Invariant tensors for simple groups

E! ELSEVIER

Nuclear Physics B 510 [PM1 (1998) 657-687

Invariant tensors for simple groups J.A. de Azc~raga

l, A . J . M a c f a r l a n e 2, A . J . M o u n t a i n 3, J.C. P 6 r e z B u e n o 4

Department of Applied Mathematics and Theoretical Physics, Silver St., Cambridge CB3 9EW, UK Received 2 June 1997; accepted 4 September 1997

Abstract

The forms of the invariant primitive tensors for the simple Lie algebras AI, BI, C/ and DI are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the At algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) and su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given. (~) 1998 Elsevier Science B.V.

1. I n t r o d u c t i o n

We devote this paper to a systematic study of the symmetric and skewsymmetric primitive invariant tensors that may be constructed on a compact simple Lie algebra ~. The symmetric invariant tensors give rise to the Casimirs of G; the skewsymmetric ones determine the non-trivial cocycles for the Lie algebra cohomology (see, e.g., Ref. [ 1 ] ). It is well known [ 2 - 9 ] that there are l such invariant symmetric primitive polynomials of order mi (i = 1 . . . . . l = rank of G), which determine l independent primitive Casimir operators of the same order, as well as l skewsymmetric invariant 1 St. John's College Overseas Visiting Scholar. E-mail: [email protected]. On sabbatical leave from Departamento de Fisica Te6rica and IHC (Centro Mixto Univ. de Valencia-CSIC), E-46100 Burjassot (Valencia), Spain. 2 E-mail: [email protected] 3 E-mail: [email protected] 4 E-mail: [email protected]. On leave of absence from Departamento de Fisica Te6rica and IFIC (Centro Mixto Univ. de Valencia-CS1C), E-46100 Burjassot (Valencia), Spain. 0550-3213/98/$19.00 (~) 1998 Elsevier Science B.V. All rights reserved. PH S 0 5 5 0 - 3 2 1 3 ( 9 7 ) 0 0 6 0 9 - 3

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J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

primitive tensors J2~2mi-l~ of order ( 2 m i - 1). The latter determine the non-trivial cocycles for the Lie algebra cohomology, their order being related to the topological properties of the associated compact group manifold G which, from the point of view of the real homology, behaves as products of l S ~2m'-1~ spheres [ 10-18]. The lowest examples of these tensors/polynomials (ml = 2) are the Killing tensor (which is a multiple of 6~i for a compact algebra), the quadratic Casimir operator and the fully skewsymmetric structure constants of the simple algebra G, which determine a threecocycle on G (see Example 3.1 below). The study of the invariant primitive tensors on (and especially in the most interesting case G = s u ( n ) ) is not only mathematically relevant; their properties determine essential aspects of the physical theories based on the associated group G. These range from the form of the vertices in Feynman diagrams to the presence or absence of non-abelian anomalies in gauge theories (see in this last respect the articles in [ 19] and references therein; see also Ref. [20] ). They are also relevant in other instances as, e.g., in (higher order) Yang-Mills duality problems [21 ], WZWN terms (see, e.g., Ref. [22] and references therein), W-symmetry in conformal field theory [23], the construction of effective actions [24], etc. It is then convenient to have an explicit expression for the primitive tensors of the different orders as well as a convenient basis for the vector spaces of the invariant tensors of a given order. The simplest way of obtaining an invariant symmetric tensor k ~r~ on G is by computing the trace of the symmetrised product of r generators. This symmetric trace (which may give a zero result depending on r and on the specific algebra G being considered) will not vanish, however, for arbitrary r. As a result, the trace will give rise to arbitrarily higher order tensors that cannot be primitive and independent of those of lower order mi (i = 1. . . . . l), and the same will apply to the Casimir operators constructed from them. This indicates that it is convenient to introduce a new family of tensors which is free of this problem. We shall do this in Section 3 by introducing a new family of symmetric invariant tensors t ('') from the l primitive (2mi - 1 )-cocycles /2 (20'- l), (i = 1. . . . . l). These tensors will turn out to be 'orthogonal' in a precise sense (see Lemma 3.3). We shall also show how the l primitive Casimirs may be equivalently obtained from the t (n'~) tensors or from the cocycles s2(2m'-l). For the s u ( n ) algebras many results and techniques are already available. For instance, we can construct recursively [25] (see also Refs. [26,4] ) the so-called d-family of symmetric invariant tensors of order m starting from the symmetric dijk (for s u ( n ) , n > 2) and symmetrising the result d(i~...im). This family (see Section 6.1) has a status like the k family, i.e. for n fixed and m large enough the d im) tensors may be expressed as linear combinations of products of lower order ones. For fixed m and n (in s u ( n ) ) sufficiently large (in fact, for n >~ m) it is known how to define a basis of the vector spaces "1,)(m) of invariant symmetric tensors of order m, and via known identities for the d ("0, how this basis reduces when n < m. We use these properties in our discussion of the t (m~) family. In particular we see that a formal attempt to construct t-tensors of higher rank than is allowed from their definition necessarily yields an identically vanishing result (Section 6.2). In proving this, we have needed to extend the set of identities for the d and f s u ( n ) tensors in the literature known to us.

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The paper is organised as follows. After a short general discussion of the invariance properties of tensors and Casimir operators on G in Section 2 the expression of the (2mi - l)-cocycles is given in Section 3, where the t (mi) family of invariant symmetric tensors is introduced. Section 4 illustrates these general considerations for the su(3) and su(4) algebras. Section 5 provides a general discussion of the primitivity of the invariant symmetric tensors and Casimir operators for the four infinite series At, Bz, CI and Dz and shows explicitly, if not systematically, how a given primitive polynomial becomes algebraically dependent (non-primitive) when the rank of the algebra G is reduced sufficiently. Due to the special relevance of the su(n) algebras, Section 6 is devoted to illustrate these ideas for the At case and, in particular, the usefulness of the t (m') family of tensors. Section 7 discusses, again for the four infinite series, the properties of the ( 2 m i -- 1)-cocycles. The topological properties underlying the compact group manifolds provide a clue to establish, using the Hodge star * operator, duality properties for the Lie algebra cocycles. This is done in Section 8, and some of the formulae are illustrated by using the explicit results in Section 4. Finally an appendix develops some properties of the d and f su(n) tensors for arbitrary n, and collects a number of new expressions which are needed for the derivation of crucial results in the main text.

2. Invariant symmetric polynomials and Casimir operators Let G be a simple algebra of rank 1 with basis {Xi} , [Xi,Xj] cik.jXk, i= 1. . . . . r = dim G, and let G be its (compact) associated Lie group.5 Let {w J} be the dual basis in G*, wJ (Xi) = ~!, and consider a G-invariant symmetric tensor h of order m =

(2.1)

h = h i L i , , Of 1 @ • . . @ tO L''.

The G-invariance of h means that p

(2.2)

Cui, hi, ...i,~pi, ~1...i,, = O. A'= 1

This is the case of the symmetric tensors k (m) given by the coordinates 6

1

kij...i,,, = ~.T s T r ( X i , . . . X i , , , )

~ T r ( X ( i , . . . Xi,,)),

(2.3)

5 In the general discussions we adopt the 'mathematical' convention and take antihermitian generators Xi and hence negative definite Killing tensor Kij = Tr(ad Xiad Xj) cx - 6 0 since G is compact. When we consider explicit su(n) examples, we follow the 'physical' convention and use hermitian generators T/, Xi = -iTi. In this case, an i accompanies the C/~ in the r.h.s of the commutators. When the generators are assumed to be in matrix form, we take them in the defining representation of the algebra. We always use unit metric and hence there is no distinction among upper and lower indices, their position being dictated by notational convenience. 6 Indices inside round brackets (it . . . . . ira) will always be understood as symmetrised with unit weight, i.e. with a factor l / m L We use the same unit weight convention to antisymmetrise indices inside square brackets [ il . . . . . im I.

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J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 6 5 7 - 6 8 7

where sTr is the symmetric trace, sTr(Xi, . . . Xi,,) = ~ e s , , , Tr(Xi~,~ . . . Xi~,,~ ), which is clearly ad-invariant.7 In particular,

kij = Tr( XiXj ) = K6O,

(2.4)

where, for instance, K = - 1 / 2 for the generators Xi of the defining representation of

su(n). Since G "~ Te(G), the tangent space at the identity of G, we may use a left translation Lg, g E G, to obtain a left-invariant (LI) m-tensor h(g) on the group manifold G from the m-linear mapping h : G × ."?. ×G ~ R. Its expression is the same as (2.1) where now the {~oJ} are replaced by the LI one-forms wi(g) on G, and (2.2) now follows from the fact that (see, e.g., Ref. [20] )

Lx,(g)~OP(g) = -C~toi(g),

v , p , i = 1 . . . . . r,

(2.5)

where Lx,(g) is the Lie derivative with respect to the vector field X~(g) obtained by applying the (tangent) left translation map Lgr to X~ (e) = X~; clearly the duality relation is maintained for the vector fields {Xi(g)} and one-forms {~oJ(g)}. In this context, the G-invariance condition (2.2) reads Lx~(g)h(g) = O. Let G moreover be compact so that the Killing tensor may be taken as the unit matrix and let hfi...i, ' be an arbitrary symmetric invariant tensor. Then the order m element in the enveloping algebra/g(G) defined by C (m) = hil...i,,xil . . . Xi,,

(2.6)

commutes with all elements in G. This is so because the commutator [Xp,C (m) ] may be written as m

[Xp, C (m) ] = Z

A

is ij ...isv...i,, Cp~h

(2.7)

X i l . . . X i , , = O,

s=l

which is indeed zero as a result of the invariance condition (2.2). In fact, the only conditions for the m-tensor h to generate a Casimir operator C ("0 of G of order m are its symmetry (non-symmetric indices would allow us to reduce the order m of C (m) by replacing certain products of generators by commutators) and its invariance (Eq. (2.7)); h does not need to be obtained from a symmetric trace (2.3). This leads to

Lemma 2.1. ( Casimirs and G-invariant symmetric polynomials) Let h be an invariant symmetric tensor of order m. Then, C ~m) =

h i > " i ' x i l . . . Xi,,

is a

Casimir of ~ of the same order m. 7 We denote by k ~m) the invariant symmetric tensors coming from the symmetric trace (2.3). In fact (see Ref. 15] ) a complete set of l primitive (see below) invariant tensors may be constructed in this way by selecting suitable representations. Other families of symmetric invariant tensors will be identified by an appropriate letter (e.g. t, d, ~,). Generic symmetric invariant polynomials are denoted by h.

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J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

It is well known [ 2 - 9 ] that a simple algebra of rank l has l independent (primitive) Casimir-Racah operators of order ml . . . . . mr, the first of them given by the standard Casimir [27] operator K i j x i x j obtained from the Killing tensor (ml = 2) Thus, there must be (Cayley-Hamilton) relations among the invariant tensors obtained from (2.3) for m > mt or otherwise one would obtain an arbitrary number of primitive Casimirs. We shall study this problem in Section 5 and apply our results to the s u ( n ) algebras in Section 6.

