τ ). What has it happened at the end of such a procedure? A surprising miracle! The little field theory that we have constructed on the world-sheet has the following marvelous properties:

•It is supersymmetric since it is invariant under a set of appropriate transformations that exchange the bosonic fields Xμ(σ, τ ) with the fermionic ones Ψ μ(σ, τ ).

•It is anomaly free and quantum consistent no longer in a D = 26 space-time, rather in a significantly smaller one D = 10.

•The states encompassed in the spectrum of this quantum theory divide in two sectors, one named NS realizes the Neveu Schwarz algebra (6.1.2)+(6.1.3), the second named R realizes the Ramond algebra (6.1.2)+(6.1.4). Both sectors are

necessary to construct Veneziano amplitudes for bosonic and fermionic particles.

218	6 Supergravity: The Principles

6.1.2 Supersymmetry

While trying to insert fermions into the structure of Veneziano amplitudes, a new algebraic structure had been discovered, supersymmetry, which was destined to mark heavily the development of Theoretical Physics in the subsequent years. Actually in the same 1971 year, the supersymmetry algebra had been constructed in a completely different context by two Russian scientists, Golfand and Likhtman [1, 2], whose almost tragic personal story is a sort of emblem of the incredible contradictions of Soviet times, also in relation with pure science.3 Results similar to those of Golfand and Likhtman were obtained also in Kharkov, by other two Soviet scientists, Volkov and Akulov [5] who constructed a non-linear field theoretical realization of the same super Poincaré Lie algebra discovered in Moscow.

If we analyse the meaning of the operators in the Virasoro algebra and in its extensions we understand the following: the operators Ln correspond to a modeexpansion of the stress-energy tensor, namely of the Noether current of translations Pμ, while the fermionic operators Gn correspond to the mode-expansion of a spinor-vector current Jμα . Which symmetry is such a Noether current associated with? The answer is unique: to some new symmetry generator Qα which transforms as a spinor under the Lorentz group and whose anti-commutator with itself must be proportional to the translation generator Pμ.

In D = 4 the super Lie Algebra corresponding to such a symmetry, contains N of such spinor generators, named supercharges, and has the following general structure:

[Jμν , Jρσ ] = −ημρ Jνσ + ηνρ Jμσ − ηνσ Jμρ + ημσ Jνρ

[Jμν , Pρ ] = −ημρ Pν + ηνρ Pμ

					1
[Jμν , QAβ ] = −									QAα (γμν )αβ		(6.1.5)
							4
{		Aα ,				Bβ } = i C γ μ αβ PμδAB − Cαβ ZAB ;					(A, B = 1, . . . , N )
	Q			Q
[ZAB , anything] = 0

3Born in Kharkov in 1922, Yuri Abramovich Golfand got his mathematical-physical education in that Ukrainian city. Later, since 1951, he joined the Tamm group at the Lebedev Physical Institute of the Soviet Academy of Sciences in Moscow (FIAN), an institution that collected seven Nobel Prizes in Physics in the course of sixty years. Golfand and his student Likhtman conducted there, at the end of the 1960s, the studies that led them to discover the super Poincaré Lie algebra and to construct its first field theoretical realizations, published in 1971 after a long procedure of checks and inspections by the Soviet censorship authorities (see [10] for a detailed account of these facts). The next year, in the course of a routine campaign of personnel cuts, Yuri Golfand was fired from FIAN and decided to apply for an exit visa to Israel. This put him in a very bad light in front of Soviet authorities who refused the visa and treated him as a renegate. For 7 years he lived unemployed and was readmitted to FIAN only in 1980. Golfand obtained permission to emigrate to Israel only in 1990 and there he lived his last four years in Haifa, where he died in 1994. Because of his association with the renegate Golfand, also Likhtman had very difficult times with Soviet authorities and could never get a proper academic position.

