- Email: [email protected]
On a new class of gauge transformations
Volume 144B, number 3,4
PHYSICS LETTERS
30 August 1984
ON A NEW CLASS OF GAUGE TRANSFORMATIONS Laurent BAULIEU Laboratoire de Physique Theorique et Hautes Energies Paris, Universit~ Paris VI, 4 place Jussieu, 75230 Paris, France
and Jean THIERRY-MIEG Groupe d'Astrophysique Relativiste, Observatoire, 92190 Meudon, France Received 14 October 1983
We construct a non-abelian symmetry involving a vector Yang-Mills field and a charged non-abelian skew-tensor field of arbitrary rank. It unifies the usual Yang-Mills symmetry with the Kalb-Ramond skew-tensor symmetry and does not commute with the Lorentz symmetry. In order to determine the gauge transformations we require the closure and integrabflity of the gauge algebra, and the existence of covariant field strengths. We also build an invariant classical lagrangian and display the full set of BRS equations associated with the gauge transformations.
1. Introduction. In this letter we examine the problem of building gauge transformations involving a nonabelian vector fieldAg and a charged skew tensor i • • fieldBD1 .-- up l 1. Our construction rehes on the requirements that (i) the gauge algebra is closed and satisfies a Jacobi identity and (ii) there exist field strengths linear in the derivatives of fields and transforming tensorially. We shall see that the fulfillment of requirement (ii) implies the introduction of other fields than A and B, and that the final gauge transformations do not commute with the Lorentz algebra. Our construction leads to a non-abelian extension of K a l b - R a m o n d symmetry [1 ] and we recover results obtained in collaboration with Yuval Ne'eman in the context of internal super symmetry [2]. We call this new class of symmetry: exterior gauge symmetry. In section 2, we construct the gauge'algebra. In section 3, we analyse an invariant classical action and in section 4 we give the BRS quantum extension of the classical gauge algebra. We use the very convenient notations of exterior calculus. The 1-form A =A~ dxUTa, valued in the Lie algebra ~, and the p-form Bp = B[u 1 u ] dxUl ^ "'" ^ dx#PRi valued in an arbitrary representation R of ~, denote respectively the vector Yang-Mills field and
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Divison)
the non-abelian generalization of the K a l b - R a m o n d skew-tensor field of rank p. Here Ta and R i are respectively basis of the Lie algebra ~ and of the representation R of ~ (with matrix elements R/~-). The symbol ^ stands for the wedge product, the exterior differential is d = dxUa u and the exterior covariant derivative is D = d + AaTa. Throughout the paper we shall need the following identities: d2 = 0 ,
DD=F =dA+AA,
DF=0.
(1.1)
In these notations, since all space-time indices are suppressed, the dimension of space-time is arbitrary but is supposed to be at least as large as p. 2. The gauge algebra. Following Yang and Mills, we parametrize the gauge transformations of the 1-form A as
6cA a = -(De) a - -de a - fgcA b e c ,
(2.1)
where fgc denote the structure constants and e a a Lie algebra valued scalar parameter. In a similar way, we parametrize the gauge transformation of the p-form B~ as
+
(2.2) 221
Volume 144B, number 3,4
PHYSICS LETTERS
where ep denotes anR-valued (p - 1) form corresponding to a skew-tensor infinitesimal parameter of rank p - 1, 1 ei[Ul ..... Up] R i dxU~ ^ ... A dxUP . ep = ~..
(2.3)
The first term in the rhs of eq. (2.2) generalizes the Kalb-Ramond abelian gauge transformation, the second term insures Yang-Mills covariance and the third term X~ stands for an unknownR-valued p-form function of the fields A and Bp.and possibly of other fields. Our aim is to determine X~ by requiring the closure and integrabflity (Jacobi identity) of the gauge algebra and the existence of a field strength. It is known since Cartan [3] that the problem of building a closed algebra with a Jacobi identity (i.e. a Lie algebra) is equivalent to that of building what is known in field theory as the corresponding nilpotent BRS algebra (i.e. a Cartan integrable system). One has just has to replace all infinitesimal gauge parameters by ghost fields with the same quantum numbers, but opposite statistics, and to define a nilpotent graded differential operator s with ghost number one acting on all fields. For convenience, we shall assume that s and d anticommute. The method is fully general [4]. This construction can be simply illustrated in the case of the ordinary Yang-Mills theory. Let us substitute in eq. (2.1) an anticommuting scalar ghost c a in place of e a and define the generator s. Since the space is graded by ghost number and Lorentz exterior degree, dimensionality requires
30 August 1984
method for obtaining the gauge transformation o f B p . We introduce an R-valued ghost (p - 1)-form ~p-1 in place of e p - l : ~p-1
_
(p-
1
1)!
~i
[#1 ..... U p - l ]
dx ul A
"'"
(2.6)
^ dxuP-lRi .
Here the p - 1 tensor ~ _ 1 carries Fermi statistics. We also introduce a ghost function Yfi to replace Xfi. Consequently, we rewrite (2.2) as sBi~ = - ( D ~ p _ X ) i - Ria.icaBjp + Y ~ ,
(2.7)
and we generalize (2.5) into S~fi_ 1 = - R a ij c a ~j- 1 +Z/~_ i 1•
(2.8)
The p-form Y~ and (p - 1) form Zfi-1 are as yet unknown but we will determine them by requiring the nilpotency relations s2Bp = 0, S2~p_l = 0 and the existence of a field strength transforming tensorially. We start from the requirement s2Bp = 0. By using the usual eqs. (2.4), (2.5) (with f ' = f " = f which ensures s2A = s2c = 0) and the Jacobi identity a i_ i k i k f~cgaj - RbkRc.i - RckRb]
(2.9)
we find from eqs. (2.7), (2.8) s2B~ = sY~ + Riajcay]p + (OZp-1) i.
(2.10)
The simplest solution which ensures s2B~ = 0 is
Z~_a=0 ,
sY~=-R~jcay~.