3. Invariant skewsymmetric tensors and cocycles Let O(g) = o)i(g)Xi be the LI canonical form on a simple and compact group G, and consider the q-form Tr(0 A. q. A0). Due to the cyclic property of the trace and the anticommutativity of one-forms, this form is zero for q even. Let q be odd. Then,

d'~2(q)(g)

=

q~Tr(0 A .q. A0)

(3.1)

is a closed form on the group manifold G, since d O cx Tr(0 A .q+! A0) = 0 on account of the Maurer-Cartan equations dO = - 0 A 0. Since S2(g) is not exact (it cannot be the exterior differential of the (q - 1)-form Tr(0 A q._! A0) which is zero because q - 1 is even) it defines a Chevalley-Eilenberg [ 1 ] Lie algebra q-cocycle. If we set q = 2m - I, we find that S2~2m_l) (g) =

1 Tr(Xi,...Xi2.,_l )co il (g) A . . . A ¢0i2"-' (g) ( 2 m - 1)!

1

1

= (2m -- l ) ! 2m-------STr( [ Xi~, Xi: ] [ Xi3, Xi4 ] . . . [ Xi2,,,_3, Xi2°, 2 ] Xi:,,,_~ )

x w i~ (g) A . . . A w i2'' 1 (g) 1 1 - (2m - 1)! 2 m-I C/~iz'" " C ~ i ~ 3 i 2 " - J r ( X l " "

.X,,,, ~X,f)

x w i' (g) A . . . A w i2"-2 (g) A w " ( g ) .

(3.2)

The trace may in fact be replaced by a unit weight symmetric trace, i.e. by Tr ( 1 / m ! ) s T r in (3.2), since the skewsymmetry in il-..i2m-2 results in symmetry in 11... Ira-1. The factor 2,,1_~ is unimportant 8 and will be ignored from now on. Hence, we may define the ( 2 m - 1)-form on G representing a Lie algebra ( 2 m - 1)-cocycle by s,2~2.... l)(g) _

1 1)!$'2il...i2,,_2o-¢oq(g) A . . . A w'""-2 (g) A w~'(g), (2m-

(3.3)

the coordinates of which are given by the constant skewsymmetric tensor 8 We recall that the cohomology space is a vector space, and that numerical factors, although they relate inequivalent cocycles (different vectors in a given cohomology space H ~2m- i) (G, JR) ), are unimportant here; they determine only the normalisation of the different tensors.

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662

1

E.jl . . . j 2 m _ 2 P c l I

-0i~...i2°,_2~ - ( 2 m -

Ira--1

1)! i,...6,, 2~ j~j2 ""Ci2,,-s.i2,,-2 kh...t,,-~p"

(3.4)

where kll...l,,_lp is the symmetric tensor of (2.3) 9 and the e tensor is defined by

e/31...13,, ~ (_ 1]~r('r)SB~ll Oil ...o~ n

~"

/

oz 1

,~fl~,,I

. . . .

(3.5)

o~ n

rrES,

where 7r(o-) is the parity of the permutation o-. Although (3.4) is what follows naturally from (3.1) (setting aside the ignored factor 1/2 (m-l)) it is convenient to notice that part of the antisymmetrisation carried out by ej~...j:,,-2p 11...12m_2 o" is unnecessary, since (cf. (3.4)) may be rewritten as l Ej2...j2m_ 2 CI I . . . (2m - 3)! /2.../2m-2 Pj2

/2pi2...i2,,_2~

f~ I I t

('7(m--l~j2,,--sjkll'''lm-ltr 2m--2

"~ lm - I

(3.6)

~"~p[i2 " " ' ~"i2,,,-3i2m-2] kll...l,,,_l~r

due to the skewsymmetry in p and o- of its r.h.s. This follows from the invariance of the symmetric polynomial ktl...t°,_w and is a generalisation of the simple m = 2 case for which (3.6) gives

~2pj~=k([Xp,Xj],X~)

=k(Xp,[Xj,X~])

=-k([X,,,Xj],Xp)

=-J2,ap.

(3.7)

Indeed, for an arbitrary invariant symmetric tensor h on G of order m we have from (3.6) (2m - 3) ! f)pi2...i2m-2o-

=

i2...i2m-2 h.

Ej2...J2m--2

( [ X. p ,

X. j 2 ] ,. [ X .j3 , X .j 4 ]

. ... .

[ X j2 m 3, Xj2m--2 ]

So)

m-- 1

_- -Eiz...i2°,_ 2hj2"-~h(Xp,[Xjs,Xj4],. 2

.

• . . , [[Xj2,_,,Xj2,],Xjz],.

•., [Xj2,, _s,Xjam_:],Xo - )

s=2

__ej2...j2,,-2 h ( i2...i2m 2

Xp, [XJ3,XJ4]

=eJ2""J2"'-2h(Xp,[Xis i2...i2m-2 " .-

--eJ2"J2"-2hrrX -i2...i2m--2 \ t

0",

~

Xj4]

'

. . . . . . . .

[Xj2m_3,Xj2,

" " ' ~

21

[Xtr,Xj2])

[Xj2,,-s,Xj2°, 2] ' [XJ2 Xo-]) , '

Xj.,] - ~ [Xjs,Xj,,]

~" • " ,

[Xh,,, 3 , XJ2.,-2] ,

Xp)

= - ( 2 m - 3)![2,~i2...i2,,_2p, where we have used the invariance of h in the second equality, the Jacobi identity in the third (to see that every term in the summation symbol is zero) and the symmetry of h in the fourth one. The fact that the skewsymmetric tensors /2 (2mi-1) expressed by their coordinates (3.4) or (3.6) are indeed ( 2 m i - 1)-cocycles follows from the Chevalley-Eilenberg approach to Lie algebra cohomology already mentioned [ 1 ]; for a direct proof which uses only the symmetry and invariance properties of the tensor used in its definition (3.6) see Ref. [28]. 9 The invariance properties of (3.4) follow from the invariance of kll..dm_l p and thus do not depend on the fact that it is expressed by a symmetric trace (2.3).

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663

Example 3.1. Let m = 2. Using 6ij (rather than kij) as the lowest order invariant polynomial, (3.6) gives ( 3.8 )

/2il i2tr = C[,li2 61, tr = C i I i2o-,

i.e. the three-cocycle /2ili,.i3 is determined by the structure constants of G. It follows that the third Lie algebra cohomology group H3(G) is non-zero for G simple, as is well known.

Example 3.2. Let kili2i3 be a third-order invariant symmetric polynomial. Such a thirdorder polynomial exists only for su(n), n > 2 (this is the reason why in four dimensions only these groups are unsafe for non-abelian anomalies). Then, the su(n) five-cocycle is given by (3.6) cl 2 k ; ;. lll2o" " pg," ~2,,

1 Ejlj2j3cll

/2pili2i3(r

=

" " i~ 3! .,2

--

(3.9)

The coordinates o f / 2 (3), /2(5) for su(3) and for su(4) are given in Tables 1, 3 and 4, 6 respectively. The expression of the three-cocycle follows directly from the structure constants, Eq. (3.8); for five-cocycles we have used the symmetric Gell-Mann tensor d(ik in (3.9) rather than kijk [ kijk = (1/3!) ( - i / 2 )3sTr( AiAi,,lk ) =( i/4)dijk, s e e (4.2) below]. Since the ( 2 m - l)-forms /2(2m-l)(g) (3.3) are invariant, the coordinates of any (2m - 1)-cocycle on the simple Lie algebra also satisfy the relation (cf. (2.2)) 2m- I

Z

P Cpis

/2it .,.ispi,~ A ~I...i2~,- I =0.

(3.10)

s=l

The simplest case corresponds to Example 3.1 for which P

cP.iCpi2i3 ~- CP2 Cilpi3 Jl- Cl~i3 Cili2 p .= 0

is the Jacobi identity Cff[ilfi2i3]p su(n), n > 2) satisfies p. c PlJl I / 2 (p l52 1)3 1 4 1"5 " -~ C~i,

=

(3.11 )

0. Similarly, the five-cocycle (which only exists for

/2(5) p 0 tll2Pl415 !5).. + C Ppl 4 O l!5). • + C 1its p./2!5 ).. =0. ilpi3i4is_ "~- C ui3 lll2t314P lt213pl5 _

(3.12)

Lemma 3.1. Let hh...l,, be a symmetric G-invariant polynomial. Then, e.Jl...J2,,,cll . CI,, t. il...i2,, -.Jlj2 " " j2,, ij2,, tql...l., = 0 .

Proof. By replacing CJ2'i,,_lj2,,hlb..lm we get

il...i2,,,

'- • jl.J2 " " " C .12,,-3./2,~-2 \ s=l

(3.13)

in

the l.h.s, of (3.13) by the other terms in (2.2)

j2,,_d, hll...l,-ikL~ b--l,,-t J2. . . .

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J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

s Csj,, k antiwhich vanishes since all terms in the sum include products of the form C/j, symmetrised in j, j ' , j ' , which are zero due to the Jacobi identity. []

Note that if h6...t., is identified with kl~...l., in (2.3) the equality follows immediately from the fact that 1.h.s. (3.13) are the coordinates of Oi,...i2°,, and O (2"') oc Tr(0/~ .2.,,. A0) = 0.

Corollary 3.1. Let h now be a non-primitive symmetric G-invariant polynomial, i.e. such that its coordinates are given by h(q) hil...iFjj...j," = h(p) (il...it,.,jl..../,l),

(3.14)

where in the r.h.s. ( . . . ) indicates unit weight symmetrisation (see footnote 6) and h (p) and h (q) are symmetric invariant polynomials. Then the cocycle /22(t'+q)-I associated to (3.14) by (3.4) is zero.

Proof ~?2(l'+q)-I is given by 2''+'~-' c t ' . . . O,...~z~,~q._ ~2(,+q)-I = ( 2 ( p + 1q) - 1)! e~'J. ,,...,2,e,,~,-, -•hJz

xh(P)

c! ~

... .,2p-,J2ec"." J2,,,,Jz,,+2

c " 'q-'

.,2,p~q,-3.,2,,,,~,-2

/a(q)

"(/l.-./e'~ml.--nlq-l.j2(p~ql

1 )

[]

and is zero by virtue of (3.13).

By definition, primitive tensors are not the product of lower order tensors, but may contain non-primitive terms. It follows from Corollary 3.1 that only the primitive term in k contributes to the cocycle (3.6). As a result, different families of symmetric tensors differing in non-primitive terms lead to proportional cocycles. Thus, and as far as the construction of the cocycles is concerned, we may use any family h ~"0 of symmetric invariant primitive tensors in (3.6), not necessarily that given by (2.3); for instance, for su(n) we may use the d family of Section 6.1. We shall use (3.6) with the definition (2.3) unless otherwise indicated. We introduce now a new type of symmetric invariant polynomials by using the cocycles. Their interest will be made explicit in Section 6.

