Superstrings with spontaneously broken supersymmetry and their effective theories

Nuclear Physics B318 (1989) 75-105 North-Holland, Amsterdam

S U P E R S T R I N G S WITH S P O N T A N E O U S L Y BROKEN S U P E R S Y M M E T R Y AND THEIR EFFECTIVE THEORIES* Sergio FERRARA

CERN, Geneva, Switzerland and

Department of Pt~vsics, UCLA, Los Angeles, USA Costas KOUNNAS

LPT, Eeole Normale Sup~rieure, Paris, France Massimo PORRATI and Fabio ZWIRNER**

Department of Plo,sics, UCB, Berkeley, USA and

Lawrence Berkeley Laboratot~v, Berkeley, USA Received 21 September 1988

We discuss superstring models with spontaneous breaking of N = 1, 2 or 4 space-time supersymmetry, via coordinate-dependent compactifications from five to four dimensions. We provide a description of the supersymmetry breaking mechanism in the formulation with complex world-sheet fermions, and we reinterpret it in terms of special deformations of the lorentzian charge lattice. Considering a representative string model with spontaneously broken N = 1 supersymmetry and massless chiral fermions, we show that the knowledge of the spectrum of states and of the flat directions completely specifies its low-energy effcctive lagrangian, which turns out to be a new no-scale supergravity model. We outline the qualitative difference between these models and other scenarios of gaugino (gravitino) condensation or general non-pcrturbative phenomena.

1. Introduction One of the most important problems in superstring models is the search for a way of promoting their beautiful mathematical structure to a candidate theory for elementary particles. To achieve this goal, it is evidently necessary to break many of * This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098, and in part by NSF under grant PHY-8515857. ** Also at Istituto Nazionale di Fisica Nuclearc, Sezione di Padova, Italy. 0550-3213/89/$03.50©Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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the string symmetries. Consistent string theories [1,2] show a general tendency to have too many exact symmetries, which seem to persist at any order in string perturbation theory. On the other hand, many of these symmetries are of main importance in order to understand the string universality. In four space-time dimensions, for instance, the number of consistent string solutions is enormous [3-14]. In a large class of cases, different string vacua can be continuously connected by varying the expectation values of some massless scalar fields, i.e. via a string Higgs phenomenon [3-5, 9,12,15]. At present it is not yet known if all string models (or at least all those having the same local super-reparametrization properties of the world sheet) are based on a unique string field theory, although there are some general indications which justify this conjecture. In this work we study a restricted class of superstring solutions in four space-time dimensions. We connect theories with exact and spontaneously broken space-time supersymmetry via a super-Higgs phenomenon, and we derive the effective lowenergy lagrangian in the broken phase. We focus our interest mainly on N = 1 superstrings, because of the possibility of a realistic spectrum with a net non-zero number of chiral fermions. Our choice of superstring models with space-time supersymmetry as a starting point is justified by consistency arguments. Namely, the presence of unbroken space-time supersymmetry guarantees the absence of tachyonic states and the stability of the flat background, at least up to two loops and probably to any order in the string loop expansion [16]. The hope is then that, for a non-empty set of models, the consistency of the theory will persist in a spontaneously broken phase. A string solution with spontaneously broken supersymmetry is in general obtained by acting on a supersymmetric solution with an operator defined on the world sheet. Such an operator must respect Lorentz invariance and commute with the left- and right-moving energy-momentum tensor and the supercurrent(s). Moreover, it has to be globally defined. Any such operator generates a one-to-one correspondence between the symmetric and the broken phase, with a pairing of bosons and fermions inside supermultiplets, up to mass splittings proportional to the gravitino mass. Restricting ourselves to perturbative breaking, the knowledge of all possible deformations (discrete or continuous) would allow us to construct all the different spontaneously broken phases connected to a given supersymmetric solution. At present, we do not know the complete set of allowed deformation operators. An interesting class of string models with manifestly spontaneous breaking of supersymmetry is that corresponding to generalized coordinate-dependent torus [T(XS)] and orbifold [T(X5)/Z2] compactifications from five to four dimensions. The torus compactifications T ( X 5) are the string generalization [17,18] of the Scherk-Schwarz mechanism [19] in field theory. The orbifold compactifications T ( X s ) / Z 2 , introduced in refs. [20, 21], yield N = 1 models with chiral families. The T ( X 5) and T ( X s ) / Z 2 compactifications use as deformation operators some internal

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SO(5) rotations, which give a non-trivial U(1) R-charge to the Ramond string vacuum. However, globally the internal SO(5) symmetry is broken by the compactification from ten to five dimensions, and only some discrete symmetries survive. As a consequence, the parameters of the deformation associated to supersymmetry breaking are quantized. General arguments barring the possibility of space-time supersymmetry breaking with continuous parameters have been given in refs. [22,23]. As we shall show here, the same phenomenon of discretization of the possible deformations occurs also in the fermionic formulation, where one uses fermions instead of compactified bosons (see also refs. [21, 24]). In the class of string models under consideration, the gravitino masses of an N = 4 theory are given by 1

'"3/2

-

2R

1 ,,,3/2

el - e2),

(1.1)

where e 1 and e 2 are the two possible independent quantized charges in the Cartan subalgebra of SO(5) and R is the radius of the fifth dimension. In the N = 2 case there are only two gravitinos [w(3'4) t,,,3/2 are absent], while in the N - - 1 case only ~(1) " ' 3/2 remains. The fact that e 1 and e 2 are quantized with values of O(1) means that small gravitino masses are possible only when the radius R is much larger than the string scale f ~ - , which also means that the theory approximately decompactifies its fifth dimension [21]. Differently stated, for a light gravitino mass many other states with masses around that scale are present in the spectrum [23]. In order to bypass the decompactification problem, one has to assume that R = f a 7, which also implies that m 3 / 2 ~ O ( m p ) . All the other existing string models with broken space-time supersymmetry are also plagued by the quantization-decompactification problem mentioned above. As discussed in more detail in the conclusions, this applies even to the conjectured non-perturbative mechanism involving gaugino condensation [25] (which, the reader should bear in mind, has not yet been formulated at the level of two-dimensional conformal field theory). Since in the string models under consideration m3/2 -- O(Mp), one could naively conclude that the effective theory, obtained by integrating over the string states of mass O(Mp) or higher, looks like any other effective theory in the absence of supersymmetry. However, the knowledge that this solution corresponds to the spontaneous breaking of a supersymmetric theory has many important implications. Indeed, the geometrical structure and the interactions of the effective theory look like those of a supersymmetric theory with some missing states. Such a theory also contains some "accidental" non-linear symmetries, which are fully understandable only with the information that there is spontaneous breakdown of supersymmetry. A more efficient way of studying these theories is to integrate over the massive modes supersymmetrically. This implies the unavoidable presence in the effective

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theory of all the superpartners of the massless states (whatever their mass), so that supersymmetry transformations can be defined in the broken phase of the theory. The knowledge of the whole spectrum, both in the unbroken and in the broken phase, the constraints of supersymmetry and some exact string properties, allow us to fix unambiguously the complete effective theory in the presence of supersymmetry breakdown. In fact, as we will show here, the N = 1 theory in the broken phase differs from its supersymmetric version only by a specific superpotential modification, which breaks its trilinear homogeneity. In a convenient parametrization for the manifold of the scalar fields y, to be described in detail in sect. 4, the superpotential modification is given in terms of the two breaking parameters e I and e 2 discussed before: g = k0 + / , y 2 + gsusy,

(1.2)

where the constant k 0 = (e 1 + e2)/2 is responsible for supersymmetry breaking, = (e 1 - e 2 ) / 2 is a supersymmetric mass parameter and gsusy is the superpotential of the corresponding theory with unbroken supersymmetry. We will show explicitly in the following that eq. (1.2) describes in the effective theory a spontaneous breaking of N = 1 supersymmetry with vanishing tree-level cosmological constant and a gravitino mass given by m3/2 = k o exp cbi) = k o / R ,

