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The supersymmetric non-linear σ-model in four dimensions and its coupling to supergravity
Volume 74B, number 4, 5
PHYSICS LETTERS
17 April 1978
THE SUPERSYMMETRIC NON-LINEAR o-MODEL IN FOUR DIMENSIONS AND ITS COUPLING TO SUPERGRAVITY E. CREMMER and J. SCHERK Laboratoire de