Lemma 3.2. ( lnvariant symmetric polynomials from primitive cocycles) Let S2~2''-J) be a primitive cocycle. The l polynomials t¢m) given by til"'i"'= •[D(2m-l)]jl'"J2"-2i"C(l'J JIJ2

lm -- 1

" " "

Cj2,n-3j2,,-2

(3.15)

are G-invariant, symmetric and primitive. Moreover, they are traceless for m > 2.

Proof By construction

t it'''i"' is an invariant polynomial (it is obtained by contracting the invariant tensors C and S2). It follows from (3.15) that it is symmetric under interchange of the (il • • • i,,-l) indices. Now

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J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

til'"i,,, = f.o(2m-I)]jl...J2m-2imcil • -J j,j2 ci2J~;4 " " C~:°,'~J2,,,-~. 2m-2 A

=_

.

.

( [ J'2(2m-l ) ]ilj2""J~P'"J2m-2i'cJi/2)C~j4

.. " C'..12,,, °'-~3.12,,-2

s=2

= -

•f

)1

• -. Cir.,

=

" " "

J~m-.

J.m-2

= ti,,i2...i,,- 1il

where we have used the invariance (3.10) of s'2{2m-I) in the second equality, the Jacobi identity in the third and the skewsymmetry of/'2 (2m-l) in the fourth. Thus t is invariant under the change il +-+ im and hence it is a completely symmetric tensor. For m = 2 Eq. (3.15) is proportional to the unit matrix since .

.

.

.

C.I'J2teC t'

.h j2

.

.

(3.16)

= K qt2

is the Killing tensor K. If we contract (3.15) with 8i, i2 for m > 2 we see that tf mu''

=0

(3.17)

by using the Jacobi identity for Ccrjmj2CS~j4.

[]

Since kili2 o( ~fii2, the tracelessness of (3.15) may be seen a s t il'''i'' having zero contraction with k(j. This extends to the full contractions with all higher order symmetric invariant tensors by means of the following Lemma

3.3. Let t iu''i" be the symmetric invariant polynomial given by (3.15). Then

t '''''itii+''''i't"~11 . . , l "I = O,

Proof

Vl < m.

Eq. (3.18) is a consequence of Lemma 3.1.

(3.18)

[]

N o t e . Eq. (3.18) implies the 'orthogonality' of the different polynomials t iiu') obtained ...i,,

from ( 2 m - 1)-cocycles. Thus, a basis for the space V ¢"') of invariant symmetric polynomials of order m is given by the symmetrised products of the primitive symmetric invariant polynomials (or their powers) leading to a symmetric invariant tensor of order m. We may express this as a C o r o l l a r y 3.2. The symmetric invariant primitive polynomial t era) and the symmetrised products t
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666

Example 3.3. For su(n) the vector space of symmetric invariant polynomials of order six is given by fil..i6,'(6) t((4) i4~isi6) ' t(3)(ili2i3~i4i5i6) ~(ili2~i4i5~i5i6) "(t~3")"~'2is'3 proportional to .

.

.

.

di, i2ia, see Section 6.2).

As a consequence of L e m m a 3.2, it follows that we can obtain Casimir operators C t from cocycles by means of

Corollary 3.3. ( Generalised Casimirs from primitive cocycles) The operator

C,(.,) = [ n(2.,-~) ] J, J2.,-,

xj,

...

xj2.,_,

(3.19)

is an m-order Casimir operator for G.

Proof Using the skewsymmetry of/'2 (2m-1) we rewrite (3.19) in the form C t(m) = [ aQ(2m-1)] j''''j2"-' Xj, . . . Xj2.,_l

= 2 m-ll [/2(2m_l)]j~...j2,,_:[Xj, . .Xj2] .. 1 -

2 m- 1 l

2 m- 1

[f~(2m_l)]jl...j2.,_2crfi •

[XJ2°,-3,Xj2,,-2]X~"

I ira--1 Jl j2 " " " C j 2 m - 3 j2 n , -2 X i l

" " " Xin,- i Xo"

• • t l l ...tm--I tY X i I . . . X i m _ 1 X o -

and it follows that it is a Casimir as a consequence of L e m m a 2.1.

(3.20) []

4. The case of su(n): coordinates of the su(3)- and su(4)-cocycles Let us take as a basis {T/} of the su(n) algebra the (n 2 - 1) traceless and hermitian n x n matrices of the defining representation of su(n) satisfying the relations

[Ti, Tj] =i fijk. Tk,

"rr(r, rj) = ½,~.,

{Ti,Tj) = c6ij + dijk.Tk,

TiT; = ~Sij + -~d;j•r, 1 k + 2fij~.rk,

(4.1)

where i,j, k = 1 . . . . . n 2 - 1 and c ~_ ~ (for a study of the su(3) tensors, see Ref. [29] ). In the 'physical' (Gell-Mann) basis, it is customary to use the A-matrices Ai = 2T/ for which the above relations trivially become [Ai, Aj] = 2ifijk. Ak,

T r ( a ; a / ) = 2,~/j,

{Ai, Aj} = 4C¢~ij -~- 2dijkAk,

~i~j = 2CSij + (d(i ~. + ifijk.)Ak.

(4.2)

Using the Gell-Mann representation of su(3) we obtain Tables 1 and 2. The fij~ constitute the coordinates of the su (3) three-cocycle. If we use the structure constants f~ik and the symmetric di# to define [cf. (3.9) ] the coordinates of the fivecocycle 0 (5) by

J.A. de Azc6rraga et aL/Nuclear Physics B 510 [PM] (1998) 657-687

667

Table 1 Non-zero structure constants for s u ( 3 ) f123 = 1 .]246 = 1 / 2 f367 = -- 1/2

f147 = 1/2 f257 = 1/2 f458 = X/'3/2

fl56 = - - 1 / 2 f345 = 1/2 f678 = V ~ / 2

Table 2 Third-order invafiant symmetric polynomial for s u ( 3 )

cl228 = l/v5 d558 = - 1 / ( 2 v / 3 ) d157 = 1/2

J338=

d~8 = - 1 / ( 2 " ¢ ' 3 ) d146 = 1/2

l/v~ d668 = - l / ( 2 v ~ ) d247 = - 1 / 2

a888 = - l / V ~ d778 = - 1 / ( 2 v / 3 ) d256 = 1/2

d344 = 1/2

d355 = 1/2

d366 = - 1/2

d377 = - 1/2

au8 = l / v ~

Table 3 Non-zero coordinates of the

su ( 3 ) five-cocycle

-(-)12345 = 1 / 4 .(212678 = - V/'3/12 .(223478 = V / 3 / 1 2

.(212367 = 1 / 4 flj 3468 = - "v/'3/12 -(223568 = -- V/-3/1 2

f212458 = x/~3/12 O13578 = - v/'3/12 $'245678 = -- V/'3/6

fl.4,7 = 1 / 2 fl.~0,H = - 1 / 2 f2.9al = 1/2 f3,6.7 = - 1 / 2

ft.5,6 = - - l/2 1/2 f2,~0,12 = 1/2 f3.9.1o = 1/2 f4,9,14 = 1/2 f5,10.14 = 1/2 f6.12,13 = - - 1 / 2 f8.9.10 = l / ( 2 V / 3 ) f9,10,15 = V/2/V ~

Table 4 Non-zero structure constants for s u ( 4 ) fl.2.3 = 1 fl,9,12 = 1/2 .[2~5,7 = 1/2 f3.4,5 = 1/2 f3,11,12 = - 1 / 2 ,f4, m, 13 = - - 1 / 2 f6.7.8 = X/3/2 f T a H 3 = 1/2 J)q,ll,12 = I / ( 2 V ' ~ ) f11.12.15 = X/~/V/~

f2!5)t,,2t3t4,5 ....

fJia [i~fki3i41. .

f4.5.8

= V/'3/2

f5,9,13 = 1/2 f6.11,14 = I / 2 fT. 12a4 = I / 2 f8,13,14 = - - I / v / ~ /13,14.15 = V/'2/V/~

djki5,

f2,4,6 =

(4.3)

we obtain Table 3. Using the natural extension of the GelI-Mann labeling of generators to s u ( 4 ) in agreement with the contents of Table VI in [30] (where in its second part f should be replaced by d), we obtain Tables 4-6. Any pair of sets of non-zero coordinates (il, i2, i3) and (i4, i5, i6, i7, i8) for the s u ( 4 ) three- and five-cocycles with distinct index sets defines a non-zero coordinate of the seven-cocycle, the indices (i9 . . . . . i15) of which take the remaining available values in the set i = (I . . . . . 15) [see (8.15) below]. There are more than 400 non-zero such coordinates, which are not given here.

668

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657--687

Table 5 Third-order invariant symmetric polynomial for s u ( 4 )

d5,5,3 = 1/2

d4.4.3 = 1 / 2 69.9. 3 = 1 / 2 d1.1.8 = 1 / x / ~ d5.5.8 = - l / ( 2 V ~ ) d9,9,8 = 1 / ( 2 V " 3 ) d13.13.8 = - l / v ~ d3.3.15 = 1/~/'~ d7.7.15 = l / v c~ dll.ll.15 = - l / v ~ d15,15,15 = - 2 / V ~ d1.10.12 = 1 / 2 d2.10.n = 1 / 2 ds.m.13 = 1 / 2 d7.12.13 = 1 / 2

dl0.10.3 = 1 / 2 d2.2.8 = I/V/'3 d6.6. 8 = - 1 / ( 2 v ~ ) d10,10,8 = I / ( 2 V " 3 ) d14.14.8 = - l / x , / 3 (/4.4.15 = l / v ~ ds.s.15 = l / v ~ dl2.12.15 = - l / v ~ dl,4, 6 = 1 / 2 d2.4.7 = - 1 / 2 d4.9.13 = l / 2 d6.11.13 = 1 / 2

d6.6.3 = - 1 / 2 dll.ll.3 = - - 1 / 2 d3.3.8 = 1/V/3 d7.7. 8 = - 1 / ( 2 x / 3 ) dll,H,8 = l / ( 2 V / ' 5 ) dl.l.15 = l / v / ' 6 a5.5.15 = l / v ' 6 d9.9.15 = - l / v ' ~ d13.13.15 = - 1 / v / - ~ dl,5,7 = 1 / 2 d2.5.6 = 1 / 2 d4A0.14 = 1 / 2 d6.12.14 = l / 2

d7,7,3 = - 1/2 d12.12.3 = - 1 / 2 d4.4.8 = - 1 / ( 2 V / 3 ) d8.8.8 = - 1 / , / 3 d12,12,8 = 1/(2V/'3) d2.2.1S = l / v / ' ~ d6.6.15 = l / v ~ dlo.lo.15 = - l / v / 6 dl4.14.15 = - I / v / ~ dl,9,11 = 1 / 2 d2.9.12 = - 1/2 d5.9.14 = - 1 / 2 d7.11.14 = - 1 / 2