(1.3)

where ~D is a dilaton-like field associated to the radius R of the compactified fifth dimension, in perfect agreement with its string equivalent, eq. (1.1). Even in this "macroscopic" language it is apparent that, since k 0 is quantized, the gravitino mass is determined by the radius R of the fifth dimension appearing in the right hand side of eq. (1.3). Another main feature of the above mechanism for supersymmetry breaking is that the goldstino has a non-vanishing component along the direction of the "dilatino" ~q: this very fact, combined with the universal form )Cab = ~ab S of the N = 1 gauge kinetic function, implies that gauginos have universal masses equal to the gravitino mass. Our paper is organized as follows. In sect. 2 we review the fermionic formulation of four-dimensional string models constructed through coordinate-dependent compactifications from five to four dimensions, and we generalize it to the case of broken supersymmetry. We also re-interpret the supersymmetry breaking mechanism in terms of special deformations of the lorentzian charge lattice. In sect. 3 we introduce as representative examples some N = 2 models and N = 1 Z2-orbifolds with chiral fermions. In sect. 4 we derive the effective theories for the previously introduced string models with spontaneously broken N = 2 and N = 1 supersymmetry. We compute explicitly the N = 1 classical potential and we show that it corresponds to a generalized no-scale model. Finally, sect. 5 contains our conclusions and discusses open problems.

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breaking by coordinate-dependentcompactifications

2. Spontaneous (super-)symmetry

Our starting point is a generic five-dimensional heterotic string model, in the formulation with free world sheet fermions and mutually commuting boundary conditions [6,7,10]. Following ref. [6], we denote by ( 0 . X ~ , ~ ) ( ~ = 3 , 4 ) and (O:XS,~ 5) the left-moving supercoordinates in the light-cone gauge, by (X ~, yl, wl) ( I = 2, 3, 4, 5, 6) the remaining left-moving fermions. The supercurrent corresponds to a non-linear realization of world-sheet supersymmetry. For definiteness, we consider here the case in which (Xt, yZ, w t) ( 1 = 2 , 3 , 4 , 5 , 6 ) are in the adjoint representation of [SO(3)] 5, 6

TF=~*O:X~, +q'S OzXs + E XIy 'w'

(2.1)

1=2

The right-moving sector contains the coordinates O~X ~ (/~= 3,4), O~X 5 and the fermions ¢bA (A = 3.... ,44)*. The theory is further specified by a set of boundary conditions. For definiteness, we assume that the 18 left-moving and 42 right-moving fermions can be rearranged in pairs into a complex basis ( N L = 9, N R = 21)

f----- { f L ; f R } ~ { f i L ( i L -----1 . . . . .

N L ) ; fi R ( i R = 1 . . . . .

NR)},

(2.2)

SO that the boundary conditions associated to non-contractible loops along the 1 and r directions of the world-sheet torus can be given by the vectors

a~-- (aL; aR} -~ {aiL (iL = 1 . . . . . NL); air (iR = 1 . . . . . NR)}, b = {bL; bR} -----{bi L ( i L = 1 . . . . . NL); bi, ( i R : 1 . . . . . NR)},

(2.3)

where - 21< aiL, air , biL, biR <~~ and under parallel transport

1:

fiL(R) ~

-- e 2~rta~L(R)f/L(R)

'7":

fiL(R) ~

-- e

--

2rrlb L(R)fiL(RI z

(2.4)

* Notice that, with respect to four-dimensionalmodels, the left-movingfermions(Xa, yt w1) have been replaced by ( 3: X5, @), the right-movingfermions(~1 ~2) by 0~X5.

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To any model defined in d = 5 Minkowski space-time one can associate the one-loop partition function

Z s = f ( ~d~'d'~) Zs(~', ~)

'r/3/2 Zs('r, ~)

1

l~/(,)16 [n(~.)] NL[~(?)l N"

ra,L1 spin str. where 7/(¢)

-

ql/24

i

N. [a,R]

1

I-I°¢n=l(1 - qn), with q

O bi ('r)=-

iR = 1

-

e 2~rir,

L

is the Dedekind eta function, and

e i~'~(k+aJz+2cri(k+ai)b'. k=

(2.6)

0¢

C(~)

are pure phases and are constrained by the requirements of The coefficients modular invariance and factorization [6]. Compactifying the fifth coordinate X 5 on a torus of radius R, one obtains a d = 4 theory described by the one-loop partition function

ZT5 =

~

ZT5 ('r, ~),

• i-a 1 Z 17/(~')14 [~O(~-)] NL[~/(,~)] NR m , n = - - o o

ZTs("/', ,r ) = _ _

X spinEstr.C ( ; ) " i

10

( T ) " iRFI=l-~[bi2l(~)'

(2.7)

where (m, n) are the winding numbers of the torus and R'rl l/2

e -(~'R2/2rD}m-'nl2 .

(2.8)

Z,.,.(T, ¢) - v~ln(~.)12 In the presence of a global two-dimensional U(1) symmetry (described in terms of complex fermions on the world sheet) one may deform the previous d = 4 model to

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obtain another consistent d = 4 model described by the following partition function r{dTd~\

7/--1

1 +09 zm ZT
ai L - nell

NR

[ a i r - heir ] ( ? ) ,

spin~str. C(~) " iNLI~=lO[biLq-meiL] ('r)" iRI~=l~LbiRWmeiR. (2.9)

where e ~ {eL; eR} -- {eiL(iL = 1 . . . . . NL); eiR(iR = 1 , . . . , NR) }

(2.10)

is a vector of U(1) charges and

c(ab)-~e2Crine(b+(m/2)e)c(ab).

(2.11)

Notice that in our notation the scalar product has lorentzian signature: if U--= (UL; UR} = {U, L ( i L = 1 . . . . . NL) ; UiR (iR = 1 . . . . . NR) } and v--- {VL; VR} -----{viL (iL = 1 . . . . . NL); viR (iR = 1 . . . . . NR)}, then u. v-= u L • v L - u R • v R - uiviL - uiviR. Notice also that the partition function ZT(X5) couples the spin structures (a, b) with the winding numbers (m, n) of the torus associated to X s. When the U(1) is a global symmetry of the theory, the partition function ZT(X~) corresponds to a consistent d = 4 string model with some of its gauge symmetries spontaneously broken. This is the string analogue of the coordinate-dependent Scherk-Schwarz compactifications in field theory. In the fermionic formulation, we define the charge operators Q = a + F,

(2.12)

where a is the vector of the boundary conditions along the 1 direction of the (1, T) toms. The components of F are the fermion number operators for the complex fermions jr. We now need to recall the Poisson resummation formula

d-oo E

+~o Y e-'~(v2''2-2i''x)=

~

e-('~/Y2)('+x)2

(2.13)

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and the trace representation of the O-functions for arbitrary boundary conditions

re,L]

= Tr

e 2rribiLQiL

= e2~ria@,L

× ~

q H'L

q (~2L/2-1/24)

(1

1/2e2~rib'L)(1 q-qn--a'c--1/2e--2rrib'L)~(2.14)

q-qn+aiL

n=l

and analogously for the right-movers. In eq. (2.14), HiL is the hamiltonian for the twisted fermion fiL and can be expressed as

HiL

=

E (n+aiL ½)""bn+a,L-1/2bn+aiL * 1/2'"+ 2" --

24 '

(2.15)

rt= --oo

where the constant term comes from normal ordering, and (2.16) Using eqs. (2.13-2.15), we can rewrite the partition function (2.9) as [18] +oo

ZT(X%($ , 'F) =

Trgq[L°lT'XS'qU'°IT'x'},

E Dl, n=

(2.17)

O0

where g is the appropriate GSO projection (independent of the values of the charges e), and [Lo]T(X 5) = ½(QL -- neL) 2 + 1(

-

m + e" QR- (n/2)e2

nR)2 +5-

~1 + oscillator contributions,

[ ]7-'°]T(Xs)= ½ ( O R - heR)2 +

½( m+e'Q-(n/2)e2R

- 1 + oscillator contributions.

nR) 22 (2.18)

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It is important to note that