(2.11)
sA a = - d e a - f ~ c A b c c ,
(2.4)
From these equations the following relations are automatically satisfied
SCa = - - ~1J ,,,,,a_b~c b c V t. ,
(2.5)
s2y~ = 0 ,
where f ' and f " are yet unknown constant matrices. It is immediately seen that s2c ='0 implies on f " the Jacobi identity and that s2A = 0 implies that f ' = f " . Therefore we have verified that the nilpotency condition s2 = 0 determines the unknown matficesf' and f " in eqs. (2.4), (2.5) as Lie algebra generators f(since they satisfy the Jacobi identity) and the usual gauge transformation (2.1) is recovered from eq. (2.4). After the identification f = f ' = f " , eqs. (2.4), (2.5) are nothing else than the usual BRS equations (without the anti-ghost dependence). They will be used in what follows as the general solution to the condition s2 = 0 when applied toA and C. We now generalize the 222
S2~_1 = O.
(2.12)
To determine Y~ further, we use the third requirement announced in the introduction, namely, the existence of a field strength linear in the derivatives of fields and transforming tensorially under s. Of course, the Yang-Mills field strength 1 ,,.a ~AbAc F a = dA a + $Jbc -'1
(2.13)
satisfies the requirement: (2.14)
sFa = - f ~ c cbFC ,
but the natural choice Ghl=(DBp)i-
dB~ * TiajAaB j
(2.15)
Volume 144B, number 3,4
PHYSICS LETTERS
30 August 1984
does not since
the field strengths transform tensorially
s (DBp) i = - R i / c a ( o B p ) f + ( D O , p _ 1) i
sG~+ 1 = _Ra] aGp+l
-(DYp)
(2.16)
i .
The simplest way out is to introduce an auxiliary R-valued (p - 1)-form T p - 1 with Bose statistics. Parametrizing its s transform as sTip_l=-Ria]caTfp_ 1 + A ~ _ I ,
(2.17)
8e, ep_l Aa = - D e a '
s(DS. + ODTp_I) e =
8e, ep_l B ip = - D e p
+ [DD(~p_ 1 + Ap_l)] i - ( D Y p ) i .
(2.18)
It then appears that Gp+ 1 = D B p + DDTp_I transforms tensorially if Y p = 0 and Ae_ 1 = - ~ p - 1 . Moreover the latter equation implies s Z T p _ l = O. Summarizing all our equations we have SBpi -- - ( D ~ p _ 1) i
--
Ri ..ajt.~aoj L,p ,
sZfi - 1 = - R a jiC a T ~j - I sA a = _ ( D c ) a ,
--
i
~p-1
i
i a ~Jp-1, j
S~p_ 1 = - R a j c
~-~ a -- - - ~ 1J b.,'a c ¢~b~c ~. .
(2.19a)
-
1- R i ' ~ a Ru p/ , *'al~
i i a j i 6e, ep_l T ~ - i = - R a / e Tip_ 1 - eb_l
[Se, Cp_l, ~C',E~_ 1 ] = ~C",e~_ 1 ,
where , a - ~a
--f~C6
b 'c e
'! _ e,a@_l) e ,,i p _ l - _R a / (i e a el~_l
(2.22)
and since s 2 = 0, they satisfy the Jacobi identity and are integrable into a Lie group
Besides the field strengths are G~+ 1 = (DBp) i + ( D D T p _ I ) i ,
[6e, cp, [Se',eb, 8e",e~]] + cycl. penn. = 0 . F a = dA a + (AA) a .
(2.21)
These transformations are closed by construction. Indeed one can check
e
,
(2.20)
The boson field Tfi_ 1 has been introduced in order to enforce the tensoriality of field strengths. It is reminescent of a Higgs field by its transformation law. We can now track back to a more familiar expression of gauge transformations. In terms of the infinitesimal gauge parameters e and ep-1 of eqs. (2.1) and (2.2) we have
where Ap_ 1 is a new R-valued unknown (p - 1)-form, one gets
+ DDTp-I)J
sffa= _ f ~ c c b F C "
(2.19b)
By construction s is nilpotent on all fields *~ and ,1 The system o f equations (2.19a) is the minimal one satisfying the requirements (i) and (ii). However, when solving s2Bp = 0, s2 ~p_ 1 = 0 and the tensoriality requirement for the field strength, less stringent solutions than lip = 0 could have been used. Indeed, by introducing another form field 0p-2 (corresponding to an anti-commuting Rvalued skew-tensor o f rank (p - 2)), the equation s2Bp = 0 is still satisfied for Z p _ 1 : 0 and Yp = D D 0 p _ 2 while sop-2 = C0p-2. If furthermore A p _ 1 = ~p-1 + DOp-2, both requirements S2~p_ 1 = 0 and sGp+ 1 = -cGp+ 1 are still satisfied. The Op_ 2 field is however completely spurious classically, since it can be completely eliminated o f all equations, without spoiling requirements (i) and (ii), by a redefinition o f ~p_ 1 -~ ~p-1 + DOp-2 which corresponds to a mere redef'mition o f infinitesimal gauge parameters At the quantum level, however, Op_ 2 becomes an auxiliary ghost, and is necessary in order to define a tensorial field strength associated to the ghost ~p_ 1-
(2.23)
The transformations (2.21) do not commute with the Lorentz algebra since the @ - 1 are in fact Lorentz skew tensors. The full symmetry group of the model is the semi direct product (G * L) × Rp-1, where G denotes the underlying Lie group, L the Lorentz group and R their shaded representation ,2. If, for example,
~2 The system of transformations (2.21) is fully self consistent, and could be used in order to generalize the K a l b Ramond point o f view in the non-abelian case by interpreting the Bp form as the connection defining the parallel transport o f a "non-abelian string" o f dimension p - 1. In order to have a possible insight of the geometrical nature of field strength Gp+l we have investigated whether the Chern theorem could be extended for exterior products of G and F. We have checked that Tr(G A G) = d Tr(B A dB) and Tr(G A F) = d Tr(B A F). However, in general, Tr(G A ... G) is not a pure exterior derivative when there are more than two G's in the trace.
223
Volume 144B, number 3,4
PHYSICS LETTERS
p = 2, the commutators with an infinitesimal Lorentz transformation of parameter Auv read:
[~ Auv, 6 e, e[po]] = guv6e, eU -- gup6 e, ev .
(2.24)
Since the field strengths F and G, of eqs. (2.19b) transform tensorially, we can immediately write a gauge invariant lagrangian ,3: £ = _!tpa ~2
4~" uv.,
l (G~v...) 2 2(p + 1)!