Table 6 Non-zero coordinates o f the s u ( 4 ) five-cocycle .(21.2.3.4. 5 = 1/4

O1,2,3A1,12 = 1/4 ~QI.2.4.10.13 = -- 1 / 12 OI.2.6.7.8 = - v/'3/12 O1.2.7.11.13 = -1/12 OL2,S,H,12 = -- V'-5/36 OI.3.4.6. 8 = --V"3/12

OI,3,5.7,8 = -- V/'3/12 OI.3.6.9.13 = - 1 / 1 2 -(21.3.7.10.13 = --1/12

-(21.2.3.6.7 = 1 / 4 "01,2,4,5,8 = v / 3 / 1 2 -(21.2.5.9.13 = 1/12 O1.2.6.11.14 = - 1 / 1 2

/21,2,7,12,14 = - 1 / 1 2 OI.2,9,10,15 = V/-6/18 -(21,3,4,11A3 = 1/12 /21,3,5,11,14 = --1/12 ~'21,3.6,10,14 = --1/12

OI.3.9.11.15 = -- V/'6/18

171.3.8.9.l l = - v/3/36 OI.3.10.12.15 = --V/'6/18

J'2L4,S,10AI = --1/12 OI.4.7.13.14 = 1/12

Ol,n&9,1o = 1/12 OI,4,8,11,J3 = --V/'3/36

OI.4.11.13.15 =

v~/36

~21,5,6,1Ll2 = --1/12 Ol,5,s,12,J3 = -- V"3/36 OI.6.7.9.12 = 1/12 "QI.6.8.10.14 = - x/'3/36 -(21.7.8.9.14 = v"3/36 OI.7.10.13.15 = ,¢~/36 g'22.3.4.7.8 = V ~ / 1 2 $'22.3.S.6. 8 = --V/'3/12 O2.3.6.9.14 = 1/12 $'22.3.7.10.14 = 1/12 .Q2.3.9.12.15 = ~v/6/18

O2,4,5,m,12 = 1/12 l]2,4,6,13,14 = 1/12 2"22.4.11.14.15 = v ~ / 3 6 "¢12,5,7,1132 = 1/12

OI.4.12.14.15 = v/'6/36 O1.5.6.13.14 = - - 1 / 1 2 O1,5,u,ta,]5 = -- x/'6/36 Oi.6.7.10.11 = - 1 / 1 2 O1.6.9.13.15 = ~/~/36 O1.7.8.10.13 = -v/'5/36 Oi.9.12.13.14 = - - 1 / 1 2 122.3.4.11.14 = 1/12 O2.3.5.11.13 = 1/12 O2.3.6.10.13 = - 1 / 1 2 O2.3.8.9.12 = "v/3/36 02,3,10,11,15 = - - V ~ / 1 8 02.4.6.9.10 = 1/12 &,4,s,] 1.14 = - x/-5/36 .Q2.4.12.13.15 = - v/'6/36 2"22.5.7.13.14 = 1/12

On,2,3,9,1o = 1/4 OI.2.4.9.14 = 1/12 Oi.2.5.10.14 = 1/12 /21.2.6.12.13 = 1/12 .(21.2.8.9.10 = v/3/36 -(/k2.11.12A5 = - - V ~ / 1 8 01.3,4,12,14 = 1/12 .(21,3,5,12,13 = 1/12 J~1,3,7,9,14 = 1/12 01.3.8.10.1 = = - v ~ / 3 6 O1.4.5.9.12 = 1/12 Oi.4.7.11.t 2 = 1/12 121.4.8.12.14 = -x/-3/36 o(21.5.6.9.10 = - 1 / 1 2 .(21,5.8,11.14 = v/3/36

-(~1,5A2,13,15 = V'6/36 O1.6.8.9.13 = --

v/'3/36

OI,630,14,15 = V~/36 O1.7.9.14.15 = --v/6/36 OI,IOA1,13,14 = 1/12 O2.3.4.12.13 = - 1 / 1 2 122.3.5.12.14 = 1/12 -(22.3.7.9.13 = 1/12 -(22.3.8.10.11 = -x/-3/36 $'22.4.5.9.11 = 1/12 O2.4.6.11.12 = 1/12 O2.4.8.12.13 = v/3/36 ~(22.5.7.9.10 = 1/12 -Q2.5.8.11.13 = - V/'3/36

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

669

Table 6 - - continued -(22,5,8,12,14 = - V ~ / 3 6 -(22,6,7,9,11 = 1/12 ,(22,6,8,10,13 = - x / 3 / 3 6 O2,7,8,9,13 = V'~/36 .(22,7,10,14,15 -- - v / g / 3 6 $'23,4,5,9,10 = 1/6 3'23,4,8,10,14 = - x / ~ / 3 6 3'23,5.8,9,14 = X/3/36 3'23,5,10,13,15 = V~/36 -(23,6,8,11,)3 = v/3/36

-(13,6,J2,14,15 = - x/'6/36 O3,7.11,14,15 = X/6/36 3'23,11,12.13,14 = 1/12 3'24,5,6,12,13 = 1/12 -(24.5,8,9,10 = v'~/9 3'24,5,13.14A 5 = --X/~/18 ~(24,6,8,9,11 = V/3/18

g/2,5,11A3,15 = x/6/36

&,5,12,14,15 = X/-6/36

,O2,6,7,10,) 2 = 1/12 ~(22,6,9,14,15 = --V/'6/36

~2,6,8,9,14 = x/3/36 3'22,6,10,13,15 -----V/'6/36 ,(22,7,9,13,15 = - x / 6 / 3 6

3'22,7,8,m,14 = x/3/36 122,9,11,13,14 = --1/12 -(23,4,5,13,14 = 1/12 ~Q3,4,9,13,15 = V/6/36

1`22,1o,12,13,14= - 1 / 1 2 1"23,4,8,9,13 = -- x/3/36

-(13,5.8,1o.13 = - x/3/36

-(23,5,9,14.15 = - x / ~ / 3 6

O3,6,7,11,12 = -- 1/6 "Q3,6,8,12,)4 = v ~ / 3 6 -(~3,7.8,11,14 = -- X/~/36

.(23,6,7,13,14 = --1/12 f23,6,11,13,15 = --X/~/36 ,(23,7,8,12,13 = x/3/36 1"23,9,10,13,14 =--1/12 1"24,5,6,11,14 = --I/12 ~(24,5,7,12,14 = --1/12 1"24,5.9,10.15 = V~/18 3"24,6.7.10.13 = 1/12 .(24,6,9,11,15 = V/"6/36 ~Q4,7,8,10,11 = --V/'3/18 1"24,8,9,13,15 = --V~/12 ,O4,10,11,12,13 = 1/12 -(25,6,8,9,12 = --V~3/18

-(23.7.12,13,15 = - x/'6/36

3'24,7,9,12,15 = X/'6/36 -(24,8,10,14.15 = -v~/12 -(25,6,7.9A 3 = --1/12 3'25,6,8,10,11 = X/~/18 -(25,7,8,9,11 = V/'3/18 -(25,7,111,12,15 = v/-6/36

-(24,5,63,8 = --V/'3/6 ~O4,5,7,11,13 = --1/12 1"24,5.8,13.14 = 5V~/36 3'24,6,7,9,14 : --1/12 O4,6,8,11,1,12 : V~/18 ,(24.7,8.9,12 -----V~/18 ,(24,7,10,11,15 = -v~/36 ,.Q4,9,I1,12,14 : -1/12 -(25,6,7,10,14 = -1/12 ~(25,6,9A2,15 = - x / 6 / 3 6 -(25,7,8,10,12 = V/'3/18 ,(25,8,9,14,l 5 = V~/12

-(25,9,11,12.13= --1/12

1"25,1o,11.12.14 = --1/12

-(26.7,8,13,14 = 5X/~/36 -(26,8.11,13,15 = --X/~/12 -(26,9,10,12,13 = 1/12

O6,7,11.12,15 = X/~/18

-(24,6.1o,12.15 = x/~/36

1"27,9,1.o,11,13 = - 1 / 1 2 3'28,9,1(I,13,14 = V'~/36

g29,1o,13,14,15 = --x/g/9

-(26,8.12,14,15 = -v/2/12 1"27,8,I1,14,15 = V/2/12 -(27.9.10,12,14 = -- 1/12 ,(28,11.12,13,14 = x/3/36 1211,12,13,14,15 = --V/'6/9

1"23,4,10,14,15 = X/6/36

-(2s,6,1o,11,a5 = x/6/36 1"25,7,9.11,15 = v ~ / 3 6

3"15,8,10,13,15= --'¢~/12 .(26,7,8, ll,12 = V/3/9 f16,7,13.14,15 = -- X/~/l S

s26,9,m,l 1,14 = - 1/12 .(J7,8,12,13,15 = -x/~/12 S&9,10,H,12 = --V~/18 J?9,10,11,12,15 = -- V/6/9

5. Primitive invariant symmetric polynomials, Casimirs and cocycles 5.1. General considerations

and the case o f s u ( n )

The polynomial ring of commuting operators in the enveloping algebra H ( ~ ) of a simple algebra ~ is freely generated by 1 Casimir-Racah operators [ 2 - 9 ] of orders ml . . . . . mr. As a result, the k (m) polynomials for su(n), say, will be expressible in terms of the lower order primitive ones if m > n = l + 1. For instance, kili2i3i4 = lsTr(XijXi2Xi3Xk) for su(4) is primitive and generates a (non-trivial) fourth-order Casimir, but it turns out to be proportional to (~(ili2~i3i4) (and does not lead to a fourthorder primitive Casimir) when the T/E su(3) (see Example 5.1 below and ( 6 . 9 ) ) . The case of su(l + ! ) has been considered before (see, e.g., Refs. [31,26,25] ) but to our knowledge there is no unified treatment available in the literature for the four simple infinite series. Let us start by recalling the case of su(n) [31] and consider

J.A. de Azcdrraga et aL/Nuclear Physics B 510 [PM] (1998) 657-687

670 4, ...... +,

)'~1 . . . . . . . . ,,.~,,,, = 0 .

