Nuclear Physics B70 (1974) 39 50. North-Holland Publishing Company
SUPERGAUGE TRANSFORMATIONS
IN F O U R D I M E N S I O N S
J. WESS
Karlsruhe University
B. ZUMINO
CERN, Geneva
Received 5 October 1973
Abstract: Supergauge transformations are defined in four space-time dimensions. Their commutators are shown to generate 75 transformations and conformal transformations. Various
kinds of multiplets are described and examples of their combinations to new representations
are given. The relevance of supergauge transformations for Lagrangian field theory is explained. Finally, the abstract group theoretic structure is discussed.
1. Introduction
Supergauge transformations have been studied until now in dual models, especially in their formulation as two-dimensional field theories [ 1 - 4 ] . They transform
scalar (in general tensor) fields into spinors and boson fields into fermion fields.
This is possible because the parameters of the supergauge transformation are themselves totally anticommuting spinors. The commutator of two infinitesimal supergauge transformation is a conformal transformation in two dimensions. Invariance
under supergauge transformations is closely connected to the absence of ghost
states in the two dimensional field theory.
It is natural to ask whether one can define supergauge transformations in four
dimensional space-time. In this paper we show that this is indeed possible, although
the generalization from two to four dimensions is not completely straightforward
and presents some interesting new features. In four dimensions the commutator of
two infinitesimal supergauge transformations turns out to be a combination of a
conformal transformation and a 3'5 transformation. Supergauge transformations can
be represented on multiplets of fields, a given multiplet containing some tensors and
some spinors. Examples of such representations are given in sect. 3. A representation is characterized not only by a given multiplet of fields but also by a weight,
much as an ordinary tensor representation corresponds to a tensor density having a
given index structure as well as a certain weight. Representations for supergauges
can be combined into other representations, as exemplified in sect. 3. Using the
40
J. Wess, B. Zumino, Supergauge transformations
multiplet of a representation one can construct an invariant Lagrangian. More precisely, the Lagrangian transforms by a total derivative as one of the members of a
supergauge density; the four-dimensional action integral is invariant. Two examples
of this are shown in sect. 4. The examples given in this paper are the simplest cases
of a "tensor" calculus which generalizes to supergauge transformations the ordinary
tensor calculus of coordinate transformations.
From any particular representation one can abstract the "group" structure containing the supergauge transformations, the conformal transformations and the 75
transformations. Actually the corresponding algebraic structure is not a group or a
Lie algebra in the conventional sense, since the parameters of a supergauge transformation are not c-numbers but rather completely anticommuting quantities belonging to a Grassmann algebra. Nevertheless, one can find the parameter composition
law and verify that it satisfies the Jacobi identity. This is done in sect. 5.
The Lagrangian example given in this paper is a free field theory. Nevertheless it
is possible, by combining representations, to construct interacting field theories invariant under supergauge transformations and consequently under conformal and
~'5 transformations. Supergauge transformations should prove a useful tool for the
study of theories with massless particles or in the approximation in which the mass
can be neglected. They may also provide a natural way for the formulation of higher
internal symmetries linking mesons and baryons. We hope to come back to these
questions in a later publication.
In two dimensions it is possible to define [5] t generalized supergauge transformations which have as commutator general coordinate transformations, rather than
conformal transformations. It is likely that such generalized supergauge transformations exist also in four-dimensional space time. Their commutators would generate
an algebra containing that of general (Einstein) coordinate transformations. We are
planning to come back at some later time to this very interesting question.
2. Conditions on the parameters
It is convenient to study separately in this section the properties o f the infinitesimal parameter of a supergauge transformation. As we shall see, they will be
Majorana spinors a(x) subject to the condition
(7.0~ + 7~0~,
- ~l n . ~ '
X
a x ) ~ -- 0 .
(1)
This equation resembles the condition on the infinitesimal parameters ~/a of a conformal transformation
t In this paper (ref. [5]) we give a systematic description of supergauges in two dimensions and
their representations and we introduce the notion of generalised supergauges.
J. Wess, B. Zumino, Supergauge transformations
au~+a~u
1
x =0,
-~u~aX~
41
(2)
and indeed, if ix1 and o~2 satisfy (1), then
~u = 2 i~lTua2
(3)
satisfies (2). It is well known that (2) can be solved explicitly; the same is true for
(1).
First multiply (1) with 7 u obtaining
a~a = ¼vuvxaxa.
(4)
From this, multiplying by 3 u, it follows
aO~= 0 .
Then, multiplying (4) with 7rOy, we find that
OuTVOvc~ = 0 .
