Fermionic realizations of superstrings on orbifolds

Volume 221, number 3,4

PHYSICS LETTERS B

4 May 1989

FERMIONIC REALIZATIONS OF SUPERSTRINGS ON ORBIFOLDS R. P E S C H A N S K I Service de Phystque Thborlque de Saclay ~, F-91191 Glf--sur-Yvette Cedex, France

and C.A. SAVOY Dzvtston des Hautes Energws, Centre de Recherches Nuclbatres, F-6 703 7 Strasbourg, France

Received 20 January 1989

Free fermion representations of four-dimensional superstrmgs compactified o n T 6 / Z 4 and TJZ8 orblfolds are constructed The underlying boson-fermlon eqmvalence reduces to identities for products of Jacobl 0-functions with rational arguments which are derived. Some obstacles to construct the fermlonlc realizations of the remaining orbffolds consistent with superstrmg compactlficatlon are pointed out

Many solutions for superstring in four-dimensional space-time have been worked out that reahze the required symmetries in the two-dimensional world sheet, using different methods [ 1-6 ]. The bosonic formulations often admit a geometric interpretation in terms o f the compactlfiCatlon o f six coordinates on an adequate manifold, or even on orbifolds obtained by the quotient o f a torus by a discrete symmetry. The simple geometry of T6/7/N orbifolds allows for a systematic investigation of solutions [ 1,4] which are interesting as limiting cases of Calabi-Yau manifolds and also because some of them are phenomenologlcally attractive. In the fermionic approach [ 7 ], classical superstring solutions are consistent sets o f spin structures for a certain number o f fermions (64 for the heterotlc string) that encode the spin and internal degrees of freedom. World-sheet supersymmetry is non-linearly reahzed in the fermionic sector with a cubic term in the supercurrent [ 8 ]. The b o s o n - f e r m l o n equivalence is a well-known property o f a two-dimensional theory on a torus. This problem has also been studied in the case of a quotient of a torus by Z2-symmetrles (reflexlons) [9 ]. In this paper we will search for a fermlonic equivalent o f b o s o n s compactified on orbifolds and then construct the fermionic formulation of the superstrlng on T6/77N orbifolds. Our approach relies upon the derivation o f identities for products o f Jacobi 0-functions allowing to express the partition function for twisted bosons in terms of partition functions o f systems of fermlons with given spin structures on the torus. We find relations involving 2~Nboundary conditions for all values o f N, but they are not enough to provide a fermionic picture in all cases. So, among the thirteen different cases of left-right symmetric orbifolds suitable for supersymmetrlc four-dimensional compactlfiCatlon of the heterotlc superstrlng [ 10], we are able to fermionize the compactified bosons for Z2, two Z4 and one of the two •8 orbifolds. Let us first consider the partition function of left-moving free world-sheet fermlons and bosons [ 11 ] on the torus, defined by a lattice (a~, a2 ) with (complex) modulus r. They are expressed in terms o f Jacobi 0-functions in the variable z. A complex left-moving fermion is defined by its spin structure which, on the torus, corresponds to the two boundary conditions Laboratotre de l'InstltUt de Recherche Fondamentale du Commissariat h l'Energle Atomtque 276

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~/(aI +2n, a2) = - exp(2ina)~u(al, a2), ~/1(0"1,a2 +2n) = -exp(21nfl)~(al,

4 May 1989

a2),

(1)

and leads to the partition function

0.., = e x p [ 2 i n a f l + i n ( ° t 2 - ~ ) ] I~ {l+exp[2im(n-a-½)-2infl]}{l+exp[2inr(n+a-½)+2anfl]}.

n>0

(2)

A similar expression, namely (~+~), is obtained for the partition function of the winding states of a real leftmoving boson, with shifted boundary conditions X(al +2n, a2) =X(al, az) +2ha; X(al, az+2n) =X(a~, az) + 2nil. This is related to the well-known equivalence between a free complex fermion and a real "shifted" boson. Instead, the partiuon function for a complex boson with twisted boundary conditions X(al+2n, = exp(21na) X(al, 0"2+ 27~) =exp(21nfl) X(al, 0"2), is (~) -l, the inverse of that of expression (2) for a fermion. Therefore, in order to express the equivalence between a complex twisted boson and a couple of complex ferm~ons in terms of the equality of the corresponding partition functions, one has to convert an inverse of 0-function in a sum of products of two 0 functions. For this sake, we derive identities among 0-functions expressing that specific products ]~, (p~') are constant (these ~dentities are implicit in a relation obtained in ref. [ 6 ] within a different framework). Consider the identxty

0"2)

0"2),X(al,

I~ (l--X2n)(1--X2n-l) ,>o 1-x n =l=I~

n>0 (I+x")(1--Xn-1/2)(I+xn-1/2),

(3)

and take x = exp (2inNr) with N even. From definition ( 2 ), one obtains the identity

r=, (2r+N-1)/ZN

(2r+N-1)/2N

r/N

= ( - - 2 ) N / 2 r=ll--[sin((g/N)[r+½(N-1)]),

(4)

where the superscript ( # ) means that for the particular value r= N / 2 , one has to replace (~) by (~)1/2(°)t/2 Analogously for N odd, one gets r:l

r/N

r/N

r/N+

r

cos( gr/N).