(5.1)

This expression is symmetric in the generator indices i l . . . i n + l and is obviously zero since the matrix representation indices ce,/3 range from 1 to n. Using (3.5), it follows that the above expression is a sum of (n + 1 ) !-terms which may be grouped in classes, each class being a sum of terms all involving a product of the same number vl . . . . . V,+l of products of traces of products of 1 . . . . . n + 1 matrices respectively, where 1 - vl + 2 - ~,~ + . . . + (n + 1)vn+l = (n + 1). In other words, the different types of products appearing in (3.5) are characterised by the partitions of (n + 1) elements, i.e. by the Young patterns (see, e.g., Ref. [ 3 2 ] ) associated with Sn+l, [ ( n + 1) .... ' , n " ' , . . . . 2 ~2, 1"'1

(5.2)

(obviously, ~'n+l = 1 or 0). All the elements of S~+l for a given pattern (a set of fixed integers vt . . . . . V,+l ) determine products of traces of matrices with the same grouping pattern and, moreover, appear in (5.1) with the same sign (they correspond in Sn+1 to the same conjugation class). The number of SN elements associated with a given Young pattern is given by the 1844 Cauchy formula N! vl !2~2v2!3~3v3! . . . N ~u

(5.3)

and since they correspond to permutations o- E SN with (equal) parity 7r(o-), 7r(o-) = ( - 1 ) b~2-}-P4-~-P6+,

(5.4)

they all contribute to (5.1) with the same sign. The mechanism which expresses the higher-order symmetric invariant polynomials in terms o f the primitive ones is now clear. A given Young pattern determines a specific product of invariant symmetric tensors with the sign (5.4) and weighted by the factor (5.3). Since one of the terms in the sum (5.1) corresponds to the partition vl = . . . = v~V-l = 0 , VN = 1, it follows that the invariant symmetric tensor of order m = n + 1 = l + 2 will be expressed, through (5.1), in terms of the I tensors of order 2, 3 . . . . . 1 + 1 and that only these are primitive for n = 1 + 1. We are thus lead to the following

Lemma 5.1. ( Invariant symmetric polynomials on su(n) ) There are l invariant polynomials of order 2, 3 . . . . , l + 1; the others are not primitive, and may be expressed in terms of products of them. The simplest application is that di, i2i3 have the following

=

0 for SU(2). For the next higher order we

Example 5.1. Let G = s u ( 3 ) . Using the Ai matrices (i -- 1 . . . . . 8) we find, since they are tridimensional,

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

E/~l/~2/~3~4/,A ,~oq /'A -~ot,~ (

=s

{1

•

,

i2,I.b2~,Ai3

Ot3

Ot4 (Ai4) .B4

.~_~Tr(Ai, hi2)Tr(,~i3Ai4) _ ~Tr( l ~'il 'Ai2~i3 ~-i4 )

671

)

= O,

where, as before, s means symmetrisation in all indices il, i2, i3, i4. Thus, the fourth-order symmetric tensor can be expressed in terms of the Killing second-order one. Using (4.2) we find Tr(A(i, Ai2Ai 3 Ai4) ) -- 2 8 ( i.i2 (~i3i4 ) •

(5.5)

But using again (4.2) we may compute directly for su(n) 1

1

~sTr(,~g, A,2.~ Ai.) = 4.----~.~sTr({ai,, ai2}{Ai3, Ai4}) 4 = --~(ili2~i3i4) -~ 2 d p ( i l i 2 d i 3 k ) p . n

(5.6)

1 Eqs. (5.5) and (5.6) for n = 3 now give dp(ili2di3i4)p = ~(ili2~i3i4). For su(3), the t e n s o r ki,i2i3i4 [Eq. (2.3)] is given by kili2i3i 4 = ( 2 )

4 T r ( , ~ ( i l A i 2 A i 3 A i 4 ) ) = ~6(ili26i3i4) I 1 -~- gdp(ili2di3i4)p.

(5.7)

5.2. The case of so(n) and sp(l) Let us now turn to the case of the orthogonal (odd, even) (Bl, Dr) and symplectic (Ct) algebras. In the defining representation these groups preserve the n × n Euclidean or symplectic metric r/and thus the generators Xi of these algebras satisfy X m = -~X~,

(5.8)

where i = 1 . . . . . I ( 2 l + 1 ) , 1 ) 2, n = 2 1 + l ( B t ) ; i = 1 . . . . . I ( 2 l + l ) , l ~> 3, n = 2 1 (Ct); and i = 1. . . . . l ( 2 1 - 1), l ) 4, n = 2l (Dr). The symmetrised products of an odd number of generators of the orthogonal and symplectic groups also satisfies (5.8); hence, they are a member of their respective algebras. In particular, we may write (5.9)

{Xi I , X i 2 , X i 3 } = Uili2i3~r X o -,

where the bracket { . . . . . } denotes symmetrisation, i.e. it is the sum of the six possible products and Uili2i3o. is an invariant symmetric polynomial [ 181. Now we may define a new v ("') family of invariant symmetric polynomials (cf. d ('0 used in (6.1) for s u ( n ) ) by Ui~...i21, = U(iti2i3~ 1U°lli4i5 ~. 2 " " " Uat,-2i2p-2izp

With this notation (see (2.4))

1i2,,) •

(5.10)

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

672

1

kil...i2v - - (2p) ! sTr(X6 " ' " Xi2v ) ~

Tr(X(6 " ' "

Xi2pl )

1

= 6P-I Tr( { . . . {{X(i~, Xi2, Xi3 }, Xi4, Xi~ } . . . . . Xi2,,_2, Xi>_~ }Xi~, ) 1

= 6p_l V(ili2i3a. lUali4isa. 2 •

Vcee_2i2t_2i2, ,~.Yr(XcrXi2, ))

K -- 6p_

1 V( iti2i3a. 1Uetli4i5 ~.2

(5.11)

• • " Vezl,-2i2p-2i2p li2p) "

Thus, K

ki,...6,, - 6P - 1 vi, ...i2p"

(5.12)

This leads us to the following simple lemma.

Lemma 5.2. (Generation of higher order invariant symmetric polynomials for BI, Cl, DI ) Let uili2i3i4 be the second lowest symmetric invariant polynomial for BI, Cl, Dr. Then, the higher order symmetric polynomials kil...i2v m a y be written in terms of oili2i3i4 by means of (5.12) and (5.10). Let now ~ be Bt (C1) and let Xi be a basis for Bt (CD given by ( 2 1 + 1)- (2•)dimensional matrices. Then, since the symmetric trace of a product of an odd number of X's is zero, and any partition of an odd number of elements will always include a symmetric trace of an odd number of X's we have to consider an even product x~L.,.B~2( y . )':~,(Xi~)'/~2 al a2 ~O¢ I ...O'21+2 k ~ll I

• .

.

,~a~+2 ( X"'2'+~"/~2,~-2 = 0;

(5.13)

otherwise each term in the l.h.s, would be zero. Reasoning as before, but taking now into account that only traces of an even number of factors are different from zero, we are led to the following

Lemma 5.3. ( lnvariant symmetric polynomials for Bt, Cl ) The symmetric invariant polynomials for BI, Ct given by (2.3) are all of even order m = 2k and non-primitive for m > 21. The relation which expresses the lowest nonprimitive symmetric polynomial in terms of the primitive ones follows from the equality X-" Z.., partitions

( - 1 ) ~2+~4+...+~2,+2 2v21:'2!4v41.'4! . . . (2l + 2 ) vit~2

x s { [ Tr (Xi, Xi2 ) ] ~2[ Tr ( X % +,Xi2~2+2X% +3Xi2~2+4 ) ] v 4 . . .

} =

0,

(5.14)

where ~ is extended over all partitions of ( 2 l + 2 ) in (even) factors (all Young patterns of $2t+2) and there is symmetrisation over all indices il . . . . . i2/+2.

Example 5.2. For B2 we obtain that

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687 ~:EI ...E6 (~1('. ~ 1 ,.,or6 \ "~ll ) 0~1 "El

...

( Xi6 ) ~6 .E6 = S

--

q_.~3Tr(Xi,Xiz)Tr(Xi3Xi4)Tr(Si, Xi6) Tr ( Xi~ Xi2 Xi 3Xi 4 ) Tr ( Xi5 Xi6 ) - ~ Tr ( Xi~

+

673

Xi6 )

----0,

a relation which expresses the invariant polynomial of order 6, Tr(Xi~... Xi6), in terms of those of order 2 and 4.

Example 5.3. For C3 we have

{,

EEJEs,...~8 ~, trXt~ J~.EI "" " (Xi~ ) ,~8.E8= s 4.--~ Tr (Xi~ Xi2 )Tr(Xi3Xi4.

)Tr (Xis. Xi6 )Tr(Xi7Xi8 )

1

2 !224 Tr ( Xi, Xi2 ) Tr ( Xi3 Xi4 ) Tr ( Xi5 Xi6 Xi7 Xi8 ) + ~ Tr (Xi, Xi2 ) Tr ( Xi~ Xi 4 S i 5 Xi 6 Xi 7X i 8 ) +~ -

1

Tr( Xi, Xi2 Xi3 Xi4 ) Tr( Xi~ Xi6 Xi7 Xi8 )

~Tr(Xit Xi 2Xi 3Xi 4Xi 5Xi 6Xi 7Xi 8 ) I = O.

This expression relates the eighth-order invariant symmetric polynomial with those of lower degree. The case of the even orthogonal DI is different because, in this case, there is an invariant polynomial which is related to the square root of the determinant of an evendimensional matrix (the Pfaffian).

Lemma 5.4. ( lnvariant symmetric polynomials for Dl) The symmetric invariant polynomials for Dt given by (2.3) are of even order and primitive for m = 2, 4 . . . . . 2I - 2. The higher order polynomials are written in terms of those and the polynomial Pfi,..i, constructed from the Pfaffian using that

EEI...E21 ( Xi, )'Elal °'l'"°L21

"

. . ( Xi21)E2t a21 = Pf(il...i, Pfi,,,...i2,) '

(5.15)

which follows from the fact that det(,~ixi) = (Pf(Aixi))2 for the (2t) skewsymmetric matrices Xi, and that these matrices constitute a basis in the vector space of 2l x 21 skewsymmetric matrices. X

/

Example 5.4. For D4 we have EEJ...E8 (

1 ~,...,,~ Xi, ) ,'~ .E, . " " (Xi8)-~ = s { 4~.24 Tr( Xi, Xi2 ) Yr( Xi3 Xi4 ) Tr( Xis Xi~ ) Tr( XiT Xi~ ) 1

2!22 4 Tr ( Xil Xi2 ) Tr (Xi~ Xi4 ) Tr ( Xi~ Xi6 Xi7 Xi~ )

674

J.A. de A z c 6 r r a g a et a l . / N u c l e a r Physics B 5 1 0 [ P M ] (1998) 6 5 7 - 6 8 7

+ 1~Tr ( Xi, Xi2 ) Tr ( Xi3 Xi, Xis Xi6 Xi7 Xi~ ) +~

1

Tr ( Xi, Xi2 Xi3 Xi, ) Tr ( Xi~Xi, Xi7 Xi~ )

½Tr( Xi~ Xi2 Xi~ Xi, Xi5 Xie Xi7 Xi~ ) I = Pf{

il...i4Pfg...i8 ) "

6. Invariant symmetric tensors for su(n): a detailed study The case of su(n) is specially important since SU(n)-invariant tensors appear in many physical theories as, e.g., QCD. The properties of the su(n)-algebra tensors have already been discussed in [33,29,31,26,25] (and tables for the fok and the dijk for su(n) up to n = 6 have been given in [30] ), but we need to perform here a more complete and systematic study. The case of the cocycles was already discussed in Section 4. We consider now the symmetric tensors, exhibiting in particular how the relations defining them for general n produce primitive tensors up to a given order mt and non-primitive ones when this order is exceeded. To this aim, we shall use Lemma 3.3 and its Corollary 3.2.

