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Internal Einstein spaces and symmetry breaking
V o l u m e 143B, n u m b e r 4, 5, 6
PHYSICS LETTERS
16 A u g u s t 1984
I N T E R N A L E I N S T E I N S P A C E S AND S Y M M E T R Y BREAKING R. C O Q U E R E A U X Centre de Physique Theorique, C N R S - L u m i n y , Case 907, F - 1 3 2 8 8 Marseille Cedex 9, France Received 27 F e b r u a r y 1984
W e first define a generalised gauge i n v a r i a n t Y a n g - M i l l s lagrangian: the Killing metric - K=# on the group is replaced by a more general m e t r i c h,~# ( x ) ; the field h,~# ( x ) - a scalar from the s p a c e - t i m e p o i n t of view - is then c o v a r i a n t l y coupled to the gauge field A~,'~ a n d is also self-coupled via a n a t u r a l scalar p o t e n t i a l (no parameters). N o n - t r i v i a l saddle p o i n t s of this scalar p o t e n t i a l c o r r e s p o n d to n o n - s t a n d a r d Einstein metrics on the g r o u p G. The associated shifts lead to a n entirely c o m p u t a b l e mass s p e c t r u m for the gauge field.
1. The lagrangian. When one considers the usual Yang-Mills lagrangian - K(F~,, F ~ ) and thinks about possible generalisations, a simple idea comes to the mind: replace the Killing metric - K ~ t ~by a more general metric h,# i.e., replace a bi-invariant metric (G x G) on the internal space G by another metric which is less symmetric. It is almost immediately clear that h,B has to be at least G-invariant, in other words it has to be entirely characterized by its value at the identity of G, i.e. by its action on the elements of the Lie algebra (one can say also that, by an appropriate choice of basis, h~B is independent of the coordinates on the group). Therefore, the isometry group of G will be of the kind K x G (where K is a subgroup of G, possibly trivial). However if h ~ is nothing but a symmetric matrix of constant real numbers, it is easy to see that gauge invariance (with respect to G) is broken, unless in the case where h ~ is proportional to K,, a. To cure this disease, the obvious solution is to decide that h ~a will be a field h ~ a ( x ) and not a constant matrix; of course x is a point in space-time. From the intuitive point of view, we associate to each point x of space-time, a quantity h ~ a ( x ) measuring the shape of the group G at this point. Now of course, h ~ a ( x ) transforms according to some representation of G (the bilinear symmetric) and we have to couple h ~ a ( x ) covariantly to the gauge field. It can
be checked that
- ¼F~,Ft'~h,q~(x) - ¼h"~(x)hVa(x)D~h,~v(x)D~'hl~n(x) +Dl, h , , ~ ( x ) O g h v s ( x ) , with
D~,h,, B ( x ) = 01,h,~#( x ) - Av~C~,,h~l ~ - A~C~vh~a is indeed a gauge invariant quantity (we discard total divergences). Now, we have a Yang-Mills field and a kinematic term for the scalar field (h,~p(x) is indeed a scalar field from the space-time point of view); however we would like to use a kind of potential term for the scalar field h,,B(x ). Being a bilinear symmetric quantity, the most obvious choice for this potential is V ( h ) = - [ s c a l a r curvature associated to h~#(x)]. Explicitly, when G is unimodular (in particular when G is compact),
"r° = - V [ h ] = h a a ' ( ½ C ~7, C ~ , v - z h1 a
aa"hvv,C~#C~,#,), 7 7"
C~aV being the structure constants associated to a basis ( X, }, orthonormal for the Killing form. The minus sign in front of V(h) is there for reasons of positivity. The reader may wonder why we choose such a complicated kinetic energy term for h~/~ and not
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
403
Volume 143B, number 4, 5, 6
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s o m e t h i n g like (D,h,BD"hv8)8"v6a~; it should then be noticed that b y m a k i n g a linear a p p r o x i m a t i o n : h ~B( x ) = 6oB + ~ ,a( x ), the first term in the e x p a n sion w o u l d i n d e e d be of the a b o v e kind. The m o t i v a t i o n for such a kinetic energy term a n d for the scalar p o t e n t i a l comes from the following: when the a b o v e l a g r a n g i a n is c o v a r i a n t l y c o u p l e d to Einstein gravity, the whole lagrangian can be r e i n t e r p r e t e d as a theory of pure gravity (without m a t t e r fields) in d i m e n s i o n 4 + d i m ( G ) ; see ref. [1] where a m o r e general situation is investigated). Moreover, this kinetic energy term is a k i n d of generalisation of the n o n - l i n e a r o model: for a fixed volume of G (det h = const.), the field h~a varies in the m a n i f o l d S L ( n ) / S O ( n ) , n = d i m G, this is i n d e e d the m a n i f o l d of G - i n v a r i a n t metrics o n G. The Q F T expert will have also noticed that, b y p o w e r counting arguments, the a b o v e lagrangian looks b a d f r o m the r e n o r m a l i s a t i o n p o i n t of view; however, if one r e m e m b e r s that it can be w r i t t e n as a theory of pure gravity (in m o r e d i m e n sions), even if one expects strong divergences in the associated q u a n t u m field theory, one also expects strong cancellations; therefore the subject deserves a m o r e d e t a i l e d study.
