Covariant quantization of non-Abelian antisymmetric tensor gauge theories

Nuclear Physics B228 (1983) 259-284 © North-Holland Publishing Company

COVARIANT QUANTIZATION OF NON-ABELIAN ANTISYMMETRIC TENSOR GAUGE THEORIES Jean THIERRY-MIEG Groupe d'Astrophysique Relativistic, Observatoire de Meudon 92190 Meudon, France

Laurent BAULIEU Laboratoire de Physique Theorique et Hautes Energies, Paris, France*

Received 23 November 1982

The non-Abelian Freedman-Townsend gauge tensor model is quantized in a large class of covariant gauges using the geometrical reinterpretation of the BRS equations. In addition to the now usual pyramid of gauge and ghost states, a pyramid of auxiliary fields is found in our construction. These fields enforce the consistency of equations of motion and the integrability of the BRS equations.

1. Introduction A n t i s y m m e t r i c tensor gauge fields [1] p r o v i d e a n a t u r a l generalization of the usual vector gauge fields. F r o m the m a t h e m a t i c a l p o i n t of view, they a p p e a r as an extension of the c o n c e p t of c o n n e c t i o n fields [2]. F r o m the p h y s i c a l p o i n t of view, they are c o n j e c t u r e d as the m e d i a t o r of string i n t e r a c t i o n [3] a n d they p l a y a key role in s u p e r g r a v i t y [4]. T h e y have also been c o n j e c t u r e d as characterizing defects in solid state physics [5] a n d as a w a y of gauging the a p p a r e n t i n t e r n a l s u p e r s y m m e t r y of the w e a k i n t e r a c t i o n s [6]. A s such they p r e s e n t a p r o m i s i n g challenge a n d deserve a t h o r o u g h study. T h e classical models, a b e l i a n [1, 3] a n d n o n - a b e l i a n [7, 8], are quite well u n d e r s t o o d a n d turn o u t to be a d u a l f o r m of the scalar o-model. T h e i r c o v a r i a n t q u a n t i z a t i o n is however subject to two difficulties which have rendered, up to now, the F a d d e e v - P o p o v c o n s t r u c t i o n inoperative: the necessity of ghosts for ghosts a n d the existence of differential constraints. In o r d e r to solve these difficulties we have used o u r g e o m e t r i c a l q u a n t i z a t i o n scheme which we have shown to w o r k in the case o f Y a n g - M i l l s theories [9]. T h e c o m p l e x i t y of the p r o b l e m s to be solved has a p p e a r e d to us as a g o o d test-case for the p o w e r a n d the validity of o u r m e t h o d . This * Laboratoire associ+ au CNRS LA 280: postal address: Universit+ Pierre et Marie Curie, Tour 16, ler +tage, 4, place Jussieu, 75230 Paris Cedex 05, France. 259

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method is based on the reinterpretation of the BRS equations as Cartan-Maurer fibration conditions [10,11]. In this scheme the gauge fields and ghosts are introduced at once and the BRS equations follow from geometrical considerations. Then the quantum lagrangian is derived from a principle of BRS invariance. Quite a long time ago [12], in the abelian case, the geometrical method has been shown to solve the first difficulty, the problem of finding the correct set of ghost fields. The solution is that the quantization of a skew p-tensor is iterative and involves in succession two ( p - 1)-tensor Fermi ghosts, three ( p - 2)-tensor boson ghosts etc.., down to p + 1 scalar ghosts. This result is in discrepancy with the naive application of the Faddeev-Popov ansatz. This ghost spectrum has been checked to insure the equivalence at the one-loop level between the p and ( D - p - 2) abelian skew-tensor models when gravity is coupled [13]. Alternative methods for reestablishing this ghost spectrum have been used by several authors [14,15]. However the corresponding derivations of the quantum lagrangian have remained complicated and unaesthetic. The second difficulty, the existence of differential constraints, is inherent in the non-abelian case and has not yet been pointed out in the literature. The main new result of this paper is the analysis of various aspects of these constraints, leading us to a system of fully integrable BRS equations. From this result we have been able to find out a consistent quantization of the Freedman-Townsend lagrangian by using a "principle of BRS symmetry" [9]. In particular we have found a large class of covariant gauges where perturbative computations could be done. The new feature which appears in our work is the necessary existence of a " p y r a m i d " of auxiliary fields, associated with the now familiar pyramid of ghosts, in order to insure all necessary integrability conditions. Our paper is organized as follows. In sects. 2 and 3 we review the classical abelian and non-abelian models. In sect. 4 we recall the geometrical construction of the BRS equations in the Yang-Mills case and apply it to the quantization of the abelian p-tensor theory in D dimension. In sect. 5 we analyse the quantization of the non-abelian ( p = 2, D = 4) model in a general covariant gauge. We end up with lagrangians which, in general, contain constraints. In sect. 6 we specialize to a particular class of gauges in which an unexpected reshuffling of fields takes place yielding to a thorough simplification of the lagrangian. In these gauges one gets an effective lagrangian without internal constraints. Following the conclusion (sect. 7) there is an appendix dealing with some technical aspects of the BRS superspace.

2. The classical abelian model

Let us consider an antisymmetric tensor ~t~p...l with p-indices in Minkowski D-dimensional space-time (1 time and D - 1 space dimensions). The most natural

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261

lagrangian for such a field is simply the square of its curl 1

2 ( p + 1)!

2

(Oi,~o...l) ,

(2.1)

where the symbol [/~, v, p... ] means a complete antisymmetrisation over the indices /~, v, p . . . . When p = 0 • is a scalar field and we recognize the Klein-Gordon lagrangian. When p = 1 we recognize Maxwell's lagrangian. When p >~ 2 we obtain the skew-tensor gauge models. For p > 1, there is an obvious gauge invariance of the lagrangian (2.1) under the transformation

8~[..o...1 = 0[.~.o...1.

