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Some remarks on anomaly cancellation in field theories derived from superstrings
Volume 169B, number 2,3
PHYSICS LETTERS
27 March 1986
S O M E REMARKS ON A N O M A L Y C A N C E L L A T I O N IN F I E L D T H E O R I E S
DERIVED FROM SUPERSTRINGS L. B O N O R A a and P. C O T T A - R A M U S I N O b " Dipartimento di Fisica, Universitlz di Padova, Via Marzolo 8, Padua, Italy and INFN, Sezione di Padova, Padua, Italy ~gipartimento di Fisica, Universit~ di Milano, Via Celoria 16, Milan, Italy and INFN, Sezione di Milano, Milan, Italy
Received 6 January 1986
We study the geometrical conditions for the existence of the fields B and H, which are necessary for the cancellation of anomalies according to the Green-Schwarz scheme. We find that a mechanism similar to the spontaneous breakdown of symmetries is required.
Cancellation of anomalies is one of the most important issues in the recent developments of (super) string theories and field theories derived as low energy limits from them [1]. A key element for this cancellation has been the introduction of the two-form B and three-form H fields linked to the gauge and 3 and coL 3 by the Lorentz Chem-Simons terms coG equation H=-dB
+ co~ - co3 L.
(1)
There has been a considerable number of papers [ 2 - 6 ] dealing with the role of these two fields. Of course any information about them should be derived from the superstring theories themselves [4,7,8]. However, it is possible to study them in the framework of the classical geometrical structures in which field theories are defined. Within this context, in this paper we would like to make a few remarks concerning the existence of the fields B and H, remarks which follow only from their role in anomaly cancellation, without explicitly invoking supersymmetry. Among other things we will see that the classical geometrical structure of gauge theories forces us to assume that the gauge connection is reducible to the spin connection (i.e. we have the "embedding" of ref. [2] ). The framework of our discussion of gauge anomalies was illustrated in refs. [9,10], where we noticed 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
that gauge anomalies and corresponding Chern-Simons terms are mathematical objects defined in the total space of a principal fiber bundle P(M, G), M being the m-dimensional space-time manifold and G the structure group supposed to be compact and semisimple. There, the coboundary (BRS) operator and the relevant "local" cohomology were considered. By "local" we mean that the cochains are polynomial expressions of the connection A, the curvature F, the ghost g and possibly other fields together with their differentials, and repeated commutators of them ,1. By solving the relevant "descent equations" ~i~ n +d~ n-1:0, d q~n +1 = 0,
~ t $ +1 + d ~ n = 0,
(2,3) (4)
where eq. (2) is the usual Wess-Zumino consistency condition and, in ~nk, n is the order of the form and k is the ghost number, it was possible to show that anomalies are in one-to-one correspondence with the non-trivial cohomology classes ofH~n +1(P)loc, the ,1 We emphasize that, generally speaking, the concept of "locality" is relative to a given theory, i.e. "local" is meant wz.t. the fields that one has (or can introduce) in a theory. 187
Volume 169B, number 2,3
PHYSICS LETTERS
"local" (m + 1)th de Rham cohomology group (i.e. not the ordinary de Rham cohomology group, but the cohomology group built over the local cochains defined above). In turn, these non-trivial cohomology classes can be expressed as linear combinations of forms of the type ~m+l O,i = Tai(A) ^ qi' (5) where the qi are basic forms (i.e. qi = rr*qi where lr : P ~ M is the projection map and the qi are globally defined [m - (2h i - 2)] -forms on M) such that dqi = O. Qi are invariant symmetric polynomials in the Lie algebra ~ of G, and the TQ i are the relevant transgressions, 1 TQi(A ) = h i f dt Qi!A, Ft ..... Ft), (6) o h i entries where F t = tF + ½(t 2 - t) [A, A ]. From eq. (5) it is then possible to go back through the descent equations and get the usual expressions for anomalies. Anomaly cancellation has two different meanings: either we refer to the mutual annihilation of anomalous contributions coming from opposite chiralities, or to the fact that an anomaly a~, in a given theory, is trivial, i.e. there are ,, local ,, forms~I'0m and a,' 1m- 1 such that _~:dxII~n-l+6xlt~ n "
