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An alternative minimal off-shell version of N = 1 supergravity
Volume 105B, number 5
PHYSICS LETTERS
15 October 1981
AN ALTERNATIVE MINIMAL OFF-SHELL VERSION OF N = 1 SUPERGRAVITY Martin F. SOHNIUS The Blackett Laboratory, Imperial College, London SW 7, UK and Peter C. WEST Department of Applied Mathematics, King's College, London WC 2, UK Received 8 July 1981
A new formulation ofN = 1 Poincar~ supergravity is presented with an axial vector A~ and an antisymmetric tensor auu as auxiliary fields. The theory has a local U(1) chiral gauge invariance with 6A~ = 0~A and 6 qJ~ = -ATs ~/s, and a vector gauge invariance with 6a~u = Og#v - Ov#~. The auxiliary fields then provide six off-shell degrees of freedom and are thus a minimal set.
Early in the development of supersymmetry it was discovered [1 ] that the supersymmetry current and the e n e r g y - m o m e n t u m tensor (stress tensor) of several different N = 1 supersymmetric models lie in a supermultiplet. It was observed that, when properly "improved", this multiplet had the same structure for all models investigated. It has also been noted [1,2] that there should be an intimate connection between this multiplet of currents and the structure o f a supersymmetric theory of gravitation. Just as the stress tensor acts as source for the gravitational field in Einstein's theory, the supercurrent is the source for the field of the spin-3/2 "gravitino". However, while the classical on-shell states of N = 1 Poincard supergravity [3] are just the graviton and the gravitino, any multiplet o f currents contains more fields than just the stress tensor and the supercurrent. With the discovery of the minimal off-shell formulation of N = 1 Poincar6 supergravity [4] it became clear that the additional fields in the current multiplet are sources for auxiliary fields: the superalgebra closes on the current multiplet subject to conservation laws for the currents and it closes on the associated set of gravitational fields subject to field-dependent gauge transformations. We expect that any multiplet of currents thus corresponds to some off-shell supergravity multiplet. 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.75 © 1981 North-HoUand
The simplest multiplet of currents [1 ] contains a conserved axial vector Cu, a conserved Majorana vector-spinor Ju and a conserved symmetric tensor Our: 8c.
=
,
6 J u = 2iTv~Ouv+i~t75~Cu + 1. ~leu.Kxv v ~a~ C X, 6Our : - ¼ i ( ~ o u ~ a x J v + govxOhSu) .
(1)
If the variation of any field q~ is interpreted as 6q~ = i [ ~ , ~ Q ] , this gives rise to the supersymmetry algebra when integrated over d3x. The stress tensor is traceless, 0u g = 0, and the super current is "7-traceless", 7uJu = 0. This implies that certain moments of these currents are also conserved, and indeed the current multiplet (1) is that of a superconformally invariant theory. The corresponding set of gravitational fields + 1, the vierbein eg a, the gravitino field qJ, and an axial vector with a pseudoscalar gauge invariance b u ~ b u + ~)/~A, forms the multiplet of N = 1 conformal supergravity [5]. Since einsteinian (and newtonian) gravity is intrin4:1 Notation: #, v, K, ~, are curved-space "world" indices; a, b, c, d are tangent space "local" vector indices. Oab = 1. 2 ~1 [3'a, "rb];~'s = -1; noo = +1.
