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Anomaly-free chiral theories in six dimensions
Nuclear Physics B254 (1985) 327-348 © North-Holland Publishing Company
A N O M A L Y - F R E E CHIRAL T H E O R I E S IN SIX D I M E N S I O N S * Michael B. GREEN**, John H. S C H W A R Z and P.C. WEST***
California Institute of Technology, Pasadena, CA 91125, USA Received 10 December 1984
The coupled N = 1 Yang-Mills plus supergravity theory in ten dimensions can be made anomaly-free for SO(32) or E 8 x E s. Only the case of SO(32) is known to correspond to a superstring theory, which is probably necessary for a fully consistent q u a n t u m theory. Anomaly-free chiral theories in lower dimensions can be obtained by considering nontrivial compactifications (involving nonzero background gauge fields) of the ten-dimensional theory that satisfy a topological consistency condition. This paper considers the compactification of four dimensions on the manifold K 3 without requiring that the equations of motion be satisfied. This leads to a large n u m b e r of anomaly-free chiral supersymmetric six-dimensional theories, corresponding to various ways of embedding U(1) factors in SO(32) or E s X E s.
1. Introduction
Superstring theories (for reviews see ref. [1]) are promising candidates for unified theories containing gravity in a quantum-mechanically satisfactory way. Until recently this program faced the dilemma that the most theoretically appealing examples (the type II theories) appeared unpromising for phenomenology, whereas the type I theories appeared too arbitrary and endangered by anomalies and infinities. The recent discovery of anomaly [2] and infinity [3] cancellations in the special case when the type I superstring gauge group is chosen to be SO(32) has dramatically changed the picture. The SO(32) superstring theory has all the esthetic advantages of a type II theory (finiteness, anomaly cancellation, uniqueness) plus some important phenomenological advantages to be discussed below. In ref. [2] anomaly cancellations were explained in terms of an effective field theory that describes the low-energy expansion of the superstring theory. It was shown that E 8 x E 8 is a second possibility for a satisfactory gauge group, although a corresponding superstring theory has not yet been constructed. The demonstration of anomaly cancellations for SO(32) and E 8 x E 8 in ten dimensions is reviewed in * Work supported in part by the US Department of Energy under contract DEAC 03-81-ER40050, the Fleischmann Foundation and the Alfred P. Sloan Foundation. ** On leave of absence from Queen Mary College, London, England, Nuffield Foundation Research Fellow. *** Permanent address: King's College, Strand, London, England. 327
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sect. 2, using a somewhat streamlined version of the proof given in ref. [2]. We wish to emphasize that the effective field theory approach, also used in this paper, is only regarded as a tool for probing certain features of the superstring theory, such as the structure of anomalies. Having found a finite and anomaly-free SO(32) superstring theory, and perhaps an E 8 × E 8 one as well, the next crucial question is whether it admits a spontaneous compactification of six dimensions that results in an effective low-energy fourdimensional theory in accord with observations. This requires discovering nontrivial solutions to the superstring field equations. At first sight, this appears quite forbidding, even at the classical level, since these are functional equations. Furthermore, they have only been formulated in a light-cone gauge. A possible approach is to invent a superstring theory in a background with six curled-up dimensions and demonstrate that it has all the requisite consistency properties. If such a scheme is found, it would presumably correspond to a solution of the equations mentioned above. One could, of course, investigate Kaluza-Klein compactifications of the minimal low-energy effective description of the superstring theories. By "minimal" we mean the " p o i n t " (as opposed to "string") field theory describing N = 1 super-Yang-Mills coupled to supergravity, without any of the additional higher-dimension terms (such as the Lorentz Chern-Simons term of eq. (2.7)) that arise in the low-energy expansion of the superstring field theory. At the classical level, it has been shown [4] that in the minimal point theory the Ricci curvature, the Yang-Mills field strength, and the third-rank field strength must all vanish in the compact dimensions as a consistency requirement on the field equation of the scalar of the supergravity multiplet. Therefore the compact manifold cannot have any nonabelian isometries. Since there can be no cosmological constant in the ten-dimensional theory, due in particular to the chirality of the gravitino, Ricci flatness of the compact manifold implies that no cosmological constant is induced in the lower-dimensional theory. The minimal theory has anomalies, however, and we do not know which of these results carry over to the superstring theory in which the anomalies cancel. In order to obtain a chiral effective theory in lower dimensions, nonzero background gauge fields are required. Therefore it is an important question whether they are possible in the superstring theory. Most dimensional-reduction schemes obtain Yang-Mills gauge symmetries in the lower dimension, in the spirit of Kaluza and Klein, as the isometries of compact dimensions. This approach not only gives an unpleasant cosmological constant, but also has difficulty giving realistic symmetries, multiplets, and chiral structure. An alternative possibility is to start with a large gauge group in ten dimensions and to obtain the four-dimensional symmetries by breaking the gauge group and supersymmetry in the compactification [5]. By giving suitable nonzero background values to Yang-Mills gauge fields it becomes possible to transmit the chirality of the tendimensional theory to the theory in lower dimensions. In this context the SO(32) and
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E 8 x E 8 superstring theories are very attractive. These groups are large enough to contain all known gauge symmetries, and their adjoint representations can describe all the observed fermion representations on reduction to subgroups. The group E s was specifically considered in the last two references of ref. [5]. The ten-dimensional superstring theory has certain topological restrictions [6], which in conjunction with index theorems, place severe constraints on the possible theories that can result from spontaneous compactification. In this paper we study compactification of four dimensions, obtaining an effective six-dimensional theory at low energies. Sect. 3 gives a general proof that for any compact four-dimensional manifold the resulting six-dimensional theory is free of all gauge and gravitational anomalies as a consequence of the anomaly freedom of the ten-dimensional theory, the topological constraint, and the index theorems. The supersymmetry breaking is also controlled by geometrical properties of the compact manifold. The present scenario for transmitting chirality to lower dimensions in anomaly-free multiplets can be contrasted with that for four-dimensional theories having extended supersymmetry. In such theories the mirror symmetry must be broken spontaneously, and this can only give masses of the order of the weak mass scale to the mirror particles. Also, such theories do not explain why the residual theory without the mirror partners should be anomaly-free. We only consider Ricci-flat manifolds so as to ensure that there is no cosmological constant in the lower dimension at the classical level. As mentioned earlier, this is required by the field equations of the minimal point theory. Maybe this result can be generalized to the quantum superstring theory. If true, it should be possible to deduce this from the structure of the full quantum effective action, perhaps with the scalar field of the supergravity multiplet playing a crucial role. Ricci-flat manifolds in four dimensions include T 4 and K 3. This paper studies reductions to six dimensions on K 3. Conventional riemannian geometry may need to be generalized to describe the geometry of superstring theories at distances of the order of the characteristic size of strings (presumably the Planck length). In this case, the description of any compact dimensions of this size would require new mathematical concepts. Such possibilities are not considered in this paper. If the compact manifold is a product space, it need not be characterized by just one length scale. One could imagine, for example, that four compact dimensions are characterized by the Planck mass Mp, breaking the gauge group down to SU(5), say. Two more could then curl up at the unification scale M x, Reaving an effective SU(3)x SU(2)x U(1) theory in four dimensions for E << M x. A more unusual idea, that (as far as we know) is not yet excluded, is that the first breaking occurs directly to SU(3)x SU(2)x U(1) at M x, and that an effective six-dimensional theory is then applicable all the way to the weak scale M w, at which point the last two dimensions curl up breaking the symmetry to SU(3)x U(1). If this were the case, the next generation of proton accelerators would be able to probe extra dimensions!
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The possibilities for compactifications based o n T 6 and related manifolds have been discussed in ref. [6]. It was found that as long as one does not require that the field equations be satisfied, there are numerous possibilities. In particular, it is possible to obtain SU(5) with any desired number of families. Here, our main purpose is to carry out an analogous analysis for the manifold K 3. Since it is four-dimensional, the theory reduces to six dimensions. A subsequent reduction to four dimensions, using a monopole background on a two-torus, could be adjoined in each case. This final reduction is not discussed in this paper. The derivation of anomaly-free chiral theories in six dimensions does not require that the compactification correspond to a solution of the ten-dimensional field equations. The manifold K 3 has certain properties that make it an especially interesting example. For instance one can show by an index theorem that half of the D = 10 supersymmetry must be broken giving rise to a six-dimensional N = 1 theory. Also, since its Pontryagin number is nonzero, the topological consistency condition requires that a nontrivial gauge-field background be chosen. This necessarily leads to a chiral theory in six dimensions. These and other general features of K 3 are reviewed in sect. 4. In sects. 5 and 6 reductions of the SO(32) and E 8 x E 8 theories, respectively, are studied. It is shown that there are numerous possibilities for embedding U(1) factors in these groups (corresponding to background gauge fields on K3) that give rise to anomaly-free theories in six dimensions. We present many examples, but the lists are by no means complete. Nonabelian background gauge fields could also be considered, but we choose not to do so in this paper. In sect. 6 we also mention a peculiar question of physical interpretation that arises because E 8 x E 8 is a product group and the fields associated with the two factors only interact with each other with gravitational strength.
2. Review of anomaly cancellations in ten dimensions This section gives a simplified demonstration of the ten-dimensional anomaly cancellation results of ref. [2]. Gauge and gravitational anomalies arising from loop diagrams (with chiral particles going around the loop) in D dimensions can be succinctly characterized by gauge-invariant ( D + 2)-forms [7, 8], which are derived from ( D + 2)-dimensional index theorems. These are purely formal expressions, since they involve more indices than are present in D dimensions. These forms are constructed out of Yang-Mills field strengths and gravitational curvatures. In the language of forms, the Yang-Mills field strength is given by F = ½F~dx ~ A d x ~ = dA + A 2 ,
(2.1)
A =A~)t~dx ~.
(2.2)
where
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The matrix ~ is antihermitian, and in this paper it is taken to be in the adjoint representation of the Yang-Mills algebra. (Thus the 2t%c are the structure constants.) Wedge products of differentials are usually left implicit. The Lorentz-curvature two-form is given in similar fashion by R _- 1 ~Rl,.dx*' A dx" = d~o + o~2,
(2.3)
where ~o = % d x ~ is a 10 x 10 antisymmetric matrix (the spin connection), corresponding to the fundamental representation of-the Lorentz algebra SO(9,1). In order to describe the structure of a type I superstring theory at low energies, we now consider N = 1 supergravity coupled to N = 1 super-Yang-Mills theory in ten dimensions, with a gauge group Gm. The chiral fields that contribute to anomalies in hexagon (and higher) one-loop diagrams are a left-handed Majorana-Weyl (MW) gravitino and a right-handed MW spinor from the supergravity multiplet, as well as n = dim G10 left-handed MW spinors from the Yang-Mills supermultiplet. From eq. (63) of ref. [7], we learn that all the Yang-Mills, gravitational, and mixed anomalies due to these loops are characterized by a 12-form proportional to 1,2= _ l T r F
6 +~4TrF4trR2_9_~TrF2[4trR4
+ 5(trRZ)q
n-496 +(~+n-496)trRZtrR4+ + {3~+ -1-3-8-~ ) (trR2)3 5-76-0
-n - 496
7560 trR6"
(2.4)
We now demonstrate the existence of a local counterterm that cancels the anomalies whenever eq. (2.4) can be factorized into an expression of the form 112 = (trR 2 + k T r F 2) )(8,
(2.5)
where X 8 is a gauge-invariant eight-form made out of F and R. The distinct trace symbols tr and Tr are used as a reminder that the R are in fundamental representation and the F are in adjoint representation. A crucial role is played in the anomaly-cancellation mechanism by a second-rank potential B = B ~ d x ~' A d x ~,
which is part of the N = I , strength [2]
(2.6)
D = 10 supergravity multiplet. A three-form field H = d B + ~03L + k~o3y
(2.7)
is formed from this potential, where the Chern-Simons forms ~03r and W3L satisfy d r 0 3 y ---. T r F
2,
d r 0 3 L = t r R z.