(2.2)

Corresponding to this gauge invariance there is a definite redundancy in the definition of the field ~t~p...l" This field has p antisymmetrized indices with values between 1 and D. Therefore it has ( ~ ) = D ! / ( D - p ) ! p ! independent components. On the other hand it can be proven by Dirac's hamiltonian method that the P

lagrangian (2.1) describes only o 2 ) independent physical degrees o f freedom. This reduction is most easily understood if one notices that no mass term is compatible with the gauge invariance (2.2). De facto, if the theory is to be Poincar6 invariant, the physical degrees of freedom contained in ~[~p...l must fill up the p-tensor representation of S O ( D - 2), the little group of a light-like translation, which is of dimension o -p2 /" For instance, in four dimensions, a massless vector / describes two degrees of freedom, and an antisymmetric 2-tensor describes one degree of freedom [2, 3]. This result suggests that the (D, p ) and (D, D - p 2) theories have the same physical content. Indeed this equivalence can be shown at the classical level by the method of duality transformations [4]. Consider the lagrangian 1

(D-p-

1 1)!(p + 1)! e,~...o ..... H~,...Oo~o, 4 2 ( D _ p _ I ) ! ( H ,

2

.... ) . (2.3)

Here ~, .... denotes again a skew p-tensor, H , .... denotes an auxiliary D - p - 1 skew tensor, t,,,.., is the antisymmetric tensor with D indices. The equations of motion are

O[~,H~o...l= Tlr

r~[~'~l

0,

(2.4a) 1 ~ ,,IK ( p + 1)! e ' ~ ° ..... %~".....

9

(2.4b)

The second equation is algebraic in H and can be substituted in the lagrangian (2.3). In the functional integral formalism this operation corresponds to a gaussian

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integration over the H field. Thereby the lagrangian (2.1) is recovered, so that the lagrangians (2.1) and (2.3) are equivalent. On the other hand eq. (2.4a) can be solved locally and H can be expressed as the curl of an arbitrary D - p - 2 skew tensor X[~v. . .1

(2.5)

H , .... = Ot~,X~... 1.

By substituting eq. (2.5) in the lagrangian (2.3) the first term vanishes up to a pure derivative. The second term yields the following expression

1

~= 2(D-p-1)!

( O[~'X"°I

)2

"

(2.6)

The theories described by the lagrangian (2.1) (2.3) and (2.6) are therefore classically equivalent. This shows the equivalence of p and D - p 2 skew tensor classical theories in D dimensions

3. The classical non-abelian model Non-abelian generalizations of these abelian lagrangians are known in three cases. (i) The o-model lagrangian t 0 generalizes the free scalar theory Eo=-½Tr(g-lO.g)

2,

(3.1)

where g denotes a matrix representation of some Lie group G of order n. This lagrangian describes n massless interacting scalar particles. (ii) The Yang-Mills lagrangian ~1 generalizes Maxwell's lagrangian.

= -¼( O,A - OvA, + [ A..

A I) 2.

(3.2)

Here the vector field A t is valued in the Lie algebra ~ of the group G. The brackets [ , ] are the Lie commutators. This lagrangian is gauge invariant and describes interacting massless vector particles. The procedure for quantizing ~1 is well understood in perturbation theory. (iii) The third generalization was first formulated by Freedman [7]. It describes the theory of a skew D - 2 tensor B~.... in a D-dimensional space. B~,.... is valued in the Lie algebra ~ of some Lie group G. This theory can be most conveniently formulated using Townsend's first-order lagrangian [7, 8]

E2

2 ( D -1 2)! %~o .... F~,~Bo.... + x5 ( H . ) 2,

(3.3)

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263

where H e is an auxiliary vector field valued in ~ and F,, is the associated field strength

+[Ue,<].

(3.4)

The same analysis as in the abelian case can be applied to ~2. However, by solving H in terms of B with the H equation of motion, one obtains a non-polynomial lagrangian. This lagrangian can hardly be analyzed. On the other hand, following Townsend, one can solve locally the B equation of motion and prove the following interesting result: lagrangians (3.1) and (3.3) are equivalent. Indeed the B equation of motion is F~, = 0,

(3.5)

H e = g - 10eg ,

(3.6)

its solution in H is locally

where g is a matrix representation of the Lie group. Substituting eqs. (3.5), (3.6) into the lagrangian (3.3) one recovers exactly the lagrangian (3.1). This shows that the lagrangians (3.1) and (3.3) describe the same classical physics. One could argue that the study of E2 should end here, since E2 can be transformed into the simpler lagrangian ~0. However ~z exhibits a non-abelian gauge invariance which is interesting in itself. The particular features of this gauge invariance render apparently inoperative the Faddeev-Popov ansatz. Consequently the problem of finding the quantum lagrangian associated with E2 was unsolved up to now. ~2 is invariant under two types of gauge transformations. The first one, established by Freedman, is parametrized by a D - 3 skew tensor ~e.... valued in 9, and reads as follows 8Bt~...1 = Dte~...1, 8H e = 0,

(3.7a) (3.7b)

Here the "covariant" derivative D is defined as

De=0 e + [ H e, ],

(3.8)

and one has (3.9) The invariance of E2 under the transformations (3.7) up to a pure divergence is

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guaranteed by the Bianchi identity

DI~,F,p ] = O.

(3.10)

The second gauge invariance of the lagrangian (3.3), yet unnoticed, is parametrized by a D - 4 skew tensor 0[~,. ..1 valued in ~. It reads as

= [e,,,o0...], 8H~ = 0.

(3.11a) (3.11b)

It is obvious that ~2 is invariant under the transformation (3.10). Indeed, from eq. (3.9) one has [F~,,,Op.]=DI~,D, Op...J, and the transformation (3.11) is formally equivalent to (3.2) with the change ~1~..1 ~ Dt,O~..1" The existence of the invariance (3.11) plays a key role in the construction of the quantum theory; it enforces the integrability of the BRS equations (sect. 5). At the same time the transformation (3.11a) is related to the closure of the classical gauge fixing condition. Indeed the natural generalization of the Landau gauge condition is

D~B[~,...] = 0.

(3.12)

For consistency one must complement eq. (3.12) with the following equation obtained by differentiation of eq. (3.12)

= ½[ F,, B,,.... 1 = 0 .

(3.13)

In our formalism 0r... is related to the differential constraint (3.13) in the same way that ~... is related to the Landau gauge condition (3.12) following the usual Faddeev-Popov procedure. All these ideas will get clearer at the end of sect. 5. An obvious idea would be to extract in turn a term like D,D, Dp~I .... from B and so on. However such term may be ignored thanks to the Bianchi identity which ensures that the system (3.12) (3.13) is closed. We end up in this section by observing that one can still generalize the lagrangians (3.1) and (3.3). Consider a subgroup G 1 of G and its subalgebra ~1 c ~. Then one introduces an auxiliary vector field A~, valued in ~ r The generalization of the o-model lagrangian is

if/o = - ½Tr( g - lD~[ A ] g ) 2 ,

(3.14)

o.IA] = 0.+[A., 1;

(3.15)

where

J. Thierry-Mieg, L. Baulieu / Covariant quantization

265

~ describes the physics of a set of fields valued in the space G 1 / G . N o kinetic energy term has been introduced for A t at the classical level. Thus the A t equation of motion is algebraic and A t can be made explicit in a function of the g field. It is however an open problem to find out whether the A t propagator acquires a pole structure through radiative corrections. Similarly one can generalize the lagrangian E2- One has simple to change everywhere the commutator [H, ] into [A + H, ]:

D = 0+[/-/, ]- DA

],

F~ --,F~=O~(A,+H,)-O,(A~+H,)+[A~+H~,A,+H~].