Therefore, always from (4),
au~va = 0,
is at most linear in x. Imposing (4) once more, it follows that
O~= a(0) + ,),uX/XO~(1) ,
(5)
where s 0 and a 1 are x-independent spinors. The forms (1) and (4) of the condition on
a are equivalent. It is easy to see that a third equivalent form, sometimes more convenient, is
7uTvauc~ + ½7vTuauc~ = 0 .
(6)
Observe that, if one assumes that cq and ot2 in (3) both have the form (5), }/a
takes the form, well known for a conformal transformation,
~u = cu + couvxV + 6 x u + aux2 -- 2x~,ax ,
where
c. : : i ~ t ° ) v . ~ ° ) ,
C O u v=- - C O v u = i ~ O ) ( T u 7
v - 7 v 7 ~ )"o~2(1) - ia(20)(TuTv
7vTo)a] 1) ,
42
J. Wess, B. Zumino, Supergauge transJorrnations
e = 2 i tZ(O)^'(l)~l ~2
au
--
a(0)a(l))
2
1
= 2 i~(1) . . . . (l)
1 I#~2 "
For later use we calculate the expression
r/= i/)uo~1T57u~2 - i0u~2T5Tuo~ 7 .
{7)
Using the explicit form for a 1 and o~2 one finds
r / = 4 f f(a1 )1~ tt 5 2c~(0)-d(1)72 5 la(0))"
Therefore r? is x-independent.
3. Supergauge transformations
Consider a multiplet consisting of a Majorana spinor ~ and four scalar fieldsA,
B, F and G. Let us define an infinitesimal supergauge transformation by
6A = i ~ ,
8 B = ia75 ~ ,
8 ~ = Ou(A
75B)TU{x + n(A
75B)TU0uc~ + F a + G75{x,
8 F = i~Tu3u~ + i(n - ~ )a u a T u ~ ,
8 G = i a 7 5 7 u 0 u ~ + i(n - ½)Oua757u ~ ,
{8)
where the parameter ~ x ) is an infinitesimal spinor which anticommutes with itself
and with ~ and commutes with the other fields. Furthermore a satisfies the differential equation discussed in the previous section. The number n is arbitrary (it
need not be an integer) and gives the weight of the multiplet. We say of a multiplet
transforming as in (8) that it belongs to a scalar representation of weight n.
The supergauge transformations generate a closed algebraic structure, similar to
a Lie algebra. To see this, let us evaluate the commutator of two infinitesimal supergauge transformations 8 t and 62, of parameters a 1 and o~2. Starting with the scalar
field A we have
8261A = i ~ 1(82 $ ) = i a l (0u(A - 3'5 B)TUa2
+ n ( A - 75B)TU3ua 2 + F a 2 + G T 5 a 2 ) .
If we use the relations
J. Wess, B. Zumino, Supergauge transformations
~1c~2 = ~2o~1 ,
43
~13`5o~2 = ~23`5c~1 ,
we see that the fields F and G drop out in the c o m m u t a t o r (62 ~ 1
the other hand, using
616 2)/1. On
o~13'uo~2 = --o~23',ao~1 ,
~l'),U0UO~2 -- ~2"),"0.ffl = 0U(~13`UOl2) ,
we obtain
(6261 -- 6162)A = ~taOuA + ½n3 u~UA + nrlB ,
where, as in the previous section,
r/= iOuo~13'5")'uot2 - 3u~23'53'U~l .
For the other fields the result is
[82,81]B=~UOuB+½nSta~UB
n~A,
[(52, 6 1] ~ = ~ " 0 , ¢ + (½n + ~)Ou~u~
[62, 61] F = ~ S uF + (½n + ¼)Ou~UF + (~ - n ) ~ a ,
[62,611G =~uouG+(½n+&)0u~uG
(,~-n)rIF,
where
R e m e m b e r that r/is i n d e p e n d e n t o f x . The c o m m u t a t o r of two supergauge transformations is a conformal transformation c o m b i n e d with a 75 transformation on t~
and a mixing o f A w i t h B and o f F with G. It is also easy to verify that the commutator of a conformal transformation with a supergauge transformation or of a 3'5
transformation with a supergauge transformation are again supergauge transformations. In this sense the algebra closes.
To explain further how the above result for the c o m m u t a t o r emerges, we indicate here as an example the evaluation of some terms of the c o m m u t a t o r on ~. One
has
44
J. Wess, B. Zumino, Supergauge transybrrnations
6261 ~ = i3,(°t2~)7"~1 - i3,(~275 ~)TST"C~l
+ in(~2~)T"3ga I
in(o~2T5 t~)V57"3,e q + i(o.27"3,~)o q
+ i(n - ½)(3,~27vtk)a 1 + i(c~2757,3,tk)75c q
+ i(n
½)(b,~1757,~k)Tsa 1 .