(2.8a) (2.8b)
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The inclusion of the Lorentz Chern-Simons term o~3/~ in eq. (2.7) is an important modification of the definition of H in ref. [9]. The parameter k is uniquely determined by the string theory, but is more easily found by considering anomaly cancellations in the effective field theory, as discussed below. There also exist two-forms w~r and 0~lL such that under an infinitesimal Yang-Mills gauge transformation [8,10] 3w3r = - d ~ v ,
(2.9a)
and under an infinitesimal local-Lorentz transformation
3w3L = - d~0~L.
(2.9b)
Substituting in eq. (2.7) it is clear that H is gauge invariant provided that
8B = ,o~L + k,o~y.
(2.10)
N o w let us return to the expression 112 in eq. (2.5) and replace X 8 by X2N SO that the analysis applies for other dimensions as well (as required in sect. 3). In analogy with ~03 and ¢o1, we can introduce forms X2N 1 and XIN_2 satisfying
X2N = dX2N_I, 3X2N
I=
-dX1N
(2.11) (2.12)
2"
It is easy to convince oneself that there are solutions of these equations* for arbitrary invariant forms X2N such that the anomaly associated with I2N+4 is proportional to
G=f(2(w12L+k~X2r)XzN+N(trRZ+kTrFz)X~N
2}-
(2.13)
The problem then is to find a local interaction S c such that (2.14)
8S~+G=0. The solution to this equation is easily seen to be given by
Sc= f [N(,%L+ k O ) 3 y ) X 2 N - I - ( N
q- 2)BX2N l .
(2.15)
This result is unique up to terms that are gauge invariant. N o w let us investigate when eq. (2.4) reduces to the form (2.5). Clearly, two necessary conditions are that n = dimG10 = 496 and that T r F 6 not be an indepen* There is a well-defined prescription for resolvingambiguities [7, 8].
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333
dent sixth-order Casimir invariant of Gx0. Both these properties are satisfied by E 8 × E 8 and SO(32)= D16. In the case of a single E 8 T r F 4 = ~-~(TrFZ) 2,
(2.16)
T r r 6 = 7z-~(Trr2) 3.
(2.17)
Neither of these is valid, of course, for E 8 × E 8. However, in that case the weaker condition [2,11] T r r 6 = 1 T r F Z T r F 4 - ~4-4-~4~(Tr F2) 3
(2.18)
is satisfied. Remarkably, eq. (2.18) is also valid for D16. Substituting this relation and n = 496 into eq. (2.4) gives a factorized expression of the form in eq. (2.5) with k --
1 30
(2.19)
X 8 = 2~XrF 4 - v - ~ ( T r F 2 ) 2 - 2~0TrF2trR 2 + -~trR 4 + 3~(trR2) 2.
(2.20)
In view of the preceding discussion, this proves the cancellation of all anomalies for both D16 and E 8 x E 8. It is easy to show that n = 496 and eq. (2.18), with precisely the coefficients given, are both necessary and sufficient for the factorization of eq.
(2.4).
3. Reduction to six dimensions
In the previous section we found that the three-form field strength is given by H = dB - ~¢03r +
0)3L.
(3.1)
As explained in ref. [6], the requirement that H be globally well-defined gives a topological condition on possible spatial compactifications. Specifically, since dH
= -
~oTrF
2+
trR 2,
(3.2)
it is necessary that background fields R o, Fo satisfy f
(trR2 - ~0TrFo2) = 0
(3.3)
M4
for any closed four-dimensional submanifold M 4 of the ten-dimensional space-time. Ref. [6] showed that compactifications of six dimensions satisfying this condition give rise to four-dimensional models that are free from Yang-Mills gauge anomalies.
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In this section the analogous theorem for reductions to six dimensions is proved. The analysis is somewhat more elaborate in this case, because six-dimensional theories can also have gravitational anomalies. The discussion applies to both the D16 and E 8 x E 8 cases and any compact manifold M 4. It is not necessary to require that R 0 and F 0 correspond to a solution of the field equations of the ten-dimensional theory. In the background specified by R 0 and F 0, the effective six-dimensional theory has a reduced gauge symmetry. Specifically, if the nonzero fields F o span a subgroup H c G10 , then there is a unique maximal subalgebra G of G~0, all of whose generators c o m m u t e with those of H, such that GloD G X H.
(3.4)
The massless gauge fields in six dimensions would appear at first sight to correspond to the generators of G plus any U(1) factors in H. As we will explain at the end of this section, the U(1) gauge fields actually acquire a mass by a Higgs mechanism. The adjoint representation of G10 can be decomposed into a sum of representations adjoint
of G10 = E(ti, i
Ci) ,
(3.5)
where L~ and C~ are irreducible representations of G and H, respectively. In particular, )-~ dim L~dimC~ = d i m G m = 496.
(3.6)
i
If X is a matrix in the adjoint representation of G10 that corresponds to a generator of the subgroup G, it can be decomposed as follows:
X = ~) (X i® li).
(3.7a)
i
Similarly, if Y corresponds to a generator of H
Y= (t) (1, ® II,,).
(3.7b)
i
F r o m these formulas it is evident that
X Y = YX= ~ (Xi® Y,),
(3.8)
i
T r ( X Y ) = Y'. tr X~tr II,, = 0,
(3.9)
i
since Gx0 is semisimple. In the following we consider anomalies associated with
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335
G-currents. Therefore we have six-dimensional two-forms F with generators restricted to G (i.e., of X type) and background fields F 0 associated with H (i.e., of Y type). Note that F and F 0 commute (eq. (3.8)). The total anomaly in six dimensions is characterized by a formal eight-form
I= n3/2I°/2 + nx/211°/2 + Y'~n[/2I;/2 .
(3.10)
i
l°/2 characterizes the anomaly due to a single left-handed spin--~
field, which in the
present case is a singlet of the gauge group G. It is given by [7, 8]
1°/2
1
(49)4
[ - 2@8(trR2)2+ ~trR4 ] .
(3.11)
The two-form R is a 6 × 6 matrix in the fundamental representation of the algebra SO(5,1).
n3/2=n}/2-n]/2
(3.12)
is the net number of left-handed gravitinos in six dimensions. This number is given by an index theorem:
n3/2-
1 1 f trR2.
891/.2 48 M4
(3.13)
This formula is a consequence of a spin-½ index theorem, which is relevant since spin-3 in six dimensions requires the internal part of the ten-dimensional gravitino field to be spin-½ [12]. It gives the number of six-dimensional gravitinos per left-handed gravitino in the D = 10 theory, but there is only one of them. Similarly, is the anomaly due to a left-handed singlet spin-½ field in six dimensions:
I°/2
io/2 -
1
(49) 4
[~(trR2)2 + 3~otrR4] .
(3.14)
The net number of these fields in six dimensions arising from one left-handed gravitino and one right-handed spinor in the D = 10 supergravity multiplet is given by a combination of the spin- ½ and spin- 3 index theorems in the internal space. The result is [13]
nl/2=n~/2_nR/2
-
1 7 fMtrR~"
892 16
(3.15)
The anomaly contribution of a multiplet of left-handed spinors in the representation
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M.B. Green et aL / Anomaly-free chiral theories
L i of G is
1 [ / 2 = (4~.) - - 1 4 [~tr L F , _ ~tr L F2trR2 + 2_~dimLi(trR2)2 + 3160dimLitrR,] ' '
,
(3.16) and the number of such multiplets is given by a spin-½ index theorem ni=nL _nR =
1 fM [--½trc F2 +~dimCitrR2°]
87/'2
4
(3.17)
'
These results can now be assembled to give the complete expression 1 of eq. (3.10). Combining eqs. (3.10)-(3.17) gives
1
I
2(4~') 6
fM4(_4Tr(F4Fo2)+½Tr(F2Fo2)trR2_T~(trR2)2TrFo 2
-
~z(tr R 2)2trR2o - ~ t r ( R 4 ) T r F 2 + ~tr( R4)tr R 2 + 1 T r ( F 4 ) t r R ~ - 7~Tr(F2)tr(R2)trR2o}.
(3.18)
To simplify this expression we note first that eq. (2.18) applied to an arbitrary linear combination of F and Fo, and eq. (3.9) imply that Tr( F4Fo2) --- 1@0Tr( F2)Tr( F2F2 ) + 7~0Tr( Fo2)Tr F 4 - 72G Tr F02(Tr F2) 2. (3.19) Using this and eq. (3.3) gives
1
I
2(4~r) 6
fM4(__~Tr(F2)Tr(F2Fo2)+a@N~(TrF2)2trR2° + ½tr(R2)Tr(F2F~) - ~Tr(F2)tr(R2)trR2 - ~2(trR2)2trR 2}
_
1 ( t r R 2 _ ~0TrF2~ /" [~Tr( F2F2)-~tr( R 2 ) T r F a - x t r ( R2)trR2] . 2(4~.)6 ~ : LM4 (3.20)
Since this is a factorized expression of the same type as in eq. (2.5), a local
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337
counterterm of the form in eq. (2.15) (now using N = 2) can be constructed, thereby showing that all anomalies cancel. Using eqs. (3.3), (3.6), (3.13), and (3.17) one can show that the net number of left-handed spin-½ fermions arising from the D = 10 Yang-Mills supermultiplet is Y[nidim L i = - 224n3/2 .
(3.21)
i
This relation provides a useful check on the arithmetic in sects. 5 and 6. Let us now consider the status of the residual U(1) gauge fields in H. The anomaly cancellation arguments do not apply to them, and, indeed, naive calculation would show anomalies in the corresponding currents. However, as mentioned in ref. [6], these U(1) symmetries are spontaneously broken in such a way that these anomalies are harmless. To see this, we note that the third-rank field strength given in eq. (3.1) contains terms
H,,j = O, B i j - ~i( F ~ + )aA . ,
+
..-
,
(3.22)
where i, j are internal indices. The square of this term occurs in the action and gives kinetic terms of B,j. However, the nonzero expectation values (F,~) imply that the corresponding A~, absorb O~B,j terms yielding massive U(1) vector fields. An alternative way to understand this is to consider the variation equation (following from eq. (2.10))
8B~j = ( F ~ ) A a + . . . ,
(3.23)
which contains a constant term indicative of spontaneous symmetry breaking. As there are only six B~j in six dimensions (there would be 15 in four dimensions), this can correspond to the spontaneous breaking of at most six U(1) factors. Presumably any U(1) factors not spontaneously broken in this way are, in fact, anomaly-free.
4. R e d u c t i o n o n K 3
The manifold K 3 (Kummer's third surface) is a four-dimensional compact closed simply-connected manifold. It is the only one with a self-dual (or anti-self-dual) curvature (and hence Ricci flat) [13], i.e., it is a gravitational instanton. Conversely, all Ricci-flat four-dimensional manifolds are self-dual if they are closed and compact. It is a (hyper)-K~hler manifold that can be obtained as a solution of a homogeneous equation of degree four in complex projective three-space (CP3). Even though the metric has never been explicitly constructed, enough is known to enable one to answer most questions of interest. For example, K 3 is known to have no isometries and to require 58 parameters for its description. From this it follows that the ten-dimensional graviton gives rise to zero modes in six dimensions consisting of
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a graviton, no vectors, and 58 scalars. Reduction of supergravity theories on K 3 has been discussed previously in refs. [14,15]. Properties of K 3 are also reviewed in ref. [16]. For example, it has the Pontryagin number p,
1 f 8"//'2
trR2 = _ 4 8 .