(3.16)

Then one obtains the lagrangian E~

1 A ÷1 2 2 ( O - 2)! e~"°.... F~,Bo .... ~(H~) .

(3.17)

This lagrangian is gauge invariant under the transformations (3.8) and (3.11) where H is changed into A + H. In order to prove the equivalence of E~ and E~ one can solve the B equation of motion as in (3.5). One has F,~ -- 0 and therefore A t + H~ = g aa~,g.

(3.18)

If we shift A t to the right-hand side and factorize g - l g we can express H~ as

Ha= -g-l(gD~g-l)g.

(3.19)

By substitution of (3.19) into (3.17) one recovers eq. (3.14), and the classical equivalence of E~ and E~ is therefore established.

4. Quantization of the abelian model

The direct application of the Faddeev-Popov quantization method to the tensor gauge lagrangian (2.1) leads to a wrong ghost spectrum and inconsistent couplings to gravity. The modifications that have been proposed by several authors [14, 15] appear to us artificial and cannot apparently be extended to the non-abelian case. To quantize the lagrangian (2.1) therefore we will apply a geometrical method that is known to work in the Yang-Mills case [9]. Let us first review our method in the (non-abelian, p = 1) Yang-Mills case. We shall afterwards generalize it to the p = 2 abelian case and in sect. 5 to the non-abelian case. The generalization to the abelian p > 2 case is obvious. The general strategy is first to introduce the ghost fields at the classical level. Then to postulate the BRS equations following a simple geometrical prescription, and only

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at the end to define the quantum lagrangian from the requirement of BRS symmetry. In order to realize this program, let us consider an extension of space-time: the ( D + 2)-dimensional superspace of Tonin and Bonora [16] P with local O S p ( D / 2 ) symmetry and local coordinates (x, y, 37), where x~ denote space-time coordinates and y and )7 a pair of scalar anticommuting coordinates. We adopt here the Bonora and Tonin superspace rather than the principal bundle space of the original formalism [11,12]. The technical reasons which motivate our choice are explained in the appendix. Over P, let us define a 1-form ~a valued in the Lie algebra ~ of the gauge group. In local coordinates ,4 a can be expanded as

Aa(x, y, y ) = A~dx ~' + A y d y + Ayd)7.

(4.1)

Here the index a stands for a group index. A~(x, 0, 0) is identified as the usual gauge field, whereas the ghost and anti-ghost fields are defined as

ca(x) =Ay(X,O,O)dy,

ca(x) = Ay (x,O,O) d)7.

(4.2)

By construction A~ is a commuting field whereas c a and ga anticommute since they have the commutation properties of exterior 1-forms. Note that the statistics of c" does not depend of that of y, and that the separation of .4 into A~, c, g is coordinate, i.e. gauge dependent. In the same coordinate system the differential operator d can be expanded as = d + s + g,

(4.3a)

where d = dx"

0

Ox"

,

s = dy

0

Oy

,

0 g = d y ~-~-=.

uy

(4.3b)

By the Leibnitz rule, the action of s and g on any polynomial in A~, c and g is defined as

s( X Y ) = ( s X ) r +_XsY, g(XY) = (gX)Y+ xgr,

(4.4)

where the minus sign occurs if there is an odd number of ghosts and anti-ghosts in X. Further we have, by (4.3b) and (4.4) S2=SSq-gS=S

on all functions of fields.

2 -----0,

(4.5)

J. Thierry-Mieg, L. Baulieu / Covariant quantization

267

We now c o n s i d e r / [ as a connection 1-form and we introduce the curvarture 2-form P = t ~ + ½[A, A].

(4.6)

The BRS and anti-BRS equations are simply derived by imposing that the geometry of the P space is integrable in y and y, i.e. we impose the following Maurer-Cartan equation on P (" horizontality" condition) aft--- 1 F ~ ( x , y, y ) dx~ A dx*;

(4.7a)

M + l [ d , d] = aA + ½[A, A],

(4.7b)

or, equivalently

where A = A ~ d x C By expanding both sides of eq. (4.7) over the pairs of 2-forms d x / x dx, d x / x dy, etc.., one gets 5 constraint equations which can be written as

sc:-½[e,c], sg+gc=

~e=-~[e,e], -[?,c],

(4.8a)

where we have combined eq. (4.7) with the definitions (4.2) (4.3) for s, g, c and ~. These equations leave undetermined the action of s on g or g on c. Thus we introduce an auxiliary scalar commuting field b valued in the Lie algebra ~ and we define s? = b.

(4.9)

The nilpotency relation (4.5) and the relation sg = - g c - [?, c] determine the values of sg, sb and gb. Thus eqs. (4.8a) can be coherently complemented by

sg=6,

~c= - b - [ e , c ] ,

sb=O,

gb=-[6,

b].

(4.8b)

Eqs. (4.8) are the well-known BRS and anti-BRS equations for the Yang-Mills theories. Note that we have called b "auxiliary" simply because it does not appear as a component in the expansion of the 1-form d over the basis dxU, dy, d)7. As a matter of fact b cannot acquire a kinetic energy term in any BRS invariant and renormalizable lagrangian made up with A,, c, g and b. This can be trivially checked by power counting arguments [9]. The BRS equations (4.8) define ~i la Cartan the usual Lie gauge symmetry. They interchange both classical and ghost fields and define algebraically the action of s

