Nuclear Physics B326 (1989) 497-543 North-Holland, Amsterdam
A UNIFIED FORMALISM FOR STRINGS IN FOUR DIMENSIONS A H CHAMSEDDINE* **
hlstztut fur Theoretlsche Physlk, Unwersltat Zurich, S6honberggasse 9, CH-8001 Zurich, Switzerland
J-P DERENDINGER Instltut fur Theoretlsche Phystk, ETH-Honggerberg, CH-8093 Zurich, Swttzerland
M QUIROS * tt lnstttuto de Estru~tura de la Materla, Serrano 119, E-28006 Madrid, Spain Received 22 May 1989
We develop a simple and umfied formahsm to construct four-dimensional stnng models consastent with modular mvanance The boundary conditions are allowed to be twisted for fermaons whale bosons are e~ther twisted or shifted We show that the generahzed GSO projection con&tlons follow from our solution to the modular lnvaraance of the total partition function The Green-Schwarz strings and the Neveu-Schwarz-Ramond strings are treated m a umfied way We give the conditions needed to preserve world-sheet supersymmetry and to characterize N = 1,2,4 space-time supersymmetry Our formahsm is general enough to contain the ferm~onlc, latuce, and orbffold constructions of four-dlmens~onal smngs and we prowde ats connection to some of them
1 Introduction In the absence of a dynamical
determination
of the true vacuum
of heterotlc
s t r i n g s [1], w e a r e f a c i n g t h e p r o b l e m o f t h e c l a s s i f i c a t i o n o f all p o s s i b l e v a c u a T h i s is a n
important
problem
if w e e x p e c t o n e
of these vacua
t o give a r e a l i s t i c
d e s c r i p t i o n o f all f u n d a m e n t a l i n t e r a c t i o n s
Large classes of vacua with four-dimen-
sional
constructed
space-ttme
symmetries
have been
using various
approaches
[2-17], but no general classification has been estabhshed yet To
describe
a strmg
theory (or vacuum)
constructs its partition function on the torus
m
four
one generally
Consistency requires modular lnvan-
* Computer mall chams@czheth5a ** Supported by the Swass National Foundation t Computer mall lmtma27@emdcslcl t t Work partly supported by CICYT under contract AE-88-0040-01 0550-3213/89/$03 50©Elsevler Scxence Pubhshers B V (North-Holland Physics Publishing Division)
dimensions,
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A H Chamseddme et a l /
Strmgs m four dtrnenstons
ance of this p a r n n o n function In addition, all consistency condlnons arising from higher order dmgrams can be expressed as constraints on the one-loop partlnon function, due to factorizanon of higher order amphtudes [18,19] The p a r n n o n function on the torus contains the m f o r m a n o n allowing us to describe the spectrum of the theory, with the space-rime and gauge quantum numbers of all states There are, however, ambiguities in the lnterpretanon of a partition funcnon in terms of closed string coordinates, which are two-dimensional (world-sheet) fields, w~th given boundary condmons Firstly, a world-sheet fernuon can always be bosonized The partmon function constructed from ferrmons with twisted boundary condlnons can always be obtained from bosons w~th shifted boundary condlnons The situation ~s much more subtle when twisted boundary condlnons are also allowed for bosons A fermxomzation of Zz-tWiStS (antIperiodlc bosons) exists, but it does not seem to be the case for general ZN, N > 2, twists These b o s o n - f e r m l o n equivalences have g~ven rise to various formahsms for the construcnon of string vacua, either purely bosomc [5,10], or purely ferrmonic [6-9], or rmxed [3,4,11-16] They describe large classes of partition functions, with considerable overlaps, and in some cases they are equivalent The ambiguity becomes important when specific propernes of a class of theories, like space-time supersymmetries, gauge groups and the massless spectra are analyzed in terms of boundary c o n d m o n s of the string fields, or when the relation with compacnfications of ten-d~mens~onal heteronc strings xs lnvesngated As an example of ambiguity, (symmetric) orbifold compacnflcanons of heterotlc strings [3, 4] lead to a lowering of s p a c e - r i m e supersymmetry from N = 4 to N = 1 through twisted boundary cond~nons of both bosons and fermlons The rank of the gauge group can be lowered below 22, for the same reason In fermaomc formalisms, both the rank of the gauge group and the order of supersymmetry can be reduced, even though no twisted bosons are a priori allowed In contrast, m the covariant lattice formahsm [10], the order of supersymmetry can be reduced, but the rank is always 22 Again no twisted boundary condmons are allowed for bosons Thus, interpreting the mechanism for reducing the rank and the order of supersymmetry m terms of b o u n d a r y condlnons requires resolving such amb~gmtles Also, s p a c e - t i m e fermlons can be obtained using either N e v e u - S c h w a r z - R a m o n d (NSR) or Green-Schwarz (GS) fermlons [20] Tins is a consequence of the fact that two parntlon functions expressed by different combinations of theta funcnons can sometimes be ~denncal because of non-trivial relanons between theta functions (Rlemann ~dennnes) In such cases, the two parntion funcnons have the same physical content, a fact which cannot be recognized from arguments related to the two-damens~onal structure of the string theory A formahsm for constructing modular lnvarlant partition funcnons is in general appropriate for both N S R and GS fermlons To help resolve such ambiguities and to understand the relanons between various formahsms, a construction describing a much more general class of string vacua in a
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simple way is necessary This is the problem we address in tins article We will derive a formahsm which certainly contains purely ferrmonlc and bosomc constructions, but also includes twisted bosons allowing the description of very large classes of orblfolds This pomt is Important since orbifolds offer the best prospects for a realistic theory [12], but an explicit construction of the full partition function for all twisted and untwisted sectors has not been given In the present article we try to reach three goals Firstly we will derwe a f o r m a h s m which is very general, includes orblfolds, but remains simple The partition function will contain a rmnlmal set of truly free parameters, analogous to the discrete torsions introduced by Vafa [18] Then we want a formalism winch describes m a umfied way N S R as well as GS stnngs, the relation being as explicit as possible Finally, we will study the relation of this new formalism with the construction presented in refs [14,15], which uses the same class of boundary conditions This previous formalism is in fact equivalent, but much more comphcated Also, GSO projections were simply postulated in refs [14,15] We show here that they are direct consequences of higher order modular mvanance In sect 2, the modular lnvarlant partition function corresponding to general abehan boundary con&tions is constructed Clearly, only constraints coming from world-sheet consistency are necessary to the derivation The partlhon function apphes then directly to both N S R and GS space-time interpretations The solution obtained m sect 2 shows an interesting discrete symmetry structure related to GSO projectors this will be the subject of sect 3 In sect 4, N S R and GS fermions are discussed They first differ by the constraints imposed by world-sheet supersymmetry on the boundary conditions They also correspond to a different set of mlmmal boundary conditions Also, the problem of translating a given model from N S R to GS f o r m a h s m (or opposite) is investigated Space-time supersymmetries are the subject of sect 5 In sect 6, we present a short proof of the equivalence of the present formahsm with that of ref [14], winch is, however, much more complicated Sect 7 contains our conclusions Three appendices give some useful formulas, a construction analogous to that m sect 2 for a fermlonic formalism, leading to the solution found by Kawax, Lewellen and Tye [7], and, as an example, a discussion of the simplest Z 3 orblfold [3]
2. Solving modular invanance Consistency of the string theory requires all order modular mvarlance of the string partition function In the factorized limit, modular invariance expresses conditions on the one-loop partition function only which IS then the basic object when constructing consistent stnng vacua The one-loop partition function is a sum of terms charactenzed by the boundary conditions of all fields of the heterotlc string along the two non-contractible loops on the torus, parametrlzed by o I and 0 2 In our construction, fermions will have
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twisted b o u n d a r y conditions and internal bosons will be twisted a n d / o r shifted T h e only restriction on the general b o u n d a r y conditions is that we use abehan twists there is a basis of the fields where all possible twists can be written as a diagonal matrix The total p a r t m o n functmn can be expressed in terms of the partition function of a c o m p l e x fermlon on the torus [21, 7] with b o u n d a r y conditions qt(0"1 + ~, 0"2) = e2"~° gt{0" \ 1, 0"2)'~ ~[t(0.1, 0"2 -t- '/7") = e 2''~°' g ' ( 0 " i ,
(2 1)
0"2)
F o r convenience, we take 0 ~< 0, 0' < 1 Defining the usual n u m b e r operator and the hamlltonlan by =
n>~l =
\ Xn+O-lxn+O-1-- qXt~n_oalln_O),
I2[(. - -
--
tt
!
n>l
+ ½(02-0+
~),
(22)
in terms of the oscillator modes, the partition function is obtmned from 8 ( 0 , O')Tr( q ~ ( ° ) e - 2"~°'~(°)),
(2 3)
where q = e 2''r~, r is the Telchmuller parameter deflmng the torus and the trace is taken over physical states A conventional phase, to be discussed below is depmted b y 8(0, 0') The trace is given b y *
~o°':q '°2-°+1/6)/2 I--I (1-q'-°e2'"°')(1-q "+°-te-2'~°')
(2 4)
n~>l
T h e minus signs are due to the trace over fern~onlc, antlcommutlng, creation operators This function is clearly periodic in 0', but ..~00,+1 = e2~(O ,
1 / 2 ) ~,,O
* In terms of theta functions, one has o~~ = e21~r(~ 1/2)(~'- 1/2) t?