(7)
In this paper we are concerned only with the second type of anomaly cancellation. We have in mind in particular the Green-Schwarz mechanism where the anomaly to be cancelled has the form M 2 ^ q, a~ 2 being the difference of two ABBJ anomalies in two dimensions; q is made out of KG ( F, F), K L (R, R ), Q4(F, F, F, F), Q4(R, R, R, R) and products of them. K G and K L are the Killing forms in ~ and in the L i e algebra of the spinor group, respectively (from now on when we write q or qi we will have generally in mind expressions of this type). One then introduces the B and H fields through eq. (1) above and, formally, the anomaly becomes trivial. The aim of this paper is to study this mechanism in particular. However, we think it is worth examining, before, the problem of anomaly cancellation in general. This will help to appreciate the peculiarities of the Green-Schwarz mechanism. 188
27 March 1986
In ref. [9] it was proven that eq. (7) is satisfied if and only if the corresponding @~n+l is trivial, i.e. ~ n +1 = d a ~ n .
(8)
ffs~n+l = - d ( T Q ( A ) A r)
(9)
because Q(F, ...,F) A r is a basic (m + 1)-form. If, in a given theory, r happens to be "local", then the anomaly a~ A q corresponding to #~n+l is trivial, as expected. In fact since 6r = 0, we have
_~ A q = 6(TQ(A) A r) + d(M A r).
(10)
These two examples should be sufficient to illustrate the problem we are faced with. However, the pattern for cancellation of anomalies described in (b), although possible, is not the one operating in field theories derived from superstrings. In the latter case, in fact, a much more stringent condition is required
TKG(A )
-
-
TKL(W) = dB +H,
(I 1)
where A and ~o are the gauge and the spin connection respectively. We examine this equation in the light of geometrical consistency only. In order to appreciate the importance of the minus sign on the LHS of eq. (11) let us start the analysis by considering first the simpler equation [12] (K = KG)
TK(A) =dB + H,
(12)
and let us assume first the generally accepted point
Volume 169B, number 2,3
PHYSICS LETTERS
27 March 1986
of view that H is a basic form, i.e. H = 7r'H, where R is globally def'med over M. Then eq. (12) cannot be satisfied. For eq. (12) would imply that d ( T K ( A ) - H ) = 0, i.e. [TK(A) - H] E H3(p). But p(TK(A) - H ) = TK(O) E H 3 (G) and, as above, [TK(O)] =/:O. Hence [TK(A) - H] :60. So there cannot exist any twoform B such that TK(A) - H = dB. At first sight, a way out would seem to exist when P is a trivial bundle. We can then pull back the equation d(TK(A) - H ) = 0 by means of a global section o : M -* P. Then d(o*TK(A) - o ' H ) = 0 and o*TK(A) - / 7 is defined in M. We can then ask ourselves if there exists in M a two-form B o, depending on o, such that dB o = o*TK(A ) - H. If this two-form exists, then, under an int'mitesimal change of section (passive gauge transformation) given by ~ : M ~ g , B ° transforms as follows:
where ~3 is another inf'mitesimal gauge transformation and ~ is the coboundary operator for the gauge Lie algebra (see ref. [i0] ). Obviously enough this is again due to the fact that the Killing form of any semisimple group G gives a non-trivial element of H3(G). In order to understand this in a different way we remark that a way to construct fields B , H , A and X as above is to identify B with the Wess-Zumino lagrangian corresponding to the two-dimensional anomaly. In conclusion the idea of defining H as TK(A) - dB does not lead to anomaly cancellation. Next let us consider a slightly more complex situation involving two connections A and A' defined in the bundles P(M, G) and Pf(M, G'), respectively, and the equation
8B a = K(o*A, d~) + dA °,
TKG(A ) + TKG,(A' ) = d B + H.