1(

[Loq'-JT,O]T(XS): [Lo-{-L,o]Tsq- ~-

o2

+ -2- (e2 + e 2 ) -

e" Q -

nt2

-£e 2

(

-- n ( e L " Q L + e R " Q R )

)

2m e ' O - ~ e n 2 ,

~T

In0- L'OIT(x5) = [Lo- LO]T5,

(2.19) (2.20)

where [L0]T, and [f~0]T~ are the Virasoro operators for the original T 5 model [whose partition function is given in eq. (2.7)], and can be immediately obtained from eq. (2.18) by putting e = 0. The above equations tell us that the two models have the same physical states but a different mass spectrum, in agreement with the definition of spontaneous symmetry breaking. In particular, for the lowest lying states in the m = n = 0 sector, which become massless in the limit e --, 0, one generates masses equal to

M = ( 1 / R ) [ e . QI,

(2.21)

exactly what one would obtain from the Scherk-Schwarz mechanism in field theory. F r o m eqs. (2.17-2.20) it is apparent that the partition function ZT(xb admits a particle interpretation, since bosons and fermions are weighted in the trace with the sign dictated by spin-statistics. We observe that, for the case of gauge symmetry breaking considered here, the charges e are arbitrary and correspond to flat directions in the scalar potential of the effective theory. For a better understanding of the above mechanism, it is convenient to reconsider it in the language of the bosonic lattice formulation. As shown in ref. [15], one can pass from the fermionic to the bosonic formulation using the key relation p = Q,

(2.22)

where p is the charge or momentum vector associated to a given state in the bosonic formulation. The above equation must be interpreted as an isomorphism between the eigenspaces of the operators p and Q, the former acting on states of the bosonic construction, the latter on states of the fermionic one. Using the trace representation (2.14) for the O-functions, one can then rewrite ZT~(r, 4) as ZT,(r,'F) =

Y'~

~

TrgqlL°ITsFt[Z°IT~,

(2.23)

P ~ INL. ~'R pO ~ l'l, 1

where g is the same as in eq. (2.17) and FNL' NR is a self-dual lorentzian lattice

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translated by the boundary condition vector a. In eq. (2.23) we have used the following definitions

v~

(pO)~

[L°]T~ - 2 - +

v~ [~01~--

1

2

2 + oscillator contributions,

(pO)2 1 + oscillator contributions,

2 + - -2

(2.24)

and we have denoted by/'1,1 the torus momentum lattice, characterized by m, n ~ Z and

po_ ~-+

,

pO_ ~

-~ .

(2.25)

The spectrum of eq. (2.18) can be obtained from eq. (2.24) by performing the following substitutions

p ~ p - ne,

m ~ m + e.p-

~nel 2.

(2.26)

Remembering eq. (2.25), this corresponds to the following deformation of the momentum lattice FNL"NR~ El,l: 0 PL-"~PL--~L(PL--pO),

pR~pR--~R(pO--pO), pO ~pO + ~ . p _ ~

1J'2/[ p L 0 _pO),

pO~pO +~.p_5

1~2/[ p L 0 _pO),

(2.27)

where we have introduced the vector of parameters -

(2.28)

e/R.

In compact notation, one can rewrite eq. (2.27) as

P---, peTr= p(1 + T+ TZ/2),

(2.29)

where P - (PL; PR) -= (PL, pO; PR, pO) and (notice that T 3 = 0) 0 T

T=

-~R

~;

(2.30)

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85

F r o m the structure of eqs. (2.29, 2.30) one can see that the deformations of eq. (2.27) correspond to a simultaneous rotation (change of base) and boost of the lorentzian charge lattice: T is a particular combination of compact and non-compact generators of S O ( N L + 1, N R + 1). This shows that in the bosonic language the Scherk-Schwarz mechanism for gauge symmetry breaking corresponds to a lorentzian lattice deformation. When we are interested in the breaking of supersymmetry, the situation is slightly more complicated. The reason is that, in the fermionic formulation, the space-time gravitinos must transform non-trivially under the action of Q, but at the same time Q must commute with the world-sheet supercurrent T F and the operator ~ - e 2~i~'q must be globally defined on the world-sheet. This implies that c~ must be an outer automorphism of the group H L, whose structure constants define the world-sheet supercurrent T F. In particular, for H L = [SO(3)] 5, the only useful automorphisms are those contained in the free product of Z 2 and ~5, where Z 2 is the outer automorphisms group of SO(3) and ~5 is the permutation group of the five (XI, yl, w I) ( I = 2,3,4,5,6). Coming back for a moment to the bosonic formulation, we notice that the only non-trivial quantum numbers of the massless gravitinos (apart from those of the five-dimensional Lorentz group) are those of the SO(5) internal Lorentz group. However, in the compactification from d = 10 to d = 5 this internal SO(5) is broken down to some discrete subgroup. Nevertheless, some permutations of the five internal coordinates can remain exact symmetries of the d = 5 theory, and some of these discrete symmetries can be regarded as special finite SO(5) rotations. To be specific, the relevant internal Lorentz currents are of the form

jMN=+M~N+½(xMozxN--xNozxM),

(M,N=6

. . . . . 10).

(2.31)

Since the X M a r e not conformal fields, the j M N do not have well defined conformal weights. However, some special SO(5) rotations, for instance those corresponding to permutations of some internal Lorentz indices, are well defined. Obviously, they leave invariant the supercurrent, now written as 10

=

(2.32)

o x. + q¢ o xs + F. M=6

where we have made the identifications

X'Ot~ M, (ytwt)~OzxM,

( • = 2 . . . . . 6;

M=I+4).

(2.33)

The above equivalence will be used in the following, whenever convenient, to pass from the bosonic to the fermionic formulation and vice versa. In summary, the only relevant symmetries of the world-sheet supercurrent Tv that are appropriate for the

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breaking of supersymmetry are discrete rotations contained in the internal Lorentz SO(5). Since in the Scherk-Schwarz mechanism one uses only the Cartan subalgebra of the internal (five-dimensional) symmetry group, one can have at most two independent charge operators acting non-trivially on the gravitino states. Some examples are now in order, to illustrate the previous statements. Consider the following cyclic permutation 503, whose action on the left-movers is given by

503:

X2 _..) X3,

y2 ___~y3,

w2 ..e, w3,

X3 ___~X4,

y3___)y4,

w3 ___)w 4,

X 4 ~ X 2,

y4 ..._~y2,

w 4 ~ w2,

xS__. x s ,

yS ~ yS,

w s __. w s,

X6____)X6,

y6.._) y6,

w6____~w 6 '

(2.34)

or, in matrix notation X2

503: X(s) --* (93X(s),

ti 1 0 0 0 0

03 -

0

0

0

1

0

0

0 0

0 1

0 0

0

0

1

X3 ,

X(s)-

X4 , (2.35) Xs X6

and similarly for the y t and w ~ ( I = 2, 3, 4, 5, 6). It is immediate to check that (93 is a finite S O ( 3 ) c SO(5) matrix [(93r(93= 1, det (93 = +1]. It is also convenient to introduce the new variables

x°--

x +x4)/Cx,

x-( g - e2~i/3),

(2.36)

and similarly for the y r and w i ( I = 2, 3, 4, 5, 6). In the new basis, the action of 503 is the following X° ~ X° ,

503:

X ~g-lx X+--* g x +,

,

y0 ~ y 0 ,

w 0 __. w 0,

y

w --*g-lw

__.g ly ,

y + ---~gy +,

,

(2.37)

w + ---~gw +.

It is therefore immediate to reinterpret 5°3 as a finite U(1) transformation contained in SO(3) c SO(5). Defining the two-dimensional charge operator

f l- dz dz

,

the action of this symmetry can be described by the finite rotation "ff3 =- e2~ri(elqO,

(2.39)

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87

where el=

! 3"

(2.40)

Another example is the following cyclic permutation

35 :

X2 _.~ X 3,

y2 __.y3,

w 2 ~ w 3,

X3 __. X 4,

y3 __~y 4,

w3 __~ w 4,

xa____)X5

y4__>yS,

w4___>w 5,

X 5 ~ X 6,

y5 __.y6,

w 5 __, w 6,

X 6 __. X 2,

y6 __~y2,

w 6 ~ w 2.