(2.25)
This lagrangian generalizes the Yang-Mills lagrangian. It implies constraints which complicate however its study and are the subject of the next section.
3. The classical theory constraints. The classical theory described by the lagrangian (2.25) presents a striking feature. To plagiate Wheeler, it is gauge invariant without gauge invariance. Indeed, if in eq. (2.25) we vary the independent fields Bp and Tp_ 1, we obtain the Euler-Lagrange equations 62/6Bu,,.." = DpGiuvp...] = 0 ,
(3. la)
6£/6 T u... = Du D o G [uvp... ] = 0 .
(3.1 b)
Eqs. (3.1) are invariant under the gauge transformations
5Bp = O e p - i ,
~ r p - i = ep-1.
(3.2)
However, if we perform the field redefinition
B~ = Bp + DTp_I ,
(3.3)
we see that the lagrangian (2.25) is equivalent to the lagrangian £, = _ l4," e n auv,~2
2(p +1 1)! (D[uB'P'"])2 "
(3.4)
Here, apart from the Usual Yang-Mills gauge transformations, no gauge invariance remains and we have on-
,3 This lagrangian was constructed in reL [2] in order to gauge an internal supersymmetry. Consider a system of forms of degree 1 to 4 over space-time. They form an exterior superalgebra. The A may be used to gauge a Lie algebra and the B, dynamically equivalent to scalar fields, to gauge supersymmetries. The higher forms earry no degree of freedom in 4 dimensions. In the ease of the superalgebra SU(2/1), o n e r e e o v e r s in this way the spectrum of the Weinberg-Salam model 224
30 August 1984
ly one equation of motion:
6£'/6B'uv... = Dp DuB[vp... 1,
(3.5a)
but we know from Cartan [5] that, to solve this eq. (3.5), we must solve at the same time its exterior differential:
Dtu6£'/6B'uv...l = 0 .
(3.5b)
Substituting back B + DT for B', we recover eqs. (3.2). The lesson is the following. Eq. (3.3) can be regarded in itself as a gauge transformation with gauge parameter Tp_ 1. Under this transformation, the lagrangian (3.4) is not invariant but is transformed into the lagrangian (2.25) which involves one more independent field. However, the two lagrangians in fact have the same equations of motion and are therefore classically equivalent. In other words, our lagrangian is not invariant in form, but is invariant in content. This argument was used in ref. [2] to derive the classical lagrangian (2.25) and the BRS algebra (4.8). At the classical level, the field Tp-1 seems spurious. It is not an auxiliary field in the sense o f supergravity, i.e. a field with an algebraic equation of motion. Indeed, Tp-1 has no equation at all since neither eq. (3. la) nor eq. (3.1b) specifies Tp-1 independently of DBp.... The role played by the field Tp-1 is to render eq. (3.1 b) explicit for the quantization of exterior gauge fields. An analogous phenomenon occurs in the context of the Freedman-Townsend model [6,7]. It is also clear from the present analysis that no further auxiliary fields are needed since the system of equation of motions is now closed, as can be checked from the relation D ^ D = F.
4. The associated quantum symmetry. The A u and B [uv...] propagators are purely transverse in lagrangian (2.25), and practical computations would require the addition of a gauge fixing lagrangian including ghost interactions and invariant under BRS symmetry. The Faddeev-Popov prescription is known to be inadequate for obtaining the gauge fixing lagrangian when skew-tensor gauge fields are involved. On the other hand, one can apply the principle of BRS symmetry which is equivalent to the Faddeev-Popov prescription for Yang-Mills theories [7], and has already solved the cases of Freedman-Townsend [6,7] and Chapline-Manton theories where non-trivial skewtensor interactions occur. The strategy is in fact the
Volume 144B, number 3,4
PHYSICS LETTERS
reverse of the usual one. It consists in determining successively: (i) the whole set of ghost fields which ensure (formally) the unitarity, (ii) the BRS anti-BRS equations defined by the action of their generators s and~ on all ghost and classical fields and (iii) only at the end the quantum lagrangian as a gauge fixed but BRS invariant lagrangian. The BRS equations are characterized by the fact that the action of generators s and~ on classical fields must be identical to infinitesimal gauge transformations (2.21) when the gauge parameters are replaced by ghost fields, and that s andg are fully nilpotent on all fields, s2 = s~+~s =~2 = 0. The set of ghosts associated with Bp and Tp-1 has been found in the context of Freedman-Townsend lagrangian [7]. One has the following pyramidal structure which enforces the quartet mechanism
30 August 1984
are expected to propagate and cancel the undesired contributions of unphysical modes in Bp [7]. The Oq ghosts are auxiliary and are necessary for the consistency of BRS equations as recalled later on [7]. Starting from equations ~ p = D~I_I + DD01_2, sTp-1 = ~1-1 + cTp-2 + D01-2 associated with classical gauge transformations (see footnote 1) one could get step by step all BRS anti-BRS equations by requiring the full nilpotency condition s2 = s~ +~s =~2 = 0 on all fields. Such a method was used in ref. [8] in order to quantize Chapline-Manton gauge symmetry. There is, however, a more direct method consisting in enlarging the space-time by associating to each point x u a pair of Grassman coordinates (y, 37). Then all fields of pyramids (4.1), (4.2) can be identified as the components of the following exterior forms, respectively of rank p, p - 1 and 1.
Bp
/\
Bp(X, y , y ) -=Bp -t-
~1__1
/
~pl 1
/
\
D dx #1 A ... A dx#P-q - R p>~q+R~ 1
^ (dy) q ^ (dy) R ~q~-.Rlap_q_R (X, y, if), \
/
Tp-I(X,y,..V) = Tp-1 +
/ /
/-,,,, ~ld-4 . . . . . .
-2
~4-p
A
A dxttP-q -R A (dy) q A (dy) g oq;R.,p_q_R(X,y,y),
~2-p,
.4(x,y,.7) =A + Ay dy + Ay dP
Tp-I
/ \ o1_ o;_1 /X/\ /
02_3 /
0
00_3
-3 .......