where the field A a is needed in order to preserve the nilpotency of the operator 6. But this field A a is going to give us problems in complete analogy to the case we are going to consider below. Next we are going to consider the case where H is a three-form on P which is not basic (H is not the pullback of any globally defined form on M). Then one might hope that the definition of the fields B and H through eq. (12) be consistent. However, another problem arises. Let us look at the way the operator 8 (BRS operator)acts on B. Anomaly cancellation requires dB = K(A, d~) + dA, where A is an auxiliary field with ghost number one. Notice that A cannot vanish, for otherwise 5 2B :/: 0. The nilpotency of depends on the transformation law of A, which must be
The LHS of this equation has an unambiguous geometrical meaning. It is the transgression with respect to the Killing form of the direct product G × G', calculated on the connection A + A ' defined in the principal fiber bundle P + P' (P + P' is a principal fiber bundle with structure group G X G' [13] ). Therefore the analysis of eq. (12) does not change when we consider eq. (15), and we find again obstructions in defining B and H. Therefore the minus sign on the LHS of eq. (11) is crucial. Let us study the LHS of eq. (11), i.e. TKG(A ) - TKL(¢O). A first remark is that if one takes any two connections A 1 andA 2 =A 1 + ¢~in the same principal fiber bundle P(M, G) one has
fiA = -K(~, d~) + dx,
TK(A 2)
-
(15)
TK(AI ) = 2K(o~,F1) + K(a, dAl oO
+ ½K(u, [o4 ~] ) + dK(o~,A1),
where X is a scalar field with ghost number 2, such that:
where F 1 is the curvature ofA 1 . So TK(A2) - TK(A1) = dB + H, provided that B = K(a, A 1) and
8X = -~K(~, [~, ~]).
H = 2K(°t, F1) + K(°t, dAI°I) + ~K( °t, [u,~]).
(13)
Now ~i2 = 0. However we know that there does not exist any local function X of the ghosts, which satisfies eq. (13). In another terminology, there does not exist any local skew functional ×(~1, ~2) of two infinitesimal gauge transformations ~1 and ~2, which satisfies the following equation: (SX) (~1, ~2, ~3) = ~K(~I, [~2' ~3]),
(14)
H i s a basic form and ~ H = 0. So a natural way to interpret eq. (11) is simply to assume that the connections A and 60 are defined on the same bundle. Notice that 8B = K(A2, d$) -- K(A1, d~). To be more definite, let us suppose that we have 189
PHYSICS LETTERS
Volume 169B, number 2,3
a ten-dimensional riemarmian manifold M 10 and let PE be an E8-principal bundle over M 10. We can consider also an E 8 × E 8 principal bundle over M 10 by setting PE + PE or more generally PE x E = PE p
8
8
8
8
+ PE8, where we assume the existence of agoundle homomorphism p : PEa PEa that reduces the Identity on M. The SO(10i-bundle of orthonormal frames o f M 10 will be denoted by O(M10). I f we require co to be a generic connection in O(M10), then the requirement of the cancellation of the anomaly through eq. (11) leads to considering the following assumption: there exists an embedding if: O(M 10) ~ PEa which induces the identity over M 10, i.e. ~ is a reduction o f the structure group E 8 to SO(10) [13]. In this framework A and ~ are both connections in PEa, but co is reducible to a connection in O(M10). In other words this is similar to a spontaneous symmetry breaking from E 8 to SO(10) [14]. Notice that if PEa x Es is defined as before, the map induces also a reduction of the group E 8 × E 8 to SO(10), so we can also consider A and co as connections in PEa x E8 and again co is reducible to a connection in O(M10). If we assume that the manifold M 10 is itself the cartesian product of two riemannian manifolds M 4 and M 6, respectively four and six dimensional, we could consider instead o f the bundle O(M 10) the bundle O(M 4) X O(M6). This point of view corresponds to allowing only changes o f coordinates separately in the two manifolds M 4 and M 6. In this case we would have the reduction of the structure group E 8 to SO(4) X S O ( 6 ) ' 2 . The choice between all the alternatives we have considered so far, depends on what we really mean by the connection co in eq. (11). Notice that to any reducible connection, there corresponds a parallel section in an associated bundle
,2 Up to now we have considered to as a connection (reducible to a connection) in O(M10) or O(M4) X O(M6). Now if M1°, M4, M6 are spin manifolds [15 ] then any connection in O(M) will give rise to a connection in the spin bundle and vice versa. So we could also consider the reduction of E8 to Spin (10) or to Spin (4) X Spin (6).