353
Volume 105B, number 5
PHYSICS LETTERS
sically not conformall,y invariant, due to the mass dimensions in the gravitational constant, we expect to need, as a source for Poincar~ supergravity, the larger multiplet of currents of a theory with broken conformal invariance. M1 supersymmetric models studied so far break the conformal invariance of (1) by inclusion of a scalar A and a pseudoscalar B, which together with 3`.J, ~uC u and OuU from a chiral submultiplet, the "trace multiplet" or "anomalies multiplet" [1]. The six fields Cu,A and B then correspond to the six auxiliary fields b u , M and N of the "minimal scheme" of off-sheU Poincar6 supersymmetry [4] (note that Cu has lost its conservation law and b u its gauge-invariance.) Motivated by an attempt to understand the anomaly situation in supersymmetric theories [6], we have investigated the most general possible structure of multiplets of currents, independent of particular model field theories and restricted only by the requirement that the charges which correspond to the currents should span the supersymmetry algebra. We have found that the scheme (1) is unique for the currents of N = 1 superconformal theories (provided one demands gaugeinvariant currents) [7], but that ways other than that of ref. [1 ] exist to enlarge the multiplet and break superconformal symmetry down to super-Poincar6 symmetry. One such way is by inclusion of a conserved, antisymmetric tensor field tuv which appears in the transformation law for Ju' The multiplet o f currents is then:
6Our
+1
+ (A u _ 3 l z . ) C u
_ l aUVtuv '
With
(3)
V U -= ~1 eUVKx~vaKx ,
(4)
we can derive the linearised transformation laws for the gravitational fields:
5guy = - 2i(~3'~ d/v + ~3`v d/u) + ~u~u + ~v~u , 6 d/u = ¼ iaKX[3Kgxu + 3`5~Au -- 3`5~Vu 1.
-~louu3,sfVU + 3ue , 8A u = -~3`uo~x3`50~ d/x + OuA, 8auv = 2i~3`ud/~ - 2i~%d/u + bu~ ~ - 3v/3u .
(5)
The transformations (5) include not only supersymmetry transformations (parameter ~') but also various gauge transformations (parameters c~u, e, A and/3u) under which the action is invariant, due to the conservation laws, and which are necessary to close the algebra. The improvement term -~ Vu in (3) is purely conventional and was chosen to render 8A u particularly simple. A linearised lagrangian, whose action is invariant under (5), is
(6)
where Euv is the linearised Einstein tensor
~ieuvKX3`v~OKC ?"
EUv -_ 1 (~uaKg w + ~v~KguK __ iS]guy __ rlUv~XgKh
--Z i(~ oux aXJv + ~°uX ~XJu) ,
6tuu _ ~ e u ~ x ~3`5 ~3`x3`. j .
(2)
With unaltered transformation laws for Cu and Ou~ , the algebra now closes without setting 3' " J and OuU to zero. The axial current is still conserved, and the trace multiplet contains only 3` • J, tuu and 0u u. In spite of the conservation law Outu~ = O, the field tu~ does not give rise to a new charge in (Q, C)}, since it can be written as the dual of a curl, in agreement with (2). The multiplet structure of the trace multiplet is well 354
01 = _1 g.VO u v + ~gj.
- 3 VuVU+4A u VU,
~Ju = 2i3`"fO uu + 2i3`u~ tuu •
known from supersymmetric gauge theories where X, *fur and D form a similar multiplet. The fields Ouu,Ju, Cu and tuv should be sources for fields gu v, d/u' Au and a u v of a corresponding supergravity theory• By demanding invariance of the action for the minimal-coupling lagrangian
J2 G = ¼gUUEuv + 2 ieuuK x t~u3`u3,5 ~ d/x
= - 3`s J . ,
+ fft3`5~Cu
15 October 1981
+ rluv[S]g~~ -- ~uOvgK K) . Mready at the linearised level, we can inspect the onshell field content of the theory: the equations of motion
O=Euu,
O=C~U=2ieUUKX3`U3`53Kd/x ,
O=4VU--eUV~XOvaKx , 0 = eUV~X(-63vV u + 4 3 v A u ) ,
(7)
Volume 105B, number 5
PHYSICS LETTERS
describe a free graviton and a free gravitino, while the gauge invariant parts of apv , namely Vp, and of A u, namely fp v - OpA v - 3vAp are set to zero. These fields become "pure gauge" and do not carry any onshell degrees of freedom. Off-shell they represent a total of six degrees of freedom, which is the minimal possible number of auxiliary fields. The task of completing the transformations (5) and the lagrangian (6) to their full non-linear form can be approached in various ways. One is the well-known method of Noether-coupling by which the lagrangian and the transformation laws can be completed order by order in the gravitational coupling. Only a few terms in the transformation laws are not uniquely fixed by the requirement of invariance of the action, and these are determined by the requirement of closure of the algebra. An alternative method is provided by the general structure of gauge theories on a curved superspace. Since the supertransformations of all geometrical fields, in particular of the connections and vielbein fields, are uniquely determined in terms of torsion and curvature [8], the linearised transformation laws are sufficient to actually read off the linearised torsion constraints of the theory. One then checks the Bianchi identities for consistency of those constraints in the full non-linear theory and for possible as yet undetermined torsion components [9]. This method, which also illuminates the superspace geometry of our theory, will be described in detail in a separate publication. The full supertransformation laws for the vierbein eua , the Rarita-Schwinger field flu, the axial vector Ap and the antisymmetric tensor apv are
6eua
=
-2i~7a~p,
~//.t
=
(~)bt~--~5~Vp
--
1.l O p a 75~'Va ,
(8)
Va=e-le --
p
aePVKX(!3vaKa --~1 4
i~vy~$x)
Q) p ~ - ap~ - k i~paboab~ + ApT5~ .