where */(r)
=
1 ( )
ql/24N,,/> 1(1 -- q") XSthe Dedelond function
(2 5)
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The disadvantage of ~0°, is that it receives comphcated phases under modular transformatmns "1"--+ r + 1
~ 0 __+ e,~(oz-o+l/6)~t,O_o ,
,r ---> - 1 / r
oq,oe, ._+ e2,~(o
1/2)(0'--1/2)~_0;
(2 6)
These phases will consxderably complicate the construction of the complete one-loop partmon functmn, even though tins analysis can be completely performed [7,14] A more elegant optmn xs to choose a non-trivial phase 8(0, 0') m expression (2 3) m such a way that modular transformations never generate phases depending on 0 and 0' Up to a constant phase, the convenient chome as [18]
d(0', 0, ~) = e - ' ~ ° ' ( ° - l > ~ ,
(2 7)
for the fermion partmon function One has then the modular transformatmns -~ • + 1
a ( o ' , o, ~) -+ 8 a ( o ' - o, o, ~),
~-~ - 1 / ~
a(o',o, ~)-+a'a(-o,o', ~),
(2 8)
with a constant phase 8 = e '~/6 only The function d(O', 0, 0-) is not perlodm d ( O ' + 1 , 0 , r ) = e '~(°+a) d ( O ' , O , r ) , d ( O ' , 0 + 1, r ) = e '=(°'+I) d ( O ' , O, r )
(2 9)
Even though the boundary conditions (2 1) are identical for 0 (0') and 0 + 1 (0' + 1), tins is not the case for a single fern-non partition function Perlodamty is spoiled by phases When all fields are taken into account, their boundary conditions must be arranged m such a way that the phases reqmred by modular invaraance m the partition function respect the periodicity propertms of boundary condmons (2 1) Tins necessary reqmrement will lead to the level-matching conditions [4,18] It as clear that the boundary condmons (2 1) can be regarded as the action of a discrete group Zu, where N is the smallest integer such that NO = 0 (modulo 1) Tins interpretation turns out to be extremely useful and suggestive to understand the physical content of the partition function For clarity of the present discussion, however, we postpone tins topic to the next section The partmon function of internal bosons can be expressed m terms of functions d(O', O, r ) We must, however, distinguish between shafted and twisted bosons Twisted bosons reqmre more care A free, left-moving, complex boson with bound-
502
A H Chamseddme et al / Stnngs m four dlmenstons
ary conditions Z ( ,r + 0" + ,rr ) = e 2'~q' Z ( ,r + 0" ) + A
(210)
(@ 4~ 0) has a mode expansion given by l
Z('r + 0") = Z o + ~ E
1
_ a . _ ~ , e - 2,(.-~,)(~+o)
(2 11)
n E Z n --
The "center of mass" coordinate is Z 0 = A(1 -- e 2'~q°) - 1
(212)
which indicates that the twisted boson is attached to a fixed point of the orblfold obtained when dividing the torus defined by A, by the discrete group generated by exp(2t~r~) [3, 4] These boundary conditions only make sense if they can be iterated, i e if one can define group elements g = (q~, A) with (213)
g Z = eZ'~q'Z q - A ,
and the group law glg2=g3=(,l+~2,e2'~lA2-k-A1)
=(~p3, A3)
(2 14)
For consistency, eE'~*lA2 + A 1 should be a point on the lattice defining the torus, 1 e exp(217r+l) should generate a point group of the lattice this is well-known from orblfold compactlficatlons [3,4] This condition is a very strong hmltation on the admissible twisted boundary conditions one can impose to bosons on a lattice Consider now a twisted boson on the torus Its boundary conditions are speofied by two group elements g = (q~, A) and h = (¢', A'), along the two basic non-contractible loops Z ( o I q- "It, 0"2) = e2'~r~ Z(0.1, 0"2) q - A ,
Z ( O l , 0.2--F ,B') = e2WN~'g(ol, 02) + A t
(2 15)
One obviously needs the commutation condition [h, g] = (0, A"), leading to e2'~q"A -- e2*~'t'A ' = A -- A' + A " ,
which follows directly from the condition that the twists define a point group of the lattice, or from the fixed point equation (2 12) The partition function of a twisted boson (0 ~ 0 a n d / o r qs'~ 0) is given by n(¢, ~') d(qs', ~, ~-) 1, where
n(q~,~')=(1-e-2'~*')3~.o+(1-e-2'~*)(1-3~o)
(216)
A H Chamseddme et al / Strings m four &menstons
503
is related to the fixed points Compared with a fermlon with same twists, the normal ordering constant is opposite, and the contribution of oscillators to the trace is inverted due to quantization by commutators instead of anticommutators Notice that the partition function does not depend explicitly on the shifts A and A', which only appear in the "center of mass" coordinates (2 12) With shifted boundary conditions only, the mode expansion contains discrete m o m e n t a P = a + n, n ~ 7] and the p a r t m o n function for a real, shifted boson ~s identical to the one of a twisted fermlon, by b o s o n - f e r m l o n eqmvalence More precisely, if we define
A=(Vx-~)+t(Vv-~),
(217)
and the analog for A', we can write the partition function of a complex boson in the following way [14] with boundary conditions specified by two triplets (9, Vx, V~) and (q¢, V,', V/), equivalent to the group elements g and h, the partition function reads
+Po+d(V;,Vx,
B (+ (+'v; ",) <') = (i -&+,).(+,+')d(+',+,
(2 18) introducing the projector P++,=6+060,0
(2 19)
Using eq (2 8), its modular transformations are easily obtained (+ v~'vv) - ~ e ( P
]B(+,v~ v,)
r--+r+ l
B(+, v-,v,,)
r---, - 1 / r
B(+,Vx,V~) _..) I n k3n(~b' Vx',V~') (+' V',V;) e~r++,) D (_+ _ vx,- v,) '
++'*
(+,-+
v-
v~,v; v,), (2 20)
with a phase given by (2 21) It is interesting to notice that modular transformations never relate the partition function of a twisted boson (with ~b or ep' non-zero), which does not depend on shifts A and A', with the partition function of an untwisted boson with non-trivial shifts This is consistent with the group law (2 14) Choosing gl = (4~, AI) and g2 = (-e~, A2) leads to
glg2 = (0, e2t~q)A2 -]-AI) = (0,0),
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A H Chamseddme et al / Stnngs m four dtmenslons
wxth the help of the fixed point equation (212) Z0=AI(1--eZz~r¢) - I = A2(1 _ e-2,~,) 1 We are now ready to assemble the contributions of the various fields to the complete partition function for given boundary condmons The boundary condmons on the torus of the set of internal fields of the heterotlc string in four dimensions can be collected in two "vectors" of the form*
m = (Oa,(•k, Vff,
¢, V?)),
W'= (0;, ( *',, U;, U )I( *;, U;, UY))
(2 22)
W e write before the bar the b o u n d a r y conditions of the right-movers 0a a n d 0,', a = 1, ,4 c o r r e s p o n d to the four complex world-sheet fermmns, a n d (q~k, Vff, VkV), k = 1, 2, 3 are related to the three right-moving complex bosons The index I = 1, ,11 refers to the 11 complex bosons of the left-moving b o s o n l c string W e disregard the s p a c e - t i m e coordinates X. whose p a r t i t i o n f u n c t i o n is b y itself m o d u l a r l n v a r l a n t R e m e m b e r that we always consider a b e h a n twists, whtch can be daagonahzed a n d written in this vectorial form The p a r t m o n f u n c t i o n for the b o u n d a r y c o n d m o n s W a n d W' is then given b y * *
D~,=
4
3
a=l
k=l
(11
t
1-[ d(O'~,Oa,'~) l-I a ~** ~;'~) * (44'U;'U{) ]
]-[ B ( * ' ' v L v ? )
)(
.it at 1=1
~b ('t,U]',U]')
where the p a r t m o n function for complex bosons is given by eq expressions (2 8) a n d (2 20), the basic m o d u l a r t r a n s f o r m a t i o n s are
r~'r+l + - 1/~"
)
(2 18)
(223)
Using
DW, ~ a D wW, _ w , D w, -~ a3DW~,
(2 24)
with a phase given b y
a = exp 21~r
~ P,,,,)I=1
P***;,
,
(2 25)
1
where P**, is given by eq (219) These transformations are then free of phase only * From now on, the components of W and W' are not reqmred to be an [0,1[ Thxsis only convement and useful for exphmt expansmns lake eq (2 2) or eq (2 4) All other propemes of the parutlon funcuons hold for arbitrary arguments ** There are however latUce partmon funcuons Z1 w~ch cannot be expressed as combinations of d functions This is m particular the case when left-nght symmetric lattices F with arbitrary radius appear The untwxsted partmon funchon should then be modified to include Z r instead of the d's This does not affect our dascussmnof modular mvanance both have the same transformations
A H Chamseddlne et al / Strmgs m four dimensions
505
when the net (1 e left minus right) number of twisted complex bosons is zero modulo 4 11
3
E P~,,4,;I=l
E P~,,0~= 0
(mod4)
(2 26)
k~l
Since the absence of phase is necessary for modular lnvariance, eq (2 26) is the first condition to be imposed on all possible boundary conditions It is clear, see eq (2 24) that the full one-loop modular invarlant partition function must be a combination, with appropriate coefficients, of the contributions of all possible boundary conditions, forming a set { W} closed under modular transformations
w,w,c{w}
( w'W)Dww,
(2 27)
The coefficients e( w) will be constrained by one-loop modular lnvarlance (two conditions, from the two generators of the modular group), and by a third condition obtained from modular lnvarlance of higher loops From the transformations (2 24), imposing a = 1, the two (one-loop) conditions are (2 28) The third condition [19], related to modular lnvarlance of higher loop contributions to the vacuum amplitude, can be translated into a condition on the one-loop amplitude, provided factorization of higher loop diagrams is assumed It is then enough to consider the two-loop vacuum amplitude, which factorizes (up to propagator terms) in two one-loop amplitudes One then obtains a condition quadratic in w)' ' which also factorize This condition has been explicitly the coefficients e w discussed in various contexts, for ten dimensional strings [19,22], orblfolds [18], fermionlc four-dimensional theories [8, 23, 9] and in the covanant lattice formalism [24] In our context, the discussion of Vafa [18] applies directly The last condition, from multlloop modular lnvarlance, is then* (2 29)
* I n s t e a d of o u r W ' s , Vafa [18] uses group elements g His definition is g = exp [2,~r d l a g ( W )1 as a m a t r i x a c t i n g on the string fields
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A H Chamseddme et al / Strings m four dtmenstons
The condlnons (2 28) and (2 29) can be combined into the simpler set of equations
~(w~
+13
WI
)
w~ '
(2 30)
which we need to solve to determine our modular lnvarlant partition function Before solving eqs (2 30)-(2 32), we need to charactenze in a concrete and possibly convenient way the set of boundary conditions ( W } in eq (2 27) This set will determine the physical content of the theory, and the free parameters remaining in the coefficients e ( ww') , after Imposing the constraints (2 30)-(2 32) It is clear from eqs (2 28)-(2 32) that if two vectors W1 and W2 are mcluded in the set, then one should also include W1 + W2, - W 1 , - W2, All linear combinations with integer coefficients are then present in the set It then makes sense to define the full set of vectors { W } m terms of a basis It will be convenient for later discussion to split the basis in two parts, the shift vectors and the twists A shift vector, by definition, contains no twist at all It is of the form (04,(0, V~, V~)[(0, Vi ~, V1V)), see eq (2 22), where 04-- (0,0,0,0) indicates that the four world-sheet fermlons are periodic For simphclty, it will be denoted by V = (Vkl Vt),
(2 33)
where now k = 1, ,6 and I = 1, ,22 label the real bosons The basis will contain some shifts Vp, and also some twist vectors W,~, of the form (2 22), containing both stufts and twists We will also use the following decomposmon of W m
W,,, = R,,, • Vm,
(2 34)
where R., ( " r o t a n o n vector") contains all twist parameters and Vm all shifts The set of boundary conditions in eq (2 27) IS then defined by a basis
{wm, and a vector W is a hnear combination
W = E a mWm + E CpWp, m
p
(2 35)
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A H Chamseddme et al / Strmgs m four dtmenstons
with integer coefficients am and Cp The total partition function is then rewritten as
A, B
B DBw '
where A = ( a . , , cp } and B = { bin, dp ) denote the sets of coefficients necessary to define a boundary con&tlon, as in eq (2 35)
A w = E.