(5)

Once a pamcular expression (4) or ( 5 ) is established for each value of N, a whole set of independent identities, denoted RZN, lS obtained by repeated use of the relations [ 12 ] (fl) (-1/z)=exp(2ino~fl)(_fl)(z),

(~)(r+l)=exp[-in(o~2-o~+~2)]

OL

(B+o~+½)(z),

(6,

which describe the behavlour of partition functions under modular transformations. The set RaN lS defined up to the well-known symmetries of 0-functions: (~) = ( -~ ) = ('~1) = exp (2in~) (a-~l). Along these lines, we find for R4:

(0) (i) (0~1/2(0~1/2 3

\½J

\0J

(~) (i) (~)1/2(00) 1/2 =exp(-ln/4)

=exp(-in/2)

(i)(i)(½~1/2(0~ I/2 \0J

\~]

=-~/2'

277

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4 May 1989

for R6:

1=-(~)

(i)(~)=exp(+-~i/g)

l=0,1,2,

(--}}l) (½ ~ }l) ( ~ l ) '

for Rs: ] ix/8=exp(

\0J

\½J

=exp(-~in)

\0J

''2 (~.,)(. ,.,)(_~,)(,_~0,)(j~,)(.,~0~"2(~ ,o~,

_,~i,,,

\½J

,__o, 1, ~.~.

Let us only quote the first relations for Rio , Rl2 , RI4-"

(o)(o) ()(~)(:)(') 0

(°)(,t)

0

~

~

=4cos½ncos2

(0) (~) (~) (~) __,.

(t)(;)( o) (i) (i) (o) (~) (i) (o) ,co~cos~co,~, Now that these relations are established, one may consider the four-dimensional superstring obtained from the heterotlc E8 X E8 superstring by compactification of six dimensions as a Zff-orbifold. The six bosomc coordinates can be grouped in three complex ones X,, such that the twisted sector (m, n ) corresponds to the following condition:

X,(a~ +2n, a2) =exp(2inma,/N)X,(al,

0"2), S~(0-1,0"2 +2n)

=exp(2inna,/N)X,(0",, 0"2),

where a, (t = 1, 2, 3 ) are integers, verifying E, a, = 0 if one requires space-time supersymmetry. There are thirteen possible ~-N(a,,a2, a3) orbifolds [ 10] to be considered, namely ZN(O, 1, -- 1 ) with N=2, 3, 4, 6 (solutions with two space-time supersymmetrles); ZN( 1, 1, --2) with N=3, 4, 6, 8; ZN( 1, 2, --3) with N=6, 7; Z8( 1, 3, - 4 ) ; 2~,2(1, 4, - 5 ) , 72~2(1, 5, - 6 ) . The partition function for the twisted sector (m, n) can be written as [ 13 ] 5

z,,,~=cm~

E

a,fl= {0,½}

n.~

fl

w,,,,,x

E

y,g={0,½}

---8

w~,,x

E

(,u={0,½}

,

(7)

where

(a+ ma,/N~ (o~+ma2/N~ f a+ ma3/N~ r {½+ma,/N~ f ½+ma2/N~ ( ~+ ma3/N~1-1 W"..,~= \ fl+na,/N ) \ #+na2/N J \ fl+na3/N J [_\ ½+na,/N / \ ½+na2/NJ \ ½+na3/NJ_J

(8)

is the contribution of the orblfold left-moving degrees of freedom which revolves three complex bosons and three complex fermlons, related by two world-sheet supersymmetries, while if'describe the similar right-moving degrees of freedom for the symmetric orbifolds. This system has an explicit (2,2) supersymmetry. The coefficients ~/.p= + 1 and C.,~ are determined by the requirement of modular lnvariance, in particular, the generalized GSO projections; C.,n is proportional to the number of fixed points of the Z~ symmetry. A necessary condinon for the fermionization of the superstrlng on an orbifold relies on the possibility of transforming the denominators of all W~m~.By inspection of the relations R2N, one finds only four possible 278