(9) (10)
The Lorentz connection COpab is the usual expression in terms of the vierbein and Rarita-Schwinger field (second-order formalism):
COpab - 51 (ea v ~pevb _ ebv 3peva) _
1 -- ~
eaKebx(O ~gxp -- 8~ gKu )
- i ( ~ ' p ~.Vb+ ~
~b -- ~.Vb~V~),
and t e a is the super-covariantised Rarita-Schwinger field strength [ ~ u is defined as in eq. (10), i.e. without a Christoffel symbol]:
Q~a = ieabcd3,b3, 5 Tcd , Tab --eaUebv( c-Ou~v -- C-~v~u) + 75 ~aVb --3'5 ~b Va 1.
1.
-- 2 l°ac 75 ~b Vc + 2 l°bc')'5 ~a V
c.
(12)
The commutator of two supertransformations closes on general coordinate transformations ~G, supertransformations 5 S, local Lorentz transformations ~L, chiral gauge transformations 5C and vector gauge transformations 6 v as follows: [8 S(~'I ), 6 S(~2)1
(13)
= ~G(~#) + ~s(e) + 5L(Xab) + 6c(A) + 6V(~3g), with parameters ~P = 2ieaP~l 7a~'2 ,
e =-~p~p,
k ab = ~ p ~ p a b - e a b c d ~ c V d ,
A =-~pAp,
flu=~u +auv ~v"
A chiral gauge transformation acts only on ffp and Ap, 6C~p=-A75q;p,
6cA p =~uA,
(14)
and a vector gauge transformation only on apv:
6vauv = ~u~v - ~v~p •
(15)
The following lagrangian gives rise to an action f d4x Z? which is invariant under all these transformations:
22=
1 -~
eeaPebVRpv ab + 2ieUV~xfjpTv75 ~ K~x
~
6A u = -½~75 7pTa cl~ a, 6a v = 2i~3'p ~v - 2i~7v~p, with
15 October 1981
(1 1)
- 3eVa Va + eUVKXApavaKx ,
(16)
with e = det ep a ,
Rpvab = ~pcovab -- ~v~pab + ¢opaccovc b -- 6JvaC~pcb This lagrangian generalises the linearised ~ G of eq. (6) to its full non-linear form. Invariance of the action under the transformations (8) can be proved rather straightforwardly by use of the so-called "1.5 order formalism" [10] which makes it unnecessary to vary wuab(e, ~). 355
Volume 105B, number 5.