6.1. d-tensors We begin our study of totally symmetric tensors of arbitrary ranks for su(n) by considering the d-tensor family [25] (see also Ref. [4]). For ranks r = 2 and 3 we have d!2),1 = 8#,. d~ ~ = diik,, where the latter is the well-known totally symmetric and traceless su(n) tensor, which exists for n /> 3. Higher order tensors are defined recursively via d!r+. 1 )

tl...tr-li,-i,.~l

dO.). ij...~.,

-d - -

(r) Jd(3) il...ir-I --jirirt

l '

r = 3,4,...,

= d i 11 t i 2 d l l12i 3 • " • dl.,_3i.,_li.,

(6.1)

i = 1 . . . . . (n 2 - 1 ). For r ~> 3, the above tensors are not symmetric in all their indices, so the required totally symmetric tensors are defined from their symmetrisation which gives d~r) ( il...i,

)"

(6.2)

Due to the fact that dijk is already symmetric, (6.2) normally represents the sum of p < r! distinct terms divided by p. The construction defines a family of symmetric invariant tensors of order r. We want to know the dimension and a basis for the vector space ~
.I.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687 d (iti2i3i4) (4~ - d (ili2i3)i4' (4) --

~(ili2~i3i4)

675 (6.3)

..~ ~ ( i l i 2 ~ i 3 ) i 4 .

Similarly, also for n large enough dim V (5~ = 2 and a basis is provided by d ( (ili2i3i4i5) 5) '

(6.4)

d ( ili2i3~i4is) "

In fact, d i m V (4) = 2, only for n ~> 4, since if (and only if) n = 3, we have from Example 5.1 d(4~ ('1i2i3i4)

(6.5)

= I ~(ili2~i3i4)"

Similarly, dim 12(5) = 2 only for n ~> 5, a basis being provided by dl5~2imi~ and d(iji2i3~i4is).

For n = 4, however,

d(5)

(ili2i3i4i5)

2

(6.6)

= "3 d ( i l i 2 i 3 6 i 4 i 5 ) '

while, for n = 3, we have as a direct consequence of (6.5) d(5) (ili2i~i4i5)

(6.7)

= 3d(ili2i3fii4is)"

These results will serve as a check on some calculations done later. They are a consequence of Corollary 3.2; Eqs. (6.5) and (6.6) also follow from work in [25] and in [31], both of which supply values for dim V (4) and suggest possible bases to use for increasing r. The recursive procedure of [25] actually supplies more information, but the work in [ 31 ] is also useful, providing direct access to results for each fixed n. One way to see that not all families of symmetric invariant tensors of a given order are equally useful arises when one uses them to construct Casimir operators by (2.6)

(6.8)

C (r) -~ " t ( r ) X il X i" ~(ib..ir) """ '

where the Xi are generators of s u ( n ) . In principle this yields an arbitrary number of Casimir operators, one of each order r ~> 2. But since s u ( n ) has ( n - 1) primitive Casimir operators of order 2,3 . . . . . n, it must be possible to express those with order r > n in terms of the primitive ones. It is well known how to do this for su(3), where (6.5), (6.7) imply

(~(4)

= ~15~ u ( i l i 2 t , ,R i 3 i 4 ) A v i t v i 2Av i ~ vAi 4A

I F... yil = 1(C(2))2 -[- 9-,''1'2t,''

C (5) ----lc(2)C(3) + ¼C (3).

X t 2"X t 3 "

= 31-(c(2))2 + 1C(2) (6.9)

Similarly, for su(4) Eq. (6.6) gives

2C(2)C(3) + .g 2C(3) • C (5) = .~

(6.10)

It is not possible in practice to treat such matters explicitly or even systematically for arbitrarily large n, r. It is thus convenient to replace the family of symmetrised d-tensors by a family for which no similar difficulties arise. This is achieved by using the t-tensors introduced in Lemma 3.2 and taking advantage of their property (3.18).

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J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

6.2. t-tensors

Working within the t family, we can build from t (m) only one non-vanishing scalar quantity, K (m) ( n ) = t (m) it'"i't!m).

(6.11)

"

II ...Inf '

all other full contractions are zero by Lemma 3.3. Since the standard s u ( n ) Gell-Mann

matrices ,,~i allow easy computation of the d-tensors, and a well-developed technology to operate with them exists (which can be completed when necessary with additional relations, see Appendix A), we give the expressions of our t-tensors in terms of the d-tensors in Section 6.1. From (3.8) we trivially obtain tili2 -~ n~il i2.

(6.12)

Similarly, for m = 3 , 4 Eqs. (3.15), (3.6) give n2

in 2

(6.13)

tili2i3 = "~ki, i2i3 = --~dijk,

tilizi3i4 = i~0[n(n 2 + l~d (4) - 2(n 2 J (iti2i3i4)

-

(6.14)

4)6(ili2~i3i4)],

and also tili2i3i4i5

=

(5) a ( n ) [ n ( n 2 + 5)d(ili2i3i4i5)

- 2(3n 2

20)d(id2i3t$i4is)],

(6.15)

by identifying tili2i3i4i5 a s the traceless linear combination of the appropriate dim V (5) = 2 tensors. The factor A ( n ) has not been determined explicitly: the evaluation of (6.14) is already time consuming. These results apply to generic n, i.e. for n such that dim V (4) = 2 = d i m V (5). It is a valuable check on the computations to insert n = 3 into (6.14) and n = 4 into (6.14) and (6.15), and employ (6.5), (6.6) and (6.7) to obtain for n = 3:

tili2i3i 4

for n = 3:

tili2i3i4is (5) l - • = 42,l(3) [ d (ili2i3i4i5) -- 3d(ili2i3~i4is)

for n = 4:

lrd(4) 1 t (ili2i3i4) -- 3~(ili2~i3i4) ]

=

tili2i3i4i5

=

[,4(5)

8 4 a ( 4 ) t~(i,i2i3i4is)

2

(6.16)

= 0,

-- ~d(ili2i~i4is)]

] = 0, =

0.

(6.17) (6.18)

These results exhibit the crucial property of the t-tensors. They are in one-to-one correspondence with the 12 tensors and hence, as we discuss below, provide an ideal way (free from problems associated with d-tensors noted above) to discuss Casimir operators. For low n, for which only ( 2 m - 1 )-cocycles with 2 m - 1 ~< 2 n - 1 can be non-zero, the generic results for t(')-tensors ( 6 . 1 2 ) - ( 6 . 1 5 ) collapse according to ( 6 . 1 6 ) - ( 6 . 1 8 ) , as they must when m > n, to give vanishing tensors. An alternative way to see the same mechanism at work stems from calculation of the numbers K ( ' ) (n) of (6.11). We have K (') =0,

m > n,

(6.19)

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677

so that s u ( n ) only defines ( n - 1) non-zero scalars. By direct computation based on ( 6 . 1 2 ) - ( 6 . 1 5 ) (using dijkdijk = (n 2 - 1 ) ( n 2 - 4 ) / n [cf. Eq. ( A . 2 ) ] and similar relations, see Appendix A) we get for m = 2, 3 and 4, with some work in the last case K (2) (n) = n2(n 2 - 1 ) ,

(6.20)

2 - 1)(n 2 - 4),

K(3)(n) = - i ~ n 3 ( n

(6.21)

g ( 4 ) ( n ) = ( 1 _ ~ ) 2 gn 2 2.tn 2 + l ) ( n 2 - 1 ) ( n

2-4)(n 2-9),

(6.22)

and likewise from (6.15) 4

K(5)(n) = [ A ( n ) ] 2n~ ( n 2 ÷ 5) r I ( n 2 - l 2) . ,J

(6.23)

/=1

Hence, it follows that K (3) (2) = K (4) (2) = K (5) (2) . . . . . K(4) (3) = K(5) (3) . . . . . K (5) (4) . . . . .

0,

0,

0.

(6.24)

It is now plausible to conclude that the pattern persists for all su(n) algebras, and that with the required modifications it remains true for other classical families. For low rank groups, for which cocycles above a certain order cannot appear, the corresponding possibly non-zero scalars K (m) that can be formed can be seen to vanish identically, as expected since the t-tensors used to define them also do so.

6.3. Relations f o r the Casimir operators 8 ~ Corollary 3.3 exhibits the one-to-one correspondence among the t (m) tensors and the Casimir invariants C (m). The properties of the t tensors in Section 6.2 show now that we do not get primitive Casimirs of undesired order, since (6.16), (6.17), for instance, yield 8 "4) = 0 = C (5). This is of course needed to make sense: there are no/2(7), /2(9) cocycles for su(3) since there are only l primitive Casimirs and cocycles for su(l + 1). Similarly (6.18) shows that C (5) = 0 for su(4) since there is no /2(9) for this algebra.

Example 6.1. For su(n) (n > 2), Eq. (3.19) gives n2 C 1(3) = [j?(5)]ilizi3i4isXilXi2XitXi4Xi s = " ~ k a b C x ,,abac v v = --~d in2 abc X,,XbXc,

(6.25)

as expected from (3.20) and (6.13). Similarly, if we compute the fourth-order Casimir from (3.19), we obtain 8t (4) = [ ~?(7) ] il i2i3i4i5i6i7Xi I Xi 2X i 3Xi 4 Xi 5Xi 6 Xi 7

_

1 ( n ( n 2 + 1)8(4) _ 2(n2 _ 4)6(i,i2~i3i4)xilxi2x6xi4) 120- 8

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J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

120-1(n(n2+l)C(4)-2(n2-4)([C(2)]2+6C(2)))8

(6.26)

If we set n = 3 and use (6.9) above we get Ct(4)

-

12.1 8(3C(4 ) _ [C(2)] 2 _ 1C(2)) = 0 .

(6.27)

Thus, with the CI family, we do not obtain Casimir operators beyond the order of the higher invariant primitive polynomial of the algebra. Also we have C~2) = fabcXaXbX c = l f a b C f a b d X d X c = ( n / 2 ) X d X d . Recall that the fourth-order Casimir C (4) is defined by C (4) - - ttP(ili2"4i3)i4y" Y " g " Y " (see (6.8)). We may conclude by stating that the l tensors t introduced by Lemma 3.2 are in a rather privileged position due to their full 'tracelessness', Eq. (3.18). In particular, Eq. (3.19) provides a definition of the primitive Casimirs CI which does not contain non-primitive terms and which hence gives zero, as it should, when their order goes beyond the maximum mt order permitted.