2. Motioation for Einstein spaces and symmetry breaking. O n one hand, we saw that the p o t e n t i a l for scalar field is i n t e r p r e t e d (or defined!) up to a sign as the scalar curvature of the internal space G at the p o i n t x, on the other hand, there is an old t h e o r e m (due to Hilbert) which says that s a d d l e p o i n t s of the total scalar curvature - c o n s i d e r e d a functional on the space of metrics - for fixed volume, coincide with the Einstein metrics. Putting these two facts together a n d r e m e m b e r i n g that the scalars h ~ ( x ) are c o v a r i a n t l y c o u p l e d to the gauge field, one m a y look for an a n a l o g u e of the Higgs mechanism.
3. Einstein metrics on groups. T h e Killing metric on a Lie group G is an Einstein metric, this is well k n o w n a n d is f o r t u n a t e in o u r context since this p r o p e r t y tells us that the usual Y a n g - M i l l s l a g r a n g i a n is i n d e e d associated to a s a d d l e p o i n t of o u r m o r e general lagrangian. There exist in general b u t n o t always - m a n y other Einstein metrics on a given Lie group G however we are only inter-
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ested here in those which are K x G invariant. As stated previously, a very old t h e o r e m assures that for a given m a n i f o l d S, the total scalar curvature f~'Sd vol, c o n s i d e r e d like a functional on the space of metrics with fixed v o l u m e a d m i t s saddle p o i n t s for all Einstein metrics. The c o n s t r a i n t of fixed v o l u m e can easily be u n d e r s t o o d : consider for i n s t a n c e a usual two-sphere of radius R, its scalar c u r v a t u r e can be m a d e a r b i t r a r i l y small or large b y m o d i f y i n g its r a d i u s (~- = 2 / R 2 ) ; this k i n d of varia t i o n is not interesting, we have to fix the volume a n d study the v a r i a t i o n of the total scalar curvature. In the special case u n d e r study, the total scalar curvature is equal to the p r o d u c t ~- x vol(G) a n d we have o n l y to l o o k at variation of ~, ~- being c o n s i d e r e d as a f u n c t i o n a l on the space of G - i n v a r i a n t metrics (of course, with such a restriction we o n l y get a necessary c o n d i t i o n but, for all the cases treated here, it can be shown to be sufficient). T h e basic strategy used to o b t a i n n o n - s t a n d a r d Einstein metrics or groups is m o r e or less always the same: one first chooses along K in G = U ~ c / K ( g K ). This a m o u n t s to consider G as a collection of copies of G / K glued together a n d p a r a m e t r i s e d b y K; one then chooses some n a t u r a l metric on g (for e x a m p l e the G × G i n v a r i a n t Killing metric) a n d begins to " d i s t o r t it" in a w a y a p p r o p r i a t e to the coset d e c o m p o s i t i o n ; in the o b t a i n e d family of new metrics, one looks for those where the Einstein c o n d i t i o n is satisfied, either b y c o m p u t i n g directly the Ricci tensor or b y l o o k i n g at s a d d l e p o i n t s of some functional. Let us d o it explicitly in the special case where G is a simple c o m p a c t Lie group a n d where m o r e o v e r K is a s u b g r o u p of G such that (G, K ) is a s y m m e t r i c pair. W e n o w start from the following b i - i n v a r i a n t ( a n d Einstein) metric on G : g = - ( K i l l i n g form). W e can now c o m p u t e the scalar curvature T G o f G in terms of the scalar curvatures ~.~/K, ~K of G / K a n d K associated to the c o r r e s p o n d i n g restrictions of g. W e find .re = ~.G/K + ~.K _ ¼rc~ab~r'~b,, or = k s + c k k - - k(1 - c ) .