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J. Thierry-Mieg, L. Baulieu / Covariant quantization

and ~ on all fields. Provided that these equations are maintained the integrability in y and )7 is guaranteed and the value of any function of the fields A,, Av, Ay is known everywhere once it is known in the section y = )7 = 0. We now define the quantum Yang-Mills lagrangian by postulating that it is the most general lagrangian function of A, c, ~ which is s and g symmetric, Lorentz and globally gauge invariant. Further, if its hermiticity under the change c ~ ~ is assumed, we proved using dimensional arguments that it can be written under the natural form [9] (4:9) where Ed(A) is the classical Yang-Mills lagrangian, and ct and fl are arbitrary gauge parameters. The lagrangian (4.9) can be expressed under a more conventional form after expansion of the sg terms by using eqs. (4.4), (4.8) and after elimination of the b field by its equation of motion. We have analyzed this lagrangian in detail in ref. [9] and we have shown that it is equivalent to the Faddeev-Popov lagrangian, at least in perturbation theory. As a matter of fact, all the perturbative properties of the theory induced by the lagrangian (4.9) can be essentially derived from the fact that it is equal to the classical lagrangian up to an s and g exact form, a modification that does not alter the physical S-matrix (neglecting infra-red problems). Note that the particular case fl = 0 yields the usual Landau gauge. The essential content of the method sketched above can be summarized as follows: the ghost and anti-ghost fields are determined from the expansion of a general 1-form in the P-space (eq. (4.1)); the BRS equations follow from the Maurer-Cartan equation restricting the geometry (eq. (4.7)); the quantum lagrangian is the most general function of A, c and ~ which is BRS and anti-BRS invariant, and which is compatible with the ordinary global invariances of the classical theory (eq. (4.9)). Our aim is now to apply the same construction for quantizing the abelian tensor theory. For the sake of simplicity we restrict ourselves to the case of p = 2 skew tensors. The method can however be straightforwardly extended to arbitrary values of p. We consider the same space P as above and s and g are still defined as

s = d y -q-s,

ay

8 g = d)7 -q-~-=,

oy

(4.10)

so that we have again by definition s 2 = s s + . ~ s = . ~ 2 = 0.

(4.11)

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J. Thierry-Mieg, L. Baulieu / Covariant quantization

To define the propagating fields of the quantum theory we generalize eq. (4.1), and define the most general 2-form B in the P-space. B is valued in the Lie algebra of U(1) [~(x, y, fi) = ½B~dx" A d x ~ + B~ydx ~ A d y + B~;dx ~ A dfi +½ByydyAdy+By;dyAdf+½B;ydyAdy.

(4.12)

By anology with the Yang-Mills case we identify in this expansion the classical gauge field B~,,, the pair of anti-commuting ghosts ~, and (~ and the three commuting ghosts m, N, n. Again the commutation properties of these fields follow from the rules of exterior calculus. We have

B~,,(x)=B~,,(x,O,O),

li.(x)=B.y(x,O,O)dy,

~.(x)=B~;(x,O,O)df,

m ( x ) = ½Byy(x,O,O) d y A d y m(x)=By;(x,O,O)dyAdfi,

m(x)=½Bg,;(x,O,O)dfiAdy.

(4.13)

One immediately observes that this field spectrum formally insures the unitarity. Indeed, following Feynman's original argument [17] the ghost fields will contribute to closed loops with the following weights, induced by their commutation relations ~

i

-4

~

i

-4

m

I

+1

n

i

+1

m,

i

+1.

(4.14)

Therefore, since B~,, has 6 independent degrees of freedom in 4 dimensions, the number of physical degrees of freedom described by the set of fields B~, ~ , (~, m, ~ , n is 6 - 4 - 4 + I + 1 + 1 = 1, in agreement with the equivalence between the theory of the skew 2-tensor in 4 dimensions and the scalar theory (sect. 2). We therefore postulate that the propagating fields necessary to quantize the lagrangian (2.3) are these given in eq. (4.14). Generalizing our Yang-Mills result we impose the "horizontality" of the exterior differential of B in order to get the BRS and anti-BRS equations cl[?= dB =1gO[~B~oldX~ A dx" A d x ° .

(4.15)

By expanding a¢/) over the basis of 3-forms dx ~/x dx ~/x dx p, dx ~ A dx ~ A d y .... we

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J. Thierry-Mieg, L. Baulieu /

Covariant quantization

get from eq. (4.18) the following 9 equations

sS~ = % ~ ,

~B~ = 0 ~ ] ,

s ~ = O, rn,

g~, = 0 ~ ,

sm=

g~=

O,

O,

s~,, + g~,, = o,,n, s~+

g n = O,

gm+

sn

= 0,

(4.16a)

where we have combined the definitions (4.13), (4.16) with eq. (4.18). In order to raise the degeneracy contained in the equations s~, + g(, = O~n, s ~ + g n = 0 and g m + s n = 0 we introduce the commuting auxiliary field b,, and the anti-commuting scalar "auxiliary" fields a and ft. We therefore complete the equations (4.16) by

s~. = 10~. - b., sm=

g~. = -~a.. + 6.,

a,

gn = -if,

sm = --or,

(4.16b)

s n = ot~

and the equations defining the action of s and ~ on the fields b., a and ff are simply given by imposing the nilpotency relation s 2 = sg + g s = g2= O. One gets

sb. = ~ a ~ , sa =ga

~b. = 1 0 ~ , =sa

=sa

(4.16c)

= O.

The BRS equations are completely determined by eqs. (4.16). They satisfy the nilpotency algebra (4.14) s 2 = s g + g s = g2= 0. The action of s and g on classical fields is equivalent to a classical gauge transformation as it is obvious by comparing eqs. (2.2) and (4.16a). Finally the Leibnitz rule defines the action of s and g on any polynomial of the above mentioned fields. Pursuing the analogy with the Yang-Mills case we now propose as a quantized lagrangian of the abelian model the following expression ~=~cl(Otxu)--Ss(1O2

u nt- ~kl~/z~tLq- ~ k 2 ~ m q- ~k~n2) ,

(4.1V)

where £c](B.~) is s and g invariant and is therefore given by eq. (2.6)

t

)2.

(4.18)

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271

Using eqs. (4.19), (4.20), (4.21) and the Leibnitz rule (4.4) we can expand all sg terms in eq. (4.22). One has ½sg

2

= 0

(4.19a)

GaE. G +

sg(~z)=-b~+~.Ozc~-~zO~S+Oz~cg.m-¼(Ozn) 2,

(4.19b)

sg( ~m) = ½sg(n 2) = Sc~.

(4.19c)

The auxiliary field dependent part of E is the following one

½Xtb2+B,.Ot~b.l+½(2tz+X'2)ffa-Xl((~O~a-~O~ff).

