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N = 2 supergravity theory with a gauged central charge
Volume 76B, number 3
PHYSICS LETTERS
5 June 1978
N = 2 S U P E R G R A V I T Y THEORY WITH A G A U G E D CENTRAL CHARGE ~
Cosmas K. ZACHOS
California Institute of Technology, Pasadena, CA 91125, USA Received 16 March 1978
A free, massive N = 2 scalar-spinor multiplet, whose supersymmetry algebra contains a central charge, is coupled to N = 2 supergravity. The resulting theory is locally invariant under supersymmetry and the central charge transformation, the latter being gauged by the vector field of the supergravity multiplet.
A simple irreducible realization o f N = 2 supersymmetry is obtained by combining two N = 1 massless multiplets * a, each involving a Majorana spinor X i, a scalar A i, and a pseudoscalar B i [ 1]. Upon the introduction of mass, the algebra must be modified to exhibit an additional generator Z, commuting with all the rest, called the central charge [2]. It shows up in the anticommutator of two supersymmetries:
{a~, 0l~ = i6klTuat3Pu + ektfa#Z
(k, l = 1, 2),
(1)
and it has the dimensions of mass, so that it may only be realized by a massive multiplet. It acts by rotating one N = 1 constituent multiplet into another. The transformation law o~"the multiplet under the two supersymmetries of Majorana spinor parameters e and ~"is:
6A i = (1/~¢~) gX i + (e i/fl¢/~) (X i,
6B i = (1/V/~) ~iTsXi + (eiJ/~¢/~) ~i,),5X/' (2)
8X i = (--i/v:2-)(~ -- im)(A i + i 7 5 B i ) e + ( i / v ~ ) eiJ(~ - i m ) ( A J - i75B/)~. The commutation of two such transformations on any of the above fields ~bi yields: [c5', 8]¢) i= i(g'yu e + ~'7u~") au(~i + m (g~" - ~e')e i/eft.
(3)
The second term on the right hand side exhibits the action of the central charge on the field, namely an 0(2) rotation between the two multiplets, with transformation parameter m ( g ~ " - ~re'). N = 2 supergravity contains a vierbein Vau, two gravitinos ¢/u and ~bu (gauging e-type and ~'-type supersymmetries, respectively), and a vector Au. The commutator of two local supersymmetry transformations acting on Au contains a gauge transformation of parameter (g~" - ( e ' ) , suggesting that A u may be used to gauge the central charge by coupling to the Noether current of the 0(2) rotation in question [3]. In this letter, it is demonstrated that this is indeed the case by coupling the matter multiplet mentioned to super gravity to produce a lagrangian invariant under both supersymmetries and the central charge gauge rotation, now realized locally. The theory is obtained iteratively, starting from the usual couplings of the gauge fields to the Noether currents. Its nonpolynomial dependence on the scalar fields is determined by the "functional methods" of Das et at. [4,5]. Work supported in part by the U.S. Department of Energy under Contract No. EY76-C-03-0068. ,1 It is easy to see that just one N = 1 scalar-spinor multiplet is not sufficient. If arbitrary coefficients are assigned to every conceivable term in the transformation of the fields, the N = 2 graded algebra constrains all coefficients to 0. 329
Volume 76B, number 3
PHYSICS LETTERS
5 June 1978
The lagrangian is £?= "/20 + £?1 + "Q2 + "/23 + "/24,
(4)
where "/20 is the pure second order N = 2 supergravity [6] ; £?6?1 is the massless sector of the globally supersymmettic matter theory, made generally covariant; £?6?2 contains the Noether current couplings and a "Pauli moment" term; £?3 contains all other terms of the massless theory, most of them quartic in the Fermi fields; and£? 4 contains all terms proportional to m or m 2. Their explicit form ,2 is:
G 0 = (-V/4K2)R - ~exouv(~xV5?uDv~ p + ~X757~Dv@) - ~ VF~vFUV 1 - K~u(VF"V - ~ i 75FuV)¢v -'~K
-lt(2V
2
~u(~v[V(~U+v - ~ , v )
_ ieuv~x ~uq, s~bxl
(5)
[(~x~,#¢P + ~xTta~bP)(~?ta f p + 2~u3'kf O + ~X3'tz~bp + 2~VTX$p) - 4 ( ~ ' 7 f f o + ~'?~0o)2],
£?1 = ( V/2a2)gUVa u Z i ~ v z i + -~1 Vy( TtaD# x ', 1
•
•
•
£?2 = -(KV/a ~/2) ~uo vzi'yuTt~xi - (K V/ax/'2)~uavzieijTv?u×i + ~K a Fuveij~iotsvxJ, £?3 = -(K 2 /2a)eXOUv['~( ~x qrf TvZ' OoZ~t~v - ~bx757u Z~ a P Z149v) + ~ u Tf Tx eq ZZ~ P Zl(~v] + 12-giK 2 euT~vo~iT5voxi(~#Tx$ v + ~uTkq~v) + ~2v2iv57pxi(~dta75"~ot~ u + ~u757o~b #)
-~rZVei/yJo~xX/(~ui,r5c~vv-leuv~:~.