W e write a, a, h for indices in G, G / K , K, where n=dimG, k=dimK, s=n-k=dimG/K and c, the index of K in G being defined as follows: Killing metric of G restricted to K = gab = - C " a a C a b ~ , K i l l i n g m e t r i c of K = gKab =
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
- - C daeC Zd a n d calling c the coefficient gKaL = cgaT,. T h e previous d e c o m p o s i t i o n of r G is a " K a l u z a - K l e i n r e d u c t i o n " (before K a l u z a - K l e i n ) where the " e x t e r n a l " space is G / K , the " i n t e r n a l " space is K a n d the field strength is C",h. Notice that when (G, K) is a symmetric pair we have indeed c = 1 - s / 2 k b u t this would not be true in the general case. Following the general recipe, we now write L i e ( G ) = L i e ( K ) • P - orthogonal dec o m p o s i t i o n for g - a n d consider the following family of metrics (t is a real parameter) on G: h = g/P + t 2 g/K. These metrics, o b t a i n e d by a scaling of g in the direction K are n o longer G × G i n v a r i a n t b u t only G × K i n v a r i a n t ; the scalar curvature of G is now
16 August 1984
4. S y m m e t r y breaking. I n the Higgs setting, one looks for a n o n - z e r o local m i n i m u m V0 of a suitable fourth degree G - i n v a r i a n t p o l y n o m i a l a n d perform the shift qb(x) = V0 + q~'(x). In our approach we write formally h , B ( x ) = h° ~ + h ' , a ( x ) , h°~B b e i n g a h o m o g e n e o u s Einstein metric for the group G. Let us analyse the situation when G is u n i m o d ular (then C ~ b = 0) a n d when h°~a is a G × K i n v a r i a n t Einstein metric o b t a i n e d b y the above m e t h o d (we even suppose that (G, K) is a symmetric pair, then the critical value of the scaling p a r a m e t e r is t 2 = (2k - s ) / ( 2 k + s). U s i n g the definition of D,, the term L = - ¼h'~Bh ~a (D.h,~,8 D~'h.y~ + D~,h,,vD~'hCa ) can be e x p a n d e d a n d we get
r ° = s/2 + (ck/4)/t
2 - k(1 - c ) t 2 / 4 .
However, when t varies, the volume of G varies; in order to keep it fixed we j u s t have to make a c o n f o r m a l rescaling a n d consider the family of metrics h = ( 1 / t 2 ) * / n h , then det h = ( 1 / t 2 ) k • (t2) k = constant. T h e associated scalar curvature reads r 6 = (,2)*/" [s/2 + (ck/4)/t
2 - k(1 - c ) t 2 / 4 ] .
W e n o w vary this expression with respect to t a n d find dcG/dt = - (s/4)(Zk
+ 2)/(k + s).t 2k/"-3
× (t 2 - 1 ) [ t 2 - ( 2 k - s ) / ( Z k
+ s)],
where we used the property c = 1 - s / 2 k , valid for a symmetric pair. We find therefore two Einstein metrics: c o r r e s p o n d i n g to the values t 2 = 1 a n d t 2 = (2k - s ) / ( 2 k + s). The first value corresponds of course to the b i - i n v a r i a n t metric g used in the beginning, the other value c o r r e s p o n d i n g to a n o n - s t a n d a r d G x K i n v a r i a n t metric on G (for example, studying G = SU(3), if we choose K = SO(3), we o b t a i n a n S U ( 3 ) x SO(3) n o n - s t a n d a r d Einstein metric on SU(3) for the value t 2 = (2 x 3 - 5 ) / ( 2 x 3 + 5) = 1/11). A general study based on the direct comput a t i o n of the Ricci tensor is carried out in ref. [2]. M a n y c o m m e n t s a n d references o n relative topics m a y be f o u n d in ref. [31.
where M~a = J . ¢ + K~¢; K~a = C~yC~a being the Killing form a n d J aB = h YY'"t~8 8 " ~ta~' g ~tf~l ~' ,'" We now make formally the shift h ( x ) = h ° + h ' ( x ) . The calculation is straightforward, we find that i AI -
2 . - . . ¢ A. .a.A1t.~ B =
_
51 .llA . . . ¢ 1A . ~. ~B
+ Rest,
where = 0,
= 0,
& b = 1 ( t 2 + 1 / , 2) - 1.