(4.20)

Up to a pure divergency E is algebraic in these fields. Therefore they can be eliminated by their equations of motion (and then they deserve the appelation of auxiliary). After the elimination of b~ the gauge is fixed for B,~, whereas the a and elimination yields a gauge fixing term for ~,. Thus collecting together eqs. (4.17), (4.19) and eliminating b~, a and ff we finally get the following lagrangian where all propagators are invertible: ,

~

(O.~t..j)

2X t

)~2

42t2 + X,2 ( 0 , ~ , ) ( 0 ~ )

~

t- O[.(.] 0[.~.]

-]- )k 1 (

O~O,m

-

~(O.n) 2).

(4.21)

As the reader has noticed we have completely skipped the Faddeev-Popov construction. However we believe that our quantization pattern leads to the correct quantum lagrangian for the following reasons. (i) The method is exactly similar to the one which has been shown to work in the Yang-Mills case. (ii) The gauge fixed lagrangian (4.17) or (4-21) is a correct starting point for a perturbative expansion since all propagators are defined. (iii) The s and g invariant quantum lagrangian (4.17) differs from the classical lagrangian by the addition of an s and g exact form. As in the Yang-Mills case such an addition should leave invariant the physical S-matrix under the change of the parameters 1~1,X2, )t~ [9]. (iv) The correct ghost spectrum insuring unitarity has been introduced at once. As shown in formula (4.14) the difference between the number of propagating bosons and fermions in lagrangian (4.17) or (4.21) is equal to the number of physical degrees of freedom of the classical lagrangian (4.18). As a matter of fact this result

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J. Thierry-Mieg, L. Baulieu / Covariant quantization

holds true if we quantized in the same way the lagrangian (2.1) describing a p-tensor in D dimensions. In this general case, the construction schematized in eq. (4.12), (4.13) yields 2 ( p - 1)-tensor anticommuting propagating ghosts etc., down to ( p + 1) scalar ghosts, leading to the effective number of physical degrees of freedom [121

p

•

N= E(-)'(i+I) i=o

( p ; i )(=

Dp -2

(4.22) "

In order to obtain the quantum, lagrangian in the general (p > 2) one has simply to follow the same pattern starting from eq. (4.12), introducing a general p-differential form in the cotangent space of P instead of a 2-form. Then the complete ghost spectrum follows. The BRS equations derive from eq. (4.15) and the gauge fixing procedure is obtained by inducing terms in Xx, X2... in the lagrangian (4.17). (v) The lagrangian (4.22) is the abelian restriction of the non-abehan coupled lagrangian studied in sect. 5.

5. Quantization of the non-abelian model In this section we generalize to the non-abelian case the method described in sect. 4. We only consider the theory of the p = 2, D = 4 non-abelian antisymmetric tensor. We postulate again that the propagating fields are those which appear in the decomposition of a general 2-form B ~ defined over P and valued in the Lie algebra of the gauge group: ,Ba(X, y , fi) = ½B~vdx~' A dx ~+ B~ydx ~ A d y + B~2dx~ A d fi

+½ByydyAdy+By~dyAdfi+½By;dfiAdfi.

(5.1)

The spectrum of propagating fields is thus analogous to that of the abelian theory (eqs. (4.12), (4.13)) apart from the group index a. From eq. (5.1) we define the a classical field B~, the vector anti-commuting ghosts ~,, ~ and the scalar anti-commuting ghosts fields m ~, ~a and n a, all valued in the Lie algebra of the gauge group

B~=B~(x,O,O), ~"=B~y(X,O,O)dy, (~=Bt,y(x,O,O)dy it ma=½Byy(X,O,O)dyAdy,

n'~=Byy(x,O,O)dyAdfi,

ma= ½Byy(x,O,O)dy A dy.

(5.2)

As in the abelian case the ghost spectrum defined in (5.2) formally insures the unitarity in perturbative theory.

J. Thierry-Mieg, L. Baufieu / Covariant quantization

273

For deriving the BRS equations associated with the gauge symmetry of the lagrangian (3.3), both kinds of gauge invariances (3.7) and (3.11) must be considered at once. As usual the ghost fields ~ and (~ are to be associated with the infinitesimal parameters appearing in the transformation law (3.7). On the other hand the existence of the invariance (3.11) should induce the appearance of a pair of anti-commuting scalar ghost fields. Such ghost fields however do not exist in the expansion (5.1) of B: m, ~ and n are commuting ghosts. Therefore we postulate that, in addition to the propagating fields (5.2) a pair of anti-commuting "auxiliary" fields 0 = and/7 = valued in the Lie algebra G must be introduced. We call them "auxiliary" because they do not belong to the expansion of B. Again they will turn out at the end not to have kinetic energy terms in the quantum lagrangian. Now, these fields are to be parametrized as 1-forms 0 ~= 7y(x,O,O)dy,

0 " = Ty(x,O,O) d r .

(5.31

On the other hand by covariance in P these fields imply the existence of an extra vector field T f ( x , y , y ) such that ~a =

T a d x , + Tyady + Ty~d)7,

(5.4)

is a general exterior 1-form valued in ~. In the gauge fixed lagrangians that we will consider, 7 ~ ( x , 0, 0) will, however, decouple. We are left now with the problem of finding the BRS equations involving all fields B,,, H,, ~ , (,, m, n, m, O,/~, T~. As previously we consider the d differential operator 3 = d + s + g,

(5.5a)

with d = dx~, __0

Ox ~ ,

O

s = d y -if-f,

0 .

g = d fi Of

(5.5b)

By construction one still has s 2 = s g + g s = g2 = O.

(5.6)

We define the "auxiliary" 1 - f o r m / t a ( x ) valued in the Lie algebra

B°(x)=H;(x)dx".

(5.7)

Thanks to H i one can build a "covariant" derivative, as in eq. (3.8)

D=d+[/L ],

(5.8a)

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274

and also D = d + [ / t , ].

(5.8b)

N o t e that we have only authorized an x dependence for H r. Therefore, by construction s H ; ( x ) = O,

g H ~ ( x ) --- 0,

(5.9a)

and D anti-commute with s and g: sD + Ds = 0.

(5.9b)

F r o m H we define the "strength field" 2-form F F---- d / t + ½ [ / t , / ) ] = D / ) = D/4,

(5.10a)

in terms of components ~ , a __ L / T a

A~,-~ A d x V = ½ ( O ~ , H :a ( x ) , O . H ~ (ax ) + [ H . ( x ) , H . ( x ) ] a ) d x ~ ' A d x --2..~'-..~

~. (5.10b)

In order to obtain the BRS equations we now impose the following constraints generalizing to the non-abelian case the constraints (4.15) D B + [_P, T] - D B + I F , T ] .