27 March 1986
[14]. Finally if we assume, following ref. [2] that M 6 is a compact, simply connected spin manifold which admits a parallel spinor, then M 6 must be a K/ihler manifold with vanishing Ricci tensor [16] and so its restricted holonomy group is (contained in) SU(3) [13,17]. Hence if the bundle O(M 4) is trivial we can consider a reduction o f E 8 to (e} × SU(3)
SU(3). We thank very much D. Kastler, M. Martellini, P. Pasti, C. Reina, R. Stora, M. Tonin for very useful discussions.
References [1 ] M.B. Green and J.H. Schwarz, Phys. Lett. 149B (1984) 117. [2] P. Candelas, G. Horowitz, A. Strominger and E. Witten, Nucl. Phys. B258 (1985) 46. [3] L.J. Romans and N.P. Warner, Some supersymmetric counterparts of the Lorentz Chern-Simons term, preprint CALT-68-1291 (1985). [4] R.I. Nepomechie, On the low-energy limit of strings, University of Washington preprint 40048-21P5; ChernSimons terms and bosonic strings, Phys. Lett. 171B (1986), to be published. [5] C.M. Hull, Phys. Lett. 167B'(1986) 51. [6] E. Witten, New issues in manifolds of SU(3) holonomy, Princeton preprint (October 1985)• [7] E.S. Fradkin and A.A. Tseytlin, Phys. Lett. 158B (1985) 316. [8] A. Sen, a-model approach to the heterotic string theory, preprint SLAC-PUB 3794 (1985); Phys. Lett. 166B (1986) 300. [9] L. Bonora and P. Cotta-Ramusino, Consistent and covariant anomalies and local cohomology, preprint DFPD 29-85• [10] L. Bonora and P. Cotta-Ramusino, Commun. Math. Phys. 87 (1983) 589. [11 ] W. Greub, S. Halperin and R. Vanstone, Connections, curvature and cohomology, Vol. III (Academic Press, New York, 1976). [12] G.F. Chapline and N.S. Manton, Phys. Lett. 120B (1983) 105. [13] S. Kobayashi and K. Nomizu, Foundations of differential geometry (Wiley, New York, 1963). [14] A. Trautman, Czech. J. Phys. B29 (1979) 107. [15] J. Milnor, L'Enseignement mathematique, IIe series Tome IX (1963) 198. [16] N. Hitchin, Adv. Math. 14 (1974) 1. [17] A. Beauville, J. Diff. Geom. 18 (1983) 755.
PHYSICS LETTERS
27 March 1986
S O M E REMARKS ON A N O M A L Y C A N C E L L A T I O N IN F I E L D T H E O R I E S
DERIVED FROM SUPERSTRINGS L. B O N O R A a and P. C O T T A - R A M U S I N O b " Dipartimento di Fisica, Universitlz di Padova, Via Marzolo 8, Padua, Italy and INFN, Sezione di Padova, Padua, Italy ~gipartimento di Fisica, Universit~ di Milano, Via Celoria 16, Milan, Italy and INFN, Sezione di Milano, Milan, Italy
Received 6 January 1986
We study the geometrical conditions for the existence of the fields B and H, which are necessary for the cancellation of anomalies according to the Green-Schwarz scheme. We find that a mechanism similar to the spontaneous breakdown of symmetries is required.