+ E .Vp,
m
p (2 37)
BW = EbnWn + EdqVq n q
It is now strmghtforward to solve the three conditions (2 30)-(2 32)* The phase e ( A ) = exp(2t~r ~ AMBueMN), "
(2 38)
M,N
where M = m, p ( N = n, q), satisfies conditions (2 30)-(2 32), provided the matrix of arbitrary coeffloents emN IS antisymmetrlc
(2 39)
EMN = -- ENM
In addition, we must perform the summation in eq (2 36) over all lnequlvalent boundary conditions only We consider in our basic set only vectors Wm and Vp with rational entries There is then only a finite number of lnequivalent boundary conditions (2 35) For the basis ( W m, Vp ), define { N m, Np } as the set of smallest Integers for which NmW,n and NpVp (no summation on rn and p) have only integer components Then, for instance, the two vectors W and W + N.,Wm correspond to the same boundary conditions (This statement is obvious for twists, see eq (2 1) For shifted bosons, recall that a shaft V leads to discrete momenta P = V - ~ + n, n Z, so that V ~s really defined only modulo 1 ) In general, a boundary vector W is eqmvalent to
W ' = W + ~ k , . N m W m + ~kpNpVp, m
(2 40)
p
with arbitrary integer coefficients k,. and kp, and should then contribute to the partition function by a single term Translated on the phases e(A], this is the \o]
* This solution was suggested by Vafa [18] See also ref [13]
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A H Chamseddme et al / Strings m four dimensions
condition
At e ( B , ) = e(AB), where A' = { a., + k.~Nm, Cp + kpNp ) and leads to
(2 41)
B' = {bm + k~Nm, dp + kpNp}, which
NMeMN = NNe~N = 0
(mod 1)
(2 42)
A p a r t from the trivial solution EMN 0, eq (2 42) admits non-trivial values when N M and N N are not copnmes In this case, ff S is the largest common dlwsor to N~4 and N N, one has =
eMN=US,
l=0,1,
,S-1
(2 43)
This solution introduces a free phase exp(2zlreMN ) in the modular mvariant partition function, which ~s an Sth root of one and is a free parameter In such cases, a set of basic vectors (Win, Vp} leads to several, in general lnequlvalent (but not always), modular mvarlant theories The condition that a vector W leads to the same physical situation as W', as given in eq (2 40), has been discussed at the level of the phases e ( ~ ) It is, however, a \
/
non-trivial condition also on the partition functions DBw, AW which are not periodic One must impose that the parutton functions for W and W' are also ~dentlcal This ts certainly necessary, since when starting in eq (2 36) by a set of lnequlvalent boundary con&tlons, modular transformations will not respect this set Start for instance with D w i~', with Wm a basis vector and W arbitrary Under • ~ r - N m, one W, obtains D~v"+N~w., which ~s not in our starting set of boundary conditions since W _ w~ and W + NmWm are equivalent We must then require that D wmDw+u~w~ This equahty wdl imply a strong restriction on the vector Him, belonging to the basis T o discuss these necessary condlUons in full generality, we proceed to write exphcatly the total p a r t m o n function Introducing some new notation will prove AW where useful We need a more exphclt expression for the partition function DBw, A W and B W are defined in eq (2 37), and in particular for the phase factor arising in the definluon of d(~, 0, ~-), eq (2 7) We introduce now a projection matrix ~ab, defined in terms of the projectors for individual twisted bosons, eq (2 19), and referring to the twists contained in the vectors A W and B W (hence the in&ces a and b since all twists come from ~2., amWm and Zm b~Wm) The projection matrix is then given by
~ a b = & a g ( 1 , 1 , 1 , 1 , ( P k - - l, Pk, Pk)l(P~-- l, Pt, Pt)), where k = 1,
,3 and I = 1,
(244)
,11 and Pk and PI stand for the projectors of the
A H Chamseddlne et a l /
Strings in four dlrnenslons
509
corresponding complex bosons, as defined in eq (2 19) Introducing also the vector 1 4
w0=
1
l'~3]tz 1
),
(2 45)
where (~)" replaces n entries equal to ½, we can then write Dffw in terms of the functions ~:0°,, which are periodic in 0', using eq (2 7) Aw = e x p [ - 2 t T r ( '7A W ~ b Dew
BW-
W o jo~b
BW)] ~Dw .4w ,
(2 46)
where ~e~Aww IS obtained by replacing in Dffw all functions d(~',qS, ~-), see eq (2 23), by functions ~ f l The "scalar products" in eq (2 46) have as usual a Mmkowskl signature, since right- and left-movers generate opposite phases V1 V2=
• I = left
VIv] -
~_,
VlkV~
(247)
k = right
The partmon funcuon ~eflwW can now be easily expressed in terms of a trace over physical states, as in expression (2 3) As in the case of fermlons, see eqs (2 2), (2 3), one needs to define the hamxltonlan and number operators appropriate to the boundary condmons of all fields The only difficulty comes from bosons, for which one must again dlstlngmsh between twisted and untwisted states, with the help of the projection matrix (2 44) We will not give a detailed derivation here (see ref [14], the necessary results can be found m appendix A), but it turns out that the convenient expression is AW__
(2 48)
The hamlltonlans for left- (~g::) and right-movers ( , ~ r ) , a s well as the number operator vector J g ' ~ are defined in terms of the oscillator modes and momenta of the fields Their expressions can be found in appendix A for bosons and in eq (2 2) for fermlons It is very convenient to restnct the boundary condlnons (twists and shifts) to be m the interval [0,1[ This is the meamng of the notation A W startmg with an arbitrary vector A W, one defines
AW=nW-a(nW),
(2 49)
where all enmes in A W are in the interval [0,1[ and all components of A(A W) are integers Using eq (2 5) one finds
(2 50) AW _ and, since ~'~w - ~ f lA~W, we do not need to explicitly introduce B W In eq (2 48), Vo
510
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is a shift vector with all shift components equal to l
v0:((1)6 (½) 22)
(251)
Its appearance in eq (2 48) is related to our definition of the shift V, with momenta P=k+V-l, k ~ 7 / Collecting eqs (2 38), (2 46), (2 48) and (2 50), the total partition function (2 36) becomes :Z= ~ T r [ e x p 2 , ~ r ( AB
I
~
AMeMNBN-- IAW ~i~ab B W +
Wo ~ b
BW
~ M,N
+ s w ~o~ ~(AW)-Wo ~'°~ ~(AW) -~w
~o~ ~-~+ ~o ~°~ ~ )
q~,~,~,T~] ( ~ )
It can also be written
where the phases are defined by
,;b= F~AN~Nm--~W~ ~°~ AW+ W~ ~°~ Wo N + Wrn ~ a b
A (A W) - W m ~ab
~A~,
(2 54)
~ = EAN~N~-Iv~ ~°~ Aw+ v~ ~°~ Wo N
+ v, ~o~ a(AW) - Vp ~°~ ~ ,
(2 55)
~ = Wo ~°~ a( A W ) - Vo ~°~ ~ w = Ro ~ o b
A(AW)
+ Vo ~ o b
[ZI(AW)-.4@w],
(2 56)
where R o is the rotation vector with all twists equal to l
Wo = 1% • Vo
(2 57)
The requirement that identical boundary condltmns lead to a unique, well-defined contribution to the partition function is now extremely simple to impose Firstly
A H Chamseddme et a l /
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changing the boundary condmon B W =Em bmWm + Zp dpVp, t o the equivalent vector w~th coefficients
B~t = BM + NM
(2 58)
should not change the partmon functmn One should then have
NMq~=O
(rood 1)
(259)
for arbitrary A W (no summauon on M), which finally gwes N.,(z~W., ~ . b A W + IV., ~ . b
W0)= 0
(modl),
(260)
Np(~Vp ~ . b A W + Vp ~ . b
Vo)= 0
(modl)
(261)
We must then choose a bas~s of vectors such that
~N,.Wm ~"~ Wo = o, l NmWm ~i~ab Vp = O,
½GG ~"~ wm= o, ~N~G ~°h Vq= o, U,.Wm y h
W0= o,
NpVe .~,,b W o = 0 '
(rood1)
(262)
These condmons generahze the "level-matching" conditions &scussed in refs [4,18] One easdy shows that these strong condmons are sufficmnt (and necessary) to ensure that all physically eqmvalent boundary condmons give ~dentlcal contrlbutxons to the partmon functmn ~ It is then enough to sum m 5 e, eq (2 52). over a single set of lneqmvalent boundary vectors, for instance O~AM,
BM
for all vectors IV,,,= (Wm, Vp} m the basis The vector V0 plays a special role It is clear that it must always be included in the basic set of vectors It will generate the sectors where all fields have penodm boundary conditions and zero stuft The coefficients A M and B M always contain c o
512
A H Chamseddme et al / Strmgs m four chmenslons
and d 0, with values 0 or 1 We rewnte then the partition functmn (2 53) in the form
where now
(~b=Ro 2z~h A(AW)+ IVo ~ b AW_Vo ~ b Vo_ y'eoMAM (2 64) M
In eq (2 63) we choose the values AM, B M
=
0,1,
, N M - 1,
(2 65)
but any other choice would lead to the same partmon function One can also obtain a more suggestive form for ~ e after a rescahng by the constant F i g N~ 1 *
~=
2Tr A
1 U.-1 I--[~-£ 2 exp[Z'~r(BM--SMO)~]e-2'~fg~q~qA--W)q~r`~-w) \ M
(2 66)
BM= 0
In this last expression, it IS apparent that each vector in the basis {Wm, Vp} generates a " G S O projector" in the partmon function of the form
i
NM-1 E exp[2'cr(BM--~MO)eP~] NM BM=0
(2 67)
It follows from eq (2 59) that this quantity is really a projector, with elgenvalue 0 or 1 depending on the number operators for a given stnng state The spectrum is then obtained directly by the condition that all states will be projected out, except those for which ~=0
(modl),
(2 68)
the number of projection conditions being equal to the number of vectors of boundary conditions in the basic set We close this section by summarizing the rules we have obtained to construct a modular invarlant string vacuum and compute Rs spectrum The choice of the basic vectors IV., and Vp ts restricted by the following conditions * Since in twisted sectors some bosons do not have momenta, it can happen that some shaft vectors whach were hnearly independent in the untwisted sector become dependent they are shielded by ~,,h Such vectors must then be removed from the basis m these twisted sectors
A H Chamseddme et a l /
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Level-matching ~N,.W,. ~"~
W. = O,
~U~Wm ~ °~ V, = O, ~N~V, ~°~ W. = O, ½UpVp ~ . b
Vq = O,
N,.W,. s~ °~ Wo = O,
Uy,
~ °~ Vo = O,
(mod 1)
(2 69)
Projector condltmn Wo .@"~ Wo = 0
(modl)
(2 70)
Ttus IS equivalent to the condition (2 26), with the help of W0 as given in eq (2 45) All previous condmons apply to all possible projectors ~ a b For any choice of vectors satisfying these conditions, there is a modular invarlant partition function, as given in eq (2 66) The only freedom is in the choice of the parameters eMN, whmh can be non-trivial when the orders N g and N N of two basic vectors are not coprime In this case, eq (2 43) apphes The set of basic vectors, and the choice, if any, of the values of egN completely detertmne the theory and speofy its physical spectrum, which can be investigated in the following way For all possible choices of boundary conditions A W = ZM AMWM in the partition functton, the physical states are those surviving the GSO projections and verifying
N
-IV., ~.h ~CA~=0 (modl, b ¢ 0 ) ,
(2 71)
q~b= ~_,Aueup_ 1Vp ~ b AW+ Vp ..@~b Wo+ Vp ~.h A(AW) N
- Vp ~ . b jffA~ = 0 (modl),
(2 72)
where J V ' ~ is the number operator vector corresponding to the state of the sector A W under consideration The mass formula for physical states is then m~_()--W) = m 2 ( ~ ) ,
(2 73)
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A H Chamseddme et al / Strmgs m four &rnenslons
where
a=l
n>~l 1I
+E
1 = 1 n>~l
11 +~ z~,
K,+<-~)12 +(K,+<-~) 2]
l~I I1
1 E [(a--~--/)2- (~//) -'}-1]
11
(2 74)
I=1
In this formula the various terms correspond respectively to the contributions of oscillators of the transverse space-time coordinates, oscillators of the complex internal bosons, with thmr two sets of creation operators, internal momenta of these bosons*, which are present only for untwisted states (hence the presence of the projector PI = 3765, 0 in this case), and the normal ordering constant 2 a=l
n>l
4
+ ~ Z [(n-~ff)5°.-~gi+(n+~-l)SP.+a~ 1] f=l
n>l
3
+ E E [(n--~--~--kk)d.~+(n+aOk-- 1)aC:+a~7-i] n~>l
k~l
3
+~ ~
12 <+~e~-~) +(,~y~+C--e~-~)2]
k=l
1 --2 + Z4 5[(aOf) -a-~y]- ~ l[(~-kk)2--a-~-~l f=l
(2 75)
k=l
The various terms are analogous to those appearing in the previous formula, with the addition of the four complex fermlons, indexed by f = 1, 2, 3, 4 The notation is explained in appendix A Shifts and twists play a complementary role m the physical content of the theory For (bosomc) left-movers, shifts define the lattice of momenta The massless modes * I n the case where some bosons are compactffled on a left-right symmetric lattme with arbitrary radius, one s~mply replaces the momenta by thmr corresponding lattme momenta [20]
A H Chamseddme et al / Strmgs m four dtmenstons
515
with shafts always form the adjoint representation of a rank 22 group When multiplied with right-movers, a theory with only shafts leads always to a rank 22 gauge group, possibly with other massless states this will depend upon the boundary conditions applied to the right-movers Twist vectors will induce GSO projections breaking the gauge group and, possibly, lowering its rank, as In m a n y orblfold compactifications The discussion of the effect of boundary conditions on right-movers is more subtle because of fermIons This is the subject of sects 4 and 5, where the various aspects of world-sheet and space-time supersymmetry are Investigated in N e v e u - S c h w a r z - R a m o n d and Green-Schwarz formalisms
3. Discrete symmetries and GSO projections In the previous section, we made little use of the group structure generated by the boundary conditions Our derivation of the partition function was essentially analytical Discrete symmetries play, however, a central role In orbIfold compactiflcations of heterotic strings [3,4] In such theories, the spectrum is obtained by retaining the states lnvarlant under the discrete symmetry of the orblfold, in the untwisted and in all twisted sectors, to preserve modular invariance We now wish to briefly discuss the group theoretical aspects of the formalism we have just estabhshed For a theory defined by the basis vectors (Win, lip }, It lS clear that one can regard each of these vectors as the generator of a discrete group Symbohcally, for a sector with b o u n d a r y conditions A W= Y~mamW., + F~pcpVp, one can define the group action by • (o + • + ,,) = 1-I ( g i n ) ° ' F l ( g p ) ~ ( m