6.1 Historical Outline and Introduction

219

Fig. 6.7 The two western founders of supersymmetric field theories. On the left Bruno Zumino, born in 1923 in Rome, graduated from the University La Sapienza in 1945. He is currently emeritus professor of Berkeley University in California. For many years he was permanent member of the Theoretical Division at CERN. Julius Wess born in 1934 in Oberwölz Stadt in Austria died in 2007 in Hamburg. Austrian by nationality, Wess graduated from Vienna University and was professor first in Karlsruhe University and then in the Ludwig Maximilians University of Munich. Zumino and Wess have given many important contributions to Theoretical Physics in several directions. Jointly they introduced the first example of a supersymmetric field theory that bears their name

Equations (6.1.5) define the N -extended super Poincaré Lie algebra; the antisymmetric generators ZAB = −ZBA which are present only for N ≥ 2 are named the central charges. The case N = 1 is the algebra introduced by Golfand and Likhtman who also tried to construct examples of field theories invariant against transformations closing such an algebra.

In the western world the date of birth supersymmetry is 1974. In that year Bruno Zumino and Julius Wess (see Fig. 6.7) published a paper [6] where they constructed the following very simple example of a field theory with supersymmetry invariance. Let A(x), F (x) be two scalar fields, B(x), G(x) two pseudo-scalar fields and let λ(x) be a Majorana spinor field.4 Consider the following very simple Lagrangian:

Ltot = Lkin + Lmass

1

i

1

2 + G2

Lkin = −

∂μA∂μA + ∂μB∂μB +

λγ μ∂μλ +

F

(6.1.6)

2

2

2

Lmass = −m(F A − GB) +

m

λλ

2

The action:

A =

Ltot d4x

(6.1.7)

is easily verified to be invariant under the following infinitesimal transformations:

δεA = 1 ε λ

2

4For the definition of Majorana spinors see (A.4.3) in Appendix A.4.

220												6 Supergravity: The Principles
δεB = −			1	i		γ5λ
					ε
			2
δελ =	1	i∂μAγ με +					1		∂μBγ μγ5ε +	1		(F + iγ5G)ε	(6.1.8)
	2						2			2

δεF = ε λ

δεG = i ε γ5λ

where εα is a constant anti-commuting spinor parameter (εα εβ = −εβ εα ). The above equations provide the explicit form of the supersymmetry transformations that for each field Φ of the theory can be thought as the result of acting on it with Qε, namely:

δε Φ = [			(6.1.9)
	Qε, Φ]

the operator Qα being the spinorial supercharge. The N = 1 case of the super Lie algebra (6.1.5) is realized, since it is immediately verified that for any field Φ we have:

δε1 δε2 Φ − δε1 δε2 Φ =

1γ με2∂μΦ

ε

)

1γ με2Pμ, Φ

(6.1.10)

[

Qε1, Qε2], Φ =

ε

and the anticommutativity of the spinor parameters implies:

[

Qε2] = ε1α ε2β {

α ,

β }

(6.1.11)

Qε1,

Q

Q

The Wess-Zumino model encoded in (6.1.6) has a very simple physical content. It contains two spin zero degrees of freedom and one spin one-half degree of freedom, which constitute the simplest supersymmetric multiplet: {0+, 0−, 12 }. Indeed the fields F and G, named auxiliary, can be eliminated through their own equations of motion which yield:

F = −mA; G = mB

(6.1.12)

After substitution of these equations into the original Lagrangian we obtain the standard action for a system composed by a free scalar A of mass m, a free pseudo-scalar B with the same mass and finally by a free spinor λ also with mass m. The interesting point is that the partial actions 4 Lkin d4x and 4 Lmass d4x are separately invariant under the transformations (6.1.8). These means that the mass term Lmass can be substituted by other more complicated but invariant functions of the four fields {F, G, A, B, λ} leading, after substitution of the new field equations for F and G, to more complicated dynamics.

In the years after 1974 a lot of work was devoted to constructing supersymmetric field theories with several multiplets that extend up to spin one and to exploring the general form of the interactions allowed by this new powerful symmetry. In parallel, representation theory of the supersymmetry algebra was worked out for all numbers