Cancellation of anomalies is one of the most important issues in the recent developments of (super) string theories and field theories derived as low energy limits from them [1]. A key element for this cancellation has been the introduction of the two-form B and three-form H fields linked to the gauge and 3 and coL 3 by the Lorentz Chem-Simons terms coG equation H=-dB
+ co~ - co3 L.
(1)
There has been a considerable number of papers [ 2 - 6 ] dealing with the role of these two fields. Of course any information about them should be derived from the superstring theories themselves [4,7,8]. However, it is possible to study them in the framework of the classical geometrical structures in which field theories are defined. Within this context, in this paper we would like to make a few remarks concerning the existence of the fields B and H, remarks which follow only from their role in anomaly cancellation, without explicitly invoking supersymmetry. Among other things we will see that the classical geometrical structure of gauge theories forces us to assume that the gauge connection is reducible to the spin connection (i.e. we have the "embedding" of ref. [2] ). The framework of our discussion of gauge anomalies was illustrated in refs. [9,10], where we noticed 0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
that gauge anomalies and corresponding Chern-Simons terms are mathematical objects defined in the total space of a principal fiber bundle P(M, G), M being the m-dimensional space-time manifold and G the structure group supposed to be compact and semisimple. There, the coboundary (BRS) operator and the relevant "local" cohomology were considered. By "local" we mean that the cochains are polynomial expressions of the connection A, the curvature F, the ghost g and possibly other fields together with their differentials, and repeated commutators of them ,1. By solving the relevant "descent equations" ~i~ n +d~ n-1:0, d q~n +1 = 0,
~ t $ +1 + d ~ n = 0,
(2,3) (4)
where eq. (2) is the usual Wess-Zumino consistency condition and, in ~nk, n is the order of the form and k is the ghost number, it was possible to show that anomalies are in one-to-one correspondence with the non-trivial cohomology classes ofH~n +1(P)loc, the ,1 We emphasize that, generally speaking, the concept of "locality" is relative to a given theory, i.e. "local" is meant wz.t. the fields that one has (or can introduce) in a theory. 187
Volume 169B, number 2,3
PHYSICS LETTERS
"local" (m + 1)th de Rham cohomology group (i.e. not the ordinary de Rham cohomology group, but the cohomology group built over the local cochains defined above). In turn, these non-trivial cohomology classes can be expressed as linear combinations of forms of the type ~m+l O,i = Tai(A) ^ qi' (5) where the qi are basic forms (i.e. qi = rr*qi where lr : P ~ M is the projection map and the qi are globally defined [m - (2h i - 2)] -forms on M) such that dqi = O. Qi are invariant symmetric polynomials in the Lie algebra ~ of G, and the TQ i are the relevant transgressions, 1 TQi(A ) = h i f dt Qi!A, Ft ..... Ft), (6) o h i entries where F t = tF + ½(t 2 - t) [A, A ]. From eq. (5) it is then possible to go back through the descent equations and get the usual expressions for anomalies. Anomaly cancellation has two different meanings: either we refer to the mutual annihilation of anomalous contributions coming from opposite chiralities, or to the fact that an anomaly a~, in a given theory, is trivial, i.e. there are ,, local ,, forms~I'0m and a,' 1m- 1 such that _~:dxII~n-l+6xlt~ n "
(7)
In this paper we are concerned only with the second type of anomaly cancellation. We have in mind in particular the Green-Schwarz mechanism where the anomaly to be cancelled has the form M 2 ^ q, a~ 2 being the difference of two ABBJ anomalies in two dimensions; q is made out of KG ( F, F), K L (R, R ), Q4(F, F, F, F), Q4(R, R, R, R) and products of them. K G and K L are the Killing forms in ~ and in the L i e algebra of the spinor group, respectively (from now on when we write q or qi we will have generally in mind expressions of this type). One then introduces the B and H fields through eq. (1) above and, formally, the anomaly becomes trivial. The aim of this paper is to study this mechanism in particular. However, we think it is worth examining, before, the problem of anomaly cancellation in general. This will help to appreciate the peculiarities of the Green-Schwarz mechanism. 188
27 March 1986
In ref. [9] it was proven that eq. (7) is satisfied if and only if the corresponding @~n+l is trivial, i.e. ~ n +1 = d a ~ n .