~ dx u' A ... p- l>q+R>l
(note that the dy and dy forms commute among themselves due to the grassmanian properties of y, y). One has in fact
c(x) =Ay(x, O, O) dy,
0p_3-2 \ \ 0 3-;
(4.3)
-d(x)= A ~ x , O, O) dy ,
~/q--t_R (X) = ~#1 q - R.." # p - q - R (X' O, O)
dxla,1 A
...
A dxUP-q -R ^ (dy) q ^ (dy) R , (4.1) 0 q - R R(X) = o q - R p p _ q _ R ( X , O, O) dx ~1 A P-q1." "'"
In this pyramid of fields the upper and lower index label respectively the ghost number and Lorentz rank. Besides c and ? are the usual ghost and anti-ghost of A
(4.4)
Besides one defines the exterior differential operator
A /
A dxtaP-q - R A (dy)q A (dfi)R .
\
(4.2)
d=d+s+~,
d = dxU~/~x u,
s=
-g =
c According to the quartet mechanism the ~q ghosts
dy
O/By,
dy
0/By,
(4.5) 225
Volume 144B, number 3,4
PHYSICS LETTERS
and the field strengths which generalize those in eq. (2.19b) Gp+I =~IBp+A~Bp+/~Tp-1 • (4.6)
F= dA +AA,
Then, in order to obtain the BRS equations, one simply imposes the horizontality condition F.v a x . A dx ,
s ~ +-gn = - 0 - c m - c n ,
~
1
dx zl A
where we have used the notation ~ = ~ , ~ = ~i-1, m = ~2, n = ~ 0 , ~ = ~ 2 0 = 0 ] , 0 =01 . Finally a completely gauge fixed, BRS and antiBRS invariant lagrangian with ghost number zero is
(4.9) , h
(4.7a)
together with its integrability condition Bp + [)7"p-l = Bp + DTp-1
1
r/~l-.* lip])
X dx ul A ,.. A dx~p.
(4.7b)
One can check that the last equation ensures that DGp+I = DGp+I, and that the whole system of equations (4.7) is closed. The consistency eq. (4.7b) shows the necessity of ghosts Oq : if they were not taken into account (i.e. Oq --- 0), eq. (4.7b) would imply that all ~q ghosts vanish. Once expanded on the basis of forms dx u A ... ^ dx v ^ (dy) R A (d~F)q, eqs. (4.7) yield all equations defining the action of s and-~ on all fields, and the nilpotency relation s2 = s~ +-gs =~2 = 0 is automatically satisfied. In order to be fully explicit one can consider the case p = 4. Then one gets from eqs. (4.7): sB = - D ~ - DD0 - c B ,
s~=-Dm-c~, sm = - c m
,
gT=--~-DO-YT,
se +
gO = ~ - - 6 0 , -sA = - D - g ,
[c,
=
,
s--e = _ 1
[-e,
=
,
= -I-e, el,
s~ +"g~ = - D n - c~ - Y~, 226
+
+
...),
with the notation [xy[ ~ X[ul " ' " u~! 3'[~1"" **ql. t/J The lagrangian (4.9) is indeed clearly BRS and antiBRS invariant from the basic properties of s andS. It is gauge fixed as can be shown by a straightforward computation. It should be a good starting point for perturbation theory, but contains, however, constraints which makes its study difficult, and which we have not been able to solve yet as we did in the F r e e d m a n Townsend lagrangian case [7] ,4 5. C o n c l u s i o n . We have exhibited a new class of gauge transformations involving a non-abelian vector field and a charged anti-symmetric tensor field. These transformations do not commute with the Lorentz transformations, but all equations are explicitly Lorentz covafiant. The gauge algebra is closed and the BRS algebra is integrable by construction. We have analysed the classical lagrangian and its BRS extension, but the presence of constraints complicates the particle interpretation of the model. These symmetries may play a role in string theory [ 1], weak interactions [2], and solid state physics.
-sB = - D ~ - DD0 - ~ B ,
sm = -cm ,
sO=m-cO,
sc =
+... +
~ =D~-~,
sT=-~-DO-cT,
sA=-De,
1
2 ( p + 1) 2 IGI2+ s-s([A [2 + IBI2
"'"
^ dxUP+l ,
-= p! (B[ul ... Upl + D[ul
(4.8 cont'd)
~ m + sn = - 0 - "Jm - cn ,
£Q = - ¼ I F [ 2 Gp+I = Gp+I = (lo + 1)! G I u l "'" Up+~]
30 August 1984
(4.8)
44 In the case of Chapline-Manton symmetry [9], the full set of associated BRS equations has also been derived by analogous considerations in the enlarged {x ta, y, y ) space [14]. The Chapline-Manton symmetry involves couplings between a non-abelian Yang-Mills field and an abelian skew-tensor field. These couplings are however non-trivial, because of the occurrence of the dimensionfull gravity coupling constant. In our analysis leading to gauge transformations (2.21) we have not considered the possibility of a dirnensionfull coupling constant, which would lead a priori to new terms breaking the homogeneity of our formula.
Volume 144B, number 3,4
PHYSICS LETTERS
References [1 ] V.I. Ogievetskii and I.V. Polubarinov, Sov. J. Nucl. Phys. 4 (1967) 156; M. Kalb and P. Ramond, Phys. Rev. D9 (1974) 2273; E. Cremmer and J. Scherk, Nucl. Phys. B72 (1974) 117. [2] Y. Ne'eman and J. Thierry-Mieg, Proc. Nat. Acad. Sci. US 79 (1982) 7068; Phys. Lett. 108B (1982) 399. [3] E. Cartan, Compt. Rend. Acad. Sci. (Paris) 182 (1926) 956.
30 August 1984
[4] J. Thierly-Mieg, J. Math. Phys. 21 (1980) 2834; Nuovo Cimento B108 (1980) 396; L. Alvarez-Gaume and L. Baulieu, Nucl. Phys. B212 (1983) 255; L. Baulieu and J. Thierly-Mieg, Nucl. Phys. B197 (1982) 477. [5] E. Caftan, Les syst~mes dffferentiels ext6rieurs (Hermann, Paris, 1971). [6] D.Z. Freedman and P.K. Townsend, Nuel. Phys. B177 (1981) 477. [7] J. Thierry-Mieg and L. Baulieu, Nucl. Phys. B228 (1983) 259. [8] L. Baulieu, Phys. Lett. B126 (1983) 455.