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It is important to note that the action is invariant under the local chiral transformations of eq. (14). This is not an invariance possessed bY either the theory without auxiliary fields [3] of by the theory with the minimal auxiliary fieldsM, N and b u [4]. The creation of a new invariance by the inclusion of auxiliary fields is an interesting phenomenon in view of the current speculations about N = 8 supergravity. I'n fact, straightforward extrapolation from the present result would lead us to expect the internal symmetry of the N = 8 superconformal group, namely U(8). Given that there are now two "minimal" formulations o f N = 1 supergravity, the question naturally arises whether they represent equivalent physical theories. For example, can they be coupled to matter in the same way? To answer this question, it is necessary to develop the tensor calculus for the new formulation as it has been done for the M , N , b u version [11 ]. We have constructed this tensor calculus and will present it elsewhere [12]. Here we report only its general features: like before, we can construct vector, chiral and linear supermultiplets and find rules for combining them; we can also find F- and D-density formulas. However, in the new tensor calculus we must assign local chiral weights to supermultiplets and the Fdensity formula exists only for chiral multiplets with a certain chiral weight. This restricts the type of coupling one can achieve; i.e. the terms XX or ~ u o u v ~ v are not invariant if X and ~ku have chiral weights. Thus both, mass-terms in conjunction with interaction terms in matter coupling and a cosmological constant are ruled out. In this respect, our formulation differs significantly from Einstein's theory and from the other formulations o f N = 1 supergravity. One can, however, construct a Maxwell action and the kinetic action for a chiral matter multiplet, as well as a Fayet-Iliopoulos term [13], the latter providing an example of the new way to break conformal invariance. In the new tensor calculus the structure of the gravity sector of the theory is described by an "Einstein multiplet" which is now of the Maxwell type [cf. the comments after eq. (2)] and begins with the spinor ~/ac]~a and by a chiral "Weyl multiplet" which begins with the spin-3/2 part of Tab. The differences between the two minimal formulations seem to indicate that they are indeed different theories, at least one would expect their topological properties to be different. The major source of the differences mentioned 356
15 October 1981
above is the local chiral invariance and one may wonder if this invariance survives in the quantum theory. Should the chiral invariance be broken by the quantum theory, the field A u would lose its gauge invariance and this would upset the off-shell Fermi-Bose balance required by supersymmetry. It would be of interest to use the closed algebra of eq. (13) to evaluate the Feynman rules of the theory using the BRS prescription. Finally, we note that the theory presented here has a radically different geometrical interpretation (details to be given elsewhere [12]) from the previously known minimal formulation. In its superspace formulation it is essential, unlike in the Wess-Zumino formulation [14], to introduce an additional local chiral gauge transformation in the tangent space with an associated gauge field. The constraints are
Tc~oa = - 2 i (Tac)a# , 0 = T~5 ~t = G a b = Taa~(qCb(1 -+ i')'5))#a , O= TabC= fa~ , where the components TAB C are the supercovariant torsions and the f i B are the supercovariant chiral field strengths. This scheme, however, does not geometrically explain the vector gauge invariance on the auv field. auv actually appears as solution to a differential constraint ~ a v a on one of the components in T a J . Geometrically, it can probably be interpreted as the remnant of a local translation which has not completely been identified with general coordinate transformations.
References [1] S. Ferrara and B. Zumino, Nucl. Phys. B87 (1975) 207. [2] V. Ogievetsky and E. Sokatchev, Nucl. Phys. B124 (1977) 309. [3] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [4] K. SteUe and P.C. West, Phys. Lett. 74B (1978) 330; S. Ferrara and P. van Nieuwenhuizen, Phys. Lett. 74B (1978) 333. [5 ] M. Kaku, P. Townsend and P. van Nieuwenhuizen, Phys. Rev. D17 (1978) 3179. [6] M. Sohnius and P.C. West, Phys. Lett. 100B (1981) 245. [7] M. Sohnius and P.C. West, in preparation. [8] F.W. Hehl, P. vonder Heyde and G.D. Gerlick, Rev. Mod. Phys. 48 (1976) 393; J. Wess and B. Zumino, Phys. Lett. 74B (1978) 51.
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PHYSICS LETTERS
[9] P. Breitenlohner and M. Sohnius, Nucl. Phys. B165 (1980) 483. [10] A.H. Chamseddine and P.C. West, Nucl. Phys. B129 (1977) 39; P. van Nieuwenhuizen and P. Townsend, Phys. Lett. 67B (1977) 439. [ 11 ] P. van Nieuwenhuizen and S. Ferrara, Phys. Lett. 76B (1978) 404; 78B (1978) 573;
15 October 1981
K. Stelle and P. West, Phys. Lett. 77B (1978) 376; Nucl. Phys. B145 (1978) 175. [12] M. Sohnius and P. West, in preparation. [13] P. Fayet and J. Iliopoulos, Phys. Lett. 5 1B (1974) 461; K. SteUe and P. West, Nucl. Phys. B145 (1978) 175. [14] J. Wess and B. Zumino, Phys. Lett. 66B (1977) 361; 74B (1978) 51.