I n other words, when t 2 = 1 (i.e., we expand a r o u n d the b i - i n v a r i a n t metric of G), the gauge field stays massless; however, when t 2 = (2k - s ) / ( 2 k + s ) (i.e. we e x p a n d a r o u n d a n o n - s t a n d a r d Einstein metric on G, G × K invariant, with the n o t a t i o n s of section 6.2.4) [3] then, the c o m p o n e n t s of the gauge field taking their value in Lie(K) stay massless b u t the c o m p o n e n t s lying in the subspace P (LIE(G)= Lie(K)+P) acquire a mass m 2 = 2 s 2 / ( 4 k 2 _ s 2) in dimensionless units. Let us assume for example that our i n t e r n a l space is G = SU(4) a n d that the volume is fixed, then if we expand the scalar fields h ~a a r o u n d the b i - i n v a r i a n t metric, we get 15 massless gauge fields b u t we can also expand h~t~ a r o u n d the following 405
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
-TGI
I
I
) t2
Fig.
G x K i n v a r i a n t Einstein metrics: (i) K = S(U(2) x U(2)) = SU(2) x SU(2) × U(1) for t 2 = 3 / 1 1 . (ii) K = USp(4) for t 2 = 3 / 5 . (iii) K = SO(4) for t 2 = 1 / 7 . I n case (i) we get k = 7 massless gauge fields a n d s = 8 massive fields of mass m 2 = 3 2 / 3 3 . In case (ii) we get k = 10 massless gauge fields a n d s = 5 massive fields of m a s s m 2 = 1 / 1 5 . In case (iii) we get k = 6 massless gauge fields a n d s = 9 massive fields of mass m 2 = 1 8 / 7 . I n the previous examples, the p a i r (SU(4), K) is s y m m e t r i c b u t there exist other s a d d l e p o i n t s (other Einstein metrics) involving n o n - s y m m e t r i c pairs. There is a difficulty which is b e t t e r e x p l a i n e d b y looking at fig. 1 which is the g r a p h of the p o t e n t i a l V(H) for the o n e - p a r a m e t e r f a m i l y of metrics a l r e a d y discussed in section 3; using typical values of the p a r a m e t e r we get the following curve" T h e p o i n t A c o r r e s p o n d s to the s t a n d a r d bi-inv a r i a n t metric on G a n d B to a n o n - s t a n d a r d Einstein metric. This " p o t e n t i a l " is therefore n o t b o u n d e d from b e l o w - even with the fixed v o l u m e restriction - m o r e o v e r the n o n - s t a n d a r d Einstein m e t r i c of this f a m i l y c o r r e s p o n d s to a local maxim u m of the curve; in a m o r e general situation,
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Einstein metrics c o r r e s p o n d to s a d d l e p o i n t s which are neither m i n i m a n o r m a x i m a o f the total scalar c u r v a t u r e function. T h e " p h y s i c a l " i n t e r p r e t a t i o n of the a b o v e results is therefore unclear: we e x p l a i n e d w h a t h a p p e n s if we e x p a n d a non trivial s a d d l e point, b u t (1) are we allowed to " e x p a n d " a r o u n d them? (2) even if we can, w h y d o it a n d are these s a d d l e p o i n t s i m p o r t a n t in a q u a n t u m m e c h a n i c a l perspective? If we w a n t to s t u d y global aspects of s y m m e t r y b r e a k i n g a n d use fiber b u n d l e techniques, there is a n o t h e r difficulty which arises when we try to define globally the shifted field h~B; however the s a m e difficulty usually o v e r l o o k e d a l r e a d y exist in the Higgs setting when we try to define the shifted field dp'; one solution is to w o r k in a trivial b u n d l e (as p e o p l e d o usually when discussing Higgs mecha n i s m - see however ref. [4]). C o m p a r e d with the usual Higgs a p p r o a c h , the a b o v e ideas have a clear geometrical a n d intuitive i n t e r p r e t a t i o n , they also lead to calculations w i t h o u t a r b i t r a r y p a r a m e t e r s . T h e y still suffer from a lack of i n t e r p r e t a t i o n b u t show a new direction in the s t u d y of " s p o n t a n e o u s s y m m e t r y b r e a k i n g " . This letter covers the last p a r t of a set of lectures given at Szczyrk ( P o l a n d ) in S e p t e m b e r 1983. M o r e details m a y be f o u n d in the set of lecture notes [5].
References [1] R. Coquereaux and A. Jadczyk, Commun. Math. Phys. 90 (1983) 79. [2] J.E. D'Atri and W. Ziller, Mem. Amer. Math. Soc. 18 (1979) 215. [3] R. Coquereaux, C E R N TH 3639, unpublished. [4] D. Bleecker, Department of Mathematics (1983). [5] R. Coquereaux, 1983 Szczyrk Lectures, Marseille CPT-83/P. 1556, to be published in Acta Phys. Polonica.