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orbifolds among the thirteen relevant ones, namely Z 2 ( 0 , 1, - 1 ), 2 4 ( 0 , l, -- 1 ), Z4 ( 1, 1, - 2) and Z8 ( 1, 3, - 4). Several comments are in order: (i) Some denominators, such as that in W?( are not present in the relations R2N for ZN with N = 3, 6, 7, 12, so that at least in this framework we are not able to fermionize the corresponding orbifold solutions. (it) Another constraint is to find among the constant product of nine 0-functions the three corresponding to denominators of W~f,, in order to keep the conformal balance of two fermions for each boson. This is an obstruction, in our approach, to the fermionization of Z8 (1, 1, - 2 ) . (in) We notice the existence of Rio while there is no Zs-orblfold. The same applies to relations R2N with N > 7. (iv) The relations R4 can be used to relate each Za-twisted (complex) boson to two complex fermions with given boundary conditions. Instead, R8 allows to transform only pairs of Zs-twisted bosons into systems of complex fermlons. (v) It is possible to put the relations R4 and R8 in a more suitable form for modular invariant systems. For instance,

(D

1_

E

e x p (V- /i~n / 4 )

3 [(0)(~)]-'_

y,,s= {o,½)

exp(-in/2)

x//~

\5+½ \ ~ J \8] '

+y'~ ,,2. ,,<~=~(o,½i(~(-l+y'~)2'\(-~½+8,]\½+8l (~Y)(½~Y)(YS)

(9)

(10)

Note that out of the four terms in the sum over configurations (~,, ~) there are only two contributions since (I) vanishes. Similar relations can be written for powers of the LHS and their moduli (appropriate for left-right symmetric twisted bosons). We now turn to the explicit construction of the free-fermion formulation of the superstring compactified on a 774 orblfold. We first recall the general framework for the fermionic description of the 4D heterotic superstring [ 2,3]. In the light cone gauge the space-time degrees of freedom are: two left-moving real scalars Xu, two leftmoving Majorana fermions !u~ and two right-moving real scalars J?~ (# = 1, 2 ). Internal degrees of freedom are represented by eighteen left-moving and fourtyfour right-moving Majorana fermions. The notation in table 1 is chosen to be suitable for the 774 symmetric orblfold, which implies, in particular, the existence of a (2,2) superconformal invariant system. With this convention the conserved supercurrent associated to the world-sheet supersymmetry of the left-moving system can be written as

G=!u~,OX~,+ ~., [Xall~.a2)~a3"t'aa(~*l']a-i'~agl*)"l-,Oa(~a#la-t'~*l']*) ],

(11)

a ~ 1,2

where ~/a, Ca ( a = 1, 2) are complex fermions and the others are all real. This supercurrent is related to the existence of a KaY-Moody algebra of fermion bilinears based on SU (2)6 because of the required space-time supersymmetry [ 3 ]. Allowed spin-structures have to leave G invarlant up to an overall sign but not necessary the KaY-Moody currents; they are automorphisms of the SU (2)6 algebra. Each modular invariant solution consists of a set E of sectors of states. A sector c ~ E is characterized by a linear unitary transformation U<~undergone by the sixtyfour fermlons for a 2n translation along a~. If under U,: G--+ - G ( G ) the states in the sector a are space-time bosons (fermions, respectively). It has been shown [2,3 ] that -= has abelian group structure (for commuting U,) and one can find a basis {b~, b2..... b,}. The neutral element (0) corresponds to Neveu-Schwarz boundary conditions for all fermions. Any element of E can be represented by bg' b~:...b~~ with 0 ~
C
1=1

]~,

,=21

(12,

fit"

The coefficients C~ are obtained by solving the lterative equations given in refs. [ 2,3 ]. In particular, they reflect 279

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Table 1 Notation for the fermlonlc content of the 4D heterotlc superstrlng The bar indicates the right-moving fermlons and the lefl-nght symmetric systems are superposed In this basis which dmgonahzes the spin structures suitable for Z4 orbffolds, r/~, ~., ~ , ~ are complex (their complex conjugates are denoted q*, ~*, 0", ~*, respectively), the others are real. The sectors discussed in the text are given m terms of the phase a, assocmted to each fermlon, see eq ( 1 )

fermlons

left-moving right-moving

~g~,

z,l Z~I

Z~2 Z~2

(gt=l,2 F

basic sectors

½

Z~3 ,~3

a=l,2

a~ #~

M=I,

,10

p~ P~

v/~ O~

~. ~

~M

IgK

0

0

K=I, .,16)