~ + ~),
(31)
p
where ~ ( o + ~-) is a vector containing all stnng fields, left- and right-movers In this expression, gM ( M = m, p ) denotes the action of the boundary condition Wm or Vp. in the basis The exphclt defimtlon of the action of the group generators g., follows directly from eq (2 1) for fermlons, and from eq (2 10) for bosons It is clearly a group the group law corresponds to the iteration of the boundary condition, o ___~- ---. o + r + n~r Since any vector A W has a finite order N (recall that we only consider rational entries in AW), ~(o_+ ~_+ N~r) = 4(o_+ r ) Then, since every vector in the basis has an order NM, the discrete group relevant in a given theory IS G = 1--IZNM ,
(3 2)
M
recalhng that Z., × Z , , - Zp, where p is the least common multiple of m and n Since we consider only abellan twists, the group G is always abehan
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A H Chamseddtne et al / Strmgs m four dtmenstons
As already mentioned in sect 2, the group law for twisted and shafted bosons implies a strong constraint on G the boundary condition (2 10), o +
= e2'"*z(, + o) +a,
says that Z is a closed string coordinate on the orblfold obtained by dividing the torus Z - Z + A by the group G O defined by g0 Z = e 2 ' ~ Z , go ~ G 0 A theory containing twisted bosons corresponds then to an orblfold, symmetric or asymmetric In tins case it is well-known [3, 4] that G O should be a point group of the lattice defining the torus It is G Owtuch enters in the full discrete group G in eq (3 1), and not the whole space group of the orblfold, whose elements are defined in eq (2 13) This is related to the fact that the partition function for twisted bosons does not depend on the shifts, see eq (2 18) It is also the point group G Owhich is relevant when discussing the spectrum of (group lnvarlant) states since only G O appears in the partition function In the partition function, each vector WM = W m or Vp from the basis, correspondmg to a discrete group ZNM, also introduces a GSO projector (2 54)-(2 56) As already discussed, a state surviving tins projection satisfies d?~ = 0 (rood 1), with ~ as gwen in eqs (2 71), (2 72) Tins condition has then a very simple interpretation in terms of the discrete group ZNM A given state In the sector A W is characterized by its vector ~V'Aw The action of the group ZNM on tins state can be represented by the quantity e 2''~wM ~°~ ~""~,
(3 3)
WM being the generator of ZN,' In the product W M ~ab . , ~ , right- and leftmovers contribute with opposite signs The conditions q ~ = 0 select then all states for which the difference of the eigenvalues under the group generator for left- and right-movers has a constant value (modulo 1), the constant depending on the sector A W in which the states appear Thus, for the case of twisted bosons, the GSO projection selects states with a specific transformation under the point group of the orblfold compactlfiCatlon Because of the projector ~ a b in eq (3 3), the action of the GSO projector for a given vector W ~ ( Win, Vp } will be &fferent when this vector contmns twists (for bosons or fermions), or sinfts For a state corresponding to an entry of W which has a twist, or when the state is twisted, the GSO projector acts on the oscillator number of this state In contrast, when only shifts are relevant, the GSO projector acts only on the internal discrete momenta, irrespective of oscillator numbers The GSO projectors for twists select oscillator levels (in fernuonlc language, they act on the charge lattice), while projectors for shifts select lattice momenta The two classes of projectors, acting on bosons, have then different and complementary implications on the spectrum
A H Chamseddme et al / Strings m four dtrnenstons
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For instance, an a model defined only with shifts, the gauge group has always rank 22 the gravlton appears an the product of a massless right-moving vector state t V~)R (whose origin differs in N e v e u - S c h w a r z - R a m o n d and in Green-Schwarz strings) wath a massless left-moving vector a L l l 0 ) L There are also 22 massless states of the form a1_ll0)L ® [V~t)R, with vanishing internal momenta With only shafts, the G S O projectors, which only feel the lattice momenta, cannot dlscrlnunate the gravlton and these 22 gauge fields, which then appear in the spectrum to form the Cartan subalgebra of the gauge group* Taking into account the discrete momenta, one can also show that the massless states in a model with shifts only are always m the adjomt representation of a gauge group This is also required by N = 4 space-tame supersymmetry, for consistency (A proof of this statement an the fermlonic formahsm can be found an ref [9] ) In general, when shafts and twasts are involved, states with fixed masses and fixed elgenvalues under the discrete symmetries generated by the basic vectors always form complete representataons of the gauge group Considering all mass levels, the G S O projectors assocmted with shifts select full conjugacy classes of the gauge group Twasts break this structure F r o m the preceding discussion, it is clear that twisting a complex boson Z I can decrease the rank of the gauge group by removing the corresponding Cartan state a / _ l [ 0 ) L ® [VtZ)R from the spectrum The GSO projector for this twist feels the oscillator number of this state, which has then a different d~screte symmetry q u a n t u m number from the gravlton The gravlton stays in the spectrum, but not the Cartan state Twastmg will also be necessary to reduce space-Ume supersymmetnes this will be discussed in sect 5 In short, the GSO conditions g,~ = 0 for twists and shifts can be completely interpreted an terms of dascrete group elgenvalues The spectrum can be analyzed using group theoretical considerations only, following a procedure analogous to the case of orblfold compactfficatlons [3, 4]
4. Green-Schwarz and Neveu-Schwarz-Ramond strings U p to this point we have not specified the nature of the world-sheet fermlons present in the right-moving sector Here, there are two posslblhties In the first case, in the light-cone gauge, we have eight world-sheet fermlons which are also space-time fermlons This corresponds to the Green-Schwarz superstrlng, and has the advantage of keeping space-time supersymmetry exphclt In the second case, also in the light-cone gauge, the eight world-sheet fermlons are space-time bosons This corresponds to the N e v e u - S c h w a r z - R a m o n d string The advantage of the N S R formulation is that it as exphcxtly covanant and world-sheet supersymmetrlc In case * A m o d e l d e f i n e d w i t h shifts o n l y h a s a l w a y s N = 4 s p a c e - t i m e s u p e r s y m m e t r y , as will b e dascussed m sect 5
518
A H Chamseddme et al / Strmgs m four dimensions
one wishes to compute string amplitudes for a promising model it is advantageous to deal with the N S R formulation where the techniques of (super)conformal theories are readily available This is not the case in the GS formulation where objects such as vertex operators in twisted sectors have not been worked out An obvious disadvantage of the N S R formulation is that space-time supersymmetry is not explicit and the concern of the next section will be to establish the conditions that insure its presence In this section, we shall show that we can accommodate both posslblhtles simultaneously when writing a modular mvarlant partition function Starting with the GS case, we first recall that for a complex space-time fermlon, with the boundary conditions as given in eq (2 1), the partition function is given by eq (2 3) [14] and the trace m that expression gives ~ f , as defined in eq (2 4) The analys~s in the previous sectmns goes through without any modification In the N S R case evaluating the trace in eq (2 3) for a complex world-sheet fermton (but space-time boson), with the boundary conditions as given in eq (2 1), gives ,,~(8°,+1/2 However, having decided to express the full partition function in terms of ~ f l , (or equivalently D °,) it is convenient to define the partition function in this case in terms of the trace Tr(q~(O)
e2,~r(0'
-
1/2)JV'(0))
(4 1)
With this deflmtion eq (2 36) becomes a unified expression for the total partition function The only difference arises in eq (2 48) which now becomes A~ _ Tr[q~eqA~qx,'r(A~ e-Z,~(BW-L) ~b X ~ ] , "~"B W--
(4 2)
where n = { V° Vo + W6
In GS, in N S R ,
(4 3)
and W6 is the twist vector
W~ ~ ((1)4, (0,0,0) 3 (0,0,0) 11)
(4 4)
The final expression for the partition function in eq (2 52) remains unchanged, except that ~ b that was defined in eq (2 56) is now replaced by ~}b= W° ~@ab A( A W ) - L ._@ab .A/.A~
(45)
Notice that the GSO conditions are identical in both cases and given by eqs (2 71), (2 72)
A H Chamseddlne et al / Strings m four dlmenslons
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One important point that must be taken into account is to insure the boundary conditions we take preserve world-sheet supersymmetry For the GS case this point has been studied in full detail in ref [15] There, it was shown that world-sheet supersymmetry is maintained provided that the twist vectors W,. satisfy 3 Y'. 0, = 0 t=0
(mod 1)
(46)
for the fermlonlc twists and ~k=00+0k,
k=1,2,3
(modl)
(47)
for the right-moving bosonIc twists We shall perform a similar analysis for the N S R formulation and determine the conditions that the boundary vectors must satisfy in order to preserve world-sheet supersymmetry The ten-dimensional right-moving fields are X M and xPM and they transform under local two-dimensional supersymmetry according to [20] 8 xM=
ixI~M,
-,pO (0o x ' - r 'xo),
(4 8)
where p~ are the two-dimensional Dlrac matrices and X~ is the gravltlno The two-dimensional spinors ,/,M and e are Majorana-Weyl and have opposite chirahties Thus they are one component splnors which we denote by ,/,_M and c . The transformation laws in eq (4 8) now read 8xM=
le+ ~-FM,
6~"M : ~+ o _ x M - ] - lC+ ( g o -- Xl)+ ~M
(4 9)
One can partition the ten-dimensional index M into /t, m where /~ = 0,1, 2, 3 is a s p a c e - t i m e index and m = 1, ,6 are internal indices The real fields X m and ,/,m are then complexlfled to Z k and ,/,k, k = 1, 2, 3, and are allowed to take twisted b o u n d a r y conditions The space-time coordinates X ~ must be penodic since the m o m e n t a are continuous The one component spmors '/'~_ and c+ are real and thus can be either periodic or antiperlodlc From the 6 X ~ transformation in eq (4 9) we deduce that 0 ( e + ) = 0 ( g ' ~ _ ) = 0 °,
20°=0,
(modl),
(4 10)
where O(~b) denotes the twist angle of a fermlon ~b Ttus then implies that the twists
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A H Chamseddlne et al / Strmgs m four dlmenstons
~k for Z k and 0 k for q,k are related by q~k= 0° + 0k
(mod 1)
(411)
Comparing with the GS case we see that eq (4 6) is replaced by eq (4 10) whale eqs (4 7) and (4 11) are identical From here on, all our boundary vectors will be assumed to satisfy eqs (4 6) and (4 7) for the GS case and eqs (4 10) and (4 11) for the N S R case We shall now make a physical assumption on all string models, that they must always include a gravlton m the spectrum This wdl imply some conditions whach we first determine for the GS case Looking at the mass formula m eq (2 75) we see that in the absence of twist vectors m 2 >/0, and it is zero only when all entries m the right-mover part of a given shaft vector are ½ The ground state in this case is massless and it is the usual Green-Schwarz supersymmetnc ground state This mchades the space-time vector la)R, space-time complex scalars IZk)R and s p a c e - t i m e spinors I S ) R and ISk_)R For the left-movers we only obtain the s p a c e - t i m e vector la)L as a massless state when we have all the entries of the left part of one shift vector to be ~ Thus we are forced to include the shift vector V0, defined in eq (2 51), among the set of boundary conditions If only the vector V0 is present, this corresponds to the N = 4 supersymmetnc four-dimensional string model with the gauge symmetry SO(44) In some way thas can be seen as our starting point Adding more vectors W m and Vp wdl change the gauge group by adding more states and projecting others with the GSO conditions Reqmrlng that the grawton la)R ® [b)L survives the new GSO conditions qS~(V0) = 0
(modl)
(4 12)
lmphes that eMo = WM ~,~b ( ~ V o _ Wo )
(modl),
(4 13)
where we have used that "f"v0 for the gravlton state xs zero The above condition (4 13) will also appear when we compare the present formulation with that in ref [14] This is to be expected since there the N = 4 supersymmetrlc model with SO(44) was taken as a starting point and the gravlton was implicitly assumed to be present We now turn our attention to the N S R case and, In analogy with our previous discussion, the vector V0 must be present among the set of boundary vectors However, this is not enough In the absence of twist vectors the ground states for nght-movers are space-time spinors (more on this later) To generate the gravlton [a)R ® Jb)L the right massless state must be a bosomc space-time vector This can happen If we twist the four complex world-sheet ferrmons by W6 defined in eq (4 4), as well as shift the bosons by V0 Our set of boundary vectors must then always