(5.11)

Expanding eq. (5.11) in components over the basis d x " A d x " A dxO, d x ~ A d x ~ A d y . . . . one finds 9 equations from (5.11). They read as sB~. = D[.~,, 1+ [ F~,,,, 0 ] ,

gB.~ = D[~,~,,1+ [ F~,,, 0 ] ,

s~.=D.m,

g~, = D,,,~,

sm = O,

g ~ = 0,

4. +

= D.n,

s ~ + gn = O, ~m + sn = 0.

(5.12)

One notices immediately that the action of s and ff on the classical field B,~ is equivalent to a general gauge transformation (3.7), (3.11). The system (5.12) however is not closed: the action of s and g upon 0, /~ and T is not defined in eq. (5.12). However, as it is usual with partial differential equations, the equation (5.11)

J. Thierry-Mieg, L. Baulieu / Covariant quantization

275

imposes its own integrability condition

b ( b [ 1 + [ L ~ 1 ) = [ L [1 + b ~ ' ] =- (. . . ) d x ~' A d x " A d x p A d x ° =* [t + D T = B + D T .

(5.13)

This condition yields the three additional equations defining sO, gO, sO + gO yielding siT+ gO = - n ,

sO = - m ,

gO= - ~ ,

(5.14a)

together with the equations defining sT. and gT. s T . = O,O + li.,

~ T . = D.O + ti..

(5.14b)

The integrability condition can be understood most simply under the form (5.14): eqs. (5.14) are indeed necessary to insure the nilpotency relation (5.5) of s and upon the fields ~. and ~.. This can be trivially checked. The relevance of the second gauge invariance (3.11) and of the associated auxiliary ghosts 0 and 0 is now quite clear. Suppose that we had not introduced 0 and iT. This is in fact equivalent to setting these fields equal to zero everywhere. Now, the integrability condition (5.14) is nevertheless necessary for consistency. Therefore one must have also rn = ~ = n = 0. Thus having set 0 = O= 0, the geometry is overconstrained, and the m, ~ , n ghost must vanish. As a consequence it is not possible to fix the gauge for f . and ~, fields. This phenomena ruined our first attempt [12] to quantize the lagrangian (3.3), since we had failed to observe the gauge invariance (3.11). The action of s on anti-ghosts and of g on the ghosts is as usual not completely specified by eqs. (5.12), (5.14). We need therefore to introduce again an auxiliary commuting vector field b~ and auxiliary anticommuting scalar fields a and ft. Then the equations s ~ + g ~ , = D u n , s ~ + g n = O and g m + s n = O can be solved. In addition we introduce a commuting scalar field t to solve gO + gO = - n . All these field b, et, ff and t are valued in the Lie algebra 9. We get

s(~ = 1D~n - b.,

sm sn

= a, ~

g~. = 1 D n + b~,

gn =

et,

sm

~

-ff, --et,

set = s f f = ga = g f = 0, sO=--½n-t, st = - ½et,

gO= - ½ n + t , gt = - ½~,

(5.15)

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J. Thierry-Mieg, L. Baulieu / Covariant quantization

where the action of s and g has been derived from the nilpotency relation (5.5). We have therefore obtained the complete set of BRS and anti-BRS equations (eqs. (5.12), (5.14), (5.15) associated with the classical symmetry (3.7) (3.11). s and g satisfy by construction the nilpotency algebra s 2 = s g + g s = g2= 0. The algebra (5.12), (5.14), (5.15) can be used now to quantize Townsend's lagrangian (3.3). In order to construct the quantum lagrangian we again skip the Faddeev-Popov construction and rely on the invariance of physical S-matrix elements under addition of an s and g total differential form. We propose the lagrangian 1

2 --

-

1

2

"{'- p ' n 2 )

(5.16)

This is the simplest s and g symmetric lagrangian which preserves the global gauge symmetry and the Lorentz symmetry and which is gauge fixed. As usual the s and g invariance of E is as usual guaranteed from the nilpotency of s and g (eq. (5.5)), and from the fact that s and g act on the classical fields B and H as classical gauge transformations. In order to check that ~ is gauge fixed one has to develop all terms of the form sg appearing in eq. (5.16). One has from eqs. (5.12), (5.14), (5.15)

+[

+ [ F,...,q) (5.17)

= D. D m - a' ( D , n ) (ram)

=

=

+ b;, + a D ~

- ~D~,

(5.18) (5.19)

It appears from these equations that the b~, a and ff fields which appear quadratically in ~ can be eliminated by gaussian integrations. The elimination of b, yields a quadratic gauge fixing term for the B~, field, of the type (0,B~p) 2, whereas the elimination of ff and a provides a gauge fixing term for the vector ghost propagator of the type ( O ~ ) ( O ~ ) . The P and P' terms in (5.16) being equivalent we may set 0' to zero and we obtain

(5.20) The propagators of the primary fields B~, ~ , ~ , m, ~ and n are invertible. However

J. Thierry-Mieg, L. Baulieu / Covariant quantization

277

the secondary fields t, 8,/~ appear as Lagrange multipliers of the constraints [ F~., B,,] = 0 ,

(5.21)

[~,,,D:,I--[e,,,D,Q--o.

(5.22)

A posteriori, an alternative derivation of the lagrangian (5.20) can be proposed. Let us forget our construction of the 0,/7, t fields and consider de novo the classical lagrangian ~cl =

--

lelxl, poBp.~,Foo -'[-1H~ •

(5.23)

This lagrangian is gauge invariant and self-contained since its system of equations of motion is closed. On the other hand the "naive" once gauge fixed lagrangian which one obtains by application of the Faddeev-Popov trick [8, 9] has not this property (5.24) Indeed the b~, ~ and ~ equations of motion

(5.25)

D~B[~,] = O,

(5.26)

D~,D[~I = D~,D[~] = O,

imply by differentiation exactly eqs. (5.21) and (5.22). Following Dirac's ideas we may inforce the constraints (5.21) and (5.22) by adding to the naive lagrangian (5.24) Lagrange multipliers. In this way we obtain the modified lagrangian

e = ec, +

+

+

+

01) ÷ ,

(5.27) and we recover exactly the correct development (5.17) of E+ sg(B~). Obviously it remains to fix the gauge for the ~ and ~ fields. It is probable that this task could be done with techniques similar to those used by the authors of refs. [14,15], and that one would get the full lagrangian (5.20) from (5.27). It is not our purpose here to do such a derivation since our principle of BRS symmetry has done the work quite simply and quite efficiently. In fact the important point to be mentioned is the

278

J. Thierry-Mieg, L. Baulieu / Covariant quantization

interpretation of the O, 0 and t fields of our BRS formalism as the Lagrange multipliers of secondary constraints. In sect. 6 we will show that there exists a class of gauge where these fields can be effectively decoupled. We end this section by noting that our method can also be used to quantize the generalized non-abelian lagrangian (3.15), involving the field A. One has simply to introduce the scalar ghosts c and g associated with A (they anti-commute and they are valued in 9). Then the s and g generators defined in (5.12) have simply to be changed into

],

],

V.=O.+[Z.,

],

(5.28)

SA~ = O~c,

fla x = 0~,

S c = O,

S g = O.

