Volume 178, number 2,3 PHYSICS LETTERS B 2 October 1986

C O N F U S I N G T H E H E T E R O T I C S T R I N G

D.L. B E N E T T

Department of Physics, The Royal Danish College of Pharmacy, Universitetsparken 2, DK-2100 Copenhagen O, Denmark

N. B R E N E

Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen 19, Denmark

Leah M I Z R A C H I

Dbpartement de Physique Thborique, Universitb de Genbve, CH-1211 Geneva 4, Switzerland

and

H.B. N I E L S E N

Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark

Received 1 July 1986

A confusion mechanism is proposed as a global modification of the heterotic string model. It envolves a confusion hy- persurface across which the two Es's of the heterotic string are permuted. A remarkable numerical coincidence is found which prevents an inconsistency in the model. The low energy limit of this theory (after compactification) is typically in- variant under one E a only, thereby removing the shadow world from the original model.

String theories *~ seem at present the most promis- ing candidates for Theories of Everything. In particu- lar, the heterotic string [2] endowed with E 8 × E 8 symmetry would appear to enjoy special favour due to considerations of low energy phenomenology [3]. Furthermore, proponents of the heterotic string claim a high degree of uniqueness as an important virtue of the theory. Here, we present a "confusion" mecha- nism which admits a broader class of theories in a manner reminiscent of confused gauge theories (gauge glass) [4].

By introducing hypersurfaces of codimension one in a space-t ime of appropriate topology, the adoption of E 8 × E 8 as the gauge group opens up the possibili- ty for exchanging the two E 8's upon crossing the

.1 Some of the works referred to here and many others can be found in ref. [1 ].

0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

hypersurfaces, which we call confusion surfaces. As a result, the gauge group typically breaks down to a

single E 8 . The usual compactification procedure [2] for ob-

taining the 4 dimensions of low energy physics from the 26 dimensions inherent to the heterotic string is performed in two independent steps: compactifica- tion of the 16 internal degrees of freedom into a torus

(T16 = T 8 X T8), followed by compactification of the ten-dimensional space of external coordinates in- to 4 dimensions. In our modified scheme we introduce an interdependence in the compactifications of inter- nal and external coordinates *2 such that the respec- tive topologies of these spaces become intertwined

.2 Compactffication procedures directly from 26 to 4 dimen- sions, which link external and internal coordinates were studied in ref. [5 ].

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Volume 178, number 2,3 PHYSICS LETTERS B 2 October 1986

globally while retaining the original structure locally (i.e. locally a direct product of T16 with the external space). This is accomplished with our confusion mechanism by imposing new boundary conditions .3 corresponding to the exchange of the two E 8's across the confusion surface.

Presented here is an investigation of the confusion mechanism for the heterotic string. In particular, a classical action is constructed from which the modi- fied boundary conditions across a confusion surface are derived. The physical state condition is used to obtain a restriction on the eigenvalues of a generator M of rotations along S 1 circles of a compactified co-

ordinate. To test the model, the quantization condi- tions for M are considered in a cylinder-like as well as a Klein bottle-like [7] compactification. In neither case did inconsistencies arise. This is taken as a hope- ful sign for the full consistency of the model which would then pose a challenge to the purported unique- ness of the heterotic string. Finally, a possible mecha- nism for the collapse of the E 8 × E 8 gauge group in- to a single E 8 is presented which removes the shadow world of the original model [3].

In characterising confusion surfaces, we limit our considerations at this stage to close hypersurfaces thereby avoiding potential interesting but trouble- some confusion "source" terms that could arise on the boundaries. As any deformation of a confusion surface can be compensated by relabelling the two E 8 groups in the region swept by the deformation, it is seen that the absence of boundaries ensures that the hypersurfaces can be continuously relocated and dis- torted without physical significance. Therefore, our restriction to closed hypersurfaces maintains the local structure of 10-space intact and any physical consequences of confusion must be sought in the global topology of the hypersurface embedding in ex- ternal space. In particular, a physically meaningful confusion surface (i.e. not removable by the above mentioned relabelling procedure), must not divide the space. This precludes contractible confusion sur- faces. Accomodation of such a hypersurface requires a non-simply connected 10-space (fig. 1), which can come about in compactifying (& la Kaluza-Klein 4:4)

t3 Different boundary conditions in string models were analysed in ref. [6].