A H Chamseddme et al / Strings mfourdlmenswns
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include
a~)W0 + coVo ,
(4 14)
where a 6 = 0,1 and c o = 0,1 The gravlton is then in the W6 + V0 sector We now require that the gravlton state is kept by the GSO conditions
eOob(W6+Vo)=0,
q~b(W6+ V o ) = 0 ,
(modl),
(4 15)
where the index 0 refers to the vector W0 Evaluating eq (4 15) using eqs (2 71), (2 72), we find that this is guaranteed if we choose e00 = 0,
(4 16)
wtuch keeps the grawton state (among other ttungs) and projects out the tachyonic ground state Choosing instead e06 = ½ lmphes that the starting point is the tachyonlc model and it is of no interest to us N o w it is not difficult to see that the model deterlmned by the set an eq (4 14) is the N = 4 s p a c e - t i m e supersymmetrlc model, with SO(44) gauge group, and will be taken as a starting point Comparing this model in the N S R and GS formulations shows that the partition functmns differ only by the fermmnlc contributions from 1 --41tol t~4 + 044 + right-movers, which factor out In the N S R case, they are given by ~*1 0 4 - 04 ) while in the GS case it as given by */ 4014 The equality of the two expressmns is none other than the famous Rlemann identity Including more vectors Wm and Vp will then alter the model, but as before we require that the GSO conditions that will arise from those vectors should not project out the gravlton state q ~ ( W 6 + V0) = 0
(modl)
(4 17)
Using that the number operator for the gravlton state 1s
gravlt°n ° =-S JV"W6+V
=
( ___1,03, ( 0 , 0 , 0 ) 3]( 0 , 0 , 0 ) il ) ,
(4 18)
we find that eq (4 17) is always satisfied provided we choose
eMO+eMO=WM JOa6 (½Wo+½Vo_Wo+S)
(modl)
(419)
To summarize this section, we have shown that m the GS formulation the set of b o u n d a r y vectors must include coV0 ( c 0 = 0 , 1 ) and that we must choose e0M to satisfy eq (4 13) The ground state is always supersymmetnc and both fermlons and bosons (if not projected out) will appear m the same sectors All vectors must satisfy eqs (4 6) and (4 7) to preserve world-sheet supersymmetry
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A H Chamseddme et al / Strmgs in four dlmenstons
In the N S R formulatmn the set of boundary vectors must include coVo + a6W6 and we must choose e0~ to vanish and e0M + e~M to satisfy eq (4 19) The role of W6 is to create the Neveu-Schwarz sectors ferrmons appear in the sectors a 6 = 0 while bosons are in sectors where a6 = 1 All vectors must satisfy eqs (4 10) and (4 11) in order to preserve world-sheet supersymmetry In both formulations the modular invarlant partition function is given by eq (2 53) with ~ as defined in eq (2 54)-(2 56) and ~ b as in eq (4 5)
5. Space-time supersymmetry The mare motivation to work in the GS formulatton is that the space-time supersymmetry is explicit String models can be characterized by their order of supersymmetry m a simple manner [14] If all the space-time fermtonic twists satisfy 0,=0,
t=0,1,2,3,
(5 1)
the model has N = 4 supersymmetry If the twists of the two fernuons vanish, say
00= 01~-O,
02*0=]=03, 02+03=0 (mod1),
(52)
then we have N = 2 supersymmetry If only one of the ferrmons has a vanishing twist which we choose to be 00=0,
0i+02+03=0
(modl),
01,82,03=/=0,
(5 3)
then we have N = 1 supersymmetry Finally if none of the ferrmonlc twists is always vanishing, then we have no supersymmetry (the only exception occurs when all 0, are 1, this has been discussed in ref [15]) On the other hand space-time supersymmetry, when present, ~s not explicit in the N S R formulation To counter this drawback we shall characterize it in a manner similar to the GS case It is also of importance to find an explicit connection between models in the two different formulations This we shall establish for the supersymmetrlc models, provided certain conditions are satisfied We have determined that in the NSR formulatmn the set of boundary vectors must be of the form a6W~ + A W, where A W includes coVo To examine space-time supersymmetry we look at the number of massless gravltlnos These can only occur when the rlght-mowng sector has zero vacuum energy and the ground state is a space-time splnor Also the left moving state must be a space-time vector, and thus the left vacuum energy must be - 1 This is possible only in the V0 sector Here the eight real fermlons (four complex) are periodic and their mode expansions contain the zero modes The eight zero modes span an eight-dimensional Clifford algebra, and can also be used to construct the SO(8) generators The right-moving ground
A H Chamseddme et al / Strings m four dtmenswns
523
state is then a ( 2 4 = 16)-&menslonal splnor that splits into two (23 = 8)-dimensional irreducible SO(8) splnors Each of those two irreducible spmors has a fixed chlrahty The number operator for the gravitmo states takes the form
xravltlnO_ vo - (ao, aI, a2, a3, (0,02) 3 ( 0 ' 0 2 ) 11) '
(5 4)
where a, ~ {0,1} Obviously the fermlontc ground state (a0, al, a2, a3) gives the 16-dimensional splnor of SO(8) that splits into the two irreducible eight-dimensional spinors of opposite chlralltles The gravltlno states will be subject to the GSO conditions q,gO(v0) = 0
(modl),
(5 5)
q~(Vo) = 0
(modl)
(5 6)
The first condition, eq (5 5), when substituted into eqs (2 54), (2 55) gwes, after using eq (5 4),
l(ao-l-al-l-a2q-a3) = 0
(modl)
(5 7)
Thus only the even chlrahty eight-dimensional splnor survives, with the states
(5 8)
(ao, al, a2, a 3 ) = (04) + (1_202) + (14),
(where underhnlng means that the corresponding value can be located at any possible entry not separated by a semi-column) Before applymg the condmons from Wm we note that, and without any loss of generahty, the twist angles 00m can all be set to zero This Is so because eq (4 10) restricts 00m to be zero or half, but since the vector W6 is already present we can always shift 00,,, to zero 00.. = 0
(5 9)
The condmon (5 6) for Wm on the gravltlno states gives the following condmons depending on the splnor state it is acting on ~ab(Vo) = __(1Vm :~ab Vo_I_R m ~ a b Ro_l_~mO ) _- _(~Vml
~ab
= _ ( ½ V m ~ab
on(04),
VO_[_Rm ~ a b RO_I_,EmO)_[_(O1..I_O2_[_O3) m 0 n ( 1 4 ) ,
Vo+R m ~.b Ro+emo)+(O,+Os )
on(1202),
t,j=O,1,2,3,,=/=j,
(5 10)
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A H Chamseddme et al / Stnngs m four &menstons
all these equations bemg valid survive we must choose e~o=
modulo 1
Thus to insure that N = 4 supersymmetry
_Rm ~ab Ro_ ½V." ~i~ab VO
0,.'=0,
(modl),
t=1,2,3
(5 11) (5 12)
For N = 2 supersymmetry to survive, we must choose eq (5 11) as well as the vanishing of one of the fermlonlc twists, say
o1.'=o
(513)
This will insure that four spmor states (two 2 hehclty splnors) will survive if
O2m+
03. ' = 0
(mod 1)
(5 14)
However, eq (5 14) is a consequence of eqs (5 11) and (5 13) because EmO should independently saUsfy
2emO = 0
(modl),
winch implies Olin + 02m "~ 03m = 0
( m o d 1)
(5 15)
N = 1 supersymmetry would survive if era0 is chosen to satisfy eq (5 11), which wdl then imply eq (5 15) None of the O,m should always be vanishing The surviving states in this case are
( a 1, a 2, a 3, a4) = (0 4 ) -4- (14 )
(5 16)
It IS quite satisfactory to see that the choice of eq (5 11) implies eq (5 15) which forces the other component of the gravltlno (14) to survive, and this is absolutely necessary for consistency If no supersymmetry is required then one can choose e" 0 so as not to satisfy eq (5 11) Examining the condition in eq (5 6) for the vectors Vp we find that if any supersymmetry ( N = 4, N = 2 or N = 1) is to survive we must choose epO= - ½Vp ~ a b
Vo ( m o d l )
(5 17)
N o t e that eqs (5 11) and (5 17) are exactly identical to eq (4 13) for eMo in the GS case These, when substituted into eq (4 19), determine eM6 to be
eM6= WM ~ab (½W~+ S)
(modl)
(5 18)
A H Chamseddme et al / Strmgs m four dlmenszons
525
This further imphes that the boundary vectors W m should satisfy 3
½E0,,.=0
(modl),
(5 19)
t=l
and this follows from the property 2era6 = 0 (mod 1) The condition (5 19) on the supersymmetrlc models in the N S R case is slightly stronger than eq (4 6) in the GS case Thus only those supersymmetrlc models in the GS case that satisfy eq (5 19) can be directly compared with the corresponding ones in the N S R case If no supersymmetry is reqmred then we can choose eq (5 17) not to be satisfied This is a preferable choice since this wall automatically project all the graxqtinos out At this point it is clear that the conditions to be satisfied by the set of vectors A W in order to insure space-ume supersymmetry ( N = 4, N = 2 or N = 1) are compatible for the GS and NSR formulations It is then possible to compare a given supersymmetrlc model m the GS formulation, with a fixed set of boundary conditions A W, with a corresponding model in the N S R formulation with the set of boundary condluons a 6 W6 + A W We shall estabhsh the conditions under which there exists a one-to-one correspondence between all the massless states in both formulations Such analysis does not seem to be possible for non-supersymmetric models This is due to the fact that we may have to use different boundary vectors m the two formulations to describe the same model The reason for this is that in the NSR case 00 is restricted to satisfy eq (4 10) while in the GS case this is replaced by eq (4 6) Many non-supersymmetric models have been found where 00 is ¼ or ~ [15] It is not clear how to obtain, systematically, the same models in the NSR formulation We are now ready to compare a given supersymmetnc model in the GS case defined by the set of vectors (A W ) with the model in the NSR case defined with the set of vectors ( a 6 W~ + A W } , where A W is the same in both cases We shall assume that the vectors Wm satisfy the stronger constraints in eq (5 19) rather than eq (4 6) First, m the GS case all states, fernuonlc and bosonlc, come together since the right-moving ground state is always supersymmetric In the NSR case the ferrmonlc states come from the sectors A W (i e a6 = 0) This is so because the right-moving vacuum energy is zero m this case as follows from eq (4 11) and that we have set 0ore to zero The massless ground state for the right-movers is then the 16-dimensional splnor (a0, al, a2, a3) , a t ~ (0, 1) The bosomc states come from the sectors W 6 + A W (ie a 6 = 1) Let us assume that a given massless state (fermlonic + bosonic) appears m the GS case in a fixed sector A W, and surwves the GSO conditions ab A W ) l ~ s = 0 ~M(
(modl)
(520)
526
A H Chamseddlne et al / Strmgs m four dzmenstons
If the same massless fermionlc state is to appear in the N S R case, m the same fixed sector A IV, then it must survive the GSO conditions ab q~M(AW)INSR=0
(modl),
(5 21)
%b(AW)INsR=0
(modl)
(5 22)
Since the expression for O~(AW) is valid for both formulations, only eq (5 22) could impose additional restrictions Evaluating eOoh(AW)from eq (2 54), after substituting for e~/~ from eq (5 18), gives
W5 ~ab [,AZA~_A(aR)] = 0
(modl)
(5 23)
However, this only picks the chlrallty of the ground state spmor In the GS case such condition can also be seen from the supersymmetrlc transformations of the ground state We thus deduce that all the massless fermlomc states m the sectors A W are completely ldent~cal for supersymmetnc models in the GS and N S R formulations provided we use the same set of vectors satisfying eq (5 19) The massless bosonlc states are m the { W6 + A W } sectors Here we can use the argument of space-time supersymmetry to prove that those, when taken all together, are the same as the massless bosonlc states m { A W } for the GS case However, we w~sh to make a connection between each of the states and deterrmne under what condmons there ~s a one-to-one correspondence between the massless b o s o m c states in the two formulations Again, assume that a given massless bosonic state appears m a fixed sector A W and satisfies eq (5 20) in the GS case Then if the same state is to appear in the sector W6 + A W in the N S R case, it must satisfy
eo~(Wo+AW)[NsR=O
(modl),
(5 24)
q~6"b(W~+ A W ) I N S R = 0
(modl)
(5 25)
The left-moving parts of the vectors A W and W6 + A W are identical and so are the left-moving parts of the state, and we can concentrate on the right-movers only Substituting eqs (5 20) and (5 24) into eqs (2 54), (2 55) and requiring that the same state surwves in both formulauons imply WM
~ab
(\ vA/'NSR , Wo+A W
,.~/'AGSW) =
WM ~ b 3(AW)+ WM ~ob S
(modl) (5 26)
527
A H Chamseddme et al / Strmgs m four chmenstons
where
3 ( A W ) = A ( A W ) - A(W O+ A W )
(5 27)
and use was made of eq (5 18) F r o m the mass formula, eq (2 75), the vacuum energy for right-movers m the sectors W6 + A W can be easily evaluated to be
( (VE)nght~--{
3
)
- 1 4- Z [ ~ - , ( 2 8 ( a 0 , ) +
1)+8(aO,)8(aO,)]
(5 28)
t~0
Looking first at the sectors W6 + cV 0 e when all the a ' s are zero), the vacuum energy is - ~ and the ground state is tachyonic while the massless state is the first excited state for the world-sheet fermlons In this case eq (5 25) gives W0 ~a~h ./Vw~+;>= ½ ( m o d l ) ,
(5 29)
projecting out the tachyomc ground state and keeping the massless state Eq (5 26) is then automatically satisfied for all left states that will couple to th~s right massless state When some of the a ' s are non-zero then the vacuum energy in eq (5 25) is zero, provided that O~
,= 1,2,3,
(530)
I .<.