227
PHYSICS LETTERS
30 August 1984
ON A NEW CLASS OF GAUGE TRANSFORMATIONS Laurent BAULIEU Laboratoire de Physique Theorique et Hautes Energies Paris, Universit~ Paris VI, 4 place Jussieu, 75230 Paris, France
and Jean THIERRY-MIEG Groupe d'Astrophysique Relativiste, Observatoire, 92190 Meudon, France Received 14 October 1983
We construct a non-abelian symmetry involving a vector Yang-Mills field and a charged non-abelian skew-tensor field of arbitrary rank. It unifies the usual Yang-Mills symmetry with the Kalb-Ramond skew-tensor symmetry and does not commute with the Lorentz symmetry. In order to determine the gauge transformations we require the closure and integrabflity of the gauge algebra, and the existence of covariant field strengths. We also build an invariant classical lagrangian and display the full set of BRS equations associated with the gauge transformations.
1. Introduction. In this letter we examine the problem of building gauge transformations involving a nonabelian vector fieldAg and a charged skew tensor i • • fieldBD1 .-- up l 1. Our construction rehes on the requirements that (i) the gauge algebra is closed and satisfies a Jacobi identity and (ii) there exist field strengths linear in the derivatives of fields and transforming tensorially. We shall see that the fulfillment of requirement (ii) implies the introduction of other fields than A and B, and that the final gauge transformations do not commute with the Lorentz algebra. Our construction leads to a non-abelian extension of K a l b - R a m o n d symmetry [1 ] and we recover results obtained in collaboration with Yuval Ne'eman in the context of internal super symmetry [2]. We call this new class of symmetry: exterior gauge symmetry. In section 2, we construct the gauge'algebra. In section 3, we analyse an invariant classical action and in section 4 we give the BRS quantum extension of the classical gauge algebra. We use the very convenient notations of exterior calculus. The 1-form A =A~ dxUTa, valued in the Lie algebra ~, and the p-form Bp = B[u 1 u ] dxUl ^ "'" ^ dx#PRi valued in an arbitrary representation R of ~, denote respectively the vector Yang-Mills field and
0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Divison)
the non-abelian generalization of the K a l b - R a m o n d skew-tensor field of rank p. Here Ta and R i are respectively basis of the Lie algebra ~ and of the representation R of ~ (with matrix elements R/~-). The symbol ^ stands for the wedge product, the exterior differential is d = dxUa u and the exterior covariant derivative is D = d + AaTa. Throughout the paper we shall need the following identities: d2 = 0 ,
DD=F =dA+AA,
DF=0.
(1.1)
In these notations, since all space-time indices are suppressed, the dimension of space-time is arbitrary but is supposed to be at least as large as p. 2. The gauge algebra. Following Yang and Mills, we parametrize the gauge transformations of the 1-form A as
6cA a = -(De) a - -de a - fgcA b e c ,
(2.1)
where fgc denote the structure constants and e a a Lie algebra valued scalar parameter. In a similar way, we parametrize the gauge transformation of the p-form B~ as
+
(2.2) 221
Volume 144B, number 3,4
PHYSICS LETTERS
where ep denotes anR-valued (p - 1) form corresponding to a skew-tensor infinitesimal parameter of rank p - 1, 1 ei[Ul ..... Up] R i dxU~ ^ ... A dxUP . ep = ~..
(2.3)
The first term in the rhs of eq. (2.2) generalizes the Kalb-Ramond abelian gauge transformation, the second term insures Yang-Mills covariance and the third term X~ stands for an unknownR-valued p-form function of the fields A and Bp.and possibly of other fields. Our aim is to determine X~ by requiring the closure and integrabflity (Jacobi identity) of the gauge algebra and the existence of a field strength. It is known since Cartan [3] that the problem of building a closed algebra with a Jacobi identity (i.e. a Lie algebra) is equivalent to that of building what is known in field theory as the corresponding nilpotent BRS algebra (i.e. a Cartan integrable system). One has just has to replace all infinitesimal gauge parameters by ghost fields with the same quantum numbers, but opposite statistics, and to define a nilpotent graded differential operator s with ghost number one acting on all fields. For convenience, we shall assume that s and d anticommute. The method is fully general [4]. This construction can be simply illustrated in the case of the ordinary Yang-Mills theory. Let us substitute in eq. (2.1) an anticommuting scalar ghost c a in place of e a and define the generator s. Since the space is graded by ghost number and Lorentz exterior degree, dimensionality requires
30 August 1984
method for obtaining the gauge transformation o f B p . We introduce an R-valued ghost (p - 1)-form ~p-1 in place of e p - l : ~p-1
_
(p-
1
1)!
~i
[#1 ..... U p - l ]
dx ul A
"'"
(2.6)
^ dxuP-lRi .
Here the p - 1 tensor ~ _ 1 carries Fermi statistics. We also introduce a ghost function Yfi to replace Xfi. Consequently, we rewrite (2.2) as sBi~ = - ( D ~ p _ X ) i - Ria.icaBjp + Y ~ ,
(2.7)
and we generalize (2.5) into S~fi_ 1 = - R a ij c a ~j- 1 +Z/~_ i 1•
(2.8)
The p-form Y~ and (p - 1) form Zfi-1 are as yet unknown but we will determine them by requiring the nilpotency relations s2Bp = 0, S2~p_l = 0 and the existence of a field strength transforming tensorially. We start from the requirement s2Bp = 0. By using the usual eqs. (2.4), (2.5) (with f ' = f " = f which ensures s2A = s2c = 0) and the Jacobi identity a i_ i k i k f~cgaj - RbkRc.i - RckRb]
(2.9)
we find from eqs. (2.7), (2.8) s2B~ = sY~ + Riajcay]p + (OZp-1) i.
(2.10)
The simplest solution which ensures s2B~ = 0 is
Z~_a=0 ,
sY~=-R~jcay~.