*4 For a review of the Kaluza-Klein approach see ref. [8].

Sz Tz

S= S: confusion confusion

surface surface

a b

Fig. 1. Confusion surface (dashed line) embedded in space- time. (a) Space-time simply connected ($2) , confusion sur- face ($1) divides the space, can be shrunk to zero. (b) Space- time not simply connected (T2) , confusion surface ($1) topologically non-trivial, does not divide the space.

down to 4 dimensions. Thus confusion reflects a ming- ling in the compactifications of internal and external spaces.

A string that crosses such a confusion surface has its two E8's permuted after passing. For a closed ,string, this "confusion" is genuine if the number of crossings is odd, but non-genuine, i.e. removable by appropriate change of notation if the number is even (fig. 2). A contractible closed string always crosses an even number of times; it stays unconfused. However, by a continuous deformation it may self interact to

produce two genuinely confused strings (fig. 3). Thus the existence of confusion surfaces necessitates the study of confused heterotic strings.

The defining equation of the confusion surface is given by

F ( x , ) = 0, (1)

where x u (/2 = 0 ... . ,9) are the external coordinates * s

, i Subscripts u = 0, ..., 9 label the 10 external (left and right moving) coordinates, and I = 1, 2 .... ,16 label the 16 (left moving) internal coordinates.

S=

a b

Fig. 2. Strings crossing topologically non-trivial confusion surface (dashed line). (a) Nullhomotopic string crossing an even number of times. (b) Non-nullhomotopic string (con- fused) crossing an odd number of times.

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Volume 178, number 2,3 PHYSICS LETTERS B 2 October 1986

r,

Fig. 3. A nullhomotopic string stretching out and self-inter- acting to produce two non-nullhomotopic strings (confused), each one crossing confusion surface once.

The t ime track of the crossing point on the string with the hypersurface is de termined by F [ x u ( o * ) ] = O, where {a*} (a = 0, 1) are the coordinates on the world sheet of the string. This defines a confusion curve on

the world sheet parametrized by {a~onf(X )} (a = 0, 1). For simplicity we take X = o 0.

The permuta t ion of the two E 8's across the con- fusion curve (fig. 4) leads to a d iscont inui ty in the in- ternal coord ina tes ,x / ( / = 1 ,2 , ..., 16). This can be expressed as a b o u n d a r y condi t ion

I . ~ + a ~ _ _ I ± 8 : _ a X I.Oconf C ) - x i .Oconf- Ca), (2)

where we take plus (minus) sign for I = 1 . . . . . 8 (I = 9, ..., 16). The infini tesimal two vector, ca, is t aken to be proport ional to the normal , na , of the confusion

curve:

% = en~ sign (n~n f ) , e > 0, (3)

Ox" g F ( x ~ na - - - - - . (4)

ao ~ g(x"(o~)) l : :O~onf The metric on the world sheet is ~ic~ -- diag(l, - i ) and sign(nc~n c~) = + ( - ) for n a time-like (space-like).

The boundary condi t ions (2) can be derived from

the act ion if a new term, Sconf , is added to the hetero- tic string action. The total act ion of the confused heterotic string is then

S = S F + S B + S I +Sconf , (5)

where SF , SB, S I are the actions of the right moving fermions, the 10 external bosonic coordinates (xu) and the 16 left moving internal coordinates ( x I ) .