Let us consider separately the terms with derivatives of ~. They are
i(c~23" ff;)7"o~'1 - io~2T53, ~ 757"°q + i(~2T" 3. ~)~1
+ i(oL2TS')'"0,~)TSCt I .
Using the rearrangement formula of the appendix, the first of these terms can be
transformed as
i ( ~ 2 3 , t~).),.a 1 = 1
i (o~2"/Ao~1)')"7 `4 0 . ~ ,
where we imply summation over all sixteen matrices 7.4- Similarly. for the other
terms,
-- i(O~2T5 0 . ~)T5 T"~I = L
4 i(d27A CZl)757"7 "4 7 5 0 . t~ ,
i(O~2T"i). ff)c~1 = - k i(~2TA ~1 ) TAT"O. t~ ,
i(~2757"3, ff)TS~x1 = _ ¼i(~27A~l)T5TAT57"3, ff .
For the c ommu tator (6261 - (51(52) l,~, in all these term s ~2"/A o~1 is replaced by
d2(TA - ~A )%" Therefore only 7.4 = 7v and 7.4 = 7v7~ (v < o) survive. If we take
the latter, the four terms cancel each other. If we take the former we obtain finally
the contribution to the commutator
i(~-27val)(')."T v + TvT")3,tk = 2i~17,~23 tk .
The other terms in the commutator on ~ are evaluated in a similar way. In the rest
of this paper we shall not give all the details of the calculations, which are often
long and rather tedious. They are based on the well known properties of the 7 matrices and on the rearrangement formula collected in the appendix.
Two supergauge representations can be combined to a third. Let the fields A 1,
B1, ~ i , F1, Gt belong to a scalar representation of weight nl, as described above,
and let the fields A 2, B2, ~2, F2, G2 belong to a scalar representation of weight
n 2. The new multiplet A, B, qJ, F, G defined by
A =A1A 2-B1B 2,
45
J. Wess, B. Zumino, Supergauge transformations
B = A 1 B 2 +B1A 2 ,
~=(A 1
v5B1)ff2 + ( A 2 - T 5 B 2 ) ~ ,
F = F 1 A 2 + F 2 A 1 + G 1 B 2 + G 2 B 1 - i~1t/.,2,
G = G1A 2 + G2A 1 - F 1 B 2
F 2 B 1 + i~1"Y5~2,
(9)
belongs to a scalar representation of weight n 1 + n2, which means that it transforms
like (8) with n = n 1 + n 2. This can be verified directly. For instance
6A = A l i ~
2 +A2io@ 1 - Bli~9,5~ 2 - B2i~75~1
= i~(A lff 2 + A z f f 1 - Bl'),5 ff 2 - B2"1,5 ~1) = i~ff .
Similarly for the other fields. For the spinor field one must use the rearrengement
formula described in the appendix. The combination of two scalar representations
into a third scalar representation just described gives only the simplest example of a
generalized tensor calculus for supergauge transformations, the theory of which we
have not yet fully developed. It is clearly the main tool for the construction of invariant interactions.
Before closing this section we give another example of a representation. Let us
consider a multiplet consisting of four scalar fields D, C, M, N, of the vector field
vu and of the two spinor fields X and X. We call it a vector multiplet, because o f the
presence in it of the vector field v u. We shall say that this vector multiplet transforms according to a vector representation of weight n if, under a supergauge transformation,
6 D = i~757uOu?t + ½i(n - 1)3u~-753,uX + i(n - l)OU~75OuX,
8 C = i~75 X,
6 M = i~X + i~'yuOuX + ~ i ( n - 3)3u~TuX,
8 N = i~75 x + i~757uauX + ,~i(n
3)Oud757u X ,
6u u = i~7u?t + i~OuX + inOuaX ,
6 X = "YUvua - auCT5TUoe - ~(n - 1)CT5TuO**a + (M + 75N)oe,
6X=
~(auv v
+k(n
(10)
avvu)yuTva+DT5a+(n-1)vUaua
1 ) ( M - 7 5 N ) T U a u a - - (n - 1)OuCY5auoe.
Just as for the scalar representation, also for the vector representation one can
calculate the commutator of two infinitesimal supergauges. One finds
[62,61 ] D = ~UOuD + k(n + l)i)u~j'UD + ½(n - 1)D,~,uc3uC,
46
J. Wess, B. Zurnino, Supergauge transformations
[62,6 1 ] C = ~u0,zC + ¼(n - 1)0,z~zC,
[ a 2 , a l ] M = ~UOuM + 2}nOu~UM + :}r?N,
[ a 2 , a l l N = UO.N + ~rnauUN - ~-r/M.