-1 1

s

(' B

~

o 0

o 0

½ 0

0 0

o 0

o 0

0

o

o

o

o

o

o

o

o

o

o

½

o

o

o

½

½

o

0 o

0 o

0 o

0 o

0 o

0 o

0 o

0 o

o

½

o

½ -1~

o o

½ ½

½ ½

o o

~ ~

~

o

o

o

0

0 o

0 o

0 o

½ ½

½ ½

½ ½

0 o

o

o

0~

0 0

0 0

0 0

0

½6~ ½6.~

½~b~ ½d~

~'~ ~, ½6~b

0 0

0

0

0

~0~ ½d~t,

½d~b ½c~~b

0 0

0 0

0 0

0 0

0 0

0

0

(b= 1, 2) Xb (b=l,2)

the presence o f generalized G S O projections, one for each element o f E or, equivalently (because o f the abelian structure), one for each element o f a basis {bl .... , b,} o f E. In this f r a m e w o r k a free-fermion equivalent o f the heterotlc string with six d i m e n s i o n s c o m p a c t i f i e d on the T6/7/4 orbifold is given by the basis B = {F, S, ~, ~', B, O}, in terms o f the sectors defined in table 1. N o t e that B IS the only Z4 a u t o m o r p h l s m in B (B 4m~-~). W i t h the definition Fy C~ Z~ = C~j Z'~ + C~ ~Z~ ~ + C~pZ~p + C~. Z~y where y is a Z z - a u t o m o r p h i s m ( y 2 = 0 ) , y = F , S, (, ( ' , O in this case, the lntegrand o f the p a r t i t i o n function can be formally written 4

"~Bo "-e. , m,n

~

(13)

l

where x = F S ( ( ' O. It can be checked that each term in ( 13 ) for given m and n is equal to Z , , , in ( 7 ) for the (m, n ) sector corresponding to the Z4 ( 1, l, - 2) orbifold, by using relations similar to ( 9 ) . Note the relevance o f the sector O in B, since for r n = n ~ O the expression FoF~Zg~ appearing in (13) does not vanish, while Z eB .n = 0 because o f the presence o f fermions periodic on the torus. The spin-structures defined by each sector in B e m b o d y some physical properties. First consider the solution given by the basis {F, S, (, (' }. The sector S defines a block o f eight real left-handed fermions, including gu, with the same spin-structure a n d results in N - - 4 s p a c e - t i m e supersymmetry. The sectors ( a n d ( ' separate two blocks o f sixteen and one o f twelve right-moving real fermions with the same spin-structures which give the E8 × E~ × O (12) gauge symmetry. By s t a n d a r d b o s o n l z a t i o n this solution corresponds to a particular T 6 c o m pactlficatlon o f the heterotic string [ 1 ]. The a d d i t i o n o f the sector O into the basis separates these twelve ferm i o n s in two blocks o f eight a n d four, corresponding to compactificatlon on T4 × T2 and the breaking o f O ( 1 2 ) 280