(4.1)
K3
Substituting this into eq. (3.13), we learn that there is one gravitino zero mode in the six-dimensional theory, which corresponds to breaking half the supersymmetry of the original N = 1, D = 10 theory. This agrees with the usual argument based on the fact that holonomy group of K 3 is SU(2), which is a direct consequence of the self-duality of the curvature. There are four D = 6, N = 1 supermultiplets to consider: (i) g,~, ~bL, B~L; (ii) A~, XL; (iii) BR, XR, % (iv) xR, 4cp. The antisymmetric potentials BE; R give rise to third-rank field strengths H,, o that are self-dual or anti-self-dual. The chirality assignments of the multiplets (ii), (iii) and (iv) are the only ones that are compatible with the chirality of the supersymmetry determined by (i). As explained in ref. [15], the K 3 reduction of the N = 1, D = 10 supergravity multiplet gives rise to the following D = 6 zero modes: ( N = 1, D = 10 SG) ~ (i) + (iii) + 20(iv).
(4.2)
These are all singlets of the Yang-Mills symmetry group. The topological consistency condition, eq. (3.3), requires that one or more of the Yang-Mills fields take a nonzero background value, F 0, on the K 3 manifold. Since K 3 has the Betti numbers b~- = 3,
b 2 = 19,
(4.3)
it admits three self-dual and nineteen anti-self-dual gauge field configurations (with the convention that R is anti-self-dual). One of the three self-dual gauge fields is given by the complex structure of the K 3 manifold. Only the anti-self-dual gauge fields can solve eq. (3.3). Furthermore, from the supersymmetry transformation law of the spinor in the Yang-Mills supermultiplet, i.e., ~?~ - y~F~e, only these nineteen gauge fields are compatible with supersymmetry. Thus a maximum of nineteen independent U(1) gauge fields can be assigned values on K 3. Since the groups D16 and E 8 × E 8 only have rank 16, however, sixteen is the largest number one could actually consider for our theories. The K 3 reduction of the N = 1, D = 10 Yang-Mills supermultiplet gives rise to the Yang-Mills supermultiplet (ii) in the adjoint representation of the group G. G is
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339
defined to consist of all generators that commute with the generators of the group H. H was defined in sect. 3 to consist of all the generators corresponding to gauge fields that take nonzero background values. In the case of K 3 the group H is taken to consist of a product of U(1) factors. This is the only case considered here, but there are also nonabelian possibilities, such as the holonomy group SU(2). Since G includes G and any U(1) factors in H, in this case G = G + H. The reduction of the D = 10 Yang-Mills supermultiplet also gives the "hypermultiplet" (iv) in some real representation R of G. Eq. (3.21) implies that dim R = dim G + 224.
(4.4)
The number of spin-½ zero modes in the Yang-Mills sector in six dimensions is given by eq. (3.17). Since only four dimensions are being compactified on a closed manifold, the topological constraint in eq. (3.3) can be applied directly. This allows us to rewrite eq. (3.17) in the form
1 (jtrc -15
j
fWrFJ t-~dimCi trR~.
(4.5)
The ratio
ri
ftrc, Fo2
(4.6)
f VrFo2 is a n u m b e r that depends only on the group theory. Hence, n i is determined entirely by the H embedding in G10 and the value of f t r R 2 o . Therefore for K 3
n, = ( - 1 5 r i + ~ d i m C i ) 1--~- /'trR 2 8~. 2 J = ( d i m C i - 720ri).
(4.7)
Clearly, reductions to four dimensions are more complicated, because one must consider the topological constraint (3.3) on all four-dimensional submanifolds of the six-dimensional compact space. Also, in general the relevant index theorem is not in a form that allows a direct substitution of eq. (3.3), since it involves terms cubic in F 0.
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5. K 3 reduction of the SO(32) theory
The general analysis described in the preceding sections is applied here to the reduction of the SO(32) theory on K 3. This enables us to construct a large number of chiral N = 1, D = 6 anomaly-free theories with gauge groups that are subgroups of SO(32). There are many possible cases. We consider some that correspond to various different ways of selecting U(1) factors inside SO(32) and assigning background values to them. For example, M ~< 16 U(1)'s can be embedded in SO(32) in such a way that =
SU(Pr
X S O ( 3 2 - 2 N ) X [U(1)] M,
(5.1)
whereP~>~P2>/ "'" > / P M > / 1 a n d M
N= ~ .
(5.2)
r=|
In this embedding the 32 of SO(32) decomposes according to M
3 2 = E [(1 . . . . . 1 , n r , 1 . . . . . 1 ; 1 ) q r + ( 1 . . . . . 1 , ~ r , 1 . . . . . 1;1)_qr ] r=l
(5.3)
+ ( 1 . . . . . 1 ; 3 2 - 2N)0.
Each of the representations carries an M-vector charge q corresponding to the eigenvalues of the M U(1) factors. For the r t h U(1) charge in eq. (5.3) we make the choice qr = (0 . . . . . O, qr, 0 . . . . , 0 ) ,
(5.4)
although one could consider more complicated possibilities. The decomposition of the 496 (adjoint) can be deduced by taking the antisymmetric product (32 x 32)A. The terms with q = 0 correspond to the adjoint of G. For the types of embeddings described by eq. (5.4) there are nonzero charges associated with each U(1) factor in G, As we will see later, this is not the case in general. For each representation that occurs in the decomposition of the 496, we can apply the index theorem of eq. (3.17) to deduce the number of spin-½ zero modes of that type that occur. Since only U(1) fields occur in F0, dimC i = 1. Using eqs. (4.6) and (4.7) it follows that for a multiplet with charge qi
n i=l
720qi "qi Tr(q.q) '
(5.5)
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341
where Tr(q. q) = Z (dim Li)q,. qi.
(5.6)
i
The only restriction on allowed embeddings is that the multiplicities n i be integers. For the adjoint fields of G, n i = 1, which implies that the adjoint of G occurs with unit multiplicity in the Yang-Mills supermultiplet (type (ii) of sect. 4), as one would expect. For all other representations in the decomposition of the 496 the multiplicities n i are zero or negative, implying an excess of left-handed spinors over righthanded ones. Clearly, these states should be assigned to the hypermultiplet (iv). Thus, even though only the spin- 1 index theorem is used in this part of the analysis, the multiplicities of all four kinds of supermultiplets are completely determined. Let us now consider some examples. Suppose that the charge qr satisfies
m r = q~,
(5.7)
where m r is a positive integer. Next define 12
n
(5.8)
ZmrPr ' and restrict the sets {Pr) and ( m r ) by the requirement that n is an integer (1, 2, 3, 4, 6,12). In this case we learn from eq. (5.5) that
R = ~ _ , ( 4 m ~ n - 1)(1... B r . . . 1 ; 1) r
+ y' [ n ( m ~ + m s ) - l ] ( ( 1 . . . n r . . . t ~ s . . . 1 ; 1 ) + ( 1 . . . t 2 r . . . E 3 s . . . 1 ; 1 ) ) r~$ + Y'~(nmr- 1 ) ( 1 . . . l ] r . . . 1; 32 - 2 / ) + c.c.
(5.9)
?,
The multiplicities are all integers, and it is easy to verify that eq. (4.4) is satisfied. This formula gives the particle content of a very large ( - 100) number of anomalyfree chiral theories in six dimensions. It is not essential that the rn themselves be integers, what is important is that the multiplicities in eq. (5.9) be integers. As a second case let us take M
Z Pr = 16,
(5.10)
r=l
so that G=
SU(Pr
×[U(1)]M
(5.11)
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In this case one has to be a little more clever to obtain integer multiplicities. A procedure that works is to choose the m r of eq. (5.7) to be a set of positive odd integers such that Y'~ m rPr = 24.
(5.12)
In this case one finds that R=~(Zm~-
1)(1... R ~ . . . 1 ) + ~ [ ½ ( m r + m s ) - I
r
]
r
× {(1 . . . [ ] r . . . n , . . . 1) + ( 1 . . . n r . . . D s . . . 1 ) } + c.c.
(5.13)
As a check, we note that eq. (4.4) is once again satisfied. When M = 1, so that CJ = SU(16) × U(1), one can take rn a = 3 to obtain R = 2(120 + 120). In general, any scheme of the m consistent with (5.12) that gives non-negative integer multiplicities for the terms in eq. (5.13) is acceptable. Because of their possible phenomenological interest, let us make eq. (5.13) more explicit for the cases SU(5) x [ U ( I ) ] 12 and SU(3) × SU(2) × [U(1)] 13. In the first case m = (1, 9,1 l°) gives R = (10 + i 0 )
+ 8(5 + 5 ) + 8 0 ( 1 ) .
(5.14)
In the second we find four possibilities: m = (1, 5,111): R = (3,1) + ( 3 , 1 ) + 4[(3,2) + (3-,2)] + 88(1,2) + 18(1,1) ;
(5.15a)
m = (1, 3, 5,110):
R=5[(3,1)+(3,1)]
+2[(3,2)+(3-,2)] +52(1,2)+90(1,1);
(5.15b)
m = (3,1, 3, 11°):
R=29[(3,1)+(3,1)[
+2[(3,2)+(3,2)]
+4(1,2)+42(1,1);
(5.15c)
m = (1, 1, 9, 11°):
R = 9[(3,1) + ( 3 , 1 ) ] + 16(1,2) + 162(1,1).
(5.15d)
Perhaps one should be encouraged that familiar multiplets occur. The list of SO(32) reductions given above is not complete. Additional ones can be obtained by associating the gauge fields with linear combinations of the U(1) subalgebras that we have considered. For example, if there is only one nonzero U(1) gauge field this gives eight new possibilities tabulated in table 1. In each case the subscript indicates the U(1) charge. There is no nonzero background gauge field for
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343
TABLE 1 S0(32) reductions corresponding to nontrivial assignments of a single U(1) charge
C,
R
S .U(2) × SO(26) X [U(1)] 2
17(2,1) 3 + 7(1,26)2 + 7(1,1)2 + (2,1)1 + (2,26 h + c.c.
SU(3) X SO(24) X [U(1)] 2 SU(4) × SU(2) × S0(20) × [U(1)] 2 SU(8) × SO(14) × [U(1)] 2
15(3,1)4 + 6(-3,1)2 + 8(1,24)3 + c.c. 15(1,1,1) 4 + 8(4,2,1) 3 + 3(1,2,20) 2 + 3(6,1,1) 2 + c.c. 8(8,1)3 + 3(1,14)2 + 3(28,1)2 + c.c.
SU(15) x [U(1)]2 SU(12) X SU(4) × [U(1)]2
7(15)4 + ( 15 )2 + (105)2 + c.c. 8(1,6)6 + 3(12,4)4 + c.c.