t=1,2,3
(531)
or
Thas can be easily verified if one notes that in the domain of eq (5 30) 8(0,) = 0 while in the domain of eq (5 31) 6(0,) = - 1 If the above conditions are not satisfied then the vacuum energy is negative and one must determine the massless states Here there does not seem to be a general way of finding the correspondence between the two formulatmns N o w assuming eqs (5 30), (5 31) are satisfied, the right-moving ground state is a massless boson, and eq (5 26) is satisfied for all left-moving states that couple to it In addition, eq (5 25) lmphes the additional constraint
W6 ~ab s f f ~ w = _ W 6
~ab A ( W 6 + A W ) _ ~ ( m o d l )
(532)
Eq (5 32) IS automatically satisfied for the right-moving massless bosomc ground state and all left-moving states associated with it Th~s is so because in this case W6 ~@ab .A/" IS zero while W6 ~ab A ( W s + A W ) 1S 21 We have thus shown that there ~s a one-to-one correspondence between all the massless bosomc states in the A W sectors m the GS formulaUon and the massless
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A H Charnseddme et al / Strings m four dimensions
b o s o m c states in W6 + A W sectors m the N S R formulation provided that the boundary vectors W., satisfy either of the eqs (5 30) or (5 31) To summarize this section, we have shown that we can insure space-time supersymmetry m the N S R formulation provided we choose eMO to satisfy eqs (5 11) and (5 17) This also lmphes that the boundary vectors Wm should satisfy eq (5 19) We further compared supersymmetric models m the GS formulation [constrained to satisfy eq (5 19)], specified by the set of boundary conditions A W. with those in the N S R formulation with the set of boundary conditions a6W~)+ A W We showed that there ~s a one-to-one correspondence between the massless fermlomc states in A W for both the GS and N S R cases The one-to-one correspondence between the massless bosomc states in A W for the GS and in W6 + A W for the N S R formalism is similar to that m the ferlrnOnlC sector if the boundary vectors Wm further satisfy eqs (5 30) or (5 31)
6. Relation with previous formulanon Starting with the same class of boundary conditions, a formahsm for constructing strings m four dlmensaons has already been established by two of us [14] Compared with the present approach, this formalism differs by two important aspects F~rstly, the c o n d m o n arising, m the factorized limit, from two-loop modular lnvarmnce was not exphcxtly considered It was assumed that an appropriate GSO projection is sufficient to take care of this condmon, and the form of the GSO projection was postulated and then tested by reconstructing known orblfold theories Secondly, th~s f o r m a h s m is not minimal ~t contains too many parameters which are subject to complicated condmons to satisfy modular lnvariance That formalism was largely inspired by the general ferrmomc construction of Kawal, Lewellen and Tye [7], where a convenient GSO projection was also postulated It was later shown that two-loop modular lnvanance is equivalent to this GSO projection [9] In this section we will indicate the relation between our new formalism and the construcuon gwen m ref [14] In principle, the comparison is rather simple one just needs to compare the expressions for the total partition function m the two cases There is, however, a difficulty The level-matching conditions on our basis vectors W., and Vp, eqs (2 69), are more restrictive than the conditions apphed in ref [14] It is then s~mple to show how from a given model in the new formalism, one constructs the same model m the "old" formalism, but the converse ~s more subtle In the formahsm of ref [14], all vectors of boundary conditions have only components m the interval [0,1[ This is the case of the basis vectors { W m, Vp }, and of the linear comblnauons describing the sectors
A W = EamW,,, + EcpVp, m
p
(6 1)
A H Chamseddme et al / Strings m four chmenswns
529
where the bar indicates that each component ~s m [0,1[ and A W= A W (mod 1) (This is the same notation as m ref [7] ) The level-matching condmons in the formalism of ref [14] are only
No,(~' ~ o, ~°~ ~,. + Wo ~°~ ~ . ) = 0
Np({Vp ~ a b
~ ..~ Vo ~ a b
(rood 1),
~fl)=0
(mod 1),
(6 2)
much weaker than eq (2 69) As a result, the partmon function contains many parameters T"ab ab and kpq,ab instead of our new parameters eMm, but these .... kmp.ab kv~ parameters are not any longer free and independent Modular mvarlance requires that
T.~o,~,,+ Ttim ~ = W,. Y ~ ~m=~Wm
Wti ab Wm -- kom,
Wm+ Wo
~ab ab = ~p ,_~ab W M, "" pM q_ k Mp
M=qorm,
po = 2 Vp ji~ab
N.,T,, m = O,
NMkpM = O,
(6 3)
(mod 1)
Clearly for two vectors, say Vp and Vq, w, th Np and Nq copnmes, k;,qb and k qapbare completely fixed The only freedom is m the case where Np and Nq are not copnmes This freedom corresponds precisely to our new parameters eMU Starting with a set of basis vectors { Win, Vp} m our new formahsm, satisfying level-matching condlnons (2 69), the set ( IV.,, Vp}, defined as m eq (2 49) certainly sansfies eq (6 2) and can then be used to construct the same theory m the formalism of ref [14] Comparing the two parntion funcnons, one gets the following relanons between the parameters of the two constructions
T;,,,,-~.,.- ~Wm
W. + Wm
Wti,
k a b =,grnp - 1Win ~ a b
G "]- Wm ~ a b
G'
k p m _ ,F.prn__ 1
kpqab----Epq -- ~-IVp ~ a b
Vq + Vp ~ a b
Vq,
(mod 1)
(6
4)
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A H Chamseddme et a l /
Strings m four dtmenslons
One must also take % = + Vo to r e _ R 0 ~ b
Wm+½V° ~ b
Vm,
(modl)
(6 5)
These last expressions lead to the phases e_21~(~b : e2t~Ro ,~ab ~ W
(6 6)
m the total partition function (2 66) One can then, without ambiguity, translate a model from the new to the old language Conversely, translating from the old to the new language is always possible, but the answer is not necessarily unique Eqs (6 4) and (6 5) lead to conditions on products of vectors It is sometimes possible to construct several sets of vectors { Win, Vp }, from a single set ( W m, Vp ), satisfying all level-matching conditions Tl'ns discussion shows that the formalism elaborated in ref [14] is in fact also modular lnvarlant beyond one loop On the other hand, it is much more complicated than the solution presented here when discussing the spectrum of a given theory
7. Conclusion In this paper we have developed a method to construct four-dimensional string theories consistent with modular lnvariance at arbitrary genus (modulo factorizatlon), with general abehan boundary conditions for bosons and fermlons (therefore covering orblfolds, fermlonlc and lattice constructions) The free parameters are fully identified they are boundary vectors, subject to some "generalized" levelmatching conditions, and phases defined by the discrete symmetry relevant to the boundary vectors For each set of free parameters, the modular invarlant partition function is completely specified Its physical features, like for instance the proper space-time statistics in the case of NSR fermlons or the GSO projections on discrete group invarlant states for orblfold-hke, twisted theories are direct consequences of two-loop modular lnvarlance Our formalism applies to fermxonlc sectors with either Green-Schwarz or N e v e u - S c h w a r z - R a m o n d fields For both cases, the conditions for space-time and world-sheet supersymmetrles are obtained For space-time supersymmetnc theories, the explicit correspondence at the level of the spectrum between the GS and NSR theories allows to take advantage of the two versions of a given string model The spectrum is much easier to study using GS fermlons, while dynamical aspects like the effective field theory are better investigated with NSR fermlons, where conformal field theory techniques can be used
531
A H Chamseddlne et al / Strmgs m four chmenslons
Our method has only been illustrated here by the simple example of the standard, Z 3 orbffold It can, however, easily be used to study large classes of models in a systematic way, and to accommodate complicated sets of boundary conditions It then provides a powerful tool to study the connections between the different string constructions, or to perform systematic searches for reahstlc string models A H C and J P D would like to thank the InstltUtO de Estructura de la Materia (IEM), where this work was mmated, for warm hosp~tahty M Q is indebted to the Institut fur Theoretische Physlk (ETH) for the warm hospitality he received during completion of this work
Appendix A FORMULAE
FOR TWISTED/SHIFTED
BOSONS
We collect here some useful formulae which were omitted in the text The partition function for a complex boson with twisted a n d / o r shafted boundary conditions has been given in eq (2 18) It can also be written in terms of a trace over physical states in the following way
2y.(eo v~,v,) = Tr(q.~(W)e_Z,~(w, (¢ v~ v,')
vo) s~ x ~ )
(A 1)
where W, W', V0 are the three-component vectors
Vy),
w, =
v;, v;),
V0=(0,½,1),
(A2)
and ~ IS a diagonal 3 × 3 matrix
(A 3)
.~ = dlag( P * * , - 1, P¢¢,, P,~q,,), as in eq (244), with P~, = 6~,o8~.o as usual The hamlltonian ~ ( W ) both shifted and twisted bosons o
,(,2_q,+~)+
(w) =
y' [(n-~)~._¢+(n+O-1)~+¢
applies to
1]
n~>l +
+
vx
1)2+ ( , c + v _
(A 4)
where Po = 6,0 and "~n-e0 ~ O~?n-4'Otn 4' '
" "~' J~nt+q}_ 1 ~ ~'* ,~n+q}_l~n+q}_l ,
(A 5)
A H Chamseddme et al / Strings m four dlmenswns
532
m terms of the oscillator modes For twisted bosons, the last term is absent and the mode expansion has been discussed m sect 2 For shafted complex bosons, this hamlltoman has the following interpretation A complex boson with boundary conditions
Z ( z + o + it) = Z ( ~ ' + o) + A, with Z = X + t Y and A = ( V x - - 1 / 2 ) + I ( V ~ - - I / 2 expansion given by
X=Xo+P~(~+o)+
(A 6)
) as in eq (217) has a mode
t 1 ~ ~-a 2.., - - a ,xe
- 2tn('r+o)
n4~O I"1
t 1 r = Y° + PY('r + ° ) + 2 n~.jO--OtYe-2tn(r+°), n "
(A 7)
with a x . = ( I x ) + and a L . = (aY) + For bosons on a (shifted) lattice, Vx(Vy ) will be replaced by K x + Vx ( K v + Vy), K x ( Ky) being an integer and Vx (Vy) in the interval [0,1[ The discrete real momenta are then 1 2,
Px=Kx+Vx
Py = K , + Vy -
(A 8)
2(px12 +
py2) The oscillator term
and the last term of the hamfltonlan (A 4) is then corresponds then to the definition (. -
¢ ) ~ . _ ~ + ( n + ¢ - 1 ) ~ ' ~ +~', _ ~
=
.