6.4. Some technical remarks

The crucial results (6.14) and (6.15), which provide explicit expressions for the fourth- and fifth-order invariant symmetric su(n)-tensors require, for their derivation, a series of expressions involving properties of the f and d tensors of s u ( n ) which at present are not available in the literature. The same applies to the calculations (6.20)(6.23) for the canonical s u ( n ) scalars K ~n), on which we have based some generalisations. In addition to early work [29] containing a modest amount of generic s u ( n ) material, we have used [33] which contains a wide class of identities for f and d tensors. To proceed further here, we make use of the fact that an arbitrary symmetric SU(n)-invariant tensor may be expanded in terms of a basis of V ~m) as explained in Section 6.1 (see also Ref. [25,31] for low values of m). We relegate the detailed treatment to Appendix A.

7. Recurrence relations for primitive cocycles Consider first the case of s u ( n ) . In the defining representation, the unit matrix and the Ti's span a basis for the space of hermitian n x n matrices. This means that, since the symmetrised product of hermitian matrices is also a hermitian matrix, we can write {T~I . . . . , T/,,, } ~ -'~ k/l ..i,,, T,~ + ki~...... i I,

(7.1)

where ~9" . and kili,, are invariant symmetric polynomials (of order (m + 1 ) and I I ,-. t m "'" m) that can be related to the k tensors defined by (2.3). However, to give recurrence

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

679

relations we want to use here the invariant symmetric tensors d [25] (see also Ref. [4] ) defined by Eq. (6.1). 10 The d (m) polynomials are not traceless, and hence differ from those of (3.15). Let us use them to define cocycles. Since the structure constants C~ themselves provide a three-cocycle, the five-cocycle may be rewritten in the form _ 1 _jl.j2j3j4.js/2(3)llcl 2 d

(5)

aQili2i~i4i5 -- ~eili2i3i4i5

jlj2.

(7.2)

.J3.J4 1,12.i5"

The next case may be treated similarly. The coordinates o f / 2 (7) are given by

l~jlJzJ3J4JsJ6JT[~ll r12 C I3 d (4)

2 7)

ili2i3i4i5i6i7 = 7~.Cili2i3i4i5i6i7

~'.hj2v'j3j4

jsJ6 Ijl213.j7"

(7.3)

Now, although d(4) l 1213J7 (see ( 6 . 1 ) )

=

s

(7.4)

dll12" dsl3j 7

is not fully symmetric, the extra skewsymmetry in (7.3)

(cf. (3.4)

and ( 3 . 6 ) ) permits us to use (7.4) in (7.3) without having to symmetrise. 1~ This means that (7.3) may be rewritten as /2(7)

_

cl 2 d scl3 d jl.J2 j3j4 1112. jsJ6 s13J7

1 Ejlj2j3j4j5J6JTcII

iti2i3i4i5i6i7--'~,

ili2i3i4i5i6i7

= L ~. !1 k2k3 k4.!5:j6j7

(5)

S 13

.

7! '~,2,3'415/6/7 ~]1 k2k3k4. Cj5j6 dsl3.tT .

( 7.5 )

Extending the procedure to an arbitrary /2(2m-t) cocycle we now get the following

L e m m a 7.1. (Recurrence relation f o r su ( n ) primitive cocycles) Given a ( 2 ( m - 1) - l)-cocycle of s u ( n ) , the coordinates of the next ( 2 m - 1)-cocycle are obtained from /2(2(m-l) - l ) by the formula f2(2m_ 1) il...i2,,,-1 --

1

(2m -

E.jl...j2,,,_13~(2lm_ll_l)S_,l ,4 ,. 1 ~jl...j2m--4 . I~.J2m--3j2m--2~St.12"--l"

1 ) ! il...i2,,,

(7.6)

Clearly, other relations may be found using again (7.6) to express the lower order /2(21m-I1-1) in terms of / 2 ( 2 [ m - 2 1 - 1 ) , etc.

cocycle

For the symplectic and orthogonal algebras B/, Ct, Dl (setting aside for Dt the case of the polynomial of order mt = l related to the Pfaffian) the primitive symmetric invariant m Other expressionssuch as (2.3) could be used here, since the non-primitiveparts in (2.3) do not contribute to the cocycle definitiondue to Corollary 3.1. i~ Note that in the above examples it is not necessary to introduce a five [seven] index e tensor in the ,(4) = r.h.s.; instead, skewsymmetryin i2i3i4 [ii ... i6 ] will suffice in (7.2) [ (7.5) ] because d(ijk) = dijk a(ijkl) , j(5) d(4)(ijk)/ " However, for the fifth-order polynomialuijkl m we cannot use such a simplification(although in the symmetrisationthere are fewer than 5! differentterms). In this case (and for higher order d-polynomials) we need the complete (2m -- 1)th order e tensor, Eq. (3.4), to remove the non-symmetricparts in (6.1).

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J.A. de Azcdrraga et aL/Nuclear Physics B 510 [PM] (1998) 657-687

p o l y n o m i a l s o f order 2, 4 . . . . . 2l (Bt, Cz) and 2, 4 . . . . ( 2 l - 2 ) (Dl) may be constructed by means o f (5.10), and they lead to primitive cocycles o f orders 3 , 7 . . . . . (4l - 1) ( B t , C z ) and o f order 3, 7 . . . . . ( 4 l - 5) ( D r ) . Thus, the first recurrence relation starts for .(~(ll), which is written as /2~"],, : 1-~eJ"'Jl'c!' CJ~jloVtl...zq H •"" 11! ~...m J~j2"'"

(7.7)

or, using (5.10),

1 1 tEjlk2...k6j7...jllEj2...J6cII C 12 C 13 u s c l 4 C 15 u - 5 ! 11! il ............... ill k2...k6 .]1.]2 j3j4 JsJ6 111213. J7J8 jgjlo slal.~jll 1 ~" ~

ll--.jll f ) ( 7 ) st'-,/4 t--,15 . . . . . eli ..-ill "l'~jl ...j6. v*""J7J8 v""JgJ,o~Sl415Jl' '

( 7.8

)

which expresses /2 (l]) in terms o f / 2 (7) and the fourth-order polynomial. This leads to the following recurrence relation: L e m m a 7.2. (Recurrence relation f o r so(21 + 1), s p ( l ) , s o ( 2 1 ) ) Let d ? ( a p - - l ) be a ( 4 p - 1)-cocycle for s o ( 2 l + 1), s p ( l ) ( p = 1 . . . . . I), s o ( 2 l ) ( p = 1 . . . . . (l - 1 ) ) . Then, _ 1 ~.jl---j4p-I z a ( 4 [ p - l l - l ) S t ~ l l t,~12 I, , , . ~/(14!Ii4pl)1 -- ( 4 p - 1 ) [ -il'''i4p-I JZ'Jl'"J4P-6 .1,.~j4p_5j4p_4~.~j4p_3j4p_2W$lll2j4p_l.

(7.9)

8. Duality relations for skewsymmetric primitive tensors The interpretation o f the primitive cocycles as closed forms on the group manifold o f a simple compact group G provides another intuitive way to obtain additional relations among them. Consider the case o f s u ( 2 ) . The identification o f the su(2)three-cocycle with the closed de Rham three-form on S U ( 2 ) ~ S 3 tells us that this form is (up to a constant) the volume form on the S U ( 2 ) group manifold. It is known [ 1,10-17] that, from the point o f view o f real homology, the compact groups behave as 'products' of spheres o f odd dimension (2mi - 1 ), i = 1 . . . . . 1. As a result, f2 = f2 ( 2 m l - l )

(8.1)

A ... A/2(2mr-l)

is proportional to the volume form on the group manifold and, indeed, ~-]I=] ( 2 m i r = d i m G for all simple groups. For instance, if G = S U ( 3 ) a(g) =/2(3)(g) A/2(5)(g)

oc

Cili2i3/2~5)si6i7i8o)il (g)

A . . . A cois(g)

1) =

(8.2)

is proportional to w t ( g ) A . . . A wS(g), the volume element on S U ( 3 ) . Algebraically (wi(g) ---+ ¢oi), we may look at (8.2) as the wedge product o f two skewsymmetric tensors defined on a vector space V o f dimension r. If we now endow V

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

681

with a metric (the unit metric for G compact) we may introduce the Hodge • operator. Then, the scalar product of two skewsymmetric tensors ce = q! l a q ...Iq 0)(i0 A . • • A W (iq) and/3 of order q is given by (0l, /3)

=

Ol

A (*/3)

1

=

(8.3)

-'~Olil...iq/3 il i,0)1 A . . . A (Dr.

q.

Consider the simplest su(2) case. There is only one cocycle (cf. (3.3)) 1

/2(3) = .~,0ili2i3(.O

i~

"

(8.4)

A 0.) t2 A (.O13.

We fix the normalisation by demanding that 1 ,0i, i2i3~/, i2is.0) 1 A 0)2 A 033 = 0)1 A 0)2 /k 0)3 ( , 0 ( 3 ) , j,~(3)) = ,0(3) A (:~,.('~(3)) = _~

(8.5) (w t (g) A w2(g) A w3(g) is the volume element on S U ( 2 ) ) , i.e. 1

~./2ili2i~ f2/1i2i3 = 1,

(8.6)

which is trivially satisfied since Oiji,6 the five cocycle be expressed as by /2(5 ) = "~../2ili2i3i4i5W 1 il A

.

.

= eili2i3

.

for su(2). Let now G = SU(3), and let

.

w '2 A o) `3 A w '4 A 0)'~.

(8.7)

We now fix now the normalisations of 0 (3) a n d / 2 (5) by requiring that J'2 (3) A ( * / 2 (3)) = 0)1 A . .. A 03 8 = ..(2(5) A ( * 0 (5)) .

(8.8)

This gives the previous relation for the coordinates o f / 2 (3) and a similar one for those of ,0 (5). Up to an irrelevant sign (which is a minus sign for even r, as is the case of SU(3), since ,2 (_l)q(r-q) where q is the order (always odd) of the cocycle) we may write =

(8.9)

,0(5) = , , 0 ( 3 )

(for a positive definite metric (ce,/3) = (*a, ,/3)) and hence (with r = 8) ,0(5)

1 l ~iljl~i2j2~i3J3E . . . . . . . . 0 ( 3 ) O.)11 O)12 o.)l 3 (.O/4 -- ( r - 3)~ 3! jij2j3tlt2t3t4ts~ili2i3 A A A A 0915 ,

(8.10) i.e,

,0(5)

1

,-~(3)Jlj2J3

l,l,~131415 = "~.e.JlJ:jsld2131415 at

.