SU(14) X [U(1)] 3
15(1) 8 + 8(14)6 + 3( 14)4 + 3(14) 4 + c.c.
the other U(1) factors. Most of the entries in table 1 correspond to solutions given previously, if one ignores the U(1) charges. The last entry is not of this type. We stress once more that the six-dimensional theories tabulated here are not expected to correspond, in general, to solutions of the ten-dimensional field equations. It was conjectured in ref. [17] that solutions of the superstring field equations that describe compactification of six dimensions give rise to low-energy effective four-dimensional theories that are ultraviolet finite. The only known example involving compactification of six dimensions in the superstring theory is the relatively trivial T 6 case. It gives rise to N = 4, D = 4 super-Yang-Mills theory, which is known to be finite. This conjecture also agrees with the result of ref. [18] on nontrivial reductions of the limiting point field theory. To each of the six-dimensional theories given here we can associate an N = 2, D = 4 theory by considering a trivial T 2 reduction from six to four dimensions. The criterion for finiteness of N = 2 theories is [19] T(R) = C2(G ) ,
(5.16)
where R is the representation of the hypermultiplet. Scanning the list of theories given here, the only one that satisfies this criterion is the last one in table 1. There might be additional such examples among the schemes that we have not constructed.
6. K 3 reduction of the E s × E s theory
This section discusses anomaly-free six-dimensional theories that can arise from K 3 reduction of the E 8 × E 8 theory in ten dimensions. The analysis is a little trickier than in the SO(32) case for two reasons. (i) The general embedding of a U(1) in E 8 is somewhat complicated, and (ii) the two E 8 factors need to be considered simultaneously. The multiplicities of the six-dimensional multiplets are again determined by eq. (5.5), which did not depend on SO(32) for its derivation. The Tr(q. q) in the denominator includes contributions from both E 8 factors.
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TABLE 2 Some embeddings of a U(1) charge in Eg
Subgroup E 7 X U1 SO14 × U 1 E 6 X SO 2 X O 1 SU 8 X U 1
SOlo x SU3 x U1
SU5 x SUn x U1
Decomposition of the 248
Tr q2
133o + 1 o + 5 6 1 + 5 6 _ 1 + 1 2 + 1 2 910 + 10 + 641 + 64 _1 + 142 + 14_2 (78,1)0 + (1,3)o + (1,1)o + (27,2)1 + ( 2 7 , 2 ) _ 1 + ( 2 7 , 1 ) 2 + (27,1)_ 2 q- (1, 2)3 + (1,2) _ 3 63o + 1 o + 5 6 1 + 5 ~ i + 2 8 - 2 + 2 ~ 2 +83 + ~ - 3 (45,1)o + (1,8)o + (1,1)o + (16,3) 1 + ( 1 6 , 3 )-1 +(10, 3 ) 2 + (10,3)-2 + (16,1)3 + ( 1~,1)-3 +(1,3)4+(1,7) 4 (24,1)o + (1,15)0 + (1,1)o + (10, 4)1 + ( 1--6,4)-1 + ( 7 , 6 ) 2 + ( 5 , 6 ) _ 2 + ( 5 , 4 ) 3 + ( 5 , 4)_3 +(1--6,1)4 + (10,1)_ 4 + (1, 4)5 + (1,4)-5
120 240 360 480 720
1200
Let us begin by considering embeddings of a single U(1) charge in E 8. Seven possibilities are shown in table 2. Since the last one has Tr q2 > 720, it cannot be used to obtain integral values of n i in eq. (5.5). Suppose, to start with, that n o U(1) charge is associated with the second E 8 factor, so that it is unbroken. In this case we find the following four possibilities arising from examples 1, 2, 3, 5 in table 2:
G = E 7X U1X E 8• R = 5 ( 5 6 1 + 5 6 _ 1 ) 4- 2 3 ( 1 2 + ] _ 2 ) ;
G
(6.1)
= 5014 X U 1 X E 8 •
R = 11(142 + 14_2) + 2 ( 6 4 1 + ~ _ 1 ) ;
(6.2)
G = E 6 x SU 2 x U 1 x E 8 : R = 1 7 ( 1 , 2 ) 3 -4- 7 ( 2 i f , 1)2 4- ( 2 7 , 2)1 q- c . c . ;
(6.3)
G = SO10 × SU(3) × U 1 × E8: R = 15(1,3)4 + 8(16,1)3 + 3(10, 3-)2 + c.c.
(6.4)
It is also possible to consider simultaneous decompositions of both E s factors whenever there exist positive integers ml, m 2 such that mlTrlq 2 + m2Tr2q 2 = 720.
(6.5)
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M.B. Green et al. / Anomaly-free chiral theories
Thus, for e x a m p l e , b r e a k i n g b o t h E 8 factors to E 7 X U 1 gives three possibilities: R l = 3(1)2 + c.c.,
(ml=l,m2=5):
R 2 = 4 ( 5 6 ) 4 f + 1 9 ( 1 ) 2 ~ + c.c. ; (m1=2, m2=4):
R 1=(56)1+7(1)2+c.c.
(ma=3, m2=3):
(6.6)
,
R 2 = 3(56)v5-+ 15(1)2v~+ c.c. ;
(6.7)
R I=R 2=2(56)1+1i(1)2+c.c.
(6.8)
T h e i r r a t i o n a l charges in eqs. (6.6) a n d (6.7) m a y suggest a violation of the D i r a c q u a n t i z a t i o n c o n d i t i o n for the " m o n o p o l e " b a c k g r o u n d gauge fields. However, this is n o t e n t i r e l y clear, because the two charges are associated with different U(1) gauge fields w h o s e n o r m a l i z a t i o n might differ b y similar i r r a t i o n a l factors. Similarly, c h o o s i n g o t h e r p a i r i n g s from table 2 one can o b t a i n G = G 1 × G 2 with the following seven possibilities (where we now suppress the charge assignments): G 1 = E 7 X U1,
G 2 = 5014 X U 1 :
R 1 = 15(1) "[- 3(56) + C.C.,
R 1 = 7 ( 1 ) + (56) + c . c . , G 1 = E 7 X U1,
R 2 = 7 ( 1 4 ) + (64) + c.c. ;
(6.9) (6.10)
G 2 = E 6 × SU 2 X U i :
R i = 11(1) + 2(56) + c . c . , G 1 = E 7 X U1,
R 2 = 3(14) + c . c . ,
R2 = 8 ( 1 , 2 ) + 3 ( 2 7 , 1 ) + c.c. ;
(6.11)
G 2 = SU 8 X Ux:
R 1 = 7(1) + (56) + c . c . , G 1 = 5014 X U1,
R x = 3(14) + c.c., G 1 = 5014 X U1, R i = 3(14) + c.c., G x = E 6 X SU2 × Ui,
R 2 = 8(8) + 3(28) + c.c. ;
(6.12)
G 2 = SU 8 x U I : R 2 = 8(8) + 3(28) + c.c. ;
(6.13)
G 2 = 5014 X U 1 : R 2 = 7(14) + (64) + c.c. ;
(6.14)
G 2 = E 6 X SU 2 X U I :
R 1 = R 2 = 8 ( 1 , 2 ) + 3 ( 2 7 , 1 ) + c.c.
(6.15)
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M.B. Green et al. / Anomaly-free chiral theories
This is only the tip of the iceberg! There are m a n y more possibilities. In the remaining discussion we consider those that arise by successive addition of U(1) gauge fields that give the breaking chain E 8 ~ E 7 ~ E 6 ~ S O 1 0 ~ S U s , since it could be of physical interest. The first step has already been discussed, so we consider E 8 D E 6 x [U1] 2 with two nontrivial background gauge fields. There turn out to be three different ways of embedding the charges that lead to integer multiplicities. Let us consider first the case in which the second E 8 factor is not broken. In this case, even though the U 1 charge assignments are different in the three cases, the E 6 multiplicities are always given by R = 33(1) + 9(27) + c.c.
(6.16)
One of the embeddings can be carried out simultaneously in each E8 factor so as to give R: = R 2 = 15(1) + 3(27) + c.c.
(6.17)
This embedding and one of the others can also be combined with other decompositions of the second E 8. The analysis described above can now be repeated for E s D SO10 × [U:] 3 with three nontrivial background gauge fields. In this case there are four embeddings that give integer multiplicities. If the second E 8 is unbroken, all four give R=42(1)+9(10)+8(16)+c.c.
(6.18)
One of the embeddings can be combined with embeddings in the second E 8. In particular, it can be performed symmetrically on the two sides to give R , = R 2 = 1 8 ( 1 ) + 3 ( 1 0 ) + 2 ( 1 6 ) + c.c.
(6.19)
Next we consider E 8 D SU 5 × U 4, using four background gauge fields. In this case we find six embeddings, none of which allows a breaking of the second E 8 factor. All six give the SU 5 multiplicities R=50(1)+26(5)+7(10)+c.c.
(6.20)
Although this paper emphasizes U(1) background gauge fields, the analysis could be repeated for reductions based on H = SU(2), the holonomy of K 3. This can be illustrated with an example: G a = E7,
G: = E 8,
R, = 10(56) + 45(1),
R 2 = 0.
(6.21)
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347
Once again, eq. (4.4) is satisfied. One could also consider simultaneous SU(2) and U(1) background gauge fields. Even though there are still many other possibilities that could be considered, let us stop here and make a few comments. First of all, it is clear that there are a very large n u m b e r of anomaly-free theories in six dimensions. An interesting question of physical interpretation arises since E 8 x E 8 is a product group. Whenever the compact manifold has no isometrics, as in the case of K 3, the gauge symmetries are those derived from the original group. In the case of E 8 × E 8 the resulting six-dimensional spectrum of zero modes contains no states with simultaneous non-singlet quantum numbers contained in the decomposition of the two E 8 factors. Therefore one could imagine that one E 8 breaks to a subgroup G x and the second to a subgroup G 2 , but that the G~ subtheory accounts for all observed nongravitational physics - strong, weak, and electromagnetic interactions. The G 2 fields would only couple with gravitational strength to ordinary matter, and would therefore be completely invisible to observations based on the other interactions. One possibility is for the symmetry breakdown on the two sides to be identical, as in (6.8), (6.15), (6.17) and (6.19). In this case, the laws of physics are identical, and there could be G 2 galaxies, etc. that are invisible to us - perhaps accounting for half the mass of the universe. Interesting questions concerning clumping, energy dissipation, separation, detectability, etc. of this dark matter arise, but will not be pursued here. We would like to acknowledge discussions with M. Gell-Mann and N. Warner.