(a.~ta~ +
aYt,',Y'~ ._.,,
( A 9)
for untwisted (4)= 0) bosons The definition of the number operator vector .V' w follows from the partition function for shifted bosons For the real part X of Z, the partmon functmn reads T r ( q ~ e-2,.(v"
Vx+l/2))
1/2)(P~-
=-q-1/z4vI(1-q')-1 n>~l
~ K~Z
q(K~+v~-l/2)2/2e 2tcr(V'-l/Z)X~=oo~fV~ v"
(A10)
Defining the number operator for stufted boson JV'v~ by
JV'v = Px - Vx + ½ = K x ,
(A 11)
~v~ = Tr( qgx e - 2,,~(v--1/2)Jv'x )
(A 12)
one can write
A H Chamseddme et a l /
Strmgs in four dtmenslons
533
The vector .A/"w is then defined by
(A a3) For twisted bosons, Jg'(q0 as given m eq (2 2) The phase in eq (A 1) corresponds with these conventmns to
(w'-Vo) ~ ~ w
=~'(1-Po,,).A/'(eO)+(Vx'-½)Pc,o,Kx+(Vy-½)P,o, Ky (A14) In the same way, the partmon function for all fields can be written in the compact form (2 48) using a number operator vector .4Pw defmed by
JV'w= (~4f(0.),(-~(q~), K~',K/)[ (-./V'(q),), K{, K{))
(A 15)
[with W as m eq (2 22)], and hamlltomans 2
ll
E n~>l
a=l
E [(n-O,)~.
,,+(n+q)t-
1)~'.+,,_a]
n~>l
I=I
11
+~ Y'.Pt[(K[+Vi~-½)2+(K}'+V]'-½) 2] I=1 11 1
1
(A16)
1=1
for left-movers, and 2
4
~ ( w ) = E E,,~:+ E E [(,-o~)s~o_.,+(.+o¢-~)s';'+oFd a=l
n>~l
f=l
n~>l
3
+ E E [('--*~)~o-,k+('+*~--l)d'+,k-a] k=l
n~>l 3
+~ E Pk[(K;+v;-~12+(K~+ v:-l) 2] k=l 4
3
+I E ( 0 } - ¢ ) - ~ E (,~- ~k) f=l
k=l
(A 17)
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A H Chamseddme et al / Strmgs m four &menslons
for right-movers The transverse space-time coordinates X a, a = 1, 2, appear in these formulae through their number operators zaTfl= ~tc~. and ~'fl = a~ta. for left- and right-movers respectively They also bring a constant contribunon - ~ to the hamiltonlans In eq (A 17), index f refers to fermaon contributions, and
=
s ° '+O~_ l = ~~'nt + O / - i %,~,, +O/-i
,
(A 18)
analogous to eq (A 5) The hamlltonlans (A 16) and (A 17) have been written for vanishing space-rime momentum They then correspond directly to the mass formulae (2 74) and (2 75)
Appendix B MODULAR INVARIANT FERMIONIC STRING MODELS
In this appendix, we will apply the general methods of sect 2 to construct modular invarlant string models in the fern~onlc formulation We will induce the GSO projections from modular invarlance at arbitrary genus, analyze the issue of space-tame supersymmetry and examine the relationship of our formalism with the previous formulanon of Kawai et al [7, 9], i e we will cover, in the fermlomc formulation, the contents of sects 2-6 above The fermlomc field content for nght-movers is (in the light-cone gauge) the transverse components of a space-time vector '/'~ (a = 1,2) and the space-time scalars '/'~' (coming from the dimensional reduction of the NSR fields q'" in ten dimensions) and ¢0~, y~' (which can be bosonized into the world-sheet scalars X~'), for m = 1, ,6 For left-movers we have the space-time scalars q.SL, J = 1, ,44 We will assume a diagonal basis for complex fermions does exist and group the real fermlons ~ , a = 1,2, (~/'~, ¢0~, y~') and (g'~+l, ~0~+1, y~,+i), m = 1,3,5, q)~ and ~ + i, j = 1, 3, ,43 into the complex ones ~/.o, (q'R, k wR, k y~), k = 1,2,3 and q)zr, I = 1, ,22 Their boundary condmons along the non-contractible loops of the world-sheet torus [as in eq (2 1)] are expressed by vectors W as W = (O °,0 k , ( S k, T k ) I U ' ) ,
(B 1)
where all the entries are, in principle, rational numbers The partition funcnon of tins set of fermions, corresponding to the boundary conditions W and W' along the
A H Chamseddme et al / Strings m four dtmenswns
535
o 1 and 0 2 cycles, respectively, can be written as
(3 3 (22 ) × 1-I d ( U " , U ' , "r) ,
)
Dw, = I-Id(o/,o,, .,-) I-I d(s'k,s k, "~)d(T'*,TL ~') * /=0
k=l
(B 2)
I=1
where the d-funcnons are defined in eq (2 7) The function (B 2) transforms under modular transformation as m eq (2 24) wxth a = 1, and the modular mvarlance of the partmon function corresponding to a set of boundary conditions ( W },
~'=
Y'~ e( W ) w (w,w'} W' D w "
(B3)
translates into the set of condmons (2 30)-(2 32)for the coefficients e( w ) We can define (following the &scusslon in sect 2) the full set of vectors ( W ) in terms of a basxs Wp such that k
aW =- ~_~apWp
I
(B 4)
In thts way we can define the total partition function with a proper normahzatlon as
~=
~
l-,1,}
e b Dbw,
(B5)
where NpWp = 0 (mod 1) and
e(;)=exp(2Crtp~,qbpepqaq)
(B 6)
The phases epq sansfy the conditions Eqp ~ -- Epq, Npl3pq = NqEpq = 0
to provide the modular lnvarlance of eq (B 5)
(mod 1),
(B 7)
536
A H Chamseddmeet al / Stringsm four dzmenswns
Now, using the relation (2 7) between
d(O',O,~) and Zff,(r) and
~bWw=Tr{q*'(7-~ae'(Y-W)exp[2~r,(Wo-bW) .Ar~] },
(B8)
where Wo ts the vector with all entries equal to 1, j~ff: and ~ r the hamlltonlans for left- and right-movers and JV'~ the occupation number operator defined in eq (2 2), we can write the total partition function (B 5) as
,o~= ~_~Trl q~e(a~r(aq~) e-2,~rxo(~) (1, 1N 2 exp[2rr'(bp+ Spo)ePl,(a)] ] {"}
(b}
L
(B 9) The phases in eq (B 9) are defined by
dpp=epqaq-- 1 %
aW+ %
Wo+ %
A(aW) -
Wp JV'A~,
(B 10)
where the dot lndmates the lorentzlan product, and
Xo=(eoq-lWo
%)aq
(Bll)
Using similar arguments to those leading to eq (2 59), one can see that Npdpp= 0 (mod 1) Is a necessary condmon for modular xnvanance and, therefore, ffp = m/Np, where m = 0 , 1 , ,Np-1 In that case eq (B9) becomes a product of (GSO) projectors lip ~p, where
1 N,,-i _
e2~r'(b?+apo)%
(B 12)
Only those states, In the aW sector, satisfying the condition ~p = 0 do in fact contribute to the partmon function (B 9) The condition Npgpp= 0 (mod 1) restncts the possible chomes of basis vectors to those satisfying
NpWp Wo = 0
(modl),
½NpW? Wq= 0
(mod 1),
(B 13)
which are generalized level-matching conditions, as m eq (2 62) The spln-stanstics connection is provided by the phase Xo(a) m eq (B 9) if we impose the (physical) condition that the gravaon belongs to the spectrum of physical states
537
A H Chamseddlne et al / Strmgs in four dimensions
The gravlton b~/t210)R ® ~/~tl0)L IS in the Wo-subsector, whose vacuum energies for right- and left-movers are - 1 and - 1 , respectively Its occupation number operator has elgenvalues ~4Pgrav"°n ) W = ( ± 1' 03' 0 (0' 03)102z
(B 14)
and therefore wp
d/" gravlton
Wo
=
0
= sp
(mod 1)
(B 15)
using the notation of ref [7]* The con&non that the gravlton state satisfies dpp= 0 is equivalent to the condition
eoe=½Wo W e + s p ( m o d l )
(B16)
Replacement of eq (B 16) into eq (B 11) translates into (B 17)
Xo = a qSq,
which provides the spin-statistics connection m the partition function (B 9) World-sheet and space-time supersymmetnes constrain the possible values of entnes in boundary vectors Wp In particular, world-sheet supersymmetry is guaranteed by imposing that the supercurrent be periodic or antipenodlc on the world-sheet torus This condlnon translates [7] into the triplet constraint O* + Sk + T k = O °
(modl)
(B18)
The xssue of space-time supersymmetrles is related to the presence of grawtmos The boundary vector
(B 19) is consistent with world-sheet supersymmetry, eq (B 18), and its presence m the set of basis vectors determines the existence of fermlonlc states and space-time supersymmetries Actually the gravltlnos are in the W0 + W6-subsector, which has vacuum energies 0 and - 1 for nght- and left-movers, respectively The number operator for the gravmno state is
)),
x,rav, t,oo ( ao, al, a2, a3, (0, 0) 3 0 22 wo+ w~ * Notice that 2Op= 0 (mod 1) and so 0° -= -Op° (modl)
(B 20)
A H Chamseddme et al / Strings in four dtmenstons
538
where a / ~ {0, 1 }, l = 0, ,3 corresponding to the reducible splnorlal representation of SO(8), with two oppostte chlrahtles The G S O condltmn 3
q~O=~ Y~ a,=O
(modl)
(B21)
/=0
will select one chlrahty, 1 e that corresponding to the eight states with an even n u m b e r of one's, corresponding to the four gravltlnos (two hehcltles each) of N = 4 s u p e r s y m m e t r y The G S O c o n d m o n s
Op=,~pO"l-Sp ~Wp W~)-l-Wp A(Wo-l- Wo)- Wp __
= 0
1
. ~ gravltlno __Wo+W 5
(modl)
(B 22)
[where we have used condition (B 16)] is satisfied provided that ep8
ep6=Sp+½W p W6-VVp A(Wo+Wg) which m turn lmphes, using
(modl),
IS
fixed to (B23)
2ep6 = 2Sp = 0 (mod 1), 3
½ Y[0~=0
(modl)
(B24)
/=0
that is a necessary condition for the existence of s p a c e - t i m e supersymmetry Notice that both W0 and W6 do satisfy condition (B 24) The n u m b e r of supersymmetrles will depend on how m a n y gravltlnos do satisfy the modular lnvariance condition 3
COp= E a,O~ = 0
(mod 1)
(B 25)
/=0
Since we are assuming condition (B 24) is satisfied, condition (B 25) always holds for the two states a t = 0 and a t = 1, l = 0, ,3, and so guarantees at least N = 1 s u p e r s y m m e t r y If 0p/ + 0p = 0 (mod 1) for l, m = 0, ,3, l =g m, then the other six states in eq (B 20) are retained giving rise to N = 4 supersymmetry This will h a p p e n if 0J = 0 or 0p/= ½, for I = 0, ,3, as was the case with W0 and W6 If 0et + 0 7 = 0 and Op/'+ 07" = 0 only for l ~ m 4= l' 4= m', then only two states out of the six remaining in eq (B 20) will stay and give nse to N = 2 supersymmetry This is the case if we Introduce, for instance, the a d d m o n a l twist (0 °, 01,02,03) =
A H Chamseddlneet al / Strmgs in four dtmenstons
539
( 0 , 0 , ~ , - ½ ) Finally, if 0I + 0 7 ~ 0 ( m o d l ) for l , m = 0 , ,3, l ~ m , then the theory is N = 1 supersymmetrlc, as would result from the mtroducnon of the addmonal twist (0 °, 01, 0 2, 0 3) = (0, ~,0,--1 ~)1 Finally we can make contact with the formulation of Kawal et al [7, 9] by noticing that one can write the phase (B 10) as dpp= k p q a q - We a W W (kop + Sp) - Wp .A/a-W,
(B 26)
where kpq IS defined by m
kpq : Epq -- 1Wp Wq + Wp Wq,
(B 27)
kop+ sp= Wo Wp
(B28)
and, usmg eq (B 16),
Using the defimtlons (B 27) and (B 16), it is straightforward to check the following identities
kpq + kqe = Wp Wq
(mod 1),
kpp+kpo+Sp= ~Wp Wp ( m o d l ) ,
(B29)
which agree with the results of Kawal et al [7, 9] Notice that, from eq (B 28), we obtain koo = ~ This is a consequence of our choice of Wo = W0 Taking W0 such that W0 14/o= ~(modl) would lead to k 0 o = 0 However, there is no loss of generahty in either choice smce it just translates mto a global phase m the total partition function