(2.11)
sA a = - d e a - f ~ c A b c c ,
(2.4)
From these equations the following relations are automatically satisfied
SCa = - - ~1J ,,,,,a_b~c b c V t. ,
(2.5)
s2y~ = 0 ,
where f ' and f " are yet unknown constant matrices. It is immediately seen that s2c ='0 implies on f " the Jacobi identity and that s2A = 0 implies that f ' = f " . Therefore we have verified that the nilpotency condition s2 = 0 determines the unknown matficesf' and f " in eqs. (2.4), (2.5) as Lie algebra generators f(since they satisfy the Jacobi identity) and the usual gauge transformation (2.1) is recovered from eq. (2.4). After the identification f = f ' = f " , eqs. (2.4), (2.5) are nothing else than the usual BRS equations (without the anti-ghost dependence). They will be used in what follows as the general solution to the condition s2 = 0 when applied toA and C. We now generalize the 222
S2~_1 = O.
(2.12)
To determine Y~ further, we use the third requirement announced in the introduction, namely, the existence of a field strength linear in the derivatives of fields and transforming tensorially under s. Of course, the Yang-Mills field strength 1 ,,.a ~AbAc F a = dA a + $Jbc -'1
(2.13)
satisfies the requirement: (2.14)
sFa = - f ~ c cbFC ,
but the natural choice Ghl=(DBp)i-
dB~ * TiajAaB j
(2.15)
Volume 144B, number 3,4
PHYSICS LETTERS
30 August 1984
does not since
the field strengths transform tensorially
s (DBp) i = - R i / c a ( o B p ) f + ( D O , p _ 1) i
sG~+ 1 = _Ra] aGp+l
-(DYp)
(2.16)
i .
The simplest way out is to introduce an auxiliary R-valued (p - 1)-form T p - 1 with Bose statistics. Parametrizing its s transform as sTip_l=-Ria]caTfp_ 1 + A ~ _ I ,
(2.17)
8e, ep_l Aa = - D e a '
s(DS. + ODTp_I) e =
8e, ep_l B ip = - D e p
+ [DD(~p_ 1 + Ap_l)] i - ( D Y p ) i .
(2.18)
It then appears that Gp+ 1 = D B p + DDTp_I transforms tensorially if Y p = 0 and Ae_ 1 = - ~ p - 1 . Moreover the latter equation implies s Z T p _ l = O. Summarizing all our equations we have SBpi -- - ( D ~ p _ 1) i
--
Ri ..ajt.~aoj L,p ,
sZfi - 1 = - R a jiC a T ~j - I sA a = _ ( D c ) a ,
--
i
~p-1
i
i a ~Jp-1, j
S~p_ 1 = - R a j c
~-~ a -- - - ~ 1J b.,'a c ¢~b~c ~. .
(2.19a)
-
1- R i ' ~ a Ru p/ , *'al~
i i a j i 6e, ep_l T ~ - i = - R a / e Tip_ 1 - eb_l
[Se, Cp_l, ~C',E~_ 1 ] = ~C",e~_ 1 ,
where , a - ~a
--f~C6
b 'c e
'! _ e,a@_l) e ,,i p _ l - _R a / (i e a el~_l
(2.22)
and since s 2 = 0, they satisfy the Jacobi identity and are integrable into a Lie group
Besides the field strengths are G~+ 1 = (DBp) i + ( D D T p _ I ) i ,
[6e, cp, [Se',eb, 8e",e~]] + cycl. penn. = 0 . F a = dA a + (AA) a .
(2.21)
These transformations are closed by construction. Indeed one can check
e
,
(2.20)
The boson field Tfi_ 1 has been introduced in order to enforce the tensoriality of field strengths. It is reminescent of a Higgs field by its transformation law. We can now track back to a more familiar expression of gauge transformations. In terms of the infinitesimal gauge parameters e and ep-1 of eqs. (2.1) and (2.2) we have
where Ap_ 1 is a new R-valued unknown (p - 1)-form, one gets
+ DDTp-I)J
sffa= _ f ~ c c b F C "
(2.19b)
By construction s is nilpotent on all fields *~ and ,1 The system o f equations (2.19a) is the minimal one satisfying the requirements (i) and (ii). However, when solving s2Bp = 0, s2 ~p_ 1 = 0 and the tensoriality requirement for the field strength, less stringent solutions than lip = 0 could have been used. Indeed, by introducing another form field 0p-2 (corresponding to an anti-commuting Rvalued skew-tensor o f rank (p - 2)), the equation s2Bp = 0 is still satisfied for Z p _ 1 : 0 and Yp = D D 0 p _ 2 while sop-2 = C0p-2. If furthermore A p _ 1 = ~p-1 + DOp-2, both requirements S2~p_ 1 = 0 and sGp+ 1 = -cGp+ 1 are still satisfied. The Op_ 2 field is however completely spurious classically, since it can be completely eliminated o f all equations, without spoiling requirements (i) and (ii), by a redefinition o f ~p_ 1 -~ ~p-1 + DOp-2 which corresponds to a mere redef'mition o f infinitesimal gauge parameters At the quantum level, however, Op_ 2 becomes an auxiliary ghost, and is necessary in order to define a tensorial field strength associated to the ghost ~p_ 1-
(2.23)
The transformations (2.21) do not commute with the Lorentz algebra since the @ - 1 are in fact Lorentz skew tensors. The full symmetry group of the model is the semi direct product (G * L) × Rp-1, where G denotes the underlying Lie group, L the Lorentz group and R their shaded representation ,2. If, for example,
~2 The system of transformations (2.21) is fully self consistent, and could be used in order to generalize the K a l b Ramond point o f view in the non-abelian case by interpreting the Bp form as the connection defining the parallel transport o f a "non-abelian string" o f dimension p - 1. In order to have a possible insight of the geometrical nature of field strength Gp+l we have investigated whether the Chern theorem could be extended for exterior products of G and F. We have checked that Tr(G A G) = d Tr(B A dB) and Tr(G A F) = d Tr(B A F). However, in general, Tr(G A ... G) is not a pure exterior derivative when there are more than two G's in the trace.
223
Volume 144B, number 3,4
PHYSICS LETTERS
p = 2, the commutators with an infinitesimal Lorentz transformation of parameter Auv read:
[~ Auv, 6 e, e[po]] = guv6e, eU -- gup6 e, ev .
(2.24)
Since the field strengths F and G, of eqs. (2.19b) transform tensorially, we can immediately write a gauge invariant lagrangian ,3: £ = _!tpa ~2
4~" uv.,
l (G~v...) 2 2(p + 1)!