(2.41)

By diagonalizing the transformation through the field redefinitions X0 ~ (X 2 +

x 3+ x 4 + x 5+

X -= (X 2 X --~- (X 2 + g2x3 + g4x4 4- gBx5 4- gSxS)/v/5 , X++~_ (X 2 4-

g3x34-

g6x4 4- g9X5 4- g 1 2 x 6 ) / ~ ,

X+~ (X 2 4- g4x3 4- gSx44- g12xS-}-g16x6)/v/5,

(g~e2~ri/5),

(2.42)

and similarly for the y i and w z ( I = 2, 3, 4, 5, 6), one can re-interpret 5~5 as a finite U(1) transformation contained in SO(5). Defining the two-dimensional charge operators _ dz Ql=-~-~iJl=~

_

d2 (X+X 4-fl+y-4-W+W ),

(2.43)

27ri

dz dz Q2=~-~iJ2-~i(x++x

+y++y

+w++w

),

(2.44)

the action of 5"5 can be described by

~5 =--e2wi(elQl+e2Q2)

,

(2.45)

"~ 5 .

(2.46)

where el=

1 5 ,

e2=

In a similar way, many other examples can be constructed. In a diagonal basis, the associated symmetry operator can always be written as 5 ~= e 2~i'~, with ~-= e . Q - ( e i Q 1 + e2Q 2 + • • • ), where Q1 and Q2 are the fermionic counterparts of operators which in the bosonic formulation lie in the Cartan subalgebra of the internal SO(5), and the dots stand for possible additional charge operators acting trivially on the gravitino states. The general structure of the corresponding string

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88

models can be characterized as follows: (i) The partition function is as in eq. (2.9), with the additional constraint that the charges e 1 and e 2 can only assume a finite number of discrete values, in contrast with the case of gauge symmetry breaking. (ii) The gravitino state is a Ramond vacuum for eight left-moving fermions (in the light-cone gauge), so that it transforms as a spinor under the internal SO(5) considered above. Thus, because of eq. (2.21), the values of the d = 4 gravitino masses are given by

m3/2 = le~ + e2l/2R.

(2.47)

(iii) The fact that we started from a five-dimensional theory with N = 1 or N = 2 supersymmetry implies that the T ( X 5) compactification can only give four-dimensional theories with an even number of exact or spontaneously broken supersymmetries. One way to obtain spontaneously broken N = 1, d = 4 models with chiral fermions is to start from a N = 1, d = 5 model and to compactify it on a Z 2 orbifold. As in the T s / Z 2 case, the T(xS)/z2 partition function is written as the sum of four contributions, corresponding to the four possible Z 2 twists along the (1, r) directions of the world-sheet torus. Let us denote by h the odd element of Z 2 and define Zo( + , + ) = T r e u,

(2.48)

Z o ( - , + ) = T r h e H,

(2.49)

Zo( + , - ) = T r e ~/~,

(2.50)

Zo(,

(2.51)

)=Yrhe

u',,

where H --- r L o + ¢T,o can be read off eqs. (2.19, 2.20) and H h is the hamiltonian in the h-twisted sector. In terms of the above quantities, the partition function of the T ( X s ) / Z 2 model reads explicitly as

ZT(xb/z 2= ½[Zo( +, + ) + Z o ( - , + ) + Zo( + , - ) + Z o ( - , - ) ] ,

(2.52)

where Z o ( + , + ) = Zv(xb and Zv(xS ) has been given in eq. (2.9). A crucial consistency condition to be satisfied by the charge .~ - e ° Q - (elQ 1 + e z Q 2 4- . . . ) in T ( X s ) / Z 2 models is

{

h } + = o.

(2 5 3 )

This equation, which must hold true for both gauge and supersymmetry breaking, derives from the fact that h, which changes the signs of the momenta and winding numbers of the torus in the fifth dimension, must commute with [Lo]T(xb and [LolT(x').

S. Ferraraet al. / Superstrings

89

3. Models with N = 1, 2 spontaneously broken supersymmetry In the following we define some representative models in which the general conditions for spontaneous breaking of space-time supersymmetry are satisfied, and we examine explicitly their mass spectrum and flat directions. Our starting point is the construction, in the fermionic formulation, of a five-dimensional heterotic string model with N = 1 supersymmetry, defined by the following boundary condition vectors F =

S =

bl =

1. all the fermions}, ~.

1.

@l~,lp5, X2, X3, X4 X5, X6 }

0:

the remaining fermions

1:

l~., lp5, X2, W3, W4, W5 W6,(i~3, ~4 ..... ~i~17,~18~

0:

the remaining fermions

'

f

(3.1)

Notice that, for the moment, we do not need to group the fermions in complex pairs, since the boundary conditions are purely periodic or antiperiodic. The boundary condition vectors (F, S, bl) define a d = 5, N = 1 model with gauge group SO(16) x SO(26). The simple torus compactification (Ts) of the latter originates a d = 4, N = 2 model with gauge group SO(16) × SO(26) × SO(2)mat × SO(2)dil and the spectrum of massless modes given in table 1. In our notation, I/t)L ~ ~b~-l/210)t.,

(/~ = 3 , 4 ) ,

I1)L = @~1/210)L, I 0 )L, II)I.- X-1/21 Ii; c~)u - l(ll~3~4ap5x2x3x4x5x6) + )L,

(I= 2,3,4,5,6), (i = 0 , 1 , 2 , 3 ) ,

] ( b l ) ± ) L ~ [(l/J31~41/)5X2)+-)L'

[(S*bl) ++-)L~- I(X3X4X5X6) +-SL ,

( S . b I --

S U b

1 -

SO bl),

(tz= 3 , 4 ) ,

15) - X llO) , Idl6) R ~

-1/2

1/210)R '

1z:~¢'26)R = t~At/2~B1/210)R, Iy/'16"¢/'26)R ~ ~A1/2q~/~a/210)R,

(A, e

26),

(A ~ "//'16, B ~ ¢/'26),

(3.2)

90

S. Ferrara et a L / Superstrings TABLE l Massless states of the T 5 model and their masses in the T( X s) model States

spin

I~)L ® IV)R

2GO

10'1; ~)L ® [P)R

32•

~

mass 0 le I +

e2[/2R

11,2)L ® [V)R

1

0

I~)k ® I5)R I0,1; ~)L ® 15)R I1,2)L ® I5)R

1 ~ 0

0 ]el + e2l/2R 0

I ~ ) k ® ( ] ~ I 6 ) R q- 1~26)R )

1

0

10,1; a)L ® (]"~16)R q- ['g~26)R) l l , 2 ) c ® ( l ~ t 6 ) R + 1~26)R)

1

le I + e21/2R 0

0

12

12,3; O/)L® [~16Y/~26)R ]3,4,5,6)L ® IYP16~26)R

0

I(bl)*)L ® 1128+,1)R

½

[(S'bl)+)L®

lel - e21/2R let [/R, le21/R 0

1128+,1)R

0

lel+e2l/2R

q ( b t ) + ) L ® L128 ,1)R I ( S ' b l ) + ) L ® [128 ,1)R

~ 0

0 [e 1 + ezl/2R

where 10)L and [0)R are the left and right Neveu-Schwarz tachyonic vacua, the state Li; a)L (i = 0, 1, 2, 3), transforming in the 4 representation of SU(4) + SO(6), is the left Ramond vacuum, and ua:n, ~ are indices in the adjoint and vectorial representations of SO(n), respectively. Finally, I(a) +) denote the vacua, with positive or negative GSO chirality, of the Ramond sector for the fermions in the set a. Beginning from the top of table 1, one finds the states of the N = 2 gravitational supermultiplet and SO(2)dil vector supermultiplet. Then one has the S O ( 2 ) r n a t v e c t o r supermultiplet, generated by the torus compactification, and the SO(16)× SO(26) vector supermultiplets. Further down one finds the hypermultiplets of the b 1untwisted sector, and finally the hypermultiplets of the hi-twisted sector, transforming in the (128 +, 1) • (128 , 1) representation of SO(16) × SO(26). By T s / Z 2 orbifold eompactification one can obtain instead d = 4, N = 1 models with gauge groups

SO(IO) x S 0 ( 6 ) x S0(5 + 8 k ) × S 0 ( 2 1 - 8 k ) ,

(k = 0 , 1 ) .