To facilitate the computa t ions , we use a first order formula t ion for SI,

1 _ i f l f l a + XI0cI)2 ], (6) SI=2-- ~ - f d 2 o [ f l f z= :

where f / ( f / = f0 / -+ f [ ) is an auxiliary field and Xla Lagrange multiplier in t roduced to ensure that xI( fI~) are left moving. It is easily seen that the act ion (6) is equivalent to the usual act ion of x I. The extra term,

Sconf , involves a g- funct ion which is required to yield the d iscont inui ty in x I across the confusion curve:

1 fd2oa [ f (xU(aO)) ] Sconf =

× n~(e~)xU(o ~ - c f )

X [f/-+8(a~ + c ~) - f I ( o ~ + e l ) ] . (7)

The equat ions of mot ion derived from S I + Sconf are then

( f l ) 2 = O, (8a)

3afJ(o f - eft) = 6 [F(xU(of))] n°~(o f )

x ir28(O onf + g(go + c )l,

ao~xI(o ~ + e ~) - f I(ot3 + e ~) + 2xIf l_(o ~ + c~)ga,+

= -5 [F(xU(o~))] noL(af )

(8b)

X [x/-+-8(a~conf -- J ) -- xI(o~conf -- Eft)]. (8C)

(o) {b) (el

Fig. 4. The two Es's of the heterotic string are depicted as two rails. If the string has an odd number of encounters with the confusion surface the rails are mingled (b) and the string is genuinely confused. In this case it takes two revolutions to return to the initial state.

The right-hand sides of (8b) and (8c) are the contr ibu- t ions resulting from varying Sconf and reflect the dis- cont inui t ies Ax I and A f / across the confusion curve, e.g.

Z2xX I = XI(O~onf _ eot) I o~ -- X (O'conf + ea). (9)

Using eq. (8) we can write z ~ I as

z2~XI = xI+-8 (aconf + ea) I a - x ( % o n f + e~ )

r~I-+8:~a = - t-~ ~Vconf - e ~ ) - x I ( O c o n f - e~)], (10)

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Volume 178, number 2,3 PHYSICS LETTERS B 2 October 1986

which implies the boundary condition (2). A similar argumentation applies to Af/ .

A priori the equations of motion o f x u are also modified in the presence of Sconf. The action S B is

1 ; d 2 o l a a x # a a x t a ' SB = -- 2--~' (11)

with metric ~uv = diag(+ . . . . . . . . . - ) in external space. Variation o f S B + Sconf with respect to xu yields

a~a'~x" = 8 [F(x~)] 6F

8(x"(oa))

x a n [ x Z ( O ~ o r e - d )M I'll. (12)

But using (8b) at the point Oconf a + e a we find a~Afa/ = 0, and similarly at the point O~con f - e a , we find from (8c) that ao~x I = f l . As a result, we can express the right-hand side of (12) in terms of f i r I~ and f l ± 8 f l a which vanish because f_/= 0, (8a), We thus conclude that x u satisfy the ordinary wave equation

ao~aC~x u = 0, (13)

even in the presence of the confusion surface. The wave equation (13) allows a mode expansion

for the external coordinates, xu; however, the in- ternal coordinates x I do not satisfy this wave equa- tion due to the singular contributions in (8). Hence, we need to find new variables appropriate to a mode expansion. For the purpose of writing such mode expansions, we note that the double transversal of a confused string yields an even number of E 8 permu- tations and is therefore equivalent to an ordinary string of double length. As the two E 8,s have lost now their original assignments there is effectively only one E 8 . This suggests the new variables (I = 1, . . . . 8):

)?I(o0, a 1 ) = x l ( a 0 ' 01 )

1 for O~onf ~< o 1 ~< rr + aconf ,

= xI+8(oO, o 1)

1 (14) for oclonf + 7r <~ o I ~< 27r + aconf ,

which are continuous across the confusion curve and

satisfy the usual equations of motion in the lineariz- ed formulation [without the singular terms in (8)]. The mode expansion is then

)?I(o0 ' O 1) = g I + (O 0 + o1)k/

6 I +'2i ~ ~n e_in(o0.a, ) (I = 1, ..., 8), ( lS)

n~0 n

where the period is doubled, 2n rather than rr for or- dinary closed strings [2]. Also )?I are left movers there by having only one set of coefficients, &/, which play the role of creation and annihilation operators in the quantum theory:

[&In, &Jm] = nSn+m,O~( J" (16)