Volume 221, number 3,4

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into O (8) × O (4) gauge symmetry. The introduction of the sector B that completes the basis B for the Z4-orbifold compactxficatlon has two main consequences: (i) GSO selection rules which project out the appropriate components of the preceding ("untwisted") sectors; (11) new ("twisted") sectors generated by multiplication by B, B 2 and B 3. Hence, among the four would-be gravitini in the sector S, only one survives this GSO projection: the solution has N = 1 supersymmetry. While all the gauge bosons (and gauginos) of the E~ symmetry remain unaffected by the proj ecUon, the E8 gauge symmetry is broken to E 6 × SU (2) × U ( 1 ) and the O (8) X O (4) one is broken to U ( 2 ) × U ( 1 )4. Therefore the rank of the gauge group is preserved. The left-handed fermions that couple to the E 6 gauge bosons as 27 (families) and 27 (antifamihes) appear in the sectors of the form Sfl and S(fl, as follows: "untwisted" ( p = 0 ), with one 27 and five 27's, "twisted" (fl= B, Bx., BO, BxO), each with four 27's "twice-twisted" (fl=B2), with two 27's and six 27's, (fl=B20) with four (27 + 27 )'s. The overall number of families and antifamllles is then: seven (27 + 27 ) and twentyfour chlra127's. For both untwisted and twisted sectors this fits with the multiplicities found in ref. [4 ]. In spite of the fact that the O sector in B is necessary for the bosonization in terms of the Z4-orblfold coordinates, as already stressed, the 4D superstring solution obtained by removing O from B is clearly consistent. The massless spectrum is different: the gauge group is now E~ X E6 X SU (2) X U ( 1 ) × O ( 4 ) × O ( 4 ) × U ( 2 ), the number of chlral famlhes is always twentyfour, but the number of 27's is reduced to fve. This is not a Z4 orblfold, since its partition function is different as &scussed above, and its geometric interpretation in terms of compactlfied bosons remains unclear. The fermionic construction of the Z4 orbifold is not unique. Indeed, the spin structures defined by the basis/3 are all consistent with pairing the real fermions into complex ones according to their subscript a = 1, 2. This (standard) complex structure does not appear in a different fermionic representation of the 774 orbifold, defined by a basis {S, ~, ~', B, Xb, 0b} ( b = 1, 2) with the sectors defined in table 1. Note that X2a = 0 a2= 0 , 0102=O, x~ x2 = x, and F = S ~ ' XlX20102. As a consequence of the new GSO projections, the gauge group is reduced to E~ × E 6 × SU (2) × U ( 1 ) × U ( 1 ) 2, so that the rank is reduced by four units. As a matter of fact, the introduction of "Z2-twxsted" sectors that split would-be Dirac fermions into pairs of Majorana ones is a general, and seemingly unique way to reduce the rank of the gauge group when the group of spin structures is assumed to be abelian. The geometric interpretation of this property is the fact that Dlrac fermions bosonize as "shifted" scalars (tori) while Majorana fermions bosonize in pairs as "shifted-twisted" scalars (Z2-0rbifolds), as discussed, e.g., in ref. [9]. The massless E6 families and anti-families are simply redistributed among the larger number of sectors w~th respect to the solution/3. Thus, the corresponding left-handed fermions appear in the sectors of the form Sfl and S~/~, as follows: (i) all 27 and 27 m the untwisted sector (/~= 0) survive the two new GSO projections; (ii) each one of the sixteen twisted sectors, fl = Bx'~' x~~07307~ (n, = 0, 1 ) contains one chira127; (lii) there are four "twicetwisted" sectors, with four chira127 for fl=B 2, and two (27+27) for each of the remaining ones, fl=B"01, B20:;, B20~02. Furthermore, there are two additional fermionic realizations of the Za-orbifold obtained by adding to the basis B either xa or Oa. The gauge symmetry U( 1 ) 4 × U ( 2 ) is broken to U( 1 ) 2 × U ( 2 ) and U( 1 )2×W( 1 )2, respectively. Finally, we give the fermionic representation of the superstring compactified on a T6/77 8( 1, 3, - 4 ) orbifold. The dmgonallzation of the spin structure requires writing the supercurrent G in a shghtly different basis for the fermlons,

G=~uOXu+ ~ Xa,Xa2L~s+[a(¢,qT+¢zrf~)+cr'(¢lrF,+~2rh)+p(~lq'~+¢'~qt+~'~rl'f+~2rl2)+h.c.

]

(14)

a=1,2

(where Z~,, a and tr' are real) and with an analogous notation for the corresponding right-moving fermlons. Then, consider the sector C, defined as shown in table 2. The basis {F, S, ~, ~', C, O}, with F, S, ¢, ¢' and O defined as in table 1, generates the set of spin structures for the fermionic realization of the T6/T8 superstrmg. The GSO projection due to C only N = 1 supersymmetry and the gauge group E~ × E 6 × U ( 1 ) 8. By the addition of the sector tel (or equivalently x2) as defined in table 1, one gets another fermionlc realization of this Z8 281

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Table 2 Fermlomc content of the sector C fermlons

basic sector C

g~,

o

~al

~a2

Za3

O"

O"

p

rh

q2

~1

~2

2a,

Za2

Za3

a

a'

fi

ql

q2

~l

~2

~M

½ 1_2

o o

½ ½

o o

½ ½

-I ~

~ ~

] ]

~ ~

]

o

orbffold solution, w i t h the gauge g r o u p b e i n g r e d u c e d by t w o U ( 1 ) factors. Instead, the i n t r o d u c t i o n o f a sector 0b in the basis w o u l d be i n c o n s i s t e n t w i t h the sector C, by the rules o f ref. [ 3 ], in a g r e e m e n t wath the fact that 7Js-twisted b o s o n s h a v e to be f e r m i o n i z e d by pairs as n o t i c e d a b o v e . We a c k n o w l e d g e a d i s c u s s i o n w i t h F. D e l d u c t h a t m o t i v a t e d th~s i n v e s t i g a t i o n a n d the help o f J.-P. D e r e n d i n g e r in its initial stage. O n e o f us ( C . A . S . ) t h a n k s the t h e o r y g r o u p o f F N A L a n d the S e r v i c e de Phys i q u e T h 6 o r i q u e at Saclay for hospitality.

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