(8)
ffs~n+l = - d ( T Q ( A ) A r)
(9)
because Q(F, ...,F) A r is a basic (m + 1)-form. If, in a given theory, r happens to be "local", then the anomaly a~ A q corresponding to #~n+l is trivial, as expected. In fact since 6r = 0, we have
_~ A q = 6(TQ(A) A r) + d(M A r).
(10)
These two examples should be sufficient to illustrate the problem we are faced with. However, the pattern for cancellation of anomalies described in (b), although possible, is not the one operating in field theories derived from superstrings. In the latter case, in fact, a much more stringent condition is required
TKG(A )
-
-
TKL(W) = dB +H,
(I 1)
where A and ~o are the gauge and the spin connection respectively. We examine this equation in the light of geometrical consistency only. In order to appreciate the importance of the minus sign on the LHS of eq. (11) let us start the analysis by considering first the simpler equation [12] (K = KG)
TK(A) =dB + H,
(12)
and let us assume first the generally accepted point
Volume 169B, number 2,3
PHYSICS LETTERS
27 March 1986
of view that H is a basic form, i.e. H = 7r'H, where R is globally def'med over M. Then eq. (12) cannot be satisfied. For eq. (12) would imply that d ( T K ( A ) - H ) = 0, i.e. [TK(A) - H] E H3(p). But p(TK(A) - H ) = TK(O) E H 3 (G) and, as above, [TK(O)] =/:O. Hence [TK(A) - H] :60. So there cannot exist any twoform B such that TK(A) - H = dB. At first sight, a way out would seem to exist when P is a trivial bundle. We can then pull back the equation d(TK(A) - H ) = 0 by means of a global section o : M -* P. Then d(o*TK(A) - o ' H ) = 0 and o*TK(A) - / 7 is defined in M. We can then ask ourselves if there exists in M a two-form B o, depending on o, such that dB o = o*TK(A ) - H. If this two-form exists, then, under an int'mitesimal change of section (passive gauge transformation) given by ~ : M ~ g , B ° transforms as follows:
where ~3 is another inf'mitesimal gauge transformation and ~ is the coboundary operator for the gauge Lie algebra (see ref. [i0] ). Obviously enough this is again due to the fact that the Killing form of any semisimple group G gives a non-trivial element of H3(G). In order to understand this in a different way we remark that a way to construct fields B , H , A and X as above is to identify B with the Wess-Zumino lagrangian corresponding to the two-dimensional anomaly. In conclusion the idea of defining H as TK(A) - dB does not lead to anomaly cancellation. Next let us consider a slightly more complex situation involving two connections A and A' defined in the bundles P(M, G) and Pf(M, G'), respectively, and the equation
8B a = K(o*A, d~) + dA °,
TKG(A ) + TKG,(A' ) = d B + H.
where the field A a is needed in order to preserve the nilpotency of the operator 6. But this field A a is going to give us problems in complete analogy to the case we are going to consider below. Next we are going to consider the case where H is a three-form on P which is not basic (H is not the pullback of any globally defined form on M). Then one might hope that the definition of the fields B and H through eq. (12) be consistent. However, another problem arises. Let us look at the way the operator 8 (BRS operator)acts on B. Anomaly cancellation requires dB = K(A, d~) + dA, where A is an auxiliary field with ghost number one. Notice that A cannot vanish, for otherwise 5 2B :/: 0. The nilpotency of depends on the transformation law of A, which must be
The LHS of this equation has an unambiguous geometrical meaning. It is the transgression with respect to the Killing form of the direct product G × G', calculated on the connection A + A ' defined in the principal fiber bundle P + P' (P + P' is a principal fiber bundle with structure group G X G' [13] ). Therefore the analysis of eq. (12) does not change when we consider eq. (15), and we find again obstructions in defining B and H. Therefore the minus sign on the LHS of eq. (11) is crucial. Let us study the LHS of eq. (11), i.e. TKG(A ) - TKL(¢O). A first remark is that if one takes any two connections A 1 andA 2 =A 1 + ¢~in the same principal fiber bundle P(M, G) one has
fiA = -K(~, d~) + dx,
TK(A 2)
-
(15)
TK(AI ) = 2K(o~,F1) + K(a, dAl oO
+ ½K(u, [o4 ~] ) + dK(o~,A1),
where X is a scalar field with ghost number 2, such that:
where F 1 is the curvature ofA 1 . So TK(A2) - TK(A1) = dB + H, provided that B = K(a, A 1) and
8X = -~K(~, [~, ~]).