To prove (16) we note that S 1 + Sconf leads to the usual momenta conjugate to x 1, and hence also to )?1; that is

~rI = (1~2no')30)? I

_ 1 ( k / + ~ ~ &Ie_in(aO+ol)) (17) 27ra' n~0

with the ordinary canonical commutation relations. The external transverse coordinates, x i (i = 1, ...,

9) satisfying (13), have a similar mode expansion, this time including both left and right movers,

x i ( a 0 , o 1) = x i + ki o 0 + 2L i o 1

+ i ~ n - l (a in e -2in(°°+°l) n4=O

+ ' a / e -2 in ( a ° - 01 )). (18)

While the term 2Lio 1 , being linear in o 1 , is consistent with eq. (13), it cannot satisfy the periodicity re- quirements of a closed string unless the external space is compactified; e.g. for a toms-like compactification {1, i} lie on some lattice.

The momentum conjugate to x I as derived from

S B + Sconf is

7r i = (1/2rra ' )30x i -- (SF/.6x i) 6 [F(xi)]

X 2I[frlCaconf + rr) - ~rI(Oeonf)]. (19)

The second term in (19) arises from varying Scorff which depends on 30x i via n o [see eq. (4)]. In (19)

1 I we expressf~) in terms of 7r according to (8) (away

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Volume 178, number 2,3 PHYSICS LETTERS B 2 October 1986

from the singularity) and change to the variables 21 , fr / [eq. (14)] thereby summing o v e r / f r o m 1 to 8. The second term in (19) prevents a simple mode ex- pansion for 7r i, however the canonical commutat ion relations and the fact that the second term in (19) does not depend on ~0 xi imply:

[x i(o0, o 1 ), (1/2 rra') 3 0 x/ ( °0 , ~'1 ) ]

= [xi(o0, ol ) , rrj(oO ,'~1)1

= (i /2a') 8(o 1 -- ~'1) 8ij. (20)

Since 30 xi has a simple mode expansion [as derived from (18)], we find trom eq. (20) that the coeffi- cients a / , a'/rn play the role of creation and annihila- tion operators in the quantum theory:

[ a i ' °Jm]= [~'/,~Jm ] =n6n+m,O 6iJ,

[a / , ~'{n ] =0 , (21)

even though a simple mode expansion for 7r i does not exist.

As the model describes closed strings, physical states should be invariant under translations in o 1 . Thus the o-translation operator Po has to vanish when acting on physical states. Like the action, the opera- tor Po has 3 contributions:

Pa = PF + PI + PB (22)

from the left moving fermions, the internal right moving coordinates and the external left and right moving bosonic coordinates.

As usual for the heterotic string [2]

PF = ~ nSan('Y-Sn) a - 8 ~ n , (23) n=l n=l

where S a are the fermionic creation and annihilation operators, 7 - = (@ - 79)/V '~-, and the last term re- suits from normal ordering.

To calculate PI we write

PI =/51 + A, (24a)

where all singularities are contained in A and

16 1 1

/5I = 2a ' ~ lirn + I=1 e-+0 0 rff+e I

Using now the definition (14) and the mode expan- sion (15), (17), we find

2n /51 = 2a ' f do 1 313¢Iir I

0

= - G & I n & / _(k/)2 _ 8 G in - - : 2 '

n=l n=l (26)

and we sum over I ; 1 ... . ,8 . Note the absence of a factor 2 in front of the sum over the normal modes. This is due to the fact that the effective length of the string (for the internal coordinates) is doubled, so the momentum is down by a factor 2. The last term in (26) results from normal ordering. To find A we in- tegrate over an infinitesimal interval [O~onf - e ~, Oconf + ec*], thus picking up only the singular terms of the integrand. Using eq. (8) we find that there is a g-function singularity in 31 xI, while 7r I ~ f d is regular in this range, therefore

a l o n f + e a

A = 2a' lira f ~lXlzr I e-+O ~ lonf _e a

= 2a'rrl(o~onf + ea)Ax I, (24b)

where the sum is o v e r / = 1, ..., 16 and z ~ I is given in (10).