(3.3)

Observe that also the choice k = 2 is allowed, but it gives a model completely equivalent to the one obtained for k = 0. The action on the two-dimensional fields

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S. Ferrara et al. / Superstrings

of the Z 2 projection, which defines the above orbifolds models, is q~g for -q~g for

gq)=

~b 2 q ~ b 2,

(3.4)

w h e r e ~ is the odd element of Z 2 and b 2 is the following set of fields (k = 0, 1):

b2 = { x s ; ~b/~, X3, xs, y2, y4, y6, ~3 . . . . . ~12, ~19 . . . . . ~23+8k }.

(3.5)

The choice of b 2, defining the Z 2 action, is consistent with the global existence of the world-sheet supercurrent ~,rI~ =

(3.6)

- Tv~,.

The massless spectrum of the N = 1, D = 4 model corresponding to the Ts/Z 2 orbifold compactifications (Zz-untwisted sector) is given in table 2, where, with

TABLE2 Massless states of the T s / Z 2 model (Z2-untwisted sector) and their masses in the T( X 5)/Z 2 model States

spin

mass

2~0

0

3 @ 21

]el + e 2 [ / 2 R

I1;~)L ®155~ 11,25L~ 15)Ft

12

0

lel + e 2 [ / 2 R 0

[/~)L ® (l~m)R + I~I6)R + I~%+**)R+ I~'21 8*)R) I0; a)L ® (Idlo)R + l~a6)R

1

0

51

lel + e 2 ] / 2 R

I1;@L®(I~0W6)R + I~+sk~21 8k)R) 11,25L®(F~,orq6)R+ I ~ s + 8 k ~ .DR)

~ 0

[el +e2[/2R 0

12;a)L®(I~0~+SX)R+I~2i 8k)R) 13,4)L®(I~0YS+8k)R+ 1~6~2X 8k)R)

1

[et-e2]/2R

0

]e I ] / R , [e21/R

51 0

]el - e 2 t / 2 R [el]/R, le2l/R

fl(bl)+)L ® 116,4)R 2" i ](S,bl)+SL ® 116,4)g

2 0

lel + e 2 1 / 2 R

[16,4)R [16,4)R

!2 0

0 lel + e21/2R

I/X)L ® IV)R

+ 1~¢5+sk)R + lae~l ~DR)

15,6)L ® (l~/'10~21_8k)R -]- 1~6~5+Sk)R)

2"

f I(~I)+)L ~ "+ ~ [ ( S . b l ) )L ~

0

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S. Ferrara et al. / Superstrings

respect to eq. (3.2) and in obvious notation

Id16) ~ 1~'10) + 1~'6) + 1 ~ 1 S 6 ) , 1d26) ~ 1~5+8k)+ 1~'2a s k ) + I ~ + s S 2 1 - s k ) , [3u'163u26) '~ 13UlO~5+8k) '[- 1~6~U'21-8k) "q- I~U'lO3U'21 8k) q- 1~6~5+8k~ -

(3.7)

For k = 1, the Z2-twisted sector of the model does not contain massless supermultiplets, whereas for k = 0 it contains two massless scalar supermultiplets in the (16,1, 4, 1) of SO(10) × SO(6) × SO(5) × SO(21). Beginning from the top of table 2, one can recognize the states of the N = 1 gravitational supermultiplet and of the scalar (gauge-singlet) supermultiplet containing the degrees of freedom of the dilaton and of the antisymmetric tensor, plus the other scalar (gauge-singlet) supermultiplet associated to the radius of the torus in the fifth dimension. Then one finds the vector supermultiplets of SO(10) × SO(6) × SO(5 + 8k) × SO(21 - 8k), the scalar (gauge non-singlet) supermultiplets from the N = 2 vector supermultiplets, two groups of scalar (gauge non-singlet) supermultiplets from the N = 2 hypermultiplets of the hi-untwisted sector, and finally the scalar (gauge non-singlet) supermultiplets from the N = 2 hypermultiplets of the bl-twisted sector, transforming as two copies of the (16,4) e ( 1 6 , 4 ) representation of SO(10)× SO(6), and singlets under the remaining factors of the gauge group. We now move to the definition of the T ( X 5) and T ( X S ) / Z 2 models corresponding to spontaneously broken versions of the previous ones. As a first step, we introduce the complex fermions

X(c1)= X 3-t-iX 4

X(c2)= X 5 q - i X 6 ,

~

y5 _~ iy6

y3 _.]_iy4

yc(1) ---=

V~-

,

'

yc 2) =

W(1) ~- (W3 -~- W4) + i ( w 5 + W6)

2

~-

'

W~2) ~ (W3 -- W4) -[- i ( w 5 -- W6)

'

(3.8)

2

The reason for this particular complexification will be clear in the following, The way in which the remaining fermions are complexified is irrelevant for the following considerations, provided that it is compatible with the boundary condition vectors (F, S, bx). Notice that the supercurrent T v and the whole theory are invariant under

S. Ferrara et al. / Superstrings and J 2 defined as follows:

the discrete symmetries J1

~1,~2:

93

X 2 ___+X 2

y2 __+y2,

w 2 ___+w 2,

X 3 ._+ _ X 4,

y3 ___+_ y 4 ,

w 3 ___+ W4,

X4 ...+ X 3 '

ya__+y3,

w4.___~w3'

xs_+ _eX 6,

yS _,, _ey6,

w5___+w 6,

X 6 [email protected],

y 6 __, eyS,

w 6 ~ w 5,

(3.9)

where e = + 1 for ,~1 and e = - 1 for F 2. In terms of the complex fermions of eq. (3.8), and with the following phase conventions J ' A f-i- + "

e2"ie!a'f'Ji,

(A = 1,2),

(3.10)

the previous transformations can be summarized by the charge vectors f l: e(1) ' e(2)=

¼e:

X(1), yc(1), X~2)' Yc~2)'

(3.11)

the remaining fermions. Therefore, defining the charges

Q 1 - - ~ dz {~(1)vo)+f~cO)Y(c 1)} 27ri ,,c ,,c

2rri

,~c ,,c

dz

Q'-

f rr----i 2 ( ~2)Wc~2'} '

(3.12)

the symmetry operators associated to F~ and J2 can be written as

~1, ~2 = e2~ri(e'Q' +e2Q2+e3Q') ,

(3.13)

4"

e, = e 2 = ¼,

(3.14)

J2:

et=-e2=

where e 3 = 1;

J,

e3=~-

(3.15)

Observe that ..@4 =2¢24 = 1: therefore they act as finite U(1) c S0(4) c S0(5) rotations corresponding to Z 4 symmetries.

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S. Ferrara et al. / Superstrings

The operators , ~ and ~2 satisfy all the consistency conditions required to define a T ( X 5) model with the fermion number projections defined by the sets (F, S, bl). Moreover, the ~ case corresponds to a spontaneously broken N = 2 model, whereas with ~ the N = 2 gravitini remain massless. This follows from the fact that both projections give masses to two of the four gravitini of the N = 4 model defined by the sets (F, S). In the ~2 N = 2 model, the two gravitini which would become massive are those already eliminated by the b I projection. We give in table 1 the mass spectrum of the T(X 5) model, characterized by a spontaneously broken N = 2, d = 4 local supersymmetry, limiting ourselves to the states that were massless in the unbroken version T s and for generic U(1) charges e 1 and e 2. As a final remark, we note that neither ~ nor ~2 satisfy eq. (2.53), and therefore they cannot define consistent T ( X s ) / Z 2 models. Consider now the operators J3 and 5¢44 corresponding to the charges J3"

e , = e 2= 1,

J44"

e1=-e2=½,

e3=O; e3 = 0 .