+ 2~uogx ~ + 2~x¢ K) -~K2V(eO2iouv×/) x
- ~K2v(2i,r5,yo×i) 2 - (KZiV/4a)2izi'~Z/x/+ (K2iV/8a) 2 i z / ~ z / × i, £?4 = -(mXV/2K Za2)( u - -] u2) - ( mV/2a)2ixi - (iKmV/a~/2)({uziT~ xi + ~ue'Tzi'ru x/) - (K2mV/2a)(~uouvZiZit~v + ~uouvZiZickv + 2~xoXvei/2iZi(ov) +-}iKmVAuei/2iTux] .. ~.~OuZI -. + (~2m2V/2a2)AuAu~izi + (K2mV/ax/~)Av(fueiiziyv.~u×] _ (KmV/2a2)AUe~IZ
- ~ uZ-i 7 v ~{~ X i )
- (i~3m V/2a)AU(2iziTue/kZk×/+ { 2 i Z k e i X y u Z / x i)
+ (K3m/4a)Apexouv(~xysTueiyzi21$ v - ~x?sq,~eijziZJCv + 2~vTsTk2izi~bv). The local supersymmetry transformations that leave the action invariant, broken up according to the above scheme (8 = 8 0 + 81 + 63 + 84), are:
() 81 B i = ~
(_ \i3'5!
)
81×t=ax/,~(~Z -i i _ei/~l~),
(6)
iY5
83X i = li~; (3'~ue'xi~ u + ')'5q'~ exi')' 5 4# + q'u~xiq~ta + 757tt~xi'y5~bts) +-~iK ei/(2~Ue~/¢ u - ?57Uex/$u - 7U~X/~b u + 3'53'U~'X@5 f . ) + (K2[2X/~)[75xix/?5 (Z/e + ek/Zk~)
-- 75xJ(g?5 z J x i + E75 Zix j + ~75 ek/ Zk xi + ~75 ekiZkxj) - x J ( e Z J x i - g Z i x J + ~ek/ Z k x i -- ~ekizkxJ)] '
64xi = _ ( m / a x / ~ ) ( z i
e _ ei/ ~ / f ) _ (iKm/a~/~) ~(ei/Z /e + Z i f ),
1
, 2 Our conventions are those of refs. [7,4,5], and Z i ~ A i + i ' y s B i , u ~ K 2 ( A i A i + B i B i ) , a =- 1 - -~u. Written outside a spinor product, ~ 2 2 i z i is interpreted as a pure number u, i.e., the unit matrix in spinor space is implicitly dropped.