Similarly PB has two contributions

PB = 2a ' j do 1 alxiTr i =fiB -- ~ (27a) 0

corresponding to the two terms of 7r i in (19). The terms

co

PB = 2 G (a in ai ~i ~i ki ? n=l - -- a-nan) + ~ d°l alxi 0

(28)

contains, in addition to the usual mode contributions, an integral which can arise in a non-trivial topology. For simplicity we consider cylindrical compactifica- tion with radius R. This path integral in the compac- tified direction can be interpreted as a product of twice a winding number L and an (external) angular momentum M about an axis, the realization of which would require an embedding in a space of extra di- mension:

X do 1 31xlrr I. (25) 183

Volume 178, number 2,3 PHYSICS LETTERS B 2 October 1986

1 Tr X 2 2 - ~ f ~ do I ~1 x i = M X 2L. kiR (29)

0

The second term, ~ , in (27a) is obtained by integrat- ing the second term of (19):

S = 20~'XI(Oconf + e°~)Arr I, (27b)

where Act I is defined as Ax I in (10). Since we sum over I = 1 . . . . . 16, we find

X = 2a'lrI(O~onf + ea)Ax I = A. (30)

Therefore, A, ~ cancel in the o translation operator

Pa = PF +/5I + A + fib -- A = PF + PI + f ib, (31)

which now has the simple mode expansion exhibited in (23), (26) and (28).

The restrictions arising from the physical state con- dition

Po Iphys} = 0 (32)

differ according to whether confusion surfaces are present and according to the string winding number L. Strings of odd L in the presence of a confusion sur- face (situated as line AA in fig. 5) are genuinely con- fused; all others including ordinary heterotic strings will be referred to as not genuinely confused.

For the latter type, Po can be calculated by ex- panding in the usual manner *6.

Po = 2( N F + NR B - NL B - N O - (k/) 2 + 2LM + 2

(without confusion), (33a)

*6 The energy-weighted number operators are used: N ~ = ~ t z etc. ~n=l O~_nO~n,

confusion s'ur foce

Fig. 5. The Klein bottle is depicted as a cylinder with points at the two ends identified as shown; it is non-orientable. Ad- ditional dimensions a~e needed to establish orientability. The operators O 1 (O 1 ) and 02 apply to the directions x 1 and x 2 .

184

where the last term is the zero point energy. This con- dition is tantamount to the requirement

LM = n E Z (not genuinely confused). (34a)

Here as well as for the genuinely confused string con- sidered below, (k/) 2 is always even.

For the genuinely confused string, P,r is given by (31) in combination with the appropriate mode ex- pansion

Po = 2( N F + NR B - NL B) --/~-I _ (k/)2 + 2LM ÷ 1

(with confusion), (33b)

the implication of which is

LM = n/2 E Z/2 (genuinely confused). (34b)

Note the difference in the zero point energy between (33a) and (33b); ~ 7 = 1.

As the possibility of winding number L 4= 0 is as- sured in a model of interacting strings, any yet to be determined quantization constraints on M, imposed by the compactification geometry, must be com- patible with the physical state condit ion requirements expressed in (34a), (34b), if the consistency of the model is to remain intact.

Consider first a cylindrical comoactification geome- w" iU x 1 • • try in conjunction lth a factor e m a stnng wave

functional. Without confusion, x 1 values modulo 27rR are to be identified. This requires

e 2~riM = + 1 (without confusion). (35a)

With confusion, however, x 1 values modulo 47rR are to be identified. This requires

e 4~riM = +1 or e 2trim = +1 (with confusion).(35b)

The relations (35a), (35b) are seen to be compatible with (34a), (34b).

In the presence of a confusion surface one may suspect the existence of a Bohm-Aharonov field which could influence a string with non-zero winding number. In fact, a non-trivial Bohm-Aharonov phase factor is implied by the shift in the zero point energy between (33a) and (33b). If there is such a phase a the conditions (35a), (35b) become

e 2~riM = e i~ (without confusion) (36a)

and

e 4~iM = 22ic~ or e 2~riM = +e ia (with confusion).