357
PHYSICS LETTERS
15 October 1981
AN ALTERNATIVE MINIMAL OFF-SHELL VERSION OF N = 1 SUPERGRAVITY Martin F. SOHNIUS The Blackett Laboratory, Imperial College, London SW 7, UK and Peter C. WEST Department of Applied Mathematics, King's College, London WC 2, UK Received 8 July 1981
A new formulation ofN = 1 Poincar~ supergravity is presented with an axial vector A~ and an antisymmetric tensor auu as auxiliary fields. The theory has a local U(1) chiral gauge invariance with 6A~ = 0~A and 6 qJ~ = -ATs ~/s, and a vector gauge invariance with 6a~u = Og#v - Ov#~. The auxiliary fields then provide six off-shell degrees of freedom and are thus a minimal set.
Early in the development of supersymmetry it was discovered [1 ] that the supersymmetry current and the e n e r g y - m o m e n t u m tensor (stress tensor) of several different N = 1 supersymmetric models lie in a supermultiplet. It was observed that, when properly "improved", this multiplet had the same structure for all models investigated. It has also been noted [1,2] that there should be an intimate connection between this multiplet of currents and the structure o f a supersymmetric theory of gravitation. Just as the stress tensor acts as source for the gravitational field in Einstein's theory, the supercurrent is the source for the field of the spin-3/2 "gravitino". However, while the classical on-shell states of N = 1 Poincard supergravity [3] are just the graviton and the gravitino, any multiplet o f currents contains more fields than just the stress tensor and the supercurrent. With the discovery of the minimal off-shell formulation of N = 1 Poincar6 supergravity [4] it became clear that the additional fields in the current multiplet are sources for auxiliary fields: the superalgebra closes on the current multiplet subject to conservation laws for the currents and it closes on the associated set of gravitational fields subject to field-dependent gauge transformations. We expect that any multiplet of currents thus corresponds to some off-shell supergravity multiplet. 0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.75 © 1981 North-HoUand
The simplest multiplet of currents [1 ] contains a conserved axial vector Cu, a conserved Majorana vector-spinor Ju and a conserved symmetric tensor Our: 8c.
=
,
6 J u = 2iTv~Ouv+i~t75~Cu + 1. ~leu.Kxv v ~a~ C X, 6Our : - ¼ i ( ~ o u ~ a x J v + govxOhSu) .
(1)
If the variation of any field q~ is interpreted as 6q~ = i [ ~ , ~ Q ] , this gives rise to the supersymmetry algebra when integrated over d3x. The stress tensor is traceless, 0u g = 0, and the super current is "7-traceless", 7uJu = 0. This implies that certain moments of these currents are also conserved, and indeed the current multiplet (1) is that of a superconformally invariant theory. The corresponding set of gravitational fields + 1, the vierbein eg a, the gravitino field qJ, and an axial vector with a pseudoscalar gauge invariance b u ~ b u + ~)/~A, forms the multiplet of N = 1 conformal supergravity [5]. Since einsteinian (and newtonian) gravity is intrin4:1 Notation: #, v, K, ~, are curved-space "world" indices; a, b, c, d are tangent space "local" vector indices. Oab = 1. 2 ~1 [3'a, "rb];~'s = -1; noo = +1.
353
Volume 105B, number 5
PHYSICS LETTERS
sically not conformall,y invariant, due to the mass dimensions in the gravitational constant, we expect to need, as a source for Poincar~ supergravity, the larger multiplet of currents of a theory with broken conformal invariance. M1 supersymmetric models studied so far break the conformal invariance of (1) by inclusion of a scalar A and a pseudoscalar B, which together with 3`.J, ~uC u and OuU from a chiral submultiplet, the "trace multiplet" or "anomalies multiplet" [1]. The six fields Cu,A and B then correspond to the six auxiliary fields b u , M and N of the "minimal scheme" of off-sheU Poincar6 supersymmetry [4] (note that Cu has lost its conservation law and b u its gauge-invariance.) Motivated by an attempt to understand the anomaly situation in supersymmetric theories [6], we have investigated the most general possible structure of multiplets of currents, independent of particular model field theories and restricted only by the requirement that the charges which correspond to the currents should span the supersymmetry algebra. We have found that the scheme (1) is unique for the currents of N = 1 superconformal theories (provided one demands gaugeinvariant currents) [7], but that ways other than that of ref. [1 ] exist to enlarge the multiplet and break superconformal symmetry down to super-Poincar6 symmetry. One such way is by inclusion of a conserved, antisymmetric tensor field tuv which appears in the transformation law for Ju' The multiplet o f currents is then:
6Our
+1
+ (A u _ 3 l z . ) C u
_ l aUVtuv '
With
(3)
V U -= ~1 eUVKx~vaKx ,
(4)
we can derive the linearised transformation laws for the gravitational fields:
5guy = - 2i(~3'~ d/v + ~3`v d/u) + ~u~u + ~v~u , 6 d/u = ¼ iaKX[3Kgxu + 3`5~Au -- 3`5~Vu 1.