330
Volume 76B, number 3
PHYSICS LETTERS
za n2
\_zi~ pzj e
Co (g')'5Zix i + f75eijzJx i) - ~
2 v 2 "rs - %
5 June 1978
"13 Co
(g'rsez72ix j + fTszix i)
K2 ~bp . . . . . 2 X/~ (_~ p )(ge'lZ'x' -~Zixi),
84(~)
=--4aa "4-aa 7peijzizJ(~e)+@aApe'lZtZ'(-~)..... K2mA ( ZiZif )" inm "YPZ'Z'(; • . ~+iKml 2a P,_ZiZie
The theory is constructed as follows: first "/~?1is coupled to .6?0 by introducing the Noether current terms of /22, and subsequently O(n) invariance determines the Pauli moment term of .6?2 and several terms in .t;?3 . Next, O(K 2) invariance determines the lowest orders in u, namely O(u 0) and O(u), for ~ 3 and .671, respectively. Although the SO(4) symmetry described below allows more complicated structures, the dependence on the scalar fields emerging from the calculation is limited to u only, the SO(4) singlet combination. Consequently, arbitrary functions of u are assigned in front of every term, and full invariance (to all orders in n) of selected sectors reveals the functional form of these coefficients. The form found is verified by checking the invarinace of further sectors. Specifically, all types of terms with one spinor, several ones with three spinors, and many with five spinors cancel, which establishes that there are no further terms in the lagrangian or the variations of the fields. Likewise, the massive sector ~ 4 , 8 4 is worked out and checked consistently in O(m) and O(m 2) variations, to all orders in n, for all one spinor terms and selected 3 spinor terms (there are no terms involving 5 spinors). The theory presented here can be restricted (~bu, ~', A~, X2, A 2 B 2 ~ 0) to a particular representative of the family of gauged massive scalar multiplet theories [4,5], with "nonpolynomiality index" [8] fixed to )t = 1/2 by the group structure of the theory. It is interesting that there are no exponentials of scalar fields in the classical potential (first term of ~4)" It is unbounded from below as u -+ 2 (a ~ 0), and, therefore, does not have an absolute minimum. However, u = 0 is a relative minimum at which the potential vanishes. The theory contains a number of different symmetries..6?0 and 6 0 are invariant under a global SU(2) ("gravity") rotating the two generators of supersymmetry into each other. In addition to this SU(2), E 1 and 81 further obey an SU(2) ("internal symmetry"), commuting with the above, which rotates the two massless (constituent) matter multiplets into each other. This larger invariance persists in/22, and consequently the entire massless sector of the theory/?0 + "6?1 + "6?2+ "23,80 + 81 + 63 possesses an SO(4) ~ SU(2) × SU(2) invariance, with respect to which Z i transforms as (1/2, 1/2), Xi as (0, 1/2), and e, Q, Cu as (1/2, 0). The explicit field transformations are, for the "gravity" SU(2):
(7)
and their commutators. For the "internal symmetry" SU(2): 7.2:
6Xi = A,2eiJxJ, fiZ- 'z= At2ei]2j,
7.3: 8X1,2 = +A3175X '. " 1,2 , 8Z1,2= +A313,5 2"1,2,
(8)
and their commutator. Finally, there is also a "duality invariance" [3], commuting with the above SO(4) with transformation laws: 331
Volume 76B, number 3 8
=
IA " 75 ~ ,
PHYSICS LETTERS
(9)
6X i = --11~ 75 x ,
\ ~ u ] = i A 75/ ~vu l,
8FUr = - -¢.. I ^ " ~ U v _ iA"Kei]2i75oUVX]
5 June 1978
A"K(euvoo~p~a ° + i~U?SCV _ i~u?5¢u).