PHYSICS LETTERS
16 A u g u s t 1984
I N T E R N A L E I N S T E I N S P A C E S AND S Y M M E T R Y BREAKING R. C O Q U E R E A U X Centre de Physique Theorique, C N R S - L u m i n y , Case 907, F - 1 3 2 8 8 Marseille Cedex 9, France Received 27 F e b r u a r y 1984
W e first define a generalised gauge i n v a r i a n t Y a n g - M i l l s lagrangian: the Killing metric - K=# on the group is replaced by a more general m e t r i c h,~# ( x ) ; the field h,~# ( x ) - a scalar from the s p a c e - t i m e p o i n t of view - is then c o v a r i a n t l y coupled to the gauge field A~,'~ a n d is also self-coupled via a n a t u r a l scalar p o t e n t i a l (no parameters). N o n - t r i v i a l saddle p o i n t s of this scalar p o t e n t i a l c o r r e s p o n d to n o n - s t a n d a r d Einstein metrics on the g r o u p G. The associated shifts lead to a n entirely c o m p u t a b l e mass s p e c t r u m for the gauge field.
1. The lagrangian. When one considers the usual Yang-Mills lagrangian - K(F~,, F ~ ) and thinks about possible generalisations, a simple idea comes to the mind: replace the Killing metric - K ~ t ~by a more general metric h,# i.e., replace a bi-invariant metric (G x G) on the internal space G by another metric which is less symmetric. It is almost immediately clear that h,B has to be at least G-invariant, in other words it has to be entirely characterized by its value at the identity of G, i.e. by its action on the elements of the Lie algebra (one can say also that, by an appropriate choice of basis, h~B is independent of the coordinates on the group). Therefore, the isometry group of G will be of the kind K x G (where K is a subgroup of G, possibly trivial). However if h ~ is nothing but a symmetric matrix of constant real numbers, it is easy to see that gauge invariance (with respect to G) is broken, unless in the case where h ~ is proportional to K,, a. To cure this disease, the obvious solution is to decide that h ~a will be a field h ~ a ( x ) and not a constant matrix; of course x is a point in space-time. From the intuitive point of view, we associate to each point x of space-time, a quantity h ~ a ( x ) measuring the shape of the group G at this point. Now of course, h ~ a ( x ) transforms according to some representation of G (the bilinear symmetric) and we have to couple h ~ a ( x ) covariantly to the gauge field. It can
be checked that
- ¼F~,Ft'~h,q~(x) - ¼h"~(x)hVa(x)D~h,~v(x)D~'hl~n(x) +Dl, h , , ~ ( x ) O g h v s ( x ) , with
D~,h,, B ( x ) = 01,h,~#( x ) - Av~C~,,h~l ~ - A~C~vh~a is indeed a gauge invariant quantity (we discard total divergences). Now, we have a Yang-Mills field and a kinematic term for the scalar field (h,~p(x) is indeed a scalar field from the space-time point of view); however we would like to use a kind of potential term for the scalar field h,,B(x ). Being a bilinear symmetric quantity, the most obvious choice for this potential is V ( h ) = - [ s c a l a r curvature associated to h~#(x)]. Explicitly, when G is unimodular (in particular when G is compact),
"r° = - V [ h ] = h a a ' ( ½ C ~7, C ~ , v - z h1 a
aa"hvv,C~#C~,#,), 7 7"
C~aV being the structure constants associated to a basis ( X, }, orthonormal for the Killing form. The minus sign in front of V(h) is there for reasons of positivity. The reader may wonder why we choose such a complicated kinetic energy term for h~/~ and not
0370-2693/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
403
Volume 143B, number 4, 5, 6
PHYSICS LETTERS
s o m e t h i n g like (D,h,BD"hv8)8"v6a~; it should then be noticed that b y m a k i n g a linear a p p r o x i m a t i o n : h ~B( x ) = 6oB + ~ ,a( x ), the first term in the e x p a n sion w o u l d i n d e e d be of the a b o v e kind. The m o t i v a t i o n for such a kinetic energy term a n d for the scalar p o t e n t i a l comes from the following: when the a b o v e l a g r a n g i a n is c o v a r i a n t l y c o u p l e d to Einstein gravity, the whole lagrangian can be r e i n t e r p r e t e d as a theory of pure gravity (without m a t t e r fields) in d i m e n s i o n 4 + d i m ( G ) ; see ref. [1] where a m o r e general situation is investigated). Moreover, this kinetic energy term is a k i n d of generalisation of the n o n - l i n e a r o model: for a fixed volume of G (det h = const.), the field h~a varies in the m a n i f o l d S L ( n ) / S O ( n ) , n = d i m G, this is i n d e e d the m a n i f o l d of G - i n v a r i a n t metrics o n G. The Q F T expert will have also noticed that, b y p o w e r counting arguments, the a b o v e lagrangian looks b a d f r o m the r e n o r m a l i s a t i o n p o i n t of view; however, if one r e m e m b e r s that it can be w r i t t e n as a theory of pure gravity (in m o r e d i m e n sions), even if one expects strong divergences in the associated q u a n t u m field theory, one also expects strong cancellations; therefore the subject deserves a m o r e d e t a i l e d study.