References [1] J.H. Schwarz, Phys. Reports 89 (1982) 223; in Supersymmetry and supergravity '84, ed. B. deWit et al. (World Scientific, Singapore, 1984) 426 M.B. Green, Surveys in High Energy Phys. 3 (1983) 127 [2] M.B. Green and J.H. Schwarz, Phys. Lett. 149B (1984) 117 [3] M.B. Green and J.H. Schwarz, Caltech preprint CALT-68-1194, Phys. Lett. B, to be published [4] D.Z. Freedman, G. Gibbons and P.C. West, Phys. Lett. 124B (1983) 491 [5] J.H. Schwarz, in Proc. Orbis Scientiae 1978, New frontiers in high-energy physics (Plenum, New York, 1978) 431; G. Chapline and R. Slansky, Nucl. Phys. B209 (1982) 461; D. Olive and P. West, Nucl. Phys. B217 (1983) 248 [6] E. Witten, Phys. Lett. 149B (1984) 351 [7] L. Alvarez-Gaum6 and E. Witten, Nucl. Phys. B234 (1983) 269 [8] B. Zumino, Les Houches lectures 1983; R. Stora, Carg~se lectures 1983; W.A. Bardeen and B. Zumino, Nucl. Phys. B244 (1984) 421; L. Alvarez-Gaum6 and P. Ginsparg, Harvard Univ. preprint HUTP-84/A016 [9] E. Bergshoeff, M. de Roo, B. de Wit and P. van Nieuwenhuizen, Nucl. Phys. B195 (1982) 97; G.F. Chapline and N.S. Manton, Phys. Lett. 120B (1983) 105 [10] B. Zumino, Y.S. Wu and A. Zee, Nucl. PhyL B239 (1984) 477; L. Baulieu, Nucl. Phys. B241 (1984) 557
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[11] J. Thierry-Mieg, preprint LBL-18464 [12] E. Witten, Nucl. Phys. B186 (1981) 412; in Proc. 1983 Shelter Island II Conf., ed. N. Khuri (MIT Press, 1984) to appear [13] S. W. Hawking and C.N. Pope, Nucl. Phys. B146 (1978) 381 [14] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Lett. 129B (1983) 39 [15] P.K. Townsend, Phys. Lett. 139B (1984) 283 [16] T. Eguchi, P.B. Gilkey and A.J. Hansen, Phys. Reports 66 (1980) 213 [17] S. Hamidi and J.H. Schwarz, Phys. Lett. 147B (1984) 301 [18] S. Thomas and P.C. West, Nucl. Phys. B245 (1984) 45 [19] P. Howe, K. Stelle and P.C. West, Phys. Lett. 124B (1983) 55
A N O M A L Y - F R E E CHIRAL T H E O R I E S IN SIX D I M E N S I O N S * Michael B. GREEN**, John H. S C H W A R Z and P.C. WEST***
California Institute of Technology, Pasadena, CA 91125, USA Received 10 December 1984
The coupled N = 1 Yang-Mills plus supergravity theory in ten dimensions can be made anomaly-free for SO(32) or E 8 x E s. Only the case of SO(32) is known to correspond to a superstring theory, which is probably necessary for a fully consistent q u a n t u m theory. Anomaly-free chiral theories in lower dimensions can be obtained by considering nontrivial compactifications (involving nonzero background gauge fields) of the ten-dimensional theory that satisfy a topological consistency condition. This paper considers the compactification of four dimensions on the manifold K 3 without requiring that the equations of motion be satisfied. This leads to a large n u m b e r of anomaly-free chiral supersymmetric six-dimensional theories, corresponding to various ways of embedding U(1) factors in SO(32) or E s X E s.
1. Introduction
Superstring theories (for reviews see ref. [1]) are promising candidates for unified theories containing gravity in a quantum-mechanically satisfactory way. Until recently this program faced the dilemma that the most theoretically appealing examples (the type II theories) appeared unpromising for phenomenology, whereas the type I theories appeared too arbitrary and endangered by anomalies and infinities. The recent discovery of anomaly [2] and infinity [3] cancellations in the special case when the type I superstring gauge group is chosen to be SO(32) has dramatically changed the picture. The SO(32) superstring theory has all the esthetic advantages of a type II theory (finiteness, anomaly cancellation, uniqueness) plus some important phenomenological advantages to be discussed below. In ref. [2] anomaly cancellations were explained in terms of an effective field theory that describes the low-energy expansion of the superstring theory. It was shown that E 8 x E 8 is a second possibility for a satisfactory gauge group, although a corresponding superstring theory has not yet been constructed. The demonstration of anomaly cancellations for SO(32) and E 8 x E 8 in ten dimensions is reviewed in * Work supported in part by the US Department of Energy under contract DEAC 03-81-ER40050, the Fleischmann Foundation and the Alfred P. Sloan Foundation. ** On leave of absence from Queen Mary College, London, England, Nuffield Foundation Research Fellow. *** Permanent address: King's College, Strand, London, England. 327
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M.B. Green et al. / Anomaly-free chiral theories
sect. 2, using a somewhat streamlined version of the proof given in ref. [2]. We wish to emphasize that the effective field theory approach, also used in this paper, is only regarded as a tool for probing certain features of the superstring theory, such as the structure of anomalies. Having found a finite and anomaly-free SO(32) superstring theory, and perhaps an E 8 × E 8 one as well, the next crucial question is whether it admits a spontaneous compactification of six dimensions that results in an effective low-energy fourdimensional theory in accord with observations. This requires discovering nontrivial solutions to the superstring field equations. At first sight, this appears quite forbidding, even at the classical level, since these are functional equations. Furthermore, they have only been formulated in a light-cone gauge. A possible approach is to invent a superstring theory in a background with six curled-up dimensions and demonstrate that it has all the requisite consistency properties. If such a scheme is found, it would presumably correspond to a solution of the equations mentioned above. One could, of course, investigate Kaluza-Klein compactifications of the minimal low-energy effective description of the superstring theories. By "minimal" we mean the " p o i n t " (as opposed to "string") field theory describing N = 1 super-Yang-Mills coupled to supergravity, without any of the additional higher-dimension terms (such as the Lorentz Chern-Simons term of eq. (2.7)) that arise in the low-energy expansion of the superstring field theory. At the classical level, it has been shown [4] that in the minimal point theory the Ricci curvature, the Yang-Mills field strength, and the third-rank field strength must all vanish in the compact dimensions as a consistency requirement on the field equation of the scalar of the supergravity multiplet. Therefore the compact manifold cannot have any nonabelian isometries. Since there can be no cosmological constant in the ten-dimensional theory, due in particular to the chirality of the gravitino, Ricci flatness of the compact manifold implies that no cosmological constant is induced in the lower-dimensional theory. The minimal theory has anomalies, however, and we do not know which of these results carry over to the superstring theory in which the anomalies cancel. In order to obtain a chiral effective theory in lower dimensions, nonzero background gauge fields are required. Therefore it is an important question whether they are possible in the superstring theory. Most dimensional-reduction schemes obtain Yang-Mills gauge symmetries in the lower dimension, in the spirit of Kaluza and Klein, as the isometries of compact dimensions. This approach not only gives an unpleasant cosmological constant, but also has difficulty giving realistic symmetries, multiplets, and chiral structure. An alternative possibility is to start with a large gauge group in ten dimensions and to obtain the four-dimensional symmetries by breaking the gauge group and supersymmetry in the compactification [5]. By giving suitable nonzero background values to Yang-Mills gauge fields it becomes possible to transmit the chirality of the tendimensional theory to the theory in lower dimensions. In this context the SO(32) and
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E 8 x E 8 superstring theories are very attractive. These groups are large enough to contain all known gauge symmetries, and their adjoint representations can describe all the observed fermion representations on reduction to subgroups. The group E s was specifically considered in the last two references of ref. [5]. The ten-dimensional superstring theory has certain topological restrictions [6], which in conjunction with index theorems, place severe constraints on the possible theories that can result from spontaneous compactification. In this paper we study compactification of four dimensions, obtaining an effective six-dimensional theory at low energies. Sect. 3 gives a general proof that for any compact four-dimensional manifold the resulting six-dimensional theory is free of all gauge and gravitational anomalies as a consequence of the anomaly freedom of the ten-dimensional theory, the topological constraint, and the index theorems. The supersymmetry breaking is also controlled by geometrical properties of the compact manifold. The present scenario for transmitting chirality to lower dimensions in anomaly-free multiplets can be contrasted with that for four-dimensional theories having extended supersymmetry. In such theories the mirror symmetry must be broken spontaneously, and this can only give masses of the order of the weak mass scale to the mirror particles. Also, such theories do not explain why the residual theory without the mirror partners should be anomaly-free. We only consider Ricci-flat manifolds so as to ensure that there is no cosmological constant in the lower dimension at the classical level. As mentioned earlier, this is required by the field equations of the minimal point theory. Maybe this result can be generalized to the quantum superstring theory. If true, it should be possible to deduce this from the structure of the full quantum effective action, perhaps with the scalar field of the supergravity multiplet playing a crucial role. Ricci-flat manifolds in four dimensions include T 4 and K 3. This paper studies reductions to six dimensions on K 3. Conventional riemannian geometry may need to be generalized to describe the geometry of superstring theories at distances of the order of the characteristic size of strings (presumably the Planck length). In this case, the description of any compact dimensions of this size would require new mathematical concepts. Such possibilities are not considered in this paper. If the compact manifold is a product space, it need not be characterized by just one length scale. One could imagine, for example, that four compact dimensions are characterized by the Planck mass Mp, breaking the gauge group down to SU(5), say. Two more could then curl up at the unification scale M x, Reaving an effective SU(3)x SU(2)x U(1) theory in four dimensions for E << M x. A more unusual idea, that (as far as we know) is not yet excluded, is that the first breaking occurs directly to SU(3)x SU(2)x U(1) at M x, and that an effective six-dimensional theory is then applicable all the way to the weak scale M w, at which point the last two dimensions curl up breaking the symmetry to SU(3)x U(1). If this were the case, the next generation of proton accelerators would be able to probe extra dimensions!
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The possibilities for compactifications based o n T 6 and related manifolds have been discussed in ref. [6]. It was found that as long as one does not require that the field equations be satisfied, there are numerous possibilities. In particular, it is possible to obtain SU(5) with any desired number of families. Here, our main purpose is to carry out an analogous analysis for the manifold K 3. Since it is four-dimensional, the theory reduces to six dimensions. A subsequent reduction to four dimensions, using a monopole background on a two-torus, could be adjoined in each case. This final reduction is not discussed in this paper. The derivation of anomaly-free chiral theories in six dimensions does not require that the compactification correspond to a solution of the ten-dimensional field equations. The manifold K 3 has certain properties that make it an especially interesting example. For instance one can show by an index theorem that half of the D = 10 supersymmetry must be broken giving rise to a six-dimensional N = 1 theory. Also, since its Pontryagin number is nonzero, the topological consistency condition requires that a nontrivial gauge-field background be chosen. This necessarily leads to a chiral theory in six dimensions. These and other general features of K 3 are reviewed in sect. 4. In sects. 5 and 6 reductions of the SO(32) and E 8 x E 8 theories, respectively, are studied. It is shown that there are numerous possibilities for embedding U(1) factors in these groups (corresponding to background gauge fields on K3) that give rise to anomaly-free theories in six dimensions. We present many examples, but the lists are by no means complete. Nonabelian background gauge fields could also be considered, but we choose not to do so in this paper. In sect. 6 we also mention a peculiar question of physical interpretation that arises because E 8 x E 8 is a product group and the fields associated with the two factors only interact with each other with gravitational strength.
2. Review of anomaly cancellations in ten dimensions This section gives a simplified demonstration of the ten-dimensional anomaly cancellation results of ref. [2]. Gauge and gravitational anomalies arising from loop diagrams (with chiral particles going around the loop) in D dimensions can be succinctly characterized by gauge-invariant ( D + 2)-forms [7, 8], which are derived from ( D + 2)-dimensional index theorems. These are purely formal expressions, since they involve more indices than are present in D dimensions. These forms are constructed out of Yang-Mills field strengths and gravitational curvatures. In the language of forms, the Yang-Mills field strength is given by F = ½F~dx ~ A d x ~ = dA + A 2 ,
(2.1)
A =A~)t~dx ~.