Upon the introduction of mass (2?4, 84) , the gravity SU(2) survives, but the duality U(1) and the chiral part (r 1 and r3) of the "internal symmetry" SU(2) are broken, while the central charge part of it (r2) is gauged, so as to become local, by the vector o f gravity ,3. Thus our lagrangian is invariant under the gauge transformation: 6 o B I =Oeq B/ \ X~ ×/
,
8oA._
(10)
1 ~uO, Km
and, hence, the derivatives of the matter fields may absorb the minimally coupled Km A u terms, turning into covar iant derivatives (8i/~ u + t~m e i / A u ) . The commutator of two local supersymmetry transformations contains a gauge transformatIon o f effectwe parameter 0 = m(e~ - ~'e ) - 1Km(e4~e + ~)~" ) = A2(x ). In conclusion, a theory has been obtained, in which the vector particle o f N = 2 supergravity gauges the central charge that occurs in the massive supersymmetry algebra. Thus, every field o f N = 2 supergravity is a gauge field. It is very likely that the massless sector o f the theory can be obtained as a consistent restriction of SO(8) supergravity theory [9]. The theory could be put in a more concise form by using the covariant derivatives mentioned above, first-order formalism, supercovariant derivatives, and a compact SO(4) notation. Nevertheless, the longer form is given here, since it exhibits powers of K and m explicitly. •
•
~
~
I
--
I
•
- -
If
--
It
I
It is a pleasure to thank Dr. P. Fayet, Prof. D. Freedman, and Dr. J. Schwarz for their help and insight. ,3 This is to be contrasted with Art being used to gauge the "gravity" r2, as presented in ref. [ 7]. References
[1] P. Eayet, Nucl. Phys. Bl13 (1976) 135. [2] R. Haag, J. Lopuszafiski and M. Sohnius, Nucl. Phys. B88 (1975) 257. [3] S. Ferrara, J. Scherk and B. Zumino, Phys. Lett. 66B (1977) 35; S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. B121 (1977) 393. [4] A. Das, M. Fischler and M. Ro~ek, Phys. Lett. 69B (1977) 186. [5] A. Das, M. Fischler and M. Ro~ek, Phys. Rev. DI6 (1977) 3427. [6] S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. Lett. 37 (1976) 1669. [7] D. Freedman and A. Das, Nucl. Phys. B120 (1977) 221. [8] E. Cremmer and J. Scherk, Phys. Lett. 69B (1977) 97. [9] B. DeWit and D. Freedman, Nucl. Phys. B (1978), to be published.
332
PHYSICS LETTERS
5 June 1978
N = 2 S U P E R G R A V I T Y THEORY WITH A G A U G E D CENTRAL CHARGE ~
Cosmas K. ZACHOS
California Institute of Technology, Pasadena, CA 91125, USA Received 16 March 1978
A free, massive N = 2 scalar-spinor multiplet, whose supersymmetry algebra contains a central charge, is coupled to N = 2 supergravity. The resulting theory is locally invariant under supersymmetry and the central charge transformation, the latter being gauged by the vector field of the supergravity multiplet.
A simple irreducible realization o f N = 2 supersymmetry is obtained by combining two N = 1 massless multiplets * a, each involving a Majorana spinor X i, a scalar A i, and a pseudoscalar B i [ 1]. Upon the introduction of mass, the algebra must be modified to exhibit an additional generator Z, commuting with all the rest, called the central charge [2]. It shows up in the anticommutator of two supersymmetries:
{a~, 0l~ = i6klTuat3Pu + ektfa#Z
(k, l = 1, 2),
(1)
and it has the dimensions of mass, so that it may only be realized by a massive multiplet. It acts by rotating one N = 1 constituent multiplet into another. The transformation law o~"the multiplet under the two supersymmetries of Majorana spinor parameters e and ~"is:
6A i = (1/~¢~) gX i + (e i/fl¢/~) (X i,
6B i = (1/V/~) ~iTsXi + (eiJ/~¢/~) ~i,),5X/' (2)
8X i = (--i/v:2-)(~ -- im)(A i + i 7 5 B i ) e + ( i / v ~ ) eiJ(~ - i m ) ( A J - i75B/)~. The commutation of two such transformations on any of the above fields ~bi yields: [c5', 8]¢) i= i(g'yu e + ~'7u~") au(~i + m (g~" - ~e')e i/eft.