H = 2K(°t, F1) + K(°t, dAI°I) + ~K( °t, [u,~]).
(13)
Now ~i2 = 0. However we know that there does not exist any local function X of the ghosts, which satisfies eq. (13). In another terminology, there does not exist any local skew functional ×(~1, ~2) of two infinitesimal gauge transformations ~1 and ~2, which satisfies the following equation: (SX) (~1, ~2, ~3) = ~K(~I, [~2' ~3]),
(14)
H i s a basic form and ~ H = 0. So a natural way to interpret eq. (11) is simply to assume that the connections A and 60 are defined on the same bundle. Notice that 8B = K(A2, d$) -- K(A1, d~). To be more definite, let us suppose that we have 189
PHYSICS LETTERS
Volume 169B, number 2,3
a ten-dimensional riemarmian manifold M 10 and let PE be an E8-principal bundle over M 10. We can consider also an E 8 × E 8 principal bundle over M 10 by setting PE + PE or more generally PE x E = PE p
8
8
8
8
+ PE8, where we assume the existence of agoundle homomorphism p : PEa PEa that reduces the Identity on M. The SO(10i-bundle of orthonormal frames o f M 10 will be denoted by O(M10). I f we require co to be a generic connection in O(M10), then the requirement of the cancellation of the anomaly through eq. (11) leads to considering the following assumption: there exists an embedding if: O(M 10) ~ PEa which induces the identity over M 10, i.e. ~ is a reduction o f the structure group E 8 to SO(10) [13]. In this framework A and ~ are both connections in PEa, but co is reducible to a connection in O(M10). In other words this is similar to a spontaneous symmetry breaking from E 8 to SO(10) [14]. Notice that if PEa x Es is defined as before, the map induces also a reduction of the group E 8 × E 8 to SO(10), so we can also consider A and co as connections in PEa x E8 and again co is reducible to a connection in O(M10). If we assume that the manifold M 10 is itself the cartesian product of two riemannian manifolds M 4 and M 6, respectively four and six dimensional, we could consider instead o f the bundle O(M 10) the bundle O(M 4) X O(M6). This point of view corresponds to allowing only changes o f coordinates separately in the two manifolds M 4 and M 6. In this case we would have the reduction of the structure group E 8 to SO(4) X S O ( 6 ) ' 2 . The choice between all the alternatives we have considered so far, depends on what we really mean by the connection co in eq. (11). Notice that to any reducible connection, there corresponds a parallel section in an associated bundle
,2 Up to now we have considered to as a connection (reducible to a connection) in O(M10) or O(M4) X O(M6). Now if M1°, M4, M6 are spin manifolds [15 ] then any connection in O(M) will give rise to a connection in the spin bundle and vice versa. So we could also consider the reduction of E8 to Spin (10) or to Spin (4) X Spin (6).
27 March 1986
[14]. Finally if we assume, following ref. [2] that M 6 is a compact, simply connected spin manifold which admits a parallel spinor, then M 6 must be a K/ihler manifold with vanishing Ricci tensor [16] and so its restricted holonomy group is (contained in) SU(3) [13,17]. Hence if the bundle O(M 4) is trivial we can consider a reduction o f E 8 to (e} × SU(3)
SU(3). We thank very much D. Kastler, M. Martellini, P. Pasti, C. Reina, R. Stora, M. Tonin for very useful discussions.
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