Appendix C AN EXAMPLE THE Z~ ORBIFOLD In this appendix we will illustrate the general method presented throughout this paper with the simplest non-trivial example, the Z3-orblfold of ref [3] The model is defined by 6
A W = aW3 + coVo + c,V1 + ~ob ~ cfl71 ' I=1
(C 1)
540
A H Chamseddme et al / Strmgs m four &mensmns
where a = 0,1, 2, c o, c 1, c~ = 0,1, and the twist vector is W3 = R 3 + V3, where R 3 = (0 ' 31-' 1 '
23 ' 3' 1 3' 1
32 131'13 '
2 , 0 8) ,
V3 = (0610i9, 1 V o ~ ((1"~61[1"~22]
~j i',~j
],
V1 = (061014, (1)8) , 1,7i = (0 ' - 1, 1 , 0 6 - 1 1 0 ' - 1, _1 2~ vfl22-"1)
(c 2)
These vectors are consistent with the level-matching c o n d m o n s (2 62) The vectors I~I are i m p o r t a n t in the compactlflcatlon of the six extra coordinates, as we will see The only free phases are e11 and els, I, J = 1, ,6, which can be given the values 0 or ~ The others are equal to zero W e will use GS fermlons In the right-moving sector to analyze the model Using N S R fermlons would lead to the same spectrum of states, as shown b y the general o n e - t o - o n e correspondence of sect 5, since R 3 in eq (C 2) satisfies condition (5 19) In the untwisted sector, the vacuum energies are 0 and - 1 for right- and left-movers, respectively If we define the occupation n u m b e r operator ~1" as ~4/'= (JV'R[.A/'L),
(C 3)
then the massless states for right-movers are the bosonlc la)R, [Zk)R, a = 1,2, k = 1, 2, 3 and fermlomc [ S t ) R , 1 = 0, ,3 zero modes, with occupation n u m b e r operators JV'R(la)R ) = ( 0 4 , 0 3 ) , . f i r ( 1 Z k ) R) = (04, 1_.02) ,
a = 1,2, .A/'R(1z ' k ) R) = (0a, -- 102),
(C4)
c o r r e s p o n d i n g to the decomposition 8_o = 3 + 3 + 1 + 1 of SO(8) under S U ( 3 ) c SO(6) - SU(4), and ~/'a(8F) = (a0, al, a2, a3) with a t ~ (0,1}, l = 0, ,3 and Et at = even, corresponding to the components of 8F* related to 80 by s p a c e - t i m e s u p e r s y m m e t r y In particular, if we decompose 8 F with respect to SU(3) × SO(2) and identify the s p a c e - t i m e hehclty X with the SO(2) entry as X = a 0 - 1, i e * At this point it is irrelevant which chlrahty, 8c or 8,, we have chosen for the spmor representation of
so(8)
A H Chamseddme et al / Strings m four dtrnenstons
541
8F = 31/2 -t'- ! - 1 / 2 "~ 11/2 "1" 1-1/2' we can write .A/'R ( [ S ° ) g ) = (1,13),
= 0,!02),
-4/'R([S°)R) = ( 0 , 0 3 ) ,
WR0S _ R) =
(c 5)
where + refers to the hehclty + ~ These states are classified by their elgenvalues under the operator W3 ~b0 JffR
(b~0),
(C6)
which appears in the GSO projection 0 °b, 3 see eq (2 54) In particular, the vector supermultiplet [a ) n ~ [S ° ) R has zero elgenvalue, the three chlral supermultlplets with hellclty ~,1 [Zk)R • [Sk+)g, elgenvalue ~, and their complex conJugates with opposite hehclty, [ Z ' k ) R ~ [Sk ) R, eigenvalue For left-movers in the untwisted sector the vacuum energy is - 1 and the only subsectors with massless states are V0, Vo + V1 and VI + F~119t In the V0-subsector, the massless states are Ib)L, b = 1,2 and [I)L, I = 7, ,22, with ~ b o jtp L = 0, [ I ) e , I = 1, ,6 with occupation number elgenvalue ~b0 ~ri. = ( + 102, 08) and 1413 ~ b o .4/-L = + ~, and K I = ( + 1 + 1,02°), which are the non-Cartan generators of SO(44) Only those states consistent with the modular invanance conditions (2 54), (2 55) remain In particular, 01 = - ¼K I = 0 implies K 1= 0, I = 1, ,6, destroying the non-Cartan generators of SO(12) c SO(44) corresponding to the first six entries of K I It is strmghtforward to check that the massless states consistent with conditions (2 54), (2 55) are the gravlton supermultlplet, a vector supermultlplet in the adjolnt representation of SO(16) X SO(10) x SU(3) x U(1), nine smglet scalar supermultiplets ~k, k, j = 1, 2, 3, and three dural supermultiplets in representation (10 + ! , 3 ) of SO(10) x SU(3) In the Vo + VI-SUbSector the massless states consistent with the modular lnvanance conditions 00 = 01 = 0 are the 128 representation of SO(16) D SO(10) × SU(3) × U(1) The condition 0i = 0 is consistent with the massless states only provided that eli = 0, that we will assume hereafter (Of course if some eli = ~ then all the massless states in this subsector are projected out ) Finally the states consistent with 03 = 0 are a vector supermultlplet in the 16 + 16 splnorlal representation of SO(10) (Together with the vector supermultlplet in the adjomt of SO(10) × U(1), they will construct the adjomt of E 6 ) , and three chlral supermultlplets in the (16,3) of SO(10) × SU(3), that will build the 3(27,_3) with respect to E 6 × SU(3) In the Vi + Y.1 Vrsubsector the massless states consistent with 00 = 01 = 0 are m the 128 representation of SO(16) For those states, 0~ = 32~ e~j = 0 and so we will take e1j = 0 The massless spectrum in this subsector is therefore a vector supermultlplet in the 128 of SO(16) that, altogether with the adjolnt representation of SO(16), will complete the adjomt of E 8
542
A H Chamseddme et al / Strings m four dlrnenstons
A t this point, at as appropriate to mention that the compactaficatlon vectors I~1 p r o d u c e the self-dual torus lattice (D1L × DIR) 6 This lattice is consistent with m o d u l a r mvarlance, produces the correct massless spectrum, but as inconsistent with the group law in eq (2 14) Our example reproduces exactly the twisted sectors of the Z s orblfold, but differs for the massive lattice states of the untwisted sector This p r o b l e m can be cured by replacing the torus partition function in the untwisted sector b y that of (SU(3)c × SU(3)R) 3, which however cannot be written in terms of d functmns I n the twisted sector ( a = 1, 2), the projectors are
= (14, (- 1)31(- 1)3,08),
= (06106, 116),
(C 7)
and then the six left + right internal coordinates are shtelded by the twist, i e
¢,= (o ,o
(c s)
I n this way the hanultonlans and GSO-projectlons in the a W 3 + cpVp-subsectors satisfy the c o n & t m n s o~g" r ( a W 3 + c pVp + c , l ~ t ) = 9fie' r ( a W 3 + CeVp)
(C 9)
T h u s the vectors 1~r are not independent m the twisted sector and must be d r o p p e d f r o m the basic set of vectors This ~s the reason for the factor ~ a b I n eq (C 1) In the twisted sector the v a c u u m energy for right-movers is zero and the ground state in the a = 1 twisted sector IS the cbaral supermultlplet, with one hehclty, ( X b, S °) while the ground state in the a = 2 twisted sector is the complex conjugate charal supermultlplet with opposite hellclty This can be more easily understood using N S R fermlons In that case the ferrmomc states appear in the a W 3 + cpVp subsectors Their fermlonic ground states for right-movers are the zero modes with o c c u p a t m n n u m b e r elgenvalues .AtR = ( a o , 0 3 , 0 3 ) where a o = 0,1 provides the two hehcities, 2~ = a o - ½ The modular invariance condltmn ¢6 = 0, corresponding to the vector W6, that should appear in the NSR-formulatlon, gives rise to the con&tlons
la_Wf ) ~
,~/'R = ~1 a - T a1o - 0 __
(modl),
(c lo)
f r o m where just one hellclty per twisted sector [ a o = a (mod2)] is selected The b o s o n l c states appear in the W~ + a W3 + %Vp subsectors The bosonlc ground states are the zero modes X b, b = 1, 2, w~th zero occupatmn n u m b e r eagenvalues, sV"R = (0 4, 0 3) The modular mvarlance condition q~6 = 0 is automatically satisfied by these states In the twisted (a = 1, 2) sectors The massless states in the twisted sector do appear in the subsectors a W 3 + Vo and a W 3+ Vo + V a In particular, m subsectors W3 + V o and W3 + Vo + V 1, the
A H Charnseddme et a l /
Strings m four dlmenszons
543
massless states consistent with the modular mvariance conditions q~0 = qh = eP3 = 0 are 27(10 + 1 , 1 ) + 81(1,3_-) and 27(16,1) chlral supermultlplets with respect to SO(10) × SU(3), respectively, making supermulnplets in the 27(27, 1) + 81(1, ~) with respect to E 6 × S U ( 3 ) Of course the antiparticles with opposite hehcitles do appear in the subsectors 2 W 3 + V0 and 214/3 + V0 + V1 The number 27 corresponds to the n u m b e r of fixed points present in the twisted sectors as given by eq (2 16)
References [1] D J Gross, J A Harvey, E Martmec and R Rohm, Phys Rev Lett 54 (1985) 502, Nucl Phys B256 (1985) 253 B267 (1986) 75 [2] P Candelas, G T Horowltz, A Strommger and E Watten, Nucl Phys B258 (1985) 46 [3] L Dixon, J A Harvey, C Vafa and E Wltten, Nucl Phys B261 (1985) 678 [4] L Dixon, J A Harvey, C Vafa and E Wltten, Nucl Phys B274 (1986) 285 [5] K S Narmn, Phys Lett B169 (1986)41, K S Narmn, M H Sarmada and E Wltten, Nucl Phys B279 (1987) 369 [6] M Mueller and E Wltten, Phys Lett B182 (1986) 28 [7] H Kawal, D C Lewellen and S -H H Tye, Phys Rev Lett 57 (1986) 1832, 58 (1987) 429(E), Nucl Phys B288 (1987) 1 [8] I Antomadls C Bachas and C Kounnas, Nucl Phys B289 (1987)87, I Antomadls and C Bachas, Nucl Phys B298 (1988) 586 [9] H Kawm, D C Lewellen, J A Schwartz and S-H H Tye, Nucl Phys B299 (1988) 431 [10] W Lerche, D Lust and A N Schellekens, Nucl Phys B287 (1987) 477, W Lerche, A N Schellekens and N P Warner, Phys Rep 177 (1989) 1 [11] K Narmn, M H Sarmadl and C Vafa, Nucl Phys B288 (1987) 551 [12] L E Ibafiez, H -P Nllles and F Quevedo, Phys Lett B187 (1987) 25, B 192 (1987) 332, L E Ibahez, J E Klm, H - P Ndles and F Quevedo, Phys Lett B191 (1987) 282, L E Ibahez, J Mas, H-P Nxlles and F Quevedo, Nucl Phys B301 (1988) 157, A Font, L E Ibgfiez, H-P Nllles and F Quevedo, Nucl Phys B307 (1988) 105 [13] A Font, L E Ib5~tez and F Quevedo, Phys Lett B217 (1989) 272 [14] A H Chamseddlne and J -P Derendmger, Nucl Phys B301 (1988) 381 [15] A H Chamseddme, J - P Derendmger and M Quxros, Nuel Phys B311 (1988)140 [16] A H Chamseddme and M Qturos, Nucl Phys B316 (1989) 101 [17] D Gepner Nucl Phys B296 (1987)757, Phys Lett B199 (1987)380 [18] C Vafa, Nucl Phys B273 (1986)592 [19] N SeLberg and E Wltten, Nucl Phys B276 (1986) 272 [20] M B Green, J H Schwarz and E Wltten, Superstrlng theory, 2 Vols (Cambndge U P, Cambndge, 1987), M Kaku, Introducnon to superstnngs (Springer, Berhn, 1988) [21] L Alvarez-Gaume, G Moore and C Vafa, Commun Math Phys 106 (1986)1 [22] S Parkes, Phys Lett B184 (1987)19 [23] Y Kakuchl and C Marzban, Phys Rev D36 (1987) 2583 [24] A N Schellekens, Phys Lett B199 (1987) 427