(3.16) (3.17)

They satisfy also eq. (2.53), and therefore can be used to define not only two other T ( X 5) models (whose spectrum is again given in table 1), but also consistent T ( X s ) / Z 2 models. The spectrum of these theories can be easily computed using eqs. (3.16-3.17) and the general discussion given above. In particular, we give in table 2 the masses for the states which were massless in the corresponding T s / Z 2 models. 5¢3 induces supersymmetry breaking, and can be thought of as ~ 2 (defining, by convention, e 2~iQ~,2,3 = 1 on all representations, including the spinorial ones); ~4 does not induce supersymmetry breaking in the d = 4 theory, since it gives masses to gravitini which are already projected away, and can be thought of as

4. The effective supergravity theories Here we derive the effective supergravity theories which describe the low-energy limit of the string models defined in the previous section. To begin with, let us consider the T 5 and T(X 5) models, corresponding to exact and spontaneously broken N = 2, d-- 4 local supersymmetry. Following the principle of supersymmetric integration, enunciated in the Introduction, the fields appearing in the effective theory of the T(X 5) model are all those in table 1, which are massless in the unbroken phase T 5. To fix the notation, we have the following N = 2, d = 4 supermultiplets: (i) The gravitational supermultiplet, whose physical degrees of freedom are the graviton g~,, two gravitini +~ (i = 1,2) and the graviphoton W~.

S. Ferrara et aL / Superstrings

95

(ii) The vector supermultiplets, whose physical degrees of freedom are one gauge boson A,, two gauginos )t i (i-- 1,2) and one complex spin-0 field V: their gauge indices, omitted here, are in the adjoint representation of SO(16)x SO(26)× SO(2)mat × SO(2)dil.

(iii) The hypermultiplets, whose physical degrees of freedom are two Weyl spinors ~i and two complex spin-0 fields H i (i = 1,2): their gauge indices, omitted here, are in the (16, 26) (hi-untwisted sector) and in the (128 +~ 128 , 1) (b~-twisted sector) representations of SO(16) × SO(26). We describe first the effective theory of the T 5 model. Since the gauge group is given and N = 2 supersymmetry is exact, we need only to specify the Kahler manifold for the spin-0 fields in the vector multiplets and the quaternionic manifold for the spin-0 fields in the hypermultiplets. If we neglect the fields in the hi-twisted sector, the spin-0 fields are described by the following coset manifold [26] SU(1,1)

SO(2,dim sO't6 + dim "5~'26q- 1)

u(1)

S0(2) × SO(dim all6 + dim d26 + 1)

x

SO(4, dim ;¢'16 x dim YF26) SO(4) x SO(dim Y/'16x dim ;¢'26) "

(4.1)

The first two factors correspond to the K~ihler manifold of the vector multiplets, the third one to the quaternionic manifold of the hi-untwisted hypermultiplets. In the general case in which also the hi-twisted fields are turned on, the quaternionic manifold parametrized by the hypermultiplets will be some non-symmetric, nonhomogeneous quaternionic quotient [27] of HP", as explained in ref. [28]. We move now to the derivation of the effective theory of the T(X 5) model. To determine it unambiguously, we can use all the informations coming from direct string computations, and precisely (i) The gauge group SO(16) × SO(26) × SO(2)mat X SO(2)dil is unbroken. (ii~ We know the mass spectrum, the vanishing of the cosmological constant at the classical level and the flat directions of the classical potential". Moreover, we know that the Scherk-Schwarz mechanism does not modify the four-dimensional kinetic terms, and therefore the manifold for the spin-0 fields in the effective theory remains the same as for the T5 model. The above properties lead to a unique N = 2 effective supergravity theory. In order to see this, let us consider first the limit in which all the hypermultiplets (hi-twisted and hi-untwisted ) are set to zero. In this case we know from refs. [29, 30] that a theory with minima of zero vacuum energy possesses an identically vanishing chiral potential. Since the manifold for the spin-0 fields in the vector multiplets has * These flat directions, which give the moduli for the deformations of the underlying conformal field theory, are in one-to-one correspondence with its truly marginal (1,1) operators.

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S. Ferrara et al. / Superstrings

a kiihlerian structure and we are mainly interested in the N = 1 truncated theory, we use N = 1 notation [31]. Adopting a parametrization compatible with the N = 2 tensor calculus, the most general theory is described by ~ = J + log Ig[ 2 ,

(4.2)

g = k o + k,,o.

(4.4)

where

T h e fields o and (~o, ~,,) parametrize the S U ( 1 , 1 ) / U ( 1 ) and SO(2, n + 1)/[SO(2) × SO(n + 1)] manifolds, respectively (n - dim d~16 q'- dim d26 ). The fields 0 and ~0 are associated to SO(2)mat X SO(2)dil , and the ~'" are in the adjoint representation of SO(16) × SO(26). Other "superpotentials" giving also vanishing chiral potentials would b r e a k the gauge group and are thus discarded. T h e T 5 m o d e l with unbroken s u p e r s y m m e t r y corresponds to k 0 = ko = 0. W h a t are the values of k o and ko corresponding to the T ( X 5) model? The answer to this question relies u p o n the observation that in the T ( X 5) model all the gaugino masses are equal to the gravitino mass. On the other hand, for the theory described by eq. (4.2) we have 2 mgaugi.o = m2/211 ÷ ( k o / k o ) •

12,

(4.5)

and we k n o w from string considerations that o and ~-0, containing the degrees of f r e e d o m of the gauge singlet scalars, are flat directions. This is sufficient to conclude that it must be k o = (e 1 + e 2 ) / 2 ,

ko = 0.

(4.6)

F o r the T 5 model, the effective N = 2 theory in the case in which the bl-untwisted hypermultiplets are turned on is easily derived from ref. [26]*. To determine its generalization to the T ( X 5) model, it is sufficient to write down the N = 2 gravitino m a s s matrix

IxHia m

3/2i

H

+

--

-

-

-

a=l\a

--

(4.7)

YH

where the conventions of ref. [33] have been used. In particular, (i, a) are indices m * The same results can be derived, as in ref. [32], using the sigma model approach.

S. Ferrara / Superstrings

97

SU(2) X S U ( 2 ) ' - S O ( 4 ) . The YH is a function of the hypermultiplets only, and plays for the quaternionic manifold a role analogous to that of Yv-- e - J for the K~ihler manifold. The case k 0 =t~ = 0 corresponds to the T 5 model, whereas the T( X 5) model corresponds to e 1 q- e 2

k° -

2

e 1 -- e 2

'

/~

2

(4.8)

In the T ( X 5) models considered before, since e I = ee 2 (e = _+1), we may have either supersymmetry breaking (k 0 4= 0), or a supersymmetric mass term for the hypermultiplets (/~ 4= 0), but not the two simultaneously. To conclude the analysis of the N = 2 effective theory, let us comment on the further modifications necessary to take into account the bl-twisted hypermultiplets. As explained in ref. [28], the expression (4.7) for the N = 2 gravitino mass matrix is modified by the addition of a term of the form H ; ~ 1 6 H ja~qo at the numerator [where 5°~6 and 5~6 are indices in the two inequivalent spinorial representations of SO(16)]. Moreover, in the denominator of eq. (4.7) one has to substitute YH with a different function 17H, mixing twisted and untwisted hypermultiplets and generating mass terms (proportional to the gravitino mass) for the spin-0 fields in the twisted hypermultiplets. The knowledge of the N = 2 case will be now used to determine the N = 1 effective theory of the T ( X 5 ) / Z 2 model. We use the fact that, since the untwisted sector of the T ( X 5 ) / Z 2 string model is a projection of the T ( X 5) model, the effective N = 1 theory of the former can be obtained by a Z z truncation of the corresponding N = 2 theory. The same result could be obtained, once again, using the sigma model approach. Following again the principle of supersymmetric integration, enunciated in the Introduction, the fields appearing in the effective theory of the T ( X 5 ) / Z 2 model are all those in table 2 (plus possible additional scalar supermultiplets from the Zz-twisted sector), which are massless in the unbroken phase T s / Z 2. To fix the notation, we have the following N = 1, d = 4 supermultiplets: (i) The gravitational supermultiplet, whose physical degrees of freedom are the graviton guy and the gravitino g,,. (ii) The vector supermultiplets, whose physical degrees of freedom are one gauge boson A t and one gaugino X: their gauge indices, omitted here, are in the adjoint representation of SO(10) x SO(6) x SO(5 + 8k) × SO(21 - 8k). (iii) The scalar supermultiplets, whose physical degrees of freedom are one Weyl spinor ~ and one complex spin-0 field q~. Some of them originate from the N = 2 vector supermultiplets: the two gauge singlets (S,S), (z °, Z°) and the gauge nonsinglets (z i~, S~), where ~ - {(Y/ion6); (~5+8k3(/'21 8k)}" Some others originate from the N - - 2 hypermultiplets of the bl-untwisted sector: they are all gauge non-singlets and we shall denote them by (y;~, yi~) and (y;~, y'~), where i 2 -=