(5.29)

= ~-2 = 0

(5.30)

One still gets that S 2 = SS +SS

and also that S and g commute with the covariant derivative D A. The construction of the quantum lagrangian is then trivial using S and S in eq. (5.16) instead of s and g.

6. Elimination of constraints

The lagrangian (5.20) is gauge fixed and BRS invariant. It contains however constraints which would prevent practical computations. In this section we will show the existence of a class of gauges where, effectively, the auxiliary fields 0, 0 and t disappear by being absorbed respectively into ~ , ~ and n. In these gauges the m and ~ scalar ghost fields also decouple. Therefore one is left with a lagrangian depending only on B,~, ~ (~ and n. Under this form, the counting of degrees of freedom is less obvious, however the unitarity and the gauge invariance of physical quantities is guaranteed by the BRS invariance. The obvious advantage of these gauges is that they should allow perturbative computations. For our purpose it is most convenient to change the field variables. Let us build the following fields from ~ , ~ , 0,/7, b~, and t

b f = b~ +_ D z t .

(6.1)

279

J. Thierry-Mieg, L. Baulieu / Covariant quantization From eqs. (5.12), (5.14), (5.15) one easily gets the BRS equations for these fields

s42 = O,

s4~ --+ = 0,

s 4 ; = 2D~m,

g4~ = 2D~m,

s~-~ = b ; ,

g~; = b ; ,

sb; = 0 ,

gb+ = 0 ,

sb~- = D~ot ,

eb;- = n : .

(6.2)

We now consider the lagrangian E=E¢I(B,H)+sg(½B~,+~4;

+X~74; + p~m),

(6.3)

where ~ is a gauge parameter, X is its complex conjugate, and O is another (real) gauge parameter. After expansion of the sg terms one gets -+

+

E = Ed + Dt~4~]D[~4~I

+ b ; ( D ~ . . + D : ' ) + (X + X)(b;) ~ --

-+

+ p f a + XSD~4£- + haD.4. ,

(6.4)

where n ' = ( X - X)n + (X + X)t. When this lagrangian is inserted in the functional integral formula the m and ~ integration is trivial: these fields do no appear in (6.4). By performing the following changes of field variables 4. -~ 42 = 4. + D : ,

b~-~b; =b. + D.t, 0~0,

0~0,

,-.,'=(x-X'),+(x+X)t. m-~ m,

~---~ ~ ,

t ~ t,

(6.5)

and by noting that the jacobian of the transformations (6.5) is field independent, one can also eliminate the 0,/~ and t field variables which do not appear explicitely in

J. Thierry-Mieg, L. Baulieu / Covariant quantization

280

(6.4). In turn b + , a and ~ can be eliminated by gaussian integrations. Consequently

the generating functional expressing the theory can be cast into the following form

e w= f

[dB..][d~-] [d~-] [dx']

xexpfdx ¢cl

1

4(X+X)

r+ ~X. ( D~,ld~, )( Dfld,+ ) P

(D~B~,.)

1 4(?~ + X)

2

-+

+

+D[~,IDI~,I

( D~,n,2)

1 n'[ F~,, B ~ ] . 4(X + X')

(6.6)

Therefore the s and g invariant lagrangian (6.3) has lead us to effective lagrangian appearing in the functional integral (6.6). This lagrangian is perfectly gauge fixed and contains no internal constraints. By analogy with the Yang-Mills case we call the class of gauges corresponding to (6.3) Feynman-Feynman gauges. These gauges should allow perturbative computations of Green functions with the usual methods, as it is explicit from the formula (6.6). In particular the quantum equivalence between Townsend's lagrangian and the non-linear o-model lagrangian could be tested. 7. Conclusion

Using a geometrical method and completely avoiding the Faddeev-Popov construction, we have been able to establish the covariant quantization of the non-abelian Freedman-Townsend tensor gauge theory. Two difficulties were resolved. We have established the correct iterative ghost spectrum at once, at the classical level, simply by defining our fundamental tensor field over a super imbedding of ordinary space-time. We have introduced a system of secondary (auxiliary) fields which serve at the same time 3 purposes: they parametrize a secondary invariance of the classical lagrangian, they act as Lagrange multiphers of the constraints implied by differentiation of the primary equations of motion and they allow the closure of the BRS algebra off shell. All those fields and two of the primary scalar ghosts decouple in the class of gauges studied in sect. 6. We hope that this construction will reinforce the interest in the geometrical formulation of the BRS equations and in tensor gauge theories. Appendix

In our earlier work [9,11,12] the total space P of the gauge and ghost fields was interpreted as an ordinary principal bundle rather than the Bonora-Tonin [16] OSp(4/2) superspace. In this paper, devoted to the quantization of the anti-symmetric tensor theory, we have used the Bonora-Tonin formalism. In this appendix we

J. Thierry-Mieg, L. Baulieu / Covariant quantization

281

compare both formalisms. In the case of Yang-Mills theories we show their equivalence. In the case of anti-symmetric tensors we show that the Bonora-Tonin superspace is better suited than the fiber bundle, and thereby we explain the choice made in this paper. Let us first consider the case of ordinary Yang-Mills theories and let us compare both formalisms. We start with the case where only the ghost field (and not the anti-ghost) is introduced. (i) In either case, given a space P with local coordinates (x, y " ) and a connection form A(x, y) the Cartan-Maurer BRS equation

F = M + ½[A, A] - ½F~.dx ~ A dx'.

(A.la)

admits the general solution

.d=T-l(x,y)(dx~'(O,+a,(x))+dy~O~)T(x,y).