2. Motioation for Einstein spaces and symmetry breaking. O n one hand, we saw that the p o t e n t i a l for scalar field is i n t e r p r e t e d (or defined!) up to a sign as the scalar curvature of the internal space G at the p o i n t x, on the other hand, there is an old t h e o r e m (due to Hilbert) which says that s a d d l e p o i n t s of the total scalar curvature - c o n s i d e r e d a functional on the space of metrics - for fixed volume, coincide with the Einstein metrics. Putting these two facts together a n d r e m e m b e r i n g that the scalars h ~ ( x ) are c o v a r i a n t l y c o u p l e d to the gauge field, one m a y look for an a n a l o g u e of the Higgs mechanism.
3. Einstein metrics on groups. T h e Killing metric on a Lie group G is an Einstein metric, this is well k n o w n a n d is f o r t u n a t e in o u r context since this p r o p e r t y tells us that the usual Y a n g - M i l l s l a g r a n g i a n is i n d e e d associated to a s a d d l e p o i n t of o u r m o r e general lagrangian. There exist in general b u t n o t always - m a n y other Einstein metrics on a given Lie group G however we are only inter-
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ested here in those which are K x G invariant. As stated previously, a very old t h e o r e m assures that for a given m a n i f o l d S, the total scalar curvature f~'Sd vol, c o n s i d e r e d like a functional on the space of metrics with fixed v o l u m e a d m i t s saddle p o i n t s for all Einstein metrics. The c o n s t r a i n t of fixed v o l u m e can easily be u n d e r s t o o d : consider for i n s t a n c e a usual two-sphere of radius R, its scalar c u r v a t u r e can be m a d e a r b i t r a r i l y small or large b y m o d i f y i n g its r a d i u s (~- = 2 / R 2 ) ; this k i n d of varia t i o n is not interesting, we have to fix the volume a n d study the v a r i a t i o n of the total scalar curvature. In the special case u n d e r study, the total scalar curvature is equal to the p r o d u c t ~- x vol(G) a n d we have o n l y to l o o k at variation of ~, ~- being c o n s i d e r e d as a f u n c t i o n a l on the space of G - i n v a r i a n t metrics (of course, with such a restriction we o n l y get a necessary c o n d i t i o n but, for all the cases treated here, it can be shown to be sufficient). T h e basic strategy used to o b t a i n n o n - s t a n d a r d Einstein metrics or groups is m o r e or less always the same: one first chooses along K in G = U ~ c / K ( g K ). This a m o u n t s to consider G as a collection of copies of G / K glued together a n d p a r a m e t r i s e d b y K; one then chooses some n a t u r a l metric on g (for e x a m p l e the G × G i n v a r i a n t Killing metric) a n d begins to " d i s t o r t it" in a w a y a p p r o p r i a t e to the coset d e c o m p o s i t i o n ; in the o b t a i n e d family of new metrics, one looks for those where the Einstein c o n d i t i o n is satisfied, either b y c o m p u t i n g directly the Ricci tensor or b y l o o k i n g at s a d d l e p o i n t s of some functional. Let us d o it explicitly in the special case where G is a simple c o m p a c t Lie group a n d where m o r e o v e r K is a s u b g r o u p of G such that (G, K ) is a s y m m e t r i c pair. W e n o w start from the following b i - i n v a r i a n t ( a n d Einstein) metric on G : g = - ( K i l l i n g form). W e can now c o m p u t e the scalar curvature T G o f G in terms of the scalar curvatures ~.~/K, ~K of G / K a n d K associated to the c o r r e s p o n d i n g restrictions of g. W e find .re = ~.G/K + ~.K _ ¼rc~ab~r'~b,, or = k s + c k k - - k(1 - c ) .