-~louu3,sfVU + 3ue , 8A u = -~3`uo~x3`50~ d/x + OuA, 8auv = 2i~3`ud/~ - 2i~%d/u + bu~ ~ - 3v/3u .
(5)
The transformations (5) include not only supersymmetry transformations (parameter ~') but also various gauge transformations (parameters c~u, e, A and/3u) under which the action is invariant, due to the conservation laws, and which are necessary to close the algebra. The improvement term -~ Vu in (3) is purely conventional and was chosen to render 8A u particularly simple. A linearised lagrangian, whose action is invariant under (5), is
(6)
where Euv is the linearised Einstein tensor
~ieuvKX3`v~OKC ?"
EUv -_ 1 (~uaKg w + ~v~KguK __ iS]guy __ rlUv~XgKh
--Z i(~ oux aXJv + ~°uX ~XJu) ,
6tuu _ ~ e u ~ x ~3`5 ~3`x3`. j .
(2)
With unaltered transformation laws for Cu and Ou~ , the algebra now closes without setting 3' " J and OuU to zero. The axial current is still conserved, and the trace multiplet contains only 3` • J, tuu and 0u u. In spite of the conservation law Outu~ = O, the field tu~ does not give rise to a new charge in (Q, C)}, since it can be written as the dual of a curl, in agreement with (2). The multiplet structure of the trace multiplet is well 354
01 = _1 g.VO u v + ~gj.
- 3 VuVU+4A u VU,
~Ju = 2i3`"fO uu + 2i3`u~ tuu •
known from supersymmetric gauge theories where X, *fur and D form a similar multiplet. The fields Ouu,Ju, Cu and tuv should be sources for fields gu v, d/u' Au and a u v of a corresponding supergravity theory• By demanding invariance of the action for the minimal-coupling lagrangian
J2 G = ¼gUUEuv + 2 ieuuK x t~u3`u3,5 ~ d/x
= - 3`s J . ,
+ fft3`5~Cu
15 October 1981
+ rluv[S]g~~ -- ~uOvgK K) . Mready at the linearised level, we can inspect the onshell field content of the theory: the equations of motion
O=Euu,
O=C~U=2ieUUKX3`U3`53Kd/x ,
O=4VU--eUV~XOvaKx , 0 = eUV~X(-63vV u + 4 3 v A u ) ,
(7)
Volume 105B, number 5
PHYSICS LETTERS
describe a free graviton and a free gravitino, while the gauge invariant parts of apv , namely Vp, and of A u, namely fp v - OpA v - 3vAp are set to zero. These fields become "pure gauge" and do not carry any onshell degrees of freedom. Off-shell they represent a total of six degrees of freedom, which is the minimal possible number of auxiliary fields. The task of completing the transformations (5) and the lagrangian (6) to their full non-linear form can be approached in various ways. One is the well-known method of Noether-coupling by which the lagrangian and the transformation laws can be completed order by order in the gravitational coupling. Only a few terms in the transformation laws are not uniquely fixed by the requirement of invariance of the action, and these are determined by the requirement of closure of the algebra. An alternative method is provided by the general structure of gauge theories on a curved superspace. Since the supertransformations of all geometrical fields, in particular of the connections and vielbein fields, are uniquely determined in terms of torsion and curvature [8], the linearised transformation laws are sufficient to actually read off the linearised torsion constraints of the theory. One then checks the Bianchi identities for consistency of those constraints in the full non-linear theory and for possible as yet undetermined torsion components [9]. This method, which also illuminates the superspace geometry of our theory, will be described in detail in a separate publication. The full supertransformation laws for the vierbein eua , the Rarita-Schwinger field flu, the axial vector Ap and the antisymmetric tensor apv are
6eua
=
-2i~7a~p,
~//.t
=
(~)bt~--~5~Vp
--
1.l O p a 75~'Va ,
(8)
Va=e-le --
p
aePVKX(!3vaKa --~1 4
i~vy~$x)
Q) p ~ - ap~ - k i~paboab~ + ApT5~ .