(2.25)
This lagrangian generalizes the Yang-Mills lagrangian. It implies constraints which complicate however its study and are the subject of the next section.
3. The classical theory constraints. The classical theory described by the lagrangian (2.25) presents a striking feature. To plagiate Wheeler, it is gauge invariant without gauge invariance. Indeed, if in eq. (2.25) we vary the independent fields Bp and Tp_ 1, we obtain the Euler-Lagrange equations 62/6Bu,,.." = DpGiuvp...] = 0 ,
(3. la)
6£/6 T u... = Du D o G [uvp... ] = 0 .
(3.1 b)
Eqs. (3.1) are invariant under the gauge transformations
5Bp = O e p - i ,
~ r p - i = ep-1.
(3.2)
However, if we perform the field redefinition
B~ = Bp + DTp_I ,
(3.3)
we see that the lagrangian (2.25) is equivalent to the lagrangian £, = _ l4," e n auv,~2
2(p +1 1)! (D[uB'P'"])2 "
(3.4)
Here, apart from the Usual Yang-Mills gauge transformations, no gauge invariance remains and we have on-
,3 This lagrangian was constructed in reL [2] in order to gauge an internal supersymmetry. Consider a system of forms of degree 1 to 4 over space-time. They form an exterior superalgebra. The A may be used to gauge a Lie algebra and the B, dynamically equivalent to scalar fields, to gauge supersymmetries. The higher forms earry no degree of freedom in 4 dimensions. In the ease of the superalgebra SU(2/1), o n e r e e o v e r s in this way the spectrum of the Weinberg-Salam model 224
30 August 1984
ly one equation of motion:
6£'/6B'uv... = Dp DuB[vp... 1,
(3.5a)
but we know from Cartan [5] that, to solve this eq. (3.5), we must solve at the same time its exterior differential:
Dtu6£'/6B'uv...l = 0 .
(3.5b)
Substituting back B + DT for B', we recover eqs. (3.2). The lesson is the following. Eq. (3.3) can be regarded in itself as a gauge transformation with gauge parameter Tp_ 1. Under this transformation, the lagrangian (3.4) is not invariant but is transformed into the lagrangian (2.25) which involves one more independent field. However, the two lagrangians in fact have the same equations of motion and are therefore classically equivalent. In other words, our lagrangian is not invariant in form, but is invariant in content. This argument was used in ref. [2] to derive the classical lagrangian (2.25) and the BRS algebra (4.8). At the classical level, the field Tp-1 seems spurious. It is not an auxiliary field in the sense o f supergravity, i.e. a field with an algebraic equation of motion. Indeed, Tp-1 has no equation at all since neither eq. (3. la) nor eq. (3.1b) specifies Tp-1 independently of DBp.... The role played by the field Tp-1 is to render eq. (3.1 b) explicit for the quantization of exterior gauge fields. An analogous phenomenon occurs in the context of the Freedman-Townsend model [6,7]. It is also clear from the present analysis that no further auxiliary fields are needed since the system of equation of motions is now closed, as can be checked from the relation D ^ D = F.
4. The associated quantum symmetry. The A u and B [uv...] propagators are purely transverse in lagrangian (2.25), and practical computations would require the addition of a gauge fixing lagrangian including ghost interactions and invariant under BRS symmetry. The Faddeev-Popov prescription is known to be inadequate for obtaining the gauge fixing lagrangian when skew-tensor gauge fields are involved. On the other hand, one can apply the principle of BRS symmetry which is equivalent to the Faddeev-Popov prescription for Yang-Mills theories [7], and has already solved the cases of Freedman-Townsend [6,7] and Chapline-Manton theories where non-trivial skewtensor interactions occur. The strategy is in fact the
Volume 144B, number 3,4
PHYSICS LETTERS
reverse of the usual one. It consists in determining successively: (i) the whole set of ghost fields which ensure (formally) the unitarity, (ii) the BRS anti-BRS equations defined by the action of their generators s and~ on all ghost and classical fields and (iii) only at the end the quantum lagrangian as a gauge fixed but BRS invariant lagrangian. The BRS equations are characterized by the fact that the action of generators s and~ on classical fields must be identical to infinitesimal gauge transformations (2.21) when the gauge parameters are replaced by ghost fields, and that s andg are fully nilpotent on all fields, s2 = s~+~s =~2 = 0. The set of ghosts associated with Bp and Tp-1 has been found in the context of Freedman-Townsend lagrangian [7]. One has the following pyramidal structure which enforces the quartet mechanism
30 August 1984
are expected to propagate and cancel the undesired contributions of unphysical modes in Bp [7]. The Oq ghosts are auxiliary and are necessary for the consistency of BRS equations as recalled later on [7]. Starting from equations ~ p = D~I_I + DD01_2, sTp-1 = ~1-1 + cTp-2 + D01-2 associated with classical gauge transformations (see footnote 1) one could get step by step all BRS anti-BRS equations by requiring the full nilpotency condition s2 = s~ +~s =~2 = 0 on all fields. Such a method was used in ref. [8] in order to quantize Chapline-Manton gauge symmetry. There is, however, a more direct method consisting in enlarging the space-time by associating to each point x u a pair of Grassman coordinates (y, 37). Then all fields of pyramids (4.1), (4.2) can be identified as the components of the following exterior forms, respectively of rank p, p - 1 and 1.
Bp
/\
Bp(X, y , y ) -=Bp -t-
~1__1
/
~pl 1
/
\
D dx #1 A ... A dx#P-q - R p>~q+R~ 1
^ (dy) q ^ (dy) R ~q~-.Rlap_q_R (X, y, if), \
/
Tp-I(X,y,..V) = Tp-1 +
/ /
/-,,,, ~ld-4 . . . . . .
-2
~4-p
A
A dxttP-q -R A (dy) q A (dy) g oq;R.,p_q_R(X,y,y),
~2-p,
.4(x,y,.7) =A + Ay dy + Ay dP
Tp-I
/ \ o1_ o;_1 /X/\ /
02_3 /
0
00_3
-3 .......