W e write a, a, h for indices in G, G / K , K, where n=dimG, k=dimK, s=n-k=dimG/K and c, the index of K in G being defined as follows: Killing metric of G restricted to K = gab = - C " a a C a b ~ , K i l l i n g m e t r i c of K = gKab =
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- - C daeC Zd a n d calling c the coefficient gKaL = cgaT,. T h e previous d e c o m p o s i t i o n of r G is a " K a l u z a - K l e i n r e d u c t i o n " (before K a l u z a - K l e i n ) where the " e x t e r n a l " space is G / K , the " i n t e r n a l " space is K a n d the field strength is C",h. Notice that when (G, K) is a symmetric pair we have indeed c = 1 - s / 2 k b u t this would not be true in the general case. Following the general recipe, we now write L i e ( G ) = L i e ( K ) • P - orthogonal dec o m p o s i t i o n for g - a n d consider the following family of metrics (t is a real parameter) on G: h = g/P + t 2 g/K. These metrics, o b t a i n e d by a scaling of g in the direction K are n o longer G × G i n v a r i a n t b u t only G × K i n v a r i a n t ; the scalar curvature of G is now
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4. S y m m e t r y breaking. I n the Higgs setting, one looks for a n o n - z e r o local m i n i m u m V0 of a suitable fourth degree G - i n v a r i a n t p o l y n o m i a l a n d perform the shift qb(x) = V0 + q~'(x). In our approach we write formally h , B ( x ) = h° ~ + h ' , a ( x ) , h°~B b e i n g a h o m o g e n e o u s Einstein metric for the group G. Let us analyse the situation when G is u n i m o d ular (then C ~ b = 0) a n d when h°~a is a G × K i n v a r i a n t Einstein metric o b t a i n e d b y the above m e t h o d (we even suppose that (G, K) is a symmetric pair, then the critical value of the scaling p a r a m e t e r is t 2 = (2k - s ) / ( 2 k + s). U s i n g the definition of D,, the term L = - ¼h'~Bh ~a (D.h,~,8 D~'h.y~ + D~,h,,vD~'hCa ) can be e x p a n d e d a n d we get
r ° = s/2 + (ck/4)/t
2 - k(1 - c ) t 2 / 4 .
However, when t varies, the volume of G varies; in order to keep it fixed we j u s t have to make a c o n f o r m a l rescaling a n d consider the family of metrics h = ( 1 / t 2 ) * / n h , then det h = ( 1 / t 2 ) k • (t2) k = constant. T h e associated scalar curvature reads r 6 = (,2)*/" [s/2 + (ck/4)/t
2 - k(1 - c ) t 2 / 4 ] .
W e n o w vary this expression with respect to t a n d find dcG/dt = - (s/4)(Zk
+ 2)/(k + s).t 2k/"-3
× (t 2 - 1 ) [ t 2 - ( 2 k - s ) / ( Z k
+ s)],
where we used the property c = 1 - s / 2 k , valid for a symmetric pair. We find therefore two Einstein metrics: c o r r e s p o n d i n g to the values t 2 = 1 a n d t 2 = (2k - s ) / ( 2 k + s). The first value corresponds of course to the b i - i n v a r i a n t metric g used in the beginning, the other value c o r r e s p o n d i n g to a n o n - s t a n d a r d G x K i n v a r i a n t metric on G (for example, studying G = SU(3), if we choose K = SO(3), we o b t a i n a n S U ( 3 ) x SO(3) n o n - s t a n d a r d Einstein metric on SU(3) for the value t 2 = (2 x 3 - 5 ) / ( 2 x 3 + 5) = 1/11). A general study based on the direct comput a t i o n of the Ricci tensor is carried out in ref. [2]. M a n y c o m m e n t s a n d references o n relative topics m a y be f o u n d in ref. [31.
where M~a = J . ¢ + K~¢; K~a = C~yC~a being the Killing form a n d J aB = h YY'"t~8 8 " ~ta~' g ~tf~l ~' ,'" We now make formally the shift h ( x ) = h ° + h ' ( x ) . The calculation is straightforward, we find that i AI -
2 . - . . ¢ A. .a.A1t.~ B =
_
51 .llA . . . ¢ 1A . ~. ~B
+ Rest,
where = 0,
= 0,
& b = 1 ( t 2 + 1 / , 2) - 1.