( 8.11 )

The previous arguments are not restricted to s u ( n ) nor to the case of two cocycles. In general we have that relation (8.1) holds and, as a result, we find up to irrelevant signs a whole series of duality relations among cocycles:

682

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687 ~(2m,-1)

= ,(o(2m1--1)

$?(2mi--1) A O ( 2 m j - 1 ) = 't~(~'~ ( 2 n q - l )

A . . . J'~(2m~-Al) A . . . O ( 2 m r - l ) )

A . . . j~(2m,--'-"-"'~l) A . . . ~'~(2mj--l) A . . . jr~(2mt--l))

(8.12) where 1 ~< i , j . . . . <~ l, etc. In general, the normalisation of the ( 2 m i - 1) cocycles may be introduced by requiring that (O(2m,--1), o(2m,-1))

= $?(2m,-1) A (.O

(2m'-l))

= W 1 A . . . A co r,

(8.13)

the volume element on G. E x a m p l e 8.1. T h e three- and five-cocycles for su(3), given in coordinates in Tables 1

and 3 respectively, satisfy the duality relation 1

........

a(~ili2i3i4i5 -~ 3[2V/~e,lt2tat4ts,617tsf

i6i7i8

(8.14)

.

E x a m p l e 8.2. For su(4), we use the definitions of the three- and five-cocycles of Ta-

bles 4 and 6 and the expression (7.5) of the seven-cocycle 0 (7) to obtain the following relationship: 1 '~/~ (7) _ 1 6 (5) 0 (3) 5"--0ili2i3i4i5i6i7 5!3[ ili2i3iaisi6iTjlj2j3j4jsklk2k30 jl.j2j3j4j5 klk2k3"

(8.15)

These relations have been computed using MAPLE and provide a further check of the cocycle tables in Section 4.

Acknowledgements

This research has been partially supported by the CICYT and the DGICYES, Spain. J.A. and J.C.EB. wish to acknowledge the kind hospitality extended to them at DAMTP and J.C.EB. wishes to thank the Spanish Ministry of Education and Culture and the CSIC for an FPI grant.

A p p e n d i x A. Traces o f products o f su(n) D and F matrices

We define the hermitian D and antihermitian F (adjoint) traceless matrices for arbitrary su ( n ) by ( E a ) bc = fbac,

( O a ) bc = dabc,

a, b, c = 1 .....

n 2 - 1,

(A. 1)

intending to present the identities that we require involving d and f tensors of S U ( n ) in terms of traces of products of D and F matrices. All the two- and three-fold traces have been known for a long time [33,29]. Explicitly,

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

683

n2

Tr FaFb = - n 6 a b ,

Tr F~Db = O,

n Tr F,,FbFc = - - ~ f

- 4 8,,b,

nd Tr FaFhDc = - - ~ ,#,c,

n2 - 4 Tr F~,DbDc -

Tr D~Db -

n 2 - 12 Tr DaDbDc - - dabc. 2n

f abc,

2n

(A.2)

The methods of [33] yield only expressions for such four-fold traces as TrF~,FbFcDd,

(A.3)

TrFaDbDcDd,

as well as all others that follow from these which involve an odd number of F and D matrices. To proceed further (to the evaluation of the traces of all four-fold products of even numbers of D and F matrices), we begin by treating Tr F(aFbFcFd)

and

(A.4)

Tr D(aDbDcDd).

Once this is done, Tr FaFbFcFd, T r D a D b D c D d , Tr FaFbDcDd (and other similar traces) can be calculated by means of further elementary procedures. We list results valid for arbitrary su (n) : n Tr FaFbFcFa = ~ab6ca + 6adSbc + ~ ( dabxdcdx + dadxdbcx -- dacxdbdx ) ,

(A.5)

Tr FaFhFcDd = - ~ n d abxfcdx _ n -~ fabx d cax,

(A.6)

TrFaFt, DcDd =

4-n2 n2

8-n2 (Sab~cd -- 8ac~bd) + ~ (

d

d abx cdx -- dacxdbdx)

n

(A.7)

4 d,,axdocx,

Tr F, DbFcDd = ~ ( d~xdbdx - dad, dbcx) -"

TrF,,DhDcDd Tr DaDt, DcDd =

4

dab~dcdx,

n 2 -- 12 n 1 4------~--f~,bxd~dx+ ~d~oxfcdx + n (f,,dxdbcx -- f~cxdbd~), n 2 -- 4 ._~(

Sab6cd "~- 6adSbc)

- nd

(A.8) (A.9)

d

"~ acx Odx

n 2 -- 1

+

4n

6(dabxdcax + d,,dxda~.x).

(A.10)

Now we illustrate the method of derivation of the above results and perform a variety of checks of their correctness. In the framework of Section 6, we set out from an expansion of an arbitrary totally symmetric fourth-rank tensor in 9 (4) and write Tr F(~,FbFcFa) = Adl4~cd) + B6(ab~cd).

(A.11)

Contracting both sides with 6,,t, and d~,be in turn and using results such as (A.2) allows us easily to find A = n / 4 , B = 2. Various checks on, e.g., (A.5) and its consequences now exist. Firstly, in [33] we find identities tbr contractions of (A.5), (A.7) and (A.10) with d,,ce and f,,~.e. Performing such contractions explicitly on our expressions for these

684

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

traces, we find total agreement for all SU(n). For the case of SU(3) a more elementary derivation of, e.g., TrFaFbFcFa is available because additional identities for SU(3) D and F matrices exist as described in [29]. One can thus get the n = 3 version of, e.g. Tr F~aFbFcFa) without using any assumptions about symmetric tensors. In particular, for SU(3), we have

Tr F(aFbFcFd) = "~6(ab6cd), 9

(A.12)

Tr D(aDbDcDd) = ~6(ab6cd), 17

(A.13)

Tr D a D b D c D d

= 5 ( ~abt~cd q_ ¢~adt~bc) _ 7 ~(abt~cd ) _ gdacxdbdx. 1

( A.14 )

At,4 (4) .... (abcd) + Btt~(ab 3cd)

(A.15)

Analogously, from Tr D(aDbDcDd)

and contracting with ~ab and dabs w e deduce that A' = (n 2 - 32)/4n and B' = 2(n 2 4 ) / n 2. Armed with the results (A.5) to (A.10) as well as those of (A.2), we can consider five-fold traces. Thus we postulate (see (6.4))

TrD(aDbDcDdDe) = a'4(5) • "~'(abcde) q- Bt~(abdcde)'

(A. 16)

Contracting this with t~ab and dabf in turn gives three linear equations for the coefficients A and B; in the latter construction equating coefficients of .'4(4) ~*(cde)f = "4(4) ~(cdef) and t~(cdt~e)f = t~(cd~ef ) has actually given rise to two equations. These three equations are consistent and give n 2 - 80 A- - - , 8n

B=

3n 2 - 20 n2

(A.17)

The major cancellations that bring (A.17) to the form displayed convince us of the correctness of our results. There are also other independent checks available because results for various five-fold traces with no more than three free indices (vertex corrections, in the diagrammatic language) are known, which we can reproduce. We also note that (A.16) simplifies, not only in the case of SU(3) when (6.7) is used to obtain Tr D ( a D b D c D d D e )

5 = ---~-~d(abct~de),

(A.18)

but also for SU(4), after use of (6.6), giving Tr D(aDbDcDdDe) = -i~d(abc6de). 5

(A.19)

Note that (A.18) and (A.19) may be obtained by contracting the expression Tr D(aDbDcDdDe) = A"d(abc6de)

(A.20)

with 6ab; the resulting equation gives A" = -2q5 in the su(3) case and A" = ~ for su(4).

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685

Finally, we note that, once (A.16) has been evaluated and further five-fold traces obtained from it by elementary procedures, we have all we need to establish the result:

TrD(aDbDcDdDeDf)=4

~ 3 4

6(ab~cd~ef)

3 3n2--64~ -'}" 4

~

+

n2 -- 192d Xd ~'d 16~--7 (ab cd" e f ) Z d x y z

x 5n2 + 4 8 (abdcd def)x + -4~2 d(abcddef).

(A.21)

This involves expansion of the left side in terms of a basis [ 25,31 ], of totally symmetric sixth-rank tensors. Various contractions give four linear equations for the coefficients involved. Regarding the expansion (A.21) we remark that the tensor d{,,bj dJkc d~t,l dtef) differs from the tensor of the second term by 2In times the difference of the tensors of the fourth minus the first term. Some further comments are now in order. Many of the procedures followed above have been guided by graphical ideas such as described in [34]. These simplify complicated expressions involving d and f tensors by reference to closed loops in their graphical representations. These loops correspond to traces and we set about the simplification of three line loops with the aid of (A.2). Then we learn how to handle in turn all loops of four and five internal lines. The calculations associated with (A.21) are organised by looking at contractions diagrammatically, preferring to simplify at all times by identifying loops of as few lines as possible. It should be noted that many of the identities of this appendix have been evaluated by different means including the use of MAPLE. Also, we emphasise that a large number of checks of our results were made by passing to subcases for which identities were given in [ 33 ]. We turn next to the use of the identities presented in this appendix in the derivation of results quoted in the body of the paper. Consider, e.g., (6.14)

a"/(4) tplP2P3P4 = "'~(PlP2P3P4) "~ B~(plP2~P3P 4)"

(A.22)

Contractions with 6ram and dp~p2q on the right are easy to do, as is the contraction of the left side with 6pw 2. The latter gives zero by direct calculation as required by Lemma 3.2. The contraction tplp2p3pndplp2q is not governed by any general argument. To compute it, we set out from (see (3.15)) e, f a b o- f c d oJf e f r , tpqrg = 12(,,7)ao . . . . . . 3"g

(A.23)

and the cocycles defined using the d-family

0(7) abcdefg

_ o~5) cz A -- ~ t l abcdd efl ~tzg,

/2~5) ¢ n tabcdJ abs = "~f ulca dtlus,

(A.24) ( A.25 )

[ (7.5), cf. (7.2) ] for SU(n). Patient evaluation of the terms involved can be completed with the aid of four-fold traces evaluated earlier in this appendix and not (it seems) without them. It is clear that the occurrence of f-tensors in the cocycle definitions is what requires us to know how to treat traces involving F as well as D matrices.

J.A. de Azcdrraga et al./Nuclear Physics B 510 [PM] (1998) 657-687

686

Eq. ( 6 . 1 4 ) e m e r g e s after typical and reassuring cancellations. To obtain ( 6 . 1 5 ) with ,~(n) left u n d e t e r m i n e d , it is necessary only to c o m p u t e the ratio o f the two scalars o c c u r r i n g in the e x p a n s i o n with respect to the basis [25,31] o f V(5~ o f the left side o f ( 6 . 1 5 ) . T h i s requires o n l y contraction with

~PlP2 w h i c h

is zero.

To p r o v e the result for K<5)(n) in ( 6 . 2 3 ) , we set out f r o m ( 6 . 1 5 ) . The only hard part i n v o l v e s d(5) A(5) = d(5) d(5) (5) (abcde)~(abcde) (abcde) abcde= d(abcde)dabxdxcydyde'

(A.26)

and at worst two equal terms o f the fifteen i n v o l v e d lead to a contracted four-fold trace o f D m a t r i c e s k n o w n f r o m [ 3 3 ] . T h e result is d(5) .4(5) ( n 2 - 4) (n 2 - 1) (5n4 _ 96n2 + 4 8 0 ) (abcde)~abcde = 15n 3

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