(2.2)
where
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331
The matrix ~ is antihermitian, and in this paper it is taken to be in the adjoint representation of the Yang-Mills algebra. (Thus the 2t%c are the structure constants.) Wedge products of differentials are usually left implicit. The Lorentz-curvature two-form is given in similar fashion by R _- 1 ~Rl,.dx*' A dx" = d~o + o~2,
(2.3)
where ~o = % d x ~ is a 10 x 10 antisymmetric matrix (the spin connection), corresponding to the fundamental representation of-the Lorentz algebra SO(9,1). In order to describe the structure of a type I superstring theory at low energies, we now consider N = 1 supergravity coupled to N = 1 super-Yang-Mills theory in ten dimensions, with a gauge group Gm. The chiral fields that contribute to anomalies in hexagon (and higher) one-loop diagrams are a left-handed Majorana-Weyl (MW) gravitino and a right-handed MW spinor from the supergravity multiplet, as well as n = dim G10 left-handed MW spinors from the Yang-Mills supermultiplet. From eq. (63) of ref. [7], we learn that all the Yang-Mills, gravitational, and mixed anomalies due to these loops are characterized by a 12-form proportional to 1,2= _ l T r F
6 +~4TrF4trR2_9_~TrF2[4trR4
+ 5(trRZ)q
n-496 +(~+n-496)trRZtrR4+ + {3~+ -1-3-8-~ ) (trR2)3 5-76-0
-n - 496
7560 trR6"
(2.4)
We now demonstrate the existence of a local counterterm that cancels the anomalies whenever eq. (2.4) can be factorized into an expression of the form 112 = (trR 2 + k T r F 2) )(8,
(2.5)
where X 8 is a gauge-invariant eight-form made out of F and R. The distinct trace symbols tr and Tr are used as a reminder that the R are in fundamental representation and the F are in adjoint representation. A crucial role is played in the anomaly-cancellation mechanism by a second-rank potential B = B ~ d x ~' A d x ~,
which is part of the N = I , strength [2]
(2.6)
D = 10 supergravity multiplet. A three-form field H = d B + ~03L + k~o3y
(2.7)
is formed from this potential, where the Chern-Simons forms ~03r and W3L satisfy d r 0 3 y ---. T r F
2,
d r 0 3 L = t r R z.
(2.8a) (2.8b)
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The inclusion of the Lorentz Chern-Simons term o~3/~ in eq. (2.7) is an important modification of the definition of H in ref. [9]. The parameter k is uniquely determined by the string theory, but is more easily found by considering anomaly cancellations in the effective field theory, as discussed below. There also exist two-forms w~r and 0~lL such that under an infinitesimal Yang-Mills gauge transformation [8,10] 3w3r = - d ~ v ,
(2.9a)
and under an infinitesimal local-Lorentz transformation
3w3L = - d~0~L.
(2.9b)
Substituting in eq. (2.7) it is clear that H is gauge invariant provided that
8B = ,o~L + k,o~y.
(2.10)
N o w let us return to the expression 112 in eq. (2.5) and replace X 8 by X2N SO that the analysis applies for other dimensions as well (as required in sect. 3). In analogy with ~03 and ¢o1, we can introduce forms X2N 1 and XIN_2 satisfying
X2N = dX2N_I, 3X2N
I=
-dX1N
(2.11) (2.12)
2"
It is easy to convince oneself that there are solutions of these equations* for arbitrary invariant forms X2N such that the anomaly associated with I2N+4 is proportional to
G=f(2(w12L+k~X2r)XzN+N(trRZ+kTrFz)X~N
2}-
(2.13)
The problem then is to find a local interaction S c such that (2.14)
8S~+G=0. The solution to this equation is easily seen to be given by
Sc= f [N(,%L+ k O ) 3 y ) X 2 N - I - ( N
q- 2)BX2N l .
(2.15)
This result is unique up to terms that are gauge invariant. N o w let us investigate when eq. (2.4) reduces to the form (2.5). Clearly, two necessary conditions are that n = dimG10 = 496 and that T r F 6 not be an indepen* There is a well-defined prescription for resolvingambiguities [7, 8].
M.B. Green et aL / Anomaly-free chiral theories
333
dent sixth-order Casimir invariant of Gx0. Both these properties are satisfied by E 8 × E 8 and SO(32)= D16. In the case of a single E 8 T r F 4 = ~-~(TrFZ) 2,
(2.16)
T r r 6 = 7z-~(Trr2) 3.
(2.17)
Neither of these is valid, of course, for E 8 × E 8. However, in that case the weaker condition [2,11] T r r 6 = 1 T r F Z T r F 4 - ~4-4-~4~(Tr F2) 3
(2.18)
is satisfied. Remarkably, eq. (2.18) is also valid for D16. Substituting this relation and n = 496 into eq. (2.4) gives a factorized expression of the form in eq. (2.5) with k --
1 30
(2.19)
X 8 = 2~XrF 4 - v - ~ ( T r F 2 ) 2 - 2~0TrF2trR 2 + -~trR 4 + 3~(trR2) 2.
(2.20)
In view of the preceding discussion, this proves the cancellation of all anomalies for both D16 and E 8 x E 8. It is easy to show that n = 496 and eq. (2.18), with precisely the coefficients given, are both necessary and sufficient for the factorization of eq.
(2.4).
3. Reduction to six dimensions
In the previous section we found that the three-form field strength is given by H = dB - ~¢03r +
0)3L.
(3.1)
As explained in ref. [6], the requirement that H be globally well-defined gives a topological condition on possible spatial compactifications. Specifically, since dH
= -
~oTrF
2+
trR 2,
(3.2)
it is necessary that background fields R o, Fo satisfy f
(trR2 - ~0TrFo2) = 0
(3.3)
M4
for any closed four-dimensional submanifold M 4 of the ten-dimensional space-time. Ref. [6] showed that compactifications of six dimensions satisfying this condition give rise to four-dimensional models that are free from Yang-Mills gauge anomalies.
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In this section the analogous theorem for reductions to six dimensions is proved. The analysis is somewhat more elaborate in this case, because six-dimensional theories can also have gravitational anomalies. The discussion applies to both the D16 and E 8 x E 8 cases and any compact manifold M 4. It is not necessary to require that R 0 and F 0 correspond to a solution of the field equations of the ten-dimensional theory. In the background specified by R 0 and F 0, the effective six-dimensional theory has a reduced gauge symmetry. Specifically, if the nonzero fields F o span a subgroup H c G10 , then there is a unique maximal subalgebra G of G~0, all of whose generators c o m m u t e with those of H, such that GloD G X H.
(3.4)
The massless gauge fields in six dimensions would appear at first sight to correspond to the generators of G plus any U(1) factors in H. As we will explain at the end of this section, the U(1) gauge fields actually acquire a mass by a Higgs mechanism. The adjoint representation of G10 can be decomposed into a sum of representations adjoint
of G10 = E(ti, i
Ci) ,
(3.5)
where L~ and C~ are irreducible representations of G and H, respectively. In particular, )-~ dim L~dimC~ = d i m G m = 496.
(3.6)
i
If X is a matrix in the adjoint representation of G10 that corresponds to a generator of the subgroup G, it can be decomposed as follows:
X = ~) (X i® li).
(3.7a)
i
Similarly, if Y corresponds to a generator of H
Y= (t) (1, ® II,,).
(3.7b)
i
F r o m these formulas it is evident that
X Y = YX= ~ (Xi® Y,),
(3.8)
i
T r ( X Y ) = Y'. tr X~tr II,, = 0,
(3.9)
i
since Gx0 is semisimple. In the following we consider anomalies associated with
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335
G-currents. Therefore we have six-dimensional two-forms F with generators restricted to G (i.e., of X type) and background fields F 0 associated with H (i.e., of Y type). Note that F and F 0 commute (eq. (3.8)). The total anomaly in six dimensions is characterized by a formal eight-form
I= n3/2I°/2 + nx/211°/2 + Y'~n[/2I;/2 .
(3.10)
i
l°/2 characterizes the anomaly due to a single left-handed spin--~
field, which in the
present case is a singlet of the gauge group G. It is given by [7, 8]
1°/2
1
(49)4
[ - 2@8(trR2)2+ ~trR4 ] .
(3.11)
The two-form R is a 6 × 6 matrix in the fundamental representation of the algebra SO(5,1).
n3/2=n}/2-n]/2
(3.12)
is the net number of left-handed gravitinos in six dimensions. This number is given by an index theorem:
n3/2-
1 1 f trR2.
891/.2 48 M4
(3.13)
This formula is a consequence of a spin-½ index theorem, which is relevant since spin-3 in six dimensions requires the internal part of the ten-dimensional gravitino field to be spin-½ [12]. It gives the number of six-dimensional gravitinos per left-handed gravitino in the D = 10 theory, but there is only one of them. Similarly, is the anomaly due to a left-handed singlet spin-½ field in six dimensions:
I°/2
io/2 -
1
(49) 4
[~(trR2)2 + 3~otrR4] .
(3.14)
The net number of these fields in six dimensions arising from one left-handed gravitino and one right-handed spinor in the D = 10 supergravity multiplet is given by a combination of the spin- ½ and spin- 3 index theorems in the internal space. The result is [13]
nl/2=n~/2_nR/2
-
1 7 fMtrR~"
892 16
(3.15)
The anomaly contribution of a multiplet of left-handed spinors in the representation
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M.B. Green et aL / Anomaly-free chiral theories
L i of G is
1 [ / 2 = (4~.) - - 1 4 [~tr L F , _ ~tr L F2trR2 + 2_~dimLi(trR2)2 + 3160dimLitrR,] ' '
,
(3.16) and the number of such multiplets is given by a spin-½ index theorem ni=nL _nR =
1 fM [--½trc F2 +~dimCitrR2°]
87/'2
4
(3.17)
'
These results can now be assembled to give the complete expression 1 of eq. (3.10). Combining eqs. (3.10)-(3.17) gives
1
I
2(4~') 6
fM4(_4Tr(F4Fo2)+½Tr(F2Fo2)trR2_T~(trR2)2TrFo 2
-
~z(tr R 2)2trR2o - ~ t r ( R 4 ) T r F 2 + ~tr( R4)tr R 2 + 1 T r ( F 4 ) t r R ~ - 7~Tr(F2)tr(R2)trR2o}.
(3.18)
To simplify this expression we note first that eq. (2.18) applied to an arbitrary linear combination of F and Fo, and eq. (3.9) imply that Tr( F4Fo2) --- 1@0Tr( F2)Tr( F2F2 ) + 7~0Tr( Fo2)Tr F 4 - 72G Tr F02(Tr F2) 2. (3.19) Using this and eq. (3.3) gives
1
I
2(4~r) 6
fM4(__~Tr(F2)Tr(F2Fo2)+a@N~(TrF2)2trR2° + ½tr(R2)Tr(F2F~) - ~Tr(F2)tr(R2)trR2 - ~2(trR2)2trR 2}
_
1 ( t r R 2 _ ~0TrF2~ /" [~Tr( F2F2)-~tr( R 2 ) T r F a - x t r ( R2)trR2] . 2(4~.)6 ~ : LM4 (3.20)
Since this is a factorized expression of the same type as in eq. (2.5), a local
M.B. Green et al. / Anomaly-free chiral theories
337
counterterm of the form in eq. (2.15) (now using N = 2) can be constructed, thereby showing that all anomalies cancel. Using eqs. (3.3), (3.6), (3.13), and (3.17) one can show that the net number of left-handed spin-½ fermions arising from the D = 10 Yang-Mills supermultiplet is Y[nidim L i = - 224n3/2 .