~ dx u' A ... p- l>q+R>l
(note that the dy and dy forms commute among themselves due to the grassmanian properties of y, y). One has in fact
c(x) =Ay(x, O, O) dy,
0p_3-2 \ \ 0 3-;
(4.3)
-d(x)= A ~ x , O, O) dy ,
~/q--t_R (X) = ~#1 q - R.." # p - q - R (X' O, O)
dxla,1 A
...
A dxUP-q -R ^ (dy) q ^ (dy) R , (4.1) 0 q - R R(X) = o q - R p p _ q _ R ( X , O, O) dx ~1 A P-q1." "'"
In this pyramid of fields the upper and lower index label respectively the ghost number and Lorentz rank. Besides c and ? are the usual ghost and anti-ghost of A
(4.4)
Besides one defines the exterior differential operator
A /
A dxtaP-q - R A (dy)q A (dfi)R .
\
(4.2)
d=d+s+~,
d = dxU~/~x u,
s=
-g =
c According to the quartet mechanism the ~q ghosts
dy
O/By,
dy
0/By,
(4.5) 225
Volume 144B, number 3,4
PHYSICS LETTERS
and the field strengths which generalize those in eq. (2.19b) Gp+I =~IBp+A~Bp+/~Tp-1 • (4.6)
F= dA +AA,
Then, in order to obtain the BRS equations, one simply imposes the horizontality condition F.v a x . A dx ,
s ~ +-gn = - 0 - c m - c n ,
~
1
dx zl A
where we have used the notation ~ = ~ , ~ = ~i-1, m = ~2, n = ~ 0 , ~ = ~ 2 0 = 0 ] , 0 =01 . Finally a completely gauge fixed, BRS and antiBRS invariant lagrangian with ghost number zero is
(4.9) , h
(4.7a)
together with its integrability condition Bp + [)7"p-l = Bp + DTp-1
1
r/~l-.* lip])
X dx ul A ,.. A dx~p.
(4.7b)
One can check that the last equation ensures that DGp+I = DGp+I, and that the whole system of equations (4.7) is closed. The consistency eq. (4.7b) shows the necessity of ghosts Oq : if they were not taken into account (i.e. Oq --- 0), eq. (4.7b) would imply that all ~q ghosts vanish. Once expanded on the basis of forms dx u A ... ^ dx v ^ (dy) R A (d~F)q, eqs. (4.7) yield all equations defining the action of s and-~ on all fields, and the nilpotency relation s2 = s~ +-gs =~2 = 0 is automatically satisfied. In order to be fully explicit one can consider the case p = 4. Then one gets from eqs. (4.7): sB = - D ~ - DD0 - c B ,
s~=-Dm-c~, sm = - c m
,
gT=--~-DO-YT,
se +
gO = ~ - - 6 0 , -sA = - D - g ,
[c,
=
,
s--e = _ 1
[-e,
=
,
= -I-e, el,
s~ +"g~ = - D n - c~ - Y~, 226
+
+
...),
with the notation [xy[ ~ X[ul " ' " u~! 3'[~1"" **ql. t/J The lagrangian (4.9) is indeed clearly BRS and antiBRS invariant from the basic properties of s andS. It is gauge fixed as can be shown by a straightforward computation. It should be a good starting point for perturbation theory, but contains, however, constraints which makes its study difficult, and which we have not been able to solve yet as we did in the F r e e d m a n Townsend lagrangian case [7] ,4 5. C o n c l u s i o n . We have exhibited a new class of gauge transformations involving a non-abelian vector field and a charged anti-symmetric tensor field. These transformations do not commute with the Lorentz transformations, but all equations are explicitly Lorentz covafiant. The gauge algebra is closed and the BRS algebra is integrable by construction. We have analysed the classical lagrangian and its BRS extension, but the presence of constraints complicates the particle interpretation of the model. These symmetries may play a role in string theory [ 1], weak interactions [2], and solid state physics.
-sB = - D ~ - DD0 - ~ B ,
sm = -cm ,
sO=m-cO,
sc =
+... +
~ =D~-~,
sT=-~-DO-cT,
sA=-De,
1
2 ( p + 1) 2 IGI2+ s-s([A [2 + IBI2
"'"
^ dxUP+l ,
-= p! (B[ul ... Upl + D[ul
(4.8 cont'd)
~ m + sn = - 0 - "Jm - cn ,
£Q = - ¼ I F [ 2 Gp+I = Gp+I = (lo + 1)! G I u l "'" Up+~]
30 August 1984
(4.8)
44 In the case of Chapline-Manton symmetry [9], the full set of associated BRS equations has also been derived by analogous considerations in the enlarged {x ta, y, y ) space [14]. The Chapline-Manton symmetry involves couplings between a non-abelian Yang-Mills field and an abelian skew-tensor field. These couplings are however non-trivial, because of the occurrence of the dimensionfull gravity coupling constant. In our analysis leading to gauge transformations (2.21) we have not considered the possibility of a dirnensionfull coupling constant, which would lead a priori to new terms breaking the homogeneity of our formula.
Volume 144B, number 3,4
PHYSICS LETTERS
References [1 ] V.I. Ogievetskii and I.V. Polubarinov, Sov. J. Nucl. Phys. 4 (1967) 156; M. Kalb and P. Ramond, Phys. Rev. D9 (1974) 2273; E. Cremmer and J. Scherk, Nucl. Phys. B72 (1974) 117. [2] Y. Ne'eman and J. Thierry-Mieg, Proc. Nat. Acad. Sci. US 79 (1982) 7068; Phys. Lett. 108B (1982) 399. [3] E. Cartan, Compt. Rend. Acad. Sci. (Paris) 182 (1926) 956.
30 August 1984
[4] J. Thierly-Mieg, J. Math. Phys. 21 (1980) 2834; Nuovo Cimento B108 (1980) 396; L. Alvarez-Gaume and L. Baulieu, Nucl. Phys. B212 (1983) 255; L. Baulieu and J. Thierly-Mieg, Nucl. Phys. B197 (1982) 477. [5] E. Caftan, Les syst~mes dffferentiels ext6rieurs (Hermann, Paris, 1971). [6] D.Z. Freedman and P.K. Townsend, Nuel. Phys. B177 (1981) 477. [7] J. Thierry-Mieg and L. Baulieu, Nucl. Phys. B228 (1983) 259. [8] L. Baulieu, Phys. Lett. B126 (1983) 455.
227