(3)
The second term on the right hand side exhibits the action of the central charge on the field, namely an 0(2) rotation between the two multiplets, with transformation parameter m ( g ~ " - ~re'). N = 2 supergravity contains a vierbein Vau, two gravitinos ¢/u and ~bu (gauging e-type and ~'-type supersymmetries, respectively), and a vector Au. The commutator of two local supersymmetry transformations acting on Au contains a gauge transformation of parameter (g~" - ( e ' ) , suggesting that A u may be used to gauge the central charge by coupling to the Noether current of the 0(2) rotation in question [3]. In this letter, it is demonstrated that this is indeed the case by coupling the matter multiplet mentioned to super gravity to produce a lagrangian invariant under both supersymmetries and the central charge gauge rotation, now realized locally. The theory is obtained iteratively, starting from the usual couplings of the gauge fields to the Noether currents. Its nonpolynomial dependence on the scalar fields is determined by the "functional methods" of Das et at. [4,5]. Work supported in part by the U.S. Department of Energy under Contract No. EY76-C-03-0068. ,1 It is easy to see that just one N = 1 scalar-spinor multiplet is not sufficient. If arbitrary coefficients are assigned to every conceivable term in the transformation of the fields, the N = 2 graded algebra constrains all coefficients to 0. 329
Volume 76B, number 3
PHYSICS LETTERS
5 June 1978
The lagrangian is £?= "/20 + £?1 + "Q2 + "/23 + "/24,
(4)
where "/20 is the pure second order N = 2 supergravity [6] ; £?6?1 is the massless sector of the globally supersymmettic matter theory, made generally covariant; £?6?2 contains the Noether current couplings and a "Pauli moment" term; £?3 contains all other terms of the massless theory, most of them quartic in the Fermi fields; and£? 4 contains all terms proportional to m or m 2. Their explicit form ,2 is:
G 0 = (-V/4K2)R - ~exouv(~xV5?uDv~ p + ~X757~Dv@) - ~ VF~vFUV 1 - K~u(VF"V - ~ i 75FuV)¢v -'~K
-lt(2V
2
~u(~v[V(~U+v - ~ , v )
_ ieuv~x ~uq, s~bxl
(5)
[(~x~,#¢P + ~xTta~bP)(~?ta f p + 2~u3'kf O + ~X3'tz~bp + 2~VTX$p) - 4 ( ~ ' 7 f f o + ~'?~0o)2],
£?1 = ( V/2a2)gUVa u Z i ~ v z i + -~1 Vy( TtaD# x ', 1
•
•
•
£?2 = -(KV/a ~/2) ~uo vzi'yuTt~xi - (K V/ax/'2)~uavzieijTv?u×i + ~K a Fuveij~iotsvxJ, £?3 = -(K 2 /2a)eXOUv['~( ~x qrf TvZ' OoZ~t~v - ~bx757u Z~ a P Z149v) + ~ u Tf Tx eq ZZ~ P Zl(~v] + 12-giK 2 euT~vo~iT5voxi(~#Tx$ v + ~uTkq~v) + ~2v2iv57pxi(~dta75"~ot~ u + ~u757o~b #)
-~rZVei/yJo~xX/(~ui,r5c~vv-leuv~:~.
+ 2~uogx ~ + 2~x¢ K) -~K2V(eO2iouv×/) x
- ~K2v(2i,r5,yo×i) 2 - (KZiV/4a)2izi'~Z/x/+ (K2iV/8a) 2 i z / ~ z / × i, £?4 = -(mXV/2K Za2)( u - -] u2) - ( mV/2a)2ixi - (iKmV/a~/2)({uziT~ xi + ~ue'Tzi'ru x/) - (K2mV/2a)(~uouvZiZit~v + ~uouvZiZickv + 2~xoXvei/2iZi(ov) +-}iKmVAuei/2iTux] .. ~.~OuZI -. + (~2m2V/2a2)AuAu~izi + (K2mV/ax/~)Av(fueiiziyv.~u×] _ (KmV/2a2)AUe~IZ
- ~ uZ-i 7 v ~{~ X i )
- (i~3m V/2a)AU(2iziTue/kZk×/+ { 2 i Z k e i X y u Z / x i)
+ (K3m/4a)Apexouv(~xysTueiyzi21$ v - ~x?sq,~eijziZJCv + 2~vTsTk2izi~bv). The local supersymmetry transformations that leave the action invariant, broken up according to the above scheme (8 = 8 0 + 81 + 63 + 84), are:
() 81 B i = ~
(_ \i3'5!