(9) (10)
The Lorentz connection COpab is the usual expression in terms of the vierbein and Rarita-Schwinger field (second-order formalism):
COpab - 51 (ea v ~pevb _ ebv 3peva) _
1 -- ~
eaKebx(O ~gxp -- 8~ gKu )
- i ( ~ ' p ~.Vb+ ~
~b -- ~.Vb~V~),
and t e a is the super-covariantised Rarita-Schwinger field strength [ ~ u is defined as in eq. (10), i.e. without a Christoffel symbol]:
Q~a = ieabcd3,b3, 5 Tcd , Tab --eaUebv( c-Ou~v -- C-~v~u) + 75 ~aVb --3'5 ~b Va 1.
1.
-- 2 l°ac 75 ~b Vc + 2 l°bc')'5 ~a V
c.
(12)
The commutator of two supertransformations closes on general coordinate transformations ~G, supertransformations 5 S, local Lorentz transformations ~L, chiral gauge transformations 5C and vector gauge transformations 6 v as follows: [8 S(~'I ), 6 S(~2)1
(13)
= ~G(~#) + ~s(e) + 5L(Xab) + 6c(A) + 6V(~3g), with parameters ~P = 2ieaP~l 7a~'2 ,
e =-~p~p,
k ab = ~ p ~ p a b - e a b c d ~ c V d ,
A =-~pAp,
flu=~u +auv ~v"
A chiral gauge transformation acts only on ffp and Ap, 6C~p=-A75q;p,
6cA p =~uA,
(14)
and a vector gauge transformation only on apv:
6vauv = ~u~v - ~v~p •
(15)
The following lagrangian gives rise to an action f d4x Z? which is invariant under all these transformations:
22=
1 -~
eeaPebVRpv ab + 2ieUV~xfjpTv75 ~ K~x
~
6A u = -½~75 7pTa cl~ a, 6a v = 2i~3'p ~v - 2i~7v~p, with
15 October 1981
(1 1)
- 3eVa Va + eUVKXApavaKx ,
(16)
with e = det ep a ,
Rpvab = ~pcovab -- ~v~pab + ¢opaccovc b -- 6JvaC~pcb This lagrangian generalises the linearised ~ G of eq. (6) to its full non-linear form. Invariance of the action under the transformations (8) can be proved rather straightforwardly by use of the so-called "1.5 order formalism" [10] which makes it unnecessary to vary wuab(e, ~). 355
Volume 105B, number 5.