(361))

Volume 178, number 2,3 PHYSICS LETTERS B 2 October 1986

Here the compactification geometry is again taken to be cylindrical. As c~ is left completely undetermined by this geometry, no additional restrictions on M are ob- tained; in fact, the unrestricted c~ could have been used to restore consistency had the constraints on M imposed by the compactification geometry been at vari- ance with the physical state requirements (34a), (34b).

The situation is different for a compactification o f more complicated topology: choosing the topology as that of the Klein bott le *7 (fig. 5) [7] will be seen to determine a up to a sign. Hence, instead of the free parameter found for the cylinder, the restricted Bohm-Aharonov phase c~ in the case of Klein bott le compactification geometry poses a new test of con- sistency.

To determine the Klein bott le constraints on a, define three global operators O1, O1, and 02 . The operators O 1 and 01 effect respectively a half revolu- tion and a full revolution along the x 1 -direction while 02 effects a full revolution in the x 2-direction. If the "length" of the Klein bott le is large compared to the "width" , the local geometry is that of a cylinder provided the x 2 variation is small: b o t h M and L can be defined locally with the result that O 1 and 01 can be defined as e i~rM and e 2~riM respectively. If no con-

fusion surface is involved, the following commutat ion relations are valid due to the special geometry of the Klein bott le (fig. 5):

O102011 = 0 2 , O 2 0 1 0 2 1 =O11

(without confusion surface). (37a)

As a consequence, O 1 = Oi -1 and e icd2 = e -ic~/2 or

e ia = +1 (without confusion surface). (38a)

If the x 1 -coordinate crosses a confusion surface, the symmetry is restricted to full revolutions along x 1 in which case

0 1 0 2 0 ~ -1 = O2, O 2 0 1 0 2 1 = 0 ] -1

(with confusion surfaces). (37b)

These relations lead to 01 = O11 and e i~ = e - i a or

.7 The Klein bottle constitutes two of the ten dimensions of the external space and must (by construction) by rendered orientable in order to accommodate Weyl particles.

e ia = +1 (with confusion surface(s)). (3819)

The Klein bott le topology leads therefore via (36a), (36b) and (38a), (38b) to the restriction that M is in- teger if no confusion surface is involved, while half in- teger as well as integer values are allowed in the pres- ence of confusion surfaces.

As the constraints on the Bohm-Aharonov phase are not in conflict with (34a), (34b), a potential inconsis- tency in the ordinary as well as the confused heterotic string is avoided. More remarkable is the not readily foreseeable integer zero point energy for the confused string which, like the fortuitous even integer for the ordinary heterotic string, is decisive for the consisten- cy of the confused model.

For low energy physics, genuinely confused strings are probably not relevant as they are expected to carry high masses unless the radius of the compacti- fled dimension is fine tuned. The mere existence of confusion surfaces does, however, have consequences for nul lhomotopic strings and thereby for low energy physics. Consider a nullhomotopic string that passes a confusion surface while orbiting a compactified di- mension. At each passage, its wave functional is acted upon by an automorphism operator A which permutes the two E8's:

A d2 [s a , x u , x I ] = ~ [s a , x ta , x I+-8 ]. (39)

This operator has eigenvalues a = -+ 1 corresponding to even and odd eigenstates. For each revolution an eigen- wave functional acquires a factor ae2rriM; hence con- structive interference requires

a = ( - 1 ) 2M. (40)

As strings of half-integer M cannot have zero momen- tum, only strings w i tha = +1 can survive in the low energy limit and subsequently give rise to Yang-Mills fields. The result is that the original E 8 × E 8 gauge symmetry is broken into a single E 8 symmetry at low energy.