)
81×t=ax/,~(~Z -i i _ei/~l~),
(6)
iY5
83X i = li~; (3'~ue'xi~ u + ')'5q'~ exi')' 5 4# + q'u~xiq~ta + 757tt~xi'y5~bts) +-~iK ei/(2~Ue~/¢ u - ?57Uex/$u - 7U~X/~b u + 3'53'U~'X@5 f . ) + (K2[2X/~)[75xix/?5 (Z/e + ek/Zk~)
-- 75xJ(g?5 z J x i + E75 Zix j + ~75 ek/ Zk xi + ~75 ekiZkxj) - x J ( e Z J x i - g Z i x J + ~ek/ Z k x i -- ~ekizkxJ)] '
64xi = _ ( m / a x / ~ ) ( z i
e _ ei/ ~ / f ) _ (iKm/a~/~) ~(ei/Z /e + Z i f ),
1
, 2 Our conventions are those of refs. [7,4,5], and Z i ~ A i + i ' y s B i , u ~ K 2 ( A i A i + B i B i ) , a =- 1 - -~u. Written outside a spinor product, ~ 2 2 i z i is interpreted as a pure number u, i.e., the unit matrix in spinor space is implicitly dropped.
330
Volume 76B, number 3
PHYSICS LETTERS
za n2
\_zi~ pzj e
Co (g')'5Zix i + f75eijzJx i) - ~
2 v 2 "rs - %
5 June 1978
"13 Co
(g'rsez72ix j + fTszix i)
K2 ~bp . . . . . 2 X/~ (_~ p )(ge'lZ'x' -~Zixi),
84(~)
=--4aa "4-aa 7peijzizJ(~e)+@aApe'lZtZ'(-~)..... K2mA ( ZiZif )" inm "YPZ'Z'(; • . ~+iKml 2a P,_ZiZie
The theory is constructed as follows: first "/~?1is coupled to .6?0 by introducing the Noether current terms of /22, and subsequently O(n) invariance determines the Pauli moment term of .6?2 and several terms in .t;?3 . Next, O(K 2) invariance determines the lowest orders in u, namely O(u 0) and O(u), for ~ 3 and .671, respectively. Although the SO(4) symmetry described below allows more complicated structures, the dependence on the scalar fields emerging from the calculation is limited to u only, the SO(4) singlet combination. Consequently, arbitrary functions of u are assigned in front of every term, and full invariance (to all orders in n) of selected sectors reveals the functional form of these coefficients. The form found is verified by checking the invarinace of further sectors. Specifically, all types of terms with one spinor, several ones with three spinors, and many with five spinors cancel, which establishes that there are no further terms in the lagrangian or the variations of the fields. Likewise, the massive sector ~ 4 , 8 4 is worked out and checked consistently in O(m) and O(m 2) variations, to all orders in n, for all one spinor terms and selected 3 spinor terms (there are no terms involving 5 spinors). The theory presented here can be restricted (~bu, ~', A~, X2, A 2 B 2 ~ 0) to a particular representative of the family of gauged massive scalar multiplet theories [4,5], with "nonpolynomiality index" [8] fixed to )t = 1/2 by the group structure of the theory. It is interesting that there are no exponentials of scalar fields in the classical potential (first term of ~4)" It is unbounded from below as u -+ 2 (a ~ 0), and, therefore, does not have an absolute minimum. However, u = 0 is a relative minimum at which the potential vanishes. The theory contains a number of different symmetries..6?0 and 6 0 are invariant under a global SU(2) ("gravity") rotating the two generators of supersymmetry into each other. In addition to this SU(2), E 1 and 81 further obey an SU(2) ("internal symmetry"), commuting with the above, which rotates the two massless (constituent) matter multiplets into each other. This larger invariance persists in/22, and consequently the entire massless sector of the theory/?0 + "6?1 + "6?2+ "23,80 + 81 + 63 possesses an SO(4) ~ SU(2) × SU(2) invariance, with respect to which Z i transforms as (1/2, 1/2), Xi as (0, 1/2), and e, Q, Cu as (1/2, 0). The explicit field transformations are, for the "gravity" SU(2):