(3.21)
i
This relation provides a useful check on the arithmetic in sects. 5 and 6. Let us now consider the status of the residual U(1) gauge fields in H. The anomaly cancellation arguments do not apply to them, and, indeed, naive calculation would show anomalies in the corresponding currents. However, as mentioned in ref. [6], these U(1) symmetries are spontaneously broken in such a way that these anomalies are harmless. To see this, we note that the third-rank field strength given in eq. (3.1) contains terms
H,,j = O, B i j - ~i( F ~ + )aA . ,
+
..-
,
(3.22)
where i, j are internal indices. The square of this term occurs in the action and gives kinetic terms of B,j. However, the nonzero expectation values (F,~) imply that the corresponding A~, absorb O~B,j terms yielding massive U(1) vector fields. An alternative way to understand this is to consider the variation equation (following from eq. (2.10))
8B~j = ( F ~ ) A a + . . . ,
(3.23)
which contains a constant term indicative of spontaneous symmetry breaking. As there are only six B~j in six dimensions (there would be 15 in four dimensions), this can correspond to the spontaneous breaking of at most six U(1) factors. Presumably any U(1) factors not spontaneously broken in this way are, in fact, anomaly-free.
4. R e d u c t i o n o n K 3
The manifold K 3 (Kummer's third surface) is a four-dimensional compact closed simply-connected manifold. It is the only one with a self-dual (or anti-self-dual) curvature (and hence Ricci flat) [13], i.e., it is a gravitational instanton. Conversely, all Ricci-flat four-dimensional manifolds are self-dual if they are closed and compact. It is a (hyper)-K~hler manifold that can be obtained as a solution of a homogeneous equation of degree four in complex projective three-space (CP3). Even though the metric has never been explicitly constructed, enough is known to enable one to answer most questions of interest. For example, K 3 is known to have no isometries and to require 58 parameters for its description. From this it follows that the ten-dimensional graviton gives rise to zero modes in six dimensions consisting of
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M.B. Green et al. / Anomaly-free ehiral theories
a graviton, no vectors, and 58 scalars. Reduction of supergravity theories on K 3 has been discussed previously in refs. [14,15]. Properties of K 3 are also reviewed in ref. [16]. For example, it has the Pontryagin number p,
1 f 8"//'2
trR2 = _ 4 8 .
(4.1)
K3
Substituting this into eq. (3.13), we learn that there is one gravitino zero mode in the six-dimensional theory, which corresponds to breaking half the supersymmetry of the original N = 1, D = 10 theory. This agrees with the usual argument based on the fact that holonomy group of K 3 is SU(2), which is a direct consequence of the self-duality of the curvature. There are four D = 6, N = 1 supermultiplets to consider: (i) g,~, ~bL, B~L; (ii) A~, XL; (iii) BR, XR, % (iv) xR, 4cp. The antisymmetric potentials BE; R give rise to third-rank field strengths H,, o that are self-dual or anti-self-dual. The chirality assignments of the multiplets (ii), (iii) and (iv) are the only ones that are compatible with the chirality of the supersymmetry determined by (i). As explained in ref. [15], the K 3 reduction of the N = 1, D = 10 supergravity multiplet gives rise to the following D = 6 zero modes: ( N = 1, D = 10 SG) ~ (i) + (iii) + 20(iv).
(4.2)
These are all singlets of the Yang-Mills symmetry group. The topological consistency condition, eq. (3.3), requires that one or more of the Yang-Mills fields take a nonzero background value, F 0, on the K 3 manifold. Since K 3 has the Betti numbers b~- = 3,
b 2 = 19,
(4.3)
it admits three self-dual and nineteen anti-self-dual gauge field configurations (with the convention that R is anti-self-dual). One of the three self-dual gauge fields is given by the complex structure of the K 3 manifold. Only the anti-self-dual gauge fields can solve eq. (3.3). Furthermore, from the supersymmetry transformation law of the spinor in the Yang-Mills supermultiplet, i.e., ~?~ - y~F~e, only these nineteen gauge fields are compatible with supersymmetry. Thus a maximum of nineteen independent U(1) gauge fields can be assigned values on K 3. Since the groups D16 and E 8 × E 8 only have rank 16, however, sixteen is the largest number one could actually consider for our theories. The K 3 reduction of the N = 1, D = 10 Yang-Mills supermultiplet gives rise to the Yang-Mills supermultiplet (ii) in the adjoint representation of the group G. G is
M.B. Green et al. / Anomalv-free chiral theories
339
defined to consist of all generators that commute with the generators of the group H. H was defined in sect. 3 to consist of all the generators corresponding to gauge fields that take nonzero background values. In the case of K 3 the group H is taken to consist of a product of U(1) factors. This is the only case considered here, but there are also nonabelian possibilities, such as the holonomy group SU(2). Since G includes G and any U(1) factors in H, in this case G = G + H. The reduction of the D = 10 Yang-Mills supermultiplet also gives the "hypermultiplet" (iv) in some real representation R of G. Eq. (3.21) implies that dim R = dim G + 224.
(4.4)
The number of spin-½ zero modes in the Yang-Mills sector in six dimensions is given by eq. (3.17). Since only four dimensions are being compactified on a closed manifold, the topological constraint in eq. (3.3) can be applied directly. This allows us to rewrite eq. (3.17) in the form
1 (jtrc -15
j
fWrFJ t-~dimCi trR~.
(4.5)
The ratio
ri
ftrc, Fo2
(4.6)
f VrFo2 is a n u m b e r that depends only on the group theory. Hence, n i is determined entirely by the H embedding in G10 and the value of f t r R 2 o . Therefore for K 3
n, = ( - 1 5 r i + ~ d i m C i ) 1--~- /'trR 2 8~. 2 J = ( d i m C i - 720ri).
(4.7)
Clearly, reductions to four dimensions are more complicated, because one must consider the topological constraint (3.3) on all four-dimensional submanifolds of the six-dimensional compact space. Also, in general the relevant index theorem is not in a form that allows a direct substitution of eq. (3.3), since it involves terms cubic in F 0.
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340
5. K 3 reduction of the SO(32) theory
The general analysis described in the preceding sections is applied here to the reduction of the SO(32) theory on K 3. This enables us to construct a large number of chiral N = 1, D = 6 anomaly-free theories with gauge groups that are subgroups of SO(32). There are many possible cases. We consider some that correspond to various different ways of selecting U(1) factors inside SO(32) and assigning background values to them. For example, M ~< 16 U(1)'s can be embedded in SO(32) in such a way that =
SU(Pr
X S O ( 3 2 - 2 N ) X [U(1)] M,
(5.1)
whereP~>~P2>/ "'" > / P M > / 1 a n d M
N= ~ .
(5.2)
r=|
In this embedding the 32 of SO(32) decomposes according to M
3 2 = E [(1 . . . . . 1 , n r , 1 . . . . . 1 ; 1 ) q r + ( 1 . . . . . 1 , ~ r , 1 . . . . . 1;1)_qr ] r=l
(5.3)
+ ( 1 . . . . . 1 ; 3 2 - 2N)0.
Each of the representations carries an M-vector charge q corresponding to the eigenvalues of the M U(1) factors. For the r t h U(1) charge in eq. (5.3) we make the choice qr = (0 . . . . . O, qr, 0 . . . . , 0 ) ,
(5.4)
although one could consider more complicated possibilities. The decomposition of the 496 (adjoint) can be deduced by taking the antisymmetric product (32 x 32)A. The terms with q = 0 correspond to the adjoint of G. For the types of embeddings described by eq. (5.4) there are nonzero charges associated with each U(1) factor in G, As we will see later, this is not the case in general. For each representation that occurs in the decomposition of the 496, we can apply the index theorem of eq. (3.17) to deduce the number of spin-½ zero modes of that type that occur. Since only U(1) fields occur in F0, dimC i = 1. Using eqs. (4.6) and (4.7) it follows that for a multiplet with charge qi
n i=l
720qi "qi Tr(q.q) '
(5.5)
M.B. Green et al. / Anomaly-free chiral theories
341
where Tr(q. q) = Z (dim Li)q,. qi.
(5.6)
i
The only restriction on allowed embeddings is that the multiplicities n i be integers. For the adjoint fields of G, n i = 1, which implies that the adjoint of G occurs with unit multiplicity in the Yang-Mills supermultiplet (type (ii) of sect. 4), as one would expect. For all other representations in the decomposition of the 496 the multiplicities n i are zero or negative, implying an excess of left-handed spinors over righthanded ones. Clearly, these states should be assigned to the hypermultiplet (iv). Thus, even though only the spin- 1 index theorem is used in this part of the analysis, the multiplicities of all four kinds of supermultiplets are completely determined. Let us now consider some examples. Suppose that the charge qr satisfies
m r = q~,
(5.7)
where m r is a positive integer. Next define 12
n
(5.8)
ZmrPr ' and restrict the sets {Pr) and ( m r ) by the requirement that n is an integer (1, 2, 3, 4, 6,12). In this case we learn from eq. (5.5) that
R = ~ _ , ( 4 m ~ n - 1)(1... B r . . . 1 ; 1) r
+ y' [ n ( m ~ + m s ) - l ] ( ( 1 . . . n r . . . t ~ s . . . 1 ; 1 ) + ( 1 . . . t 2 r . . . E 3 s . . . 1 ; 1 ) ) r~$ + Y'~(nmr- 1 ) ( 1 . . . l ] r . . . 1; 32 - 2 / ) + c.c.
(5.9)
?,
The multiplicities are all integers, and it is easy to verify that eq. (4.4) is satisfied. This formula gives the particle content of a very large ( - 100) number of anomalyfree chiral theories in six dimensions. It is not essential that the rn themselves be integers, what is important is that the multiplicities in eq. (5.9) be integers. As a second case let us take M
Z Pr = 16,
(5.10)
r=l
so that G=
SU(Pr
×[U(1)]M
(5.11)
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M.B. Green et aL / Anomaly-free chiral theories
In this case one has to be a little more clever to obtain integer multiplicities. A procedure that works is to choose the m r of eq. (5.7) to be a set of positive odd integers such that Y'~ m rPr = 24.
(5.12)
In this case one finds that R=~(Zm~-
1)(1... R ~ . . . 1 ) + ~ [ ½ ( m r + m s ) - I
r
]
r
× {(1 . . . [ ] r . . . n , . . . 1) + ( 1 . . . n r . . . D s . . . 1 ) } + c.c.
(5.13)
As a check, we note that eq. (4.4) is once again satisfied. When M = 1, so that CJ = SU(16) × U(1), one can take rn a = 3 to obtain R = 2(120 + 120). In general, any scheme of the m consistent with (5.12) that gives non-negative integer multiplicities for the terms in eq. (5.13) is acceptable. Because of their possible phenomenological interest, let us make eq. (5.13) more explicit for the cases SU(5) x [ U ( I ) ] 12 and SU(3) × SU(2) × [U(1)] 13. In the first case m = (1, 9,1 l°) gives R = (10 + i 0 )
+ 8(5 + 5 ) + 8 0 ( 1 ) .
(5.14)
In the second we find four possibilities: m = (1, 5,111): R = (3,1) + ( 3 , 1 ) + 4[(3,2) + (3-,2)] + 88(1,2) + 18(1,1) ;
(5.15a)
m = (1, 3, 5,110):
R=5[(3,1)+(3,1)]
+2[(3,2)+(3-,2)] +52(1,2)+90(1,1);
(5.15b)
m = (3,1, 3, 11°):
R=29[(3,1)+(3,1)[
+2[(3,2)+(3,2)]
+4(1,2)+42(1,1);
(5.15c)
m = (1, 1, 9, 11°):
R = 9[(3,1) + ( 3 , 1 ) ] + 16(1,2) + 162(1,1).