[62,~11 v v = ~laOuvv + av~Uv" +~(n -- 1)a#~tavv ,
[~2,61]
[~52, 81 ]
+ ~-(n
(11)
1)D~UTUX.
Observe that, except for n = 1, the transformation law f o r D and for X is not simply
that of a conformal transformation, but has additional terms containing []~u. In the
next section we shall see that, for the case of D, this corresponds to the very well
known fact that the free Lagrangian for a scalar field does not transform simply as
a density under conformal transformations. The additional term corresponds exactly to the additional term in 6 D above. These additional terms do not alter the algebraic structure described in sect. 5.
4. Two Lagrangian examples
Consider a scalar multiplet of weight ½. It consists of fields A, B, ~, F, G transforming as in (8) with n = ½. We claim that the Lagrangian
L =-½0#AOUA ½auBa#B
½i~Tuau~+½(F 2 +G 2)
(12)
gives rise to an invariant action integral. The Lagrangian is not itself invariant.
Rather, it is one of the members of a multiplet and it changes by a total derivative
under a supergauge transformation. Observe that the above Lagrangian is essentially
the sum of the free Lagrangians for the fields A, B and ~. The variational equations
for the fields F and G are simply F = 0 and G = 0. Nevertheless, the fields F and G
are essential for the transformation properties and especially for the closing of the
commutator algebra described in sect. 3.
The multiplet to which the above Lagrangian (12) belongs is simply a vector
multiplet of weight 3. Indeed one can verify that, when the fieldsA, B, ~, F a n d G
transform according to (8) with n = ½ (scalar representation of weight ½), then the
fields D, C etc. defined by
D=2L,
C=I(A2+B2),
N=BG-AF,
X=(B-75A)t~,
M=AG+BF,
vu = B ~ u A - A ~ u B - ~ i ~ T s T u ~ ,
X = {G+75 F
7uOu(B-75A)}~,
(13)
J. Wess, B. Zumino, Supergauge transformations
47
transform according to a vector representation of weight 3 (given by (10) with
n = 3). The first of these fields is twice the Lagrangian. It follows, in particular,
that
6L = ~-i~u(c~757"X + 2~uaTSX),
and the four-dimensional action integral is invariant. Here we have used the fact
that second derivatives of a vanish. The above formulas (13) give an example of
how, starting from a representation, one can construct a new one. In (13) derivatives of the fields occur, while in our earlier example (9) (in which the two representations could be taken to be identical) the fields occurred without derivatives.
Having treated the Lagrangian for a scalar multiplet, one may ask whether one
can construct a Lagrangian for a vector multiplet. The answer is affirmative. Let
the fields D, C, M, N, vu, X and X belong to a vector multiplet of weight n = 1 (representation given by (10) with n = 1). Then the fields A, B etc. defined by
A =½ixx,
B =½i~75x,
=DTsX+½
F - T- - v 1u u v U
V
vuvTuTvX
,
+ iXT"0,X - D 2
G = - ¼ e U W ~ v , u V c ~ p - iX3,57u~,X,
(14)
transform according to a scalar representation of weight ~ (given by (8) for n = }).
Here
Viav = 3 # v v -- Ovvt~,
e0123 = 1 .
Now
.2 = - ½F= _ k vuvv~" _ ½ i X T u 3 u X + ½D 2
is essentially the sum of the free Lagrangian for the vector field vu and for the
spinor field X, the extra term ½D 2 giving as variational equation simply D = 0.
According to (8) with n = 3,
8 . 2 = - ½i a . ( a ~ . ~ )
and the action integral is invariant. Observe that in (14) the other fields of the vector multiplet, namely C, N, M and ×, do not occur. This can happen because, for
n = 1, none of them enters in the transformation laws of X and D, and X enters in
that of vu but only in a gauge transformation. In fact, when n = 1, one can restrict
the vector multiplet to the fields our , ff a n d D , drop the additional fields, and write
the transformation law for the restricted vector multiplet simply as
48
J. Wess, B. Zumino, Supergauge transformations
6D = iwY5y#au~,,
av**~=iau(av~?,) ia~(dv~x).
6X=
1
-iVuvyUTV~ + D75 ~ .
Now, of course, one must impose the restriction
OoO~v + O#Ovo + OvOo~ = 0 ,
which, however, is respected by the supergauge transformation. For n 4= 1, on the
other hand, the other fields of tire vector multiplet are necessary to give the correct
definition of the vector representation and, in particular, for the closing of the commutator algebra.