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S. Ferrara et al. / Superstrings

((~/'1o~5+8k); (~6~/'21 8k)} and i 3 - {(Y/'loY/'21_sk); (Yf6~+sk)}- We also have other scalar supermultiplets originating from the hi-twisted hypermultiplets: they form two copies of the (16,4 • 4) representation of SO(10)× SO(6), are singlets under SO(5 + 8 k ) × S O ( 2 1 - 8k) and will be denoted by (x h, 2h), where, in obvious notation, 11 -- [(50~0, 506,1), (50~o, 50~-, 2), (50]o, 506,1), (50~0, 5°6,2)]. Finally, there might be some additional scalar supermultiplets from the Z2-twisted sector: apart from some side remarks, in the following we shall restrict our considerations to the case k = 1, when they are not present. We describe now the effective theory of the T ( x S ) / z 2 model. The gauge kinetic function has the universal form [34]:

f,b = 3,hS.

(4.9)

Moreover, when all the scalar fields of the hi-twisted and Z2-twisted sectors are set to zero, the N = 1 theory is described by e ~= Igl2/YoYiY2Y3,

(4.10)

where, in our parametrization: g

^

= gsusy +

•

.

A g = fl]i2i3z ily i2y i3 q- ko q- l~(yi2yi2 q- yl3y'~),

Y0 = S + S,

(4.11) (4.12)

Y1 = (z° + f o ) 2 - Y'~(z/x + 5/~ ) 2,

(4.13)

^

il

yA=l_Y'y

yiA+~ ~yiAyiA

iA

yjAy;~

(A=2,3),

(4.14)

iA

and the coefficients f~1i2i3 are proportional to the corresponding structure constants of SO(42). The connection with the N - - 2 effective theory of T(X 5) is obtained by making the following field identifications

S - io,

z °-~ i~ °,

z ' I - i f i;,

(4.15)

whilst k o and ~ are given as before by eq. (4.8). Observe that the spin-0 fields of the theory parametrize the following K~ihler manifold:

SU(l, 1) Jg

3 SO(2, x A=II--[SO(2) × SO(mA) '

(4.16)

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S. Ferrara et aL / Superstrings

where m1= m3

6 . 1 0 + (5 + 8 k ) . (21 - 8 k ) ,

m 2 = 10. (5 + Sk) + 6. (21 - 8 k ) ,

= 10. (21 - 8k) + 6. (5 + 8 k ) .

Observe also that for the SO(2, ml)/[SO(2 ) × SO(m1) ] manifold one can move to a parametrization of type (4.14) by making the following analytic field redefinitions 4 - (z°) 2 + z'~z ''

~ y'~ = a

yO = a 4+

(Z0) 2 -

z ". z q .

.

.

4iz 6 (4.17) . ( z ° ). 2 4 -]-

zllz

ll

with a = (1/v~-)(1 +yOyO +yT~y;~),

(4.18)

under which the K~ihler function ~susy - log(I&ussl2/YoY1Yff3) remains invariant. The effective T s / Z 2 theory corresponds to k0 =/* = 0: following the lines of ref. [35], one can derive its classical potential, show that it is manifestly positive semidefinite and study the possible fiat directions. We move now to the effective T ( X s ) / Z 2 theories and comment first about the case e t = - e 2, corresponding to k 0 = 0, /2 4= 0 and unbroken supersymmetry. The potential in this case is obviously positive semidefinite, since it corresponds to a truncation of the potential of a N = 2 theory in which the compensating hypermultiplets are inert under the gauge group. Since, due to the charge quantization discussed before, /~ is of order Mp, we can integrate out the massive scalar supermultiplets ( y % yi2) and (y% y,3), obtaining thus an effective theory based on the manifold SU(1,1) SO(2, m,) Me'= U ( I ~ × S O ( 2 ) × SO(mr) " (4.19) In the absence of b,-twisted scalars, the potential of this effective theory is given only by the D-term contribution, as one could have guessed from the N = 2 theory. We now concentrate on the T ( X s ) / Z 2 model where supersymmetry is broken, i.e. e I = e 2 4= 0 and therefore k 0 e 0, /~ = 0. In this case we have computed the explicit form of the N = 1 potential: v =

+ vD,

(4.20)

where VF

roYIY2Y_

Igl2+ ~-~!-~Fg~-12+ 15;,g;, - k o 1 2 +

+

Pi2gi*

ko +~.Piffi:

2

Y2(Ig~2l2 - Ikol z)

+Y3(Igi,[Z-[kol2)

ko + .Pi,g i ~ + ~ P i , V 6

1 -5-'Ta'~'z ;~ Yi2 - T ,~i~j2 y i~ Yi3Ta'3/3 y'3] 2] . il Jl + +

% = Too

r,

2

, (4.21)

(4.22)

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At the minimum

z,'l =yi2 = y 6 = O,

z °, S arbitrary,

(4.23)

corresponding to the T ( X s ) / Z 2 model considered here, the spectrum computed in the effective theory coincides with the one computed at the string level, provided that we identify the compactification radius with

R = (S-l-S) 1/2 (z° q- Zo)2 - ~ (z/"l q- Z[l)211/2 . il

(4.24)

Let us now spell out some properties of the effective potential (4.20), which can also be proven directly at the string level: (i) In the case under exam, the minimization of the potential uniquely determines yi2 =yi3 = 0, in contrast with the Ts/Z 2 model (k0 = bt = 0), where there are some flat directions in the (y~2, yi3) configuration space. (ii) The flat directions for the z '1 fields are then all those which correspond to vanishing D-terms. With the inclusion of the z ° and S fields, this is exactly the residual moduli space of the corresponding string theory. (iii) It is a matter of simple algebra to check that, for any z ;1 configuration along the flat directions, there are no tachyonic states. Needless to say, all these minima of the potential have zero cosmological constant. To complete the discussion of the T ( X s ) / Z 2 models, one has to allow for non-zero values of the bl-twisted scalar multiplets. In this case, following again ref. [28], the N = 1 effective theory can be obtained by Z 2 projection of the corresponding N = 2 theory. The manifold for the spin-0 fields in the scalar supermultiplets becomes SV(1,1) SO(Z, ml) J/= U ( I ~ X SO(Z)× SO(m1) × J / { "

(4.25)

where d t ' is a non-symmetric, non-homogeneous K~ihler space, mixing non-trivially the hi-twisted and bl-untwisted sectors. In our parametrization, this would correspond to an additional factor

[]2 x 6Y6

YTs = 1

2(}12}13)1/2

(4.26)

at the denominator of eq. (4.10), and an additional contribution to the superpotential

AgTS =

t;llljCZ,'qxllx Jl .