(A.lb)

Here T(x, y ) is a function of x and ym, valued in the group G. The field a~,(x) is valued in the Lie Algebra of G and is identified with the Yang-Mills field in the "gauge" y(x, y) = 1. One can show that the 1-form A, defined in (A.la), is valued in the Lie algebra of G. Therefore a Lie algebra index a can be put on both sides of eqs. (A.lb). (ii) Now, as recalled in sect. 3, the statistics of the ghost 1-form c a ca(x)=dy

m Caly=O =- (]t -1 dy m Om'y);=O,

(A.2)

does not depend on the statistics of the variables y. Indeed, only from the exterior calculus rules, one gets in either case cO(x)c

(x ,) = - c b ( x , ) c O ( x ) .

(A.3)

As a consequence local operators involving twice the same ghost (same group index) at the same space-time point vanish. (iii) In either case the total Grassman algebra where ghosts are defined is infinite dimensional, but the reasons are different. In the principal bundle formalism the number of y coordinates is equal to the dimension of the gauge group, and the c a and ym denote real functions. The 1-forms d y " are generators of a Grassman exterior algebra. This algebra is finite dimensional over each point x. Therefore the algebra of local ghost operators is finite. In contrast, exterior forms evaluated at different points of P, i.e. multilocal ghost operators, are linearly independent. Thus the total Grassman algebra, (more precisely the space of sections of the cotangent bundle P*) is infinite dimensional. As a consequence, ordinary Green functions may involve arbitrary numbers of ghost

282

J. Thierry-Mieg, L. Baulieu / Covariant quantization

fields evaluated at different points of space-time and the criticism of Leinaas and Olaussen (18) against the principal bundle formalism is void. In the superspace formalism, there is only one Grassman coordinate y. This time d y is a commuting entity, and ( d y ) 2 does not vanish. However, the ghost component Cy(y, 0) must be defined as a generator of an abstract Grassman algebra, finite dimensional at each point x. But again, ghost components evaluated at different points of P are assumed to be linearly independent objects and the total abstract Grassman algebra is infinite dimensional, as in the principal bundle formalism. (iv) In either case one may expand the gauge function T(x, y ) in powers o f y . To first order in y one has 3'(x, y ) =

g(x) +y'hm(x),

(A.4a)

"y-a(x, y) = g-'(x) -y'g-X(x)h,,,(x)g-l(x), Cm = g - l ( x ) h m ( x )

__yng-l(x)hn(x)g-l(x)hm(x)

(A.4b) '

(A.5a)

A~= g-l(x)( O~+ a~(x))g(x) + ym( O~(g-lhm(x)) -'F [ g-l(x)( 0p. -4- a~(x))g(x), g-l(x)hm(x)]),

(A.5b)

where g is a group element only of functions x and h(x) is valued in the Lie algebra of G. One may note that in the superspace formalism, eqs. (A.4), (A.5) are exact, since yy = O. In a recent work, Bonora, Pasti and Tonin [19] have substituted in (A.4a) the solution h m = g(x)c,,,ly=o of eq. (A.5a), obtaining to first order in y the strange looking equation

A(x,y)=e-Y~¢~(*)g l(x)(cl+dx~a~(x))g(x)eY~(x).

(A.6)

Therefore the apparition of the ghost field as a gauge parameter is not a peculiarity of the superspace formalism but corresponds to a mere change of variables. (v) In either formalism, the anti-ghost field and the anti-BRS transformation are accounted for by doubling the number of y coordinates. This step was first taken by Quiros et al. [20] in the principal bundle formalism. They have assumed the local structure P = (space-time)× G × G, and have rediscovered the Curci and Ferrari (anti-BRS) transformation [21]. More precisely the two copies of G may be considered as parametrizing the left- and right-group actions [22]. Bonora and Tonin, following [20], have also included the anti-ghost in the superspace formalism by assuming two independent Grassman coordinates y and )7. Thereby they have also recovered the anti-BRS equations. Thus, for the Yang-Mills theories, we may conclude that the superspace and principal bundle formalisms are equivalent.

J. Thierry-Mieg,L. Baulieu / Covariantquantization

283

T h e B o n o r a - T o n i n s u p e r s p a c e has, however, p r o p e r t i e s which a p p e a r crucial w h e n a n t i - s y m m e t r i c tensors are considered. I n d e e d , as shown in this p a p e r , one c a n derive within the s u p e r s p a c e formalism, the correct ghost s p e c t r u m for q u a n t i z i n g the a n t i - s y m m e t r i c p - t e n s o r theory, for any value of p a n d a n y gauge group. This is n o t the case with the p r i n c i p a l b u n d l e formalism. This can be u n d e r s t o o d in the easiest w a y b y c o n s i d e r i n g an example. C o n s i d e r for i n s t a n c e the l l - d i m e n s i o n a l space of e x t e n d e d supergravity, a n d the a b e l i a n 3-tensors At~,~pl of this theory. A c c o r d i n g to our q u a n t i z a t i o n scheme a p y r a m i d of ghosts has to be built, a m o n g which are 4 scalars C+++=

1 ~Ayyydy A dy A dy,

C+-+~

I ~Ay~ydy A dfi A d y ,

C

C-

+

~

1 7Ayy;dy A dy A dy,

1 = ~A;;yd y A d ~ A d.~.

(A.8)

I n the p r i n c i p a l b u n d l e f o r m a l i s m all these fields vanish, since, in the a b e l i a n model, there is o n l y one gauge d i r e c t i o n ym, so that d y / x d y = d y A d y = 0. O n the c o n t r a r y , in the s u p e r s p a c e f o r m a l i s m the d y a n d d)7 1-forms c o m m u t e , a n d one can p r o c e e d to the c o n s t r u c t i o n of the ghost s p e c t r u m w i t h o u t difficulty. It is for this r e a s o n that we have chosen in sect. 3 the s u p e r s p a c e of B o n o r a a n d Tonin. W e c o n c l u d e this a p p e n d i x b y the following remark. A true s u p e r s p a c e l a g r a n g i a n d e s c r i b i n g directly the field A(x, y, )7) has n o t yet b e e n constructed. This w o u l d be especially useful since, using a Parisi-Sourlas r e d u c t i o n [23], one c o u l d h o p e to o b t a i n a f o r m a l i s m directly in the q u o t i e n t p h y s i c a l space of d i m e n s i o n C( D 2 ] \

P

] "

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