The two Lagrangian examples given in this section show that Lagrangians giving
rise to action integrals invariant under supergauge transformations can belong to
different representations. Here we found that the Lagrangian for a scalar multiplet
is a member of a vector multiplet while the Lagrangian for a vector multiplet is a
member of a scalar multiplet. In both cases the weight is such that the Lagrangian
transforms by the addition of a total derivative. Presumably the same applies in the
case of invariant interactions, and they also can belong to different types of multiplets.
5. Algebraic structure
In this section we abstract from the transformation law (8) for a scalar multiplet
the composition law of the parameters. This is the analogue of finding the structure
constants of a Lie algebra, except that our 'algebraic structure is a kind of generalized Lie algebra, because of the anticommutation property of the parameters c~.
Under a combined supergauge, conformal and T5 transformation of parameters
c¢, ~ , 7), the particular members A, B and ~ of the multiplet, for instance, transform as follows
6A = i~O + ~UOuA + ~nSu~UA + nr~B ,
6 B = ia75 ~ + ~UOuB + ½nO u~UB - nr~A ,
6 ~ = 0u(A
TsB)TUa + n(A
+ U a u ~ + (½n + ~ )a ~ . ~
TsB)TUOua + F a + GT5 a
+ }(a ~
- a.~)~u.~
- (~ - ,~)~vs ~ .
Here the commuting parameters ~u and ~/are additional parameters, independent
of a. While r~ is x-independent, a and ~u satisfy respectively (1) and (2). We can
now evaluate the commutator of two such transformations, the first 61 o f parame-
J. Wess, B. Zumino, Supergauge transformations
4!)
ters a 1 , ~ , r/l , and the second 62 of parameters oe2, se~, r/2, for instance on the
field A. One sees easily that the c o m m u t a t o r [62, 61 ] is again a transformation of
the same kind, with parameters a, ~, r/given by
O/ = ~ a/.t ~ 2
+¼r/13`5a2
~p
~
1
/.t
(1 and 2 exchanged),
u
r/= iO~ct13`53`~a2 - ia~a23`53`#a 1 ,
(15)
This composition law for the parameters defines the algebraic structure and is independent of the particular representation from which we have abstracted it. We could
have used, for instance, the vector representation (12) described in sect. 3.
Clearly, if one exchanges a l , ~ , r/1 with a2, ~ , r/2, a, ~ , r/change sign. A further condition which must be satisfied is the Jacobi identity for the composition of
three transformations. For instance, if we define
r/3,12 = i ~a33`53`~a
iO~a3`53`~a3 ,
with a given by (15), it must be identically
r/3,12 + r?1,23 + r/2,31 = 0 .
The same relation must be valid for ~ , 1 2 and a3,12 , defined in an analogous manner. The Jacobi identity can naturally be expected to be satisfied, since we have
derived the composition law from a particular representation, but we have verified
it directly, again using the formulas collected in the appendix.
The fact that the conformal algebra in four dimensions can be extended to the
algebraic structure described above, including 3`5 and supergauge transformations,
and that the entire algebraic structure is generated by the supergauge transformations, does not appear to have been realized before. It is interesting in itself, irrespective of the field theoretic applications we have in mind.
Appendix
We use a Majorana representation. The four 3` matrices have real elements and
satisfy
3`#3`v + 3`v3`~ = 2 ~ u ,
with r/00 = - 1 . The sixteen basic matrices 7/1 (A = 1, 2 ..... 16) are 1,3'5 = 70")'172
3'3, 7 u, 757 u, 7uTv (/J < u). They are all real and 3`2 = - 1,3`[ = - 3`5, where T denotes the ordinary transposed of a matrix.
J. Wess, B. Zumino, Supergauge transformations
50
For any four by four matrix I" define the adjoint
~" =
,),o['T,),O.
Then, for any two hermitian a n t i c o m m u t i n g spinors o~1 and o~2,
~1I'o~2 =O~2Pal , ~ = ~ T 0 .
Now ~'A = 7.4 for 1,3'5, 75Tu , while ~A = -- 7A for 3'u, 3,u3,v (/l < u).
The sixteen matrices 7,4 have squares equal to + l. For any 7,4 define TA so
that 7,4 TA = + 1 (no summation). The following rearrangement formula is then
valid
(~-1~)O~2 = -- k ~
A
(~I"YA~2)TA~ ,
where a l , a 2 and ~ are any three spinors. The minus sign comes from the antic o m m u t a t i o n property of the spinors.
The formulas collected in this appendix are sufficient to derive all the results
given in the text of this paper.
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J.-L. Gervais and B. Sakita, Nucl. Phys. B34 (1971) 633.
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