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It is important to note that the action is invariant under the local chiral transformations of eq. (14). This is not an invariance possessed bY either the theory without auxiliary fields [3] of by the theory with the minimal auxiliary fieldsM, N and b u [4]. The creation of a new invariance by the inclusion of auxiliary fields is an interesting phenomenon in view of the current speculations about N = 8 supergravity. I'n fact, straightforward extrapolation from the present result would lead us to expect the internal symmetry of the N = 8 superconformal group, namely U(8). Given that there are now two "minimal" formulations o f N = 1 supergravity, the question naturally arises whether they represent equivalent physical theories. For example, can they be coupled to matter in the same way? To answer this question, it is necessary to develop the tensor calculus for the new formulation as it has been done for the M , N , b u version [11 ]. We have constructed this tensor calculus and will present it elsewhere [12]. Here we report only its general features: like before, we can construct vector, chiral and linear supermultiplets and find rules for combining them; we can also find F- and D-density formulas. However, in the new tensor calculus we must assign local chiral weights to supermultiplets and the Fdensity formula exists only for chiral multiplets with a certain chiral weight. This restricts the type of coupling one can achieve; i.e. the terms XX or ~ u o u v ~ v are not invariant if X and ~ku have chiral weights. Thus both, mass-terms in conjunction with interaction terms in matter coupling and a cosmological constant are ruled out. In this respect, our formulation differs significantly from Einstein's theory and from the other formulations o f N = 1 supergravity. One can, however, construct a Maxwell action and the kinetic action for a chiral matter multiplet, as well as a Fayet-Iliopoulos term [13], the latter providing an example of the new way to break conformal invariance. In the new tensor calculus the structure of the gravity sector of the theory is described by an "Einstein multiplet" which is now of the Maxwell type [cf. the comments after eq. (2)] and begins with the spinor ~/ac]~a and by a chiral "Weyl multiplet" which begins with the spin-3/2 part of Tab. The differences between the two minimal formulations seem to indicate that they are indeed different theories, at least one would expect their topological properties to be different. The major source of the differences mentioned 356
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above is the local chiral invariance and one may wonder if this invariance survives in the quantum theory. Should the chiral invariance be broken by the quantum theory, the field A u would lose its gauge invariance and this would upset the off-shell Fermi-Bose balance required by supersymmetry. It would be of interest to use the closed algebra of eq. (13) to evaluate the Feynman rules of the theory using the BRS prescription. Finally, we note that the theory presented here has a radically different geometrical interpretation (details to be given elsewhere [12]) from the previously known minimal formulation. In its superspace formulation it is essential, unlike in the Wess-Zumino formulation [14], to introduce an additional local chiral gauge transformation in the tangent space with an associated gauge field. The constraints are
Tc~oa = - 2 i (Tac)a# , 0 = T~5 ~t = G a b = Taa~(qCb(1 -+ i')'5))#a , O= TabC= fa~ , where the components TAB C are the supercovariant torsions and the f i B are the supercovariant chiral field strengths. This scheme, however, does not geometrically explain the vector gauge invariance on the auv field. auv actually appears as solution to a differential constraint ~ a v a on one of the components in T a J . Geometrically, it can probably be interpreted as the remnant of a local translation which has not completely been identified with general coordinate transformations.
References [1] S. Ferrara and B. Zumino, Nucl. Phys. B87 (1975) 207. [2] V. Ogievetsky and E. Sokatchev, Nucl. Phys. B124 (1977) 309. [3] D.Z. Freedman, P. van Nieuwenhuizen and S. Ferrara, Phys. Rev. D13 (1976) 3214; S. Deser and B. Zumino, Phys. Lett. 62B (1976) 335. [4] K. SteUe and P.C. West, Phys. Lett. 74B (1978) 330; S. Ferrara and P. van Nieuwenhuizen, Phys. Lett. 74B (1978) 333. [5 ] M. Kaku, P. Townsend and P. van Nieuwenhuizen, Phys. Rev. D17 (1978) 3179. [6] M. Sohnius and P.C. West, Phys. Lett. 100B (1981) 245. [7] M. Sohnius and P.C. West, in preparation. [8] F.W. Hehl, P. vonder Heyde and G.D. Gerlick, Rev. Mod. Phys. 48 (1976) 393; J. Wess and B. Zumino, Phys. Lett. 74B (1978) 51.
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[9] P. Breitenlohner and M. Sohnius, Nucl. Phys. B165 (1980) 483. [10] A.H. Chamseddine and P.C. West, Nucl. Phys. B129 (1977) 39; P. van Nieuwenhuizen and P. Townsend, Phys. Lett. 67B (1977) 439. [ 11 ] P. van Nieuwenhuizen and S. Ferrara, Phys. Lett. 76B (1978) 404; 78B (1978) 573;
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K. Stelle and P. West, Phys. Lett. 77B (1978) 376; Nucl. Phys. B145 (1978) 175. [12] M. Sohnius and P. West, in preparation. [13] P. Fayet and J. Iliopoulos, Phys. Lett. 5 1B (1974) 461; K. SteUe and P. West, Nucl. Phys. B145 (1978) 175. [14] J. Wess and B. Zumino, Phys. Lett. 66B (1977) 361; 74B (1978) 51.
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