(4.27)

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101

The classical potential (4.20) undergoes the following modification

V~

- Ig[ 2 + (Y1/2)lg;x[ 2 +

YOIqY##TS

I'~z,g,"l-

kol 2 + V1/2V1/2V1/21'n" 12 *2 *3 *TS 1,51t

q- y2y1/s2(Igi212- Ikol 2) q- Y1/s2[Yi2gi2q- (ko/2))5,2y,212 + Y3Y~/s2(lg,,I 2 - Ikol a) 1

-

+

Y1/s21~i3g,, q- (ko/2)~,3yi, I2, (4.28)

z,;1Taqjqz dl Yi2Ta'2j2yy2 y, T"'~,y" XllTahy, yJt + _ : ~ + . . . +. J]2. Y1 ]/2 Y3 ]/1/2171/2

(4.29)

At the minimum (4.23), supplemented now by the condition x 6 = 0, one can repeat all the previous considerations, with the additional result that the x tl scalars all get masses equal to the gravitino mass, whereas their fermionic partners y6 remain massless: this coincides with the results of table 2, obtained from direct string computations. From the structure of the potential (4.28, 4.29) one can also see that there are no additional flat directions along the spin-0 fields in the hi-twisted sector. Before concluding this section, some other comments are in order. Given the fact that in these models all the supersymmetry breaking masses are O ( M p ) , o n e may further integrate out the massive degrees of freedom. The residual effective theory will be explicitly non-supersymmetric, yet its form will still be dictated by the underlying supersymmetric model. In fact, its form will correspond to a consistent truncation of a spontaneously broken N = 1 theory. To be specific, it will contain the massless scalars of the manifold of eq. (4.19) (without their fermionic superpartners), and the massless fermions ()7 i2, y3) (without their bosonic superpartners). The possible Z2-twisted states will appear in massless boson-fermion pairs, as in the case of a supersymmetric theory, whereas the bl-twisted states will provide only fermions to the effective theory. We stress again that such a peculiar structure would look unnatural if we did not know its supersymmetric origin. A potential with a D-term structure has been also found for the untwisted sector of some non-supersymmetric four-dimensional superstring models. This suggests that some of these models might correspond to spontaneously broken versions of some supersymmetric string theory.

5. Conclusions and open problems In this paper we gave the general rules to construct four-dimensional superstring models with spontaneous breaking of N = 1, 2 or 4 space time supersymmetry, via coordinate-dependent torus or orbifold compactifications from five to four dimen-

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sions. We also interpreted the gauge and supersymmetry breaking mechanisms in terms of continuous and discrete deformations of the lorentzian charge lattice. We considered in particular a representative N = 1 model with chiral fermions, deriving its effective N = 1 supergravity theory from the effective theory of the parent N = 2 model, and examining some properties of its classical potential. We have studied in detail two representative T ( x S ) / z 2 string models, in which the charge operators 9. = e l Q a + e2Q 2 (e I = _+e2) , made out of left-moving fermions only, leave the gauge group unbroken. However, it would be straightforward to generalize the above constructions to cases in which the charge operators ~ contain also right-handed fermions with non-trivial gauge quantum numbers. Consistent T ( X 5 ) / Z 2 string solutions with broken gauge symmetry would correspond to the s a m e effective theory, but to a d i f f e r e n t vacuum field configuration in which some of the spin-0 fields z i. acquire non-vanishing VEVs along their flat directions. A general feature of the supersymmetry-breaking mechanism presented in this paper is the fact that the gauginos of the unbroken four-dimensional group are given supersymmetry-breaking masses identical to the gravitino mass. In the effective N = 1 supergravity theory of the T ( X s ) / Z 2 model considered above, with the gauge kinetic function given by eq. (4.9), this is a consequence of the fact that, at the minima of the potential (4.20), the Goldstino has a non-vanishing projection along the direction of the "dilatino" S,

(Ns)-

=

~

* O.

(5.1)

Using the standard N = 1 supergravity formalism [31] and the N function of eq. (4.9), one obtains in fact mgaugin~° =

m3/2

'~S'~S?

= 1.

(5.2)

S+ S

Notice that, since the above equation holds for any value of S, only an S-dependent superpotential modification could decouple gaugino and gravitino masses. At the perturbative level, S-dependent superpotential modifications have been considered in refs. [35,36] and shown to lead to many interesting properties, but no string model originating them in the effective theory has yet been constructed. Extending our consideration to possible non-perturbative phenomena, S-dependent superpotential modifications could describe a (conjectured) supersymmetry breaking via gaugino [37, 25] or gravitino [38] condensation. However, we should stress again that these scenarios have only been suggested in the framework of the effective field theory, and have not yet been implemented in any consistent string model. Moreover, gaugino condensation alone is not enough to induce a scalar potential with supersymmetry breaking minima of zero energy. Additional modifications that have

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103

been considered to fix this problem, like a VEV for the antisymmetric t e n s o r HMN P in the internal space, have also to face [39] the quantization-decompactification issue. For example, in the notation of the second paper of ref. [25], the gravitino mass is given by Ic + h e -3s/2h°] = m3/2 (S + S ) ' / 2 ( T + ~)3/2 ,

(5.31

where ~HMNP) = CEMNP,with c quantized and O(1). In eq. (5.3), b 0 is the one-loop coefficient for the/~-function of the gauge group whose gauginos condense, and the denominator plays the role of the compactification radius for a Calabi Yau manifold. However, in this case a large gravitino mass might coexist with small supersymmetry-breaking splittings within the supermultiplets of the observable sector, since hidden and observable sectors are decoupled at tree level, and no scalar a n d / o r gaugino masses are generated for the observable supermultiplets. Of course, one cannot exclude that some string miracles could solve the hierarchy problem even in the presence of large supersymmetry breaking masses in the observable sector. The link between gaugino and gravitino masses can be seen directly in the string formalism, remembering that the corresponding states are obtained from

(I~>L + I0; ~>L) ¢~ (1~>~ + IdG>R),

(5.4)

where ..~'~ are indices in the adjoint representation of the unbroken gauge group G and the remaining symbols have been defined in sect. 3. Since in the models under consideration the charge ~ which breaks supersymmetry acts trivially on the two-dimensional left- and right-movers with d = 4 Lorentz indices ~t, ~, it must also act trivially on the right-moving fields with indices in the unbroken gauge group G. Therefore, to give the gravitino a mass, ~ must act non-trivially on 10; c~)L, and this unavoidably induces an identical mass for the gauginos of the unbroken gauge group. A general problem of the superstring models considered in this paper (which also plagues all the non-supersymmetric four-dimensional string models constructed until now), is the stability of the flat background with respect to the string loop expansion. Barring unforeseen cancellations, one would expect the generation, 2. already at the genus one level, of a cosmological constant term of order m 32 / z M p. this is one of the crucial problems of superstring models and we do not have anything new to say in this respect. This problem affects even more severely all models with U(1) D-term supersymmetry breaking [40]: there, indeed, the supersymmetry breaking scale is O(Mp) and a huge positive cosmological constant O ( M 4) is unavoidably generated at higher genus, so that the vacuum of the theory evolves to a supersymmetric configuration of zero energy.

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E v e n if the m o d e l s c o n s i d e r e d in this p a p e r are n o t c o m p l e t e l y realistic a n d several u n s o l v e d p r o b l e m s r e m a i n , in o u r w o r k we have given the first e x a m p l e of a n o n - s u p e r s y m m e t r i c s t r i n g s o l u t i o n (with f e r m i o n s in chiral SO(10) r e p r e s e n t a t i o n s ) , w h e r e e x t e n d e d s u p e r s y m m e t r y a r g u m e n t s allow to keep the effective t h e o r y u n d e r c o m p l e t e c o n t r o l , t h a n k s to the p r i n c i p l e to s u p e r s y m m e t r i c i n t e g r a t i o n of the m a s s i v e states. T h e a u t h o r s w o u l d like to t h a n k the U C L A Physics D e p a r t m e n t a n d the L a b o r a toire d e P h y s i q u e T h e o r i q u e , Ecole N o r m a l e Sup6rieure, Paris, for the w a r m h o s p i t a l i t y o f f e r e d to s o m e of t h e m at d i f f e r e n t stages of this work.

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