(7)
and their commutators. For the "internal symmetry" SU(2): 7.2:
6Xi = A,2eiJxJ, fiZ- 'z= At2ei]2j,
7.3: 8X1,2 = +A3175X '. " 1,2 , 8Z1,2= +A313,5 2"1,2,
(8)
and their commutator. Finally, there is also a "duality invariance" [3], commuting with the above SO(4) with transformation laws: 331
Volume 76B, number 3 8
=
IA " 75 ~ ,
PHYSICS LETTERS
(9)
6X i = --11~ 75 x ,
\ ~ u ] = i A 75/ ~vu l,
8FUr = - -¢.. I ^ " ~ U v _ iA"Kei]2i75oUVX]
5 June 1978
A"K(euvoo~p~a ° + i~U?SCV _ i~u?5¢u).
Upon the introduction of mass (2?4, 84) , the gravity SU(2) survives, but the duality U(1) and the chiral part (r 1 and r3) of the "internal symmetry" SU(2) are broken, while the central charge part of it (r2) is gauged, so as to become local, by the vector o f gravity ,3. Thus our lagrangian is invariant under the gauge transformation: 6 o B I =Oeq B/ \ X~ ×/
,
8oA._
(10)
1 ~uO, Km
and, hence, the derivatives of the matter fields may absorb the minimally coupled Km A u terms, turning into covar iant derivatives (8i/~ u + t~m e i / A u ) . The commutator of two local supersymmetry transformations contains a gauge transformatIon o f effectwe parameter 0 = m(e~ - ~'e ) - 1Km(e4~e + ~)~" ) = A2(x ). In conclusion, a theory has been obtained, in which the vector particle o f N = 2 supergravity gauges the central charge that occurs in the massive supersymmetry algebra. Thus, every field o f N = 2 supergravity is a gauge field. It is very likely that the massless sector o f the theory can be obtained as a consistent restriction of SO(8) supergravity theory [9]. The theory could be put in a more concise form by using the covariant derivatives mentioned above, first-order formalism, supercovariant derivatives, and a compact SO(4) notation. Nevertheless, the longer form is given here, since it exhibits powers of K and m explicitly. •
•
~
~
I
--
I
•
- -
If
--
It
I
It is a pleasure to thank Dr. P. Fayet, Prof. D. Freedman, and Dr. J. Schwarz for their help and insight. ,3 This is to be contrasted with Art being used to gauge the "gravity" r2, as presented in ref. [ 7]. References
[1] P. Eayet, Nucl. Phys. Bl13 (1976) 135. [2] R. Haag, J. Lopuszafiski and M. Sohnius, Nucl. Phys. B88 (1975) 257. [3] S. Ferrara, J. Scherk and B. Zumino, Phys. Lett. 66B (1977) 35; S. Ferrara, J. Scherk and B. Zumino, Nucl. Phys. B121 (1977) 393. [4] A. Das, M. Fischler and M. Ro~ek, Phys. Lett. 69B (1977) 186. [5] A. Das, M. Fischler and M. Ro~ek, Phys. Rev. DI6 (1977) 3427. [6] S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. Lett. 37 (1976) 1669. [7] D. Freedman and A. Das, Nucl. Phys. B120 (1977) 221. [8] E. Cremmer and J. Scherk, Phys. Lett. 69B (1977) 97. [9] B. DeWit and D. Freedman, Nucl. Phys. B (1978), to be published.
332