As to further investigations of the consistency of the confusion mechanism, the most obvious would be an analysis of the possible anomalies in the two-di- mensional (world sheet) space, as well as the gravita- tional and non-abelian anomalies in external 10-space. Regarding the former, the analysis of conformal an- omalies by Polyakov [9] is applicable also in the pres- ence of confusion surfaces. Conformal anomalies result

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Volume 178, number 2,3 PHYSICS LETTERS B 2 October 1986

from ultraviolet divergences, which are local in origin, whereas the confusion surface, representing a global modification of the topology, can be shifted around so as not to influence local effects. In other words, since we can decompose the path integral over ~1 into a path integral over its periodic [1/2(5: I +.~I+-8)] and antiperiodic [1/2(~ I - ~I_+8)] components (with

period 7r), we can use the calculation of Polyakov, which does not depend on the boundary conditions, to show that confusion does not affect the conformal anomalies. As to gauge and gravitational anomalies [10], several lines of investigations have been consider- ed. Possible anomalies as seen from inside the world sheet could be investigated and subsequently related

to ten-dimensional anomalies [ 11 ]. Alternatively, an investigation of the possible anomalies in a con- fused ten-dimensional gauge theory as the field theory limit of the confused heterotic string model, may re- veal information as to the consistency of the confu- sion mechanism. Finally, anomalies could be analysed in lower dimensional (compactified) theories.

In conclusion, we stress that the emergence of a remarkable numerical coincidence, which prevents an inconsistency of the theory, justifies the hope that it is fully consistent. The decomposition o f ~ I into periodic and antiperiodic components will probably be useful for the proof.

Somewhat related concepts utilizing orbifolds [12] are found in ref. [13].

L.M. would like to thank the Niels Bohr Institute for hospitality, where most of this work was carried out. She would also like to thank Prof. M. Guenin for his support.

[2] D.J. Gross, J.A. Harvey, E. Martinec and R. Rohm, Phys. Rev. Lett. 54 (1985) 502; Nucl. Phys. B 256 (1985) 253; B 267 (1986) 75.

[3] E. Witten, Nucl. Phys. B 258 (1985) 75; J.H. Schwarz, Introduction to superstrings, Lectures Trieste Workshop in High energy physics and cosmology, and the Scottish Universities Summer School in Physics (July 1985), Caltech. preprint CALT-68-120.

[4] H.B. Nielsen and N. Brene, Gauge glass, in: Proc. XVII Intern. Symp. (Ahrenshoop), Institut fiar Hochenergie- physik (Akademie der Wissenschaften der DDR, Berlin- Zeuthen, 1985).

[5] K.S. Narain, Phys. Lett. B 169 (1986) 41; C.S. Lam and Da-Xi Li, McGill Univ. preprint (December 1985).

[6] S.M. Roy and Virendra Singh, preprint TIFR/TH/85-21; C. Vafa and E. Witten, Princeton preprint (1985).

[7 ] C. Nash and S. Sen, Topology and geometry for physi- cists (Academic Press, London, 1983) p. 76.

[8] W. Mecklenburg, Fortschr. Phys. 32 (1984) 207. [9] A.M. Polyakov, Phys. Lett. B 103 (1981) 207,211.

[10] P.H. Frampton and T. Kephart, Phys. Rev. Lett. 50 (1983) 1343, 1347; Phys. Rev. D 28 (1983) 1010; L. Alvarez-Gaum6 and E. Witten, Nucl. Phys. B 234 (1983) 269; E. Witten, in: Geometry, anomalies and topology, eds. W.A. Bardeen and A.R. White (World Scientific, Singapore, 1985); Commun. Math. Phys. 100 (1985) 197; M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117.

[ 11 ] C.G. Gallan, D. Friedan, E.J. Martinec and M.J. Perry, Nucl. Phys. B 262 (1985) 593.

[12] L. Dixon, J.A. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 (1985) 678; Strings on orbifolds II, Princeton preprint (February 1986); C. Vafa, Modular invariance and discrete torsion on orbifolds, Harvard Univ. preprint HUTP-86/A011.

[13] J.A. Harvey, Twisting the heterotic string, Santa Barbara Workshop on Unified string theories (1985).

References

[ 1 ] J.H. Schwarz, ed., Superstrings, the first 15 years of superstring theory, Vols. I and II (World Scientific, Singapore, 1985).

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