(5.15d)
Perhaps one should be encouraged that familiar multiplets occur. The list of SO(32) reductions given above is not complete. Additional ones can be obtained by associating the gauge fields with linear combinations of the U(1) subalgebras that we have considered. For example, if there is only one nonzero U(1) gauge field this gives eight new possibilities tabulated in table 1. In each case the subscript indicates the U(1) charge. There is no nonzero background gauge field for
M.B. Green et al. / Anomaly-free chiral theories
343
TABLE 1 S0(32) reductions corresponding to nontrivial assignments of a single U(1) charge
C,
R
S .U(2) × SO(26) X [U(1)] 2
17(2,1) 3 + 7(1,26)2 + 7(1,1)2 + (2,1)1 + (2,26 h + c.c.
SU(3) X SO(24) X [U(1)] 2 SU(4) × SU(2) × S0(20) × [U(1)] 2 SU(8) × SO(14) × [U(1)] 2
15(3,1)4 + 6(-3,1)2 + 8(1,24)3 + c.c. 15(1,1,1) 4 + 8(4,2,1) 3 + 3(1,2,20) 2 + 3(6,1,1) 2 + c.c. 8(8,1)3 + 3(1,14)2 + 3(28,1)2 + c.c.
SU(15) x [U(1)]2 SU(12) X SU(4) × [U(1)]2
7(15)4 + ( 15 )2 + (105)2 + c.c. 8(1,6)6 + 3(12,4)4 + c.c.
SU(14) X [U(1)] 3
15(1) 8 + 8(14)6 + 3( 14)4 + 3(14) 4 + c.c.
the other U(1) factors. Most of the entries in table 1 correspond to solutions given previously, if one ignores the U(1) charges. The last entry is not of this type. We stress once more that the six-dimensional theories tabulated here are not expected to correspond, in general, to solutions of the ten-dimensional field equations. It was conjectured in ref. [17] that solutions of the superstring field equations that describe compactification of six dimensions give rise to low-energy effective four-dimensional theories that are ultraviolet finite. The only known example involving compactification of six dimensions in the superstring theory is the relatively trivial T 6 case. It gives rise to N = 4, D = 4 super-Yang-Mills theory, which is known to be finite. This conjecture also agrees with the result of ref. [18] on nontrivial reductions of the limiting point field theory. To each of the six-dimensional theories given here we can associate an N = 2, D = 4 theory by considering a trivial T 2 reduction from six to four dimensions. The criterion for finiteness of N = 2 theories is [19] T(R) = C2(G ) ,
(5.16)
where R is the representation of the hypermultiplet. Scanning the list of theories given here, the only one that satisfies this criterion is the last one in table 1. There might be additional such examples among the schemes that we have not constructed.
6. K 3 reduction of the E s × E s theory
This section discusses anomaly-free six-dimensional theories that can arise from K 3 reduction of the E 8 × E 8 theory in ten dimensions. The analysis is a little trickier than in the SO(32) case for two reasons. (i) The general embedding of a U(1) in E 8 is somewhat complicated, and (ii) the two E 8 factors need to be considered simultaneously. The multiplicities of the six-dimensional multiplets are again determined by eq. (5.5), which did not depend on SO(32) for its derivation. The Tr(q. q) in the denominator includes contributions from both E 8 factors.
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TABLE 2 Some embeddings of a U(1) charge in Eg
Subgroup E 7 X U1 SO14 × U 1 E 6 X SO 2 X O 1 SU 8 X U 1
SOlo x SU3 x U1
SU5 x SUn x U1
Decomposition of the 248
Tr q2
133o + 1 o + 5 6 1 + 5 6 _ 1 + 1 2 + 1 2 910 + 10 + 641 + 64 _1 + 142 + 14_2 (78,1)0 + (1,3)o + (1,1)o + (27,2)1 + ( 2 7 , 2 ) _ 1 + ( 2 7 , 1 ) 2 + (27,1)_ 2 q- (1, 2)3 + (1,2) _ 3 63o + 1 o + 5 6 1 + 5 ~ i + 2 8 - 2 + 2 ~ 2 +83 + ~ - 3 (45,1)o + (1,8)o + (1,1)o + (16,3) 1 + ( 1 6 , 3 )-1 +(10, 3 ) 2 + (10,3)-2 + (16,1)3 + ( 1~,1)-3 +(1,3)4+(1,7) 4 (24,1)o + (1,15)0 + (1,1)o + (10, 4)1 + ( 1--6,4)-1 + ( 7 , 6 ) 2 + ( 5 , 6 ) _ 2 + ( 5 , 4 ) 3 + ( 5 , 4)_3 +(1--6,1)4 + (10,1)_ 4 + (1, 4)5 + (1,4)-5
120 240 360 480 720
1200
Let us begin by considering embeddings of a single U(1) charge in E 8. Seven possibilities are shown in table 2. Since the last one has Tr q2 > 720, it cannot be used to obtain integral values of n i in eq. (5.5). Suppose, to start with, that n o U(1) charge is associated with the second E 8 factor, so that it is unbroken. In this case we find the following four possibilities arising from examples 1, 2, 3, 5 in table 2:
G = E 7X U1X E 8• R = 5 ( 5 6 1 + 5 6 _ 1 ) 4- 2 3 ( 1 2 + ] _ 2 ) ;
G
(6.1)
= 5014 X U 1 X E 8 •
R = 11(142 + 14_2) + 2 ( 6 4 1 + ~ _ 1 ) ;
(6.2)
G = E 6 x SU 2 x U 1 x E 8 : R = 1 7 ( 1 , 2 ) 3 -4- 7 ( 2 i f , 1)2 4- ( 2 7 , 2)1 q- c . c . ;
(6.3)
G = SO10 × SU(3) × U 1 × E8: R = 15(1,3)4 + 8(16,1)3 + 3(10, 3-)2 + c.c.
(6.4)
It is also possible to consider simultaneous decompositions of both E s factors whenever there exist positive integers ml, m 2 such that mlTrlq 2 + m2Tr2q 2 = 720.
(6.5)
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Thus, for e x a m p l e , b r e a k i n g b o t h E 8 factors to E 7 X U 1 gives three possibilities: R l = 3(1)2 + c.c.,
(ml=l,m2=5):
R 2 = 4 ( 5 6 ) 4 f + 1 9 ( 1 ) 2 ~ + c.c. ; (m1=2, m2=4):
R 1=(56)1+7(1)2+c.c.
(ma=3, m2=3):
(6.6)
,
R 2 = 3(56)v5-+ 15(1)2v~+ c.c. ;
(6.7)
R I=R 2=2(56)1+1i(1)2+c.c.
(6.8)
T h e i r r a t i o n a l charges in eqs. (6.6) a n d (6.7) m a y suggest a violation of the D i r a c q u a n t i z a t i o n c o n d i t i o n for the " m o n o p o l e " b a c k g r o u n d gauge fields. However, this is n o t e n t i r e l y clear, because the two charges are associated with different U(1) gauge fields w h o s e n o r m a l i z a t i o n might differ b y similar i r r a t i o n a l factors. Similarly, c h o o s i n g o t h e r p a i r i n g s from table 2 one can o b t a i n G = G 1 × G 2 with the following seven possibilities (where we now suppress the charge assignments): G 1 = E 7 X U1,
G 2 = 5014 X U 1 :
R 1 = 15(1) "[- 3(56) + C.C.,
R 1 = 7 ( 1 ) + (56) + c . c . , G 1 = E 7 X U1,
R 2 = 7 ( 1 4 ) + (64) + c.c. ;
(6.9) (6.10)
G 2 = E 6 × SU 2 X U i :
R i = 11(1) + 2(56) + c . c . , G 1 = E 7 X U1,
R 2 = 3(14) + c . c . ,
R2 = 8 ( 1 , 2 ) + 3 ( 2 7 , 1 ) + c.c. ;
(6.11)
G 2 = SU 8 X Ux:
R 1 = 7(1) + (56) + c . c . , G 1 = 5014 X U1,
R x = 3(14) + c.c., G 1 = 5014 X U1, R i = 3(14) + c.c., G x = E 6 X SU2 × Ui,
R 2 = 8(8) + 3(28) + c.c. ;
(6.12)
G 2 = SU 8 x U I : R 2 = 8(8) + 3(28) + c.c. ;
(6.13)
G 2 = 5014 X U 1 : R 2 = 7(14) + (64) + c.c. ;
(6.14)
G 2 = E 6 X SU 2 X U I :
R 1 = R 2 = 8 ( 1 , 2 ) + 3 ( 2 7 , 1 ) + c.c.
(6.15)
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This is only the tip of the iceberg! There are m a n y more possibilities. In the remaining discussion we consider those that arise by successive addition of U(1) gauge fields that give the breaking chain E 8 ~ E 7 ~ E 6 ~ S O 1 0 ~ S U s , since it could be of physical interest. The first step has already been discussed, so we consider E 8 D E 6 x [U1] 2 with two nontrivial background gauge fields. There turn out to be three different ways of embedding the charges that lead to integer multiplicities. Let us consider first the case in which the second E 8 factor is not broken. In this case, even though the U 1 charge assignments are different in the three cases, the E 6 multiplicities are always given by R = 33(1) + 9(27) + c.c.
(6.16)
One of the embeddings can be carried out simultaneously in each E8 factor so as to give R: = R 2 = 15(1) + 3(27) + c.c.
(6.17)
This embedding and one of the others can also be combined with other decompositions of the second E 8. The analysis described above can now be repeated for E s D SO10 × [U:] 3 with three nontrivial background gauge fields. In this case there are four embeddings that give integer multiplicities. If the second E 8 is unbroken, all four give R=42(1)+9(10)+8(16)+c.c.
(6.18)
One of the embeddings can be combined with embeddings in the second E 8. In particular, it can be performed symmetrically on the two sides to give R , = R 2 = 1 8 ( 1 ) + 3 ( 1 0 ) + 2 ( 1 6 ) + c.c.
(6.19)
Next we consider E 8 D SU 5 × U 4, using four background gauge fields. In this case we find six embeddings, none of which allows a breaking of the second E 8 factor. All six give the SU 5 multiplicities R=50(1)+26(5)+7(10)+c.c.
(6.20)
Although this paper emphasizes U(1) background gauge fields, the analysis could be repeated for reductions based on H = SU(2), the holonomy of K 3. This can be illustrated with an example: G a = E7,
G: = E 8,
R, = 10(56) + 45(1),
R 2 = 0.
(6.21)
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347
Once again, eq. (4.4) is satisfied. One could also consider simultaneous SU(2) and U(1) background gauge fields. Even though there are still many other possibilities that could be considered, let us stop here and make a few comments. First of all, it is clear that there are a very large n u m b e r of anomaly-free theories in six dimensions. An interesting question of physical interpretation arises since E 8 x E 8 is a product group. Whenever the compact manifold has no isometrics, as in the case of K 3, the gauge symmetries are those derived from the original group. In the case of E 8 × E 8 the resulting six-dimensional spectrum of zero modes contains no states with simultaneous non-singlet quantum numbers contained in the decomposition of the two E 8 factors. Therefore one could imagine that one E 8 breaks to a subgroup G x and the second to a subgroup G 2 , but that the G~ subtheory accounts for all observed nongravitational physics - strong, weak, and electromagnetic interactions. The G 2 fields would only couple with gravitational strength to ordinary matter, and would therefore be completely invisible to observations based on the other interactions. One possibility is for the symmetry breakdown on the two sides to be identical, as in (6.8), (6.15), (6.17) and (6.19). In this case, the laws of physics are identical, and there could be G 2 galaxies, etc. that are invisible to us - perhaps accounting for half the mass of the universe. Interesting questions concerning clumping, energy dissipation, separation, detectability, etc. of this dark matter arise, but will not be pursued here. We would like to acknowledge discussions with M. Gell-Mann and N. Warner.
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