I n other words, when t 2 = 1 (i.e., we expand a r o u n d the b i - i n v a r i a n t metric of G), the gauge field stays massless; however, when t 2 = (2k - s ) / ( 2 k + s ) (i.e. we e x p a n d a r o u n d a n o n - s t a n d a r d Einstein metric on G, G × K invariant, with the n o t a t i o n s of section 6.2.4) [3] then, the c o m p o n e n t s of the gauge field taking their value in Lie(K) stay massless b u t the c o m p o n e n t s lying in the subspace P (LIE(G)= Lie(K)+P) acquire a mass m 2 = 2 s 2 / ( 4 k 2 _ s 2) in dimensionless units. Let us assume for example that our i n t e r n a l space is G = SU(4) a n d that the volume is fixed, then if we expand the scalar fields h ~a a r o u n d the b i - i n v a r i a n t metric, we get 15 massless gauge fields b u t we can also expand h~t~ a r o u n d the following 405
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PHYSICS LETTERS
-TGI
I
I
) t2
Fig.
G x K i n v a r i a n t Einstein metrics: (i) K = S(U(2) x U(2)) = SU(2) x SU(2) × U(1) for t 2 = 3 / 1 1 . (ii) K = USp(4) for t 2 = 3 / 5 . (iii) K = SO(4) for t 2 = 1 / 7 . I n case (i) we get k = 7 massless gauge fields a n d s = 8 massive fields of mass m 2 = 3 2 / 3 3 . In case (ii) we get k = 10 massless gauge fields a n d s = 5 massive fields of m a s s m 2 = 1 / 1 5 . In case (iii) we get k = 6 massless gauge fields a n d s = 9 massive fields of mass m 2 = 1 8 / 7 . I n the previous examples, the p a i r (SU(4), K) is s y m m e t r i c b u t there exist other s a d d l e p o i n t s (other Einstein metrics) involving n o n - s y m m e t r i c pairs. There is a difficulty which is b e t t e r e x p l a i n e d b y looking at fig. 1 which is the g r a p h of the p o t e n t i a l V(H) for the o n e - p a r a m e t e r f a m i l y of metrics a l r e a d y discussed in section 3; using typical values of the p a r a m e t e r we get the following curve" T h e p o i n t A c o r r e s p o n d s to the s t a n d a r d bi-inv a r i a n t metric on G a n d B to a n o n - s t a n d a r d Einstein metric. This " p o t e n t i a l " is therefore n o t b o u n d e d from b e l o w - even with the fixed v o l u m e restriction - m o r e o v e r the n o n - s t a n d a r d Einstein m e t r i c of this f a m i l y c o r r e s p o n d s to a local maxim u m of the curve; in a m o r e general situation,
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Einstein metrics c o r r e s p o n d to s a d d l e p o i n t s which are neither m i n i m a n o r m a x i m a o f the total scalar c u r v a t u r e function. T h e " p h y s i c a l " i n t e r p r e t a t i o n of the a b o v e results is therefore unclear: we e x p l a i n e d w h a t h a p p e n s if we e x p a n d a non trivial s a d d l e point, b u t (1) are we allowed to " e x p a n d " a r o u n d them? (2) even if we can, w h y d o it a n d are these s a d d l e p o i n t s i m p o r t a n t in a q u a n t u m m e c h a n i c a l perspective? If we w a n t to s t u d y global aspects of s y m m e t r y b r e a k i n g a n d use fiber b u n d l e techniques, there is a n o t h e r difficulty which arises when we try to define globally the shifted field h~B; however the s a m e difficulty usually o v e r l o o k e d a l r e a d y exist in the Higgs setting when we try to define the shifted field dp'; one solution is to w o r k in a trivial b u n d l e (as p e o p l e d o usually when discussing Higgs mecha n i s m - see however ref. [4]). C o m p a r e d with the usual Higgs a p p r o a c h , the a b o v e ideas have a clear geometrical a n d intuitive i n t e r p r e t a t i o n , they also lead to calculations w i t h o u t a r b i t r a r y p a r a m e t e r s . T h e y still suffer from a lack of i n t e r p r e t a t i o n b u t show a new direction in the s t u d y of " s p o n t a n e o u s s y m m e t r y b r e a k i n g " . This letter covers the last p a r t of a set of lectures given at Szczyrk ( P o l a n d ) in S e p t e m b e r 1983. M o r e details m a y be f o u n d in the set of lecture notes [5].
References [1] R. Coquereaux and A. Jadczyk, Commun. Math. Phys. 90 (1983) 79. [2] J.E. D'Atri and W. Ziller, Mem. Amer. Math. Soc. 18 (1979) 215. [3] R. Coquereaux, C E R N TH 3639, unpublished. [4] D. Bleecker, Department of Mathematics (1983). [5] R. Coquereaux, 1983 Szczyrk Lectures, Marseille CPT-83/P. 1556, to be published in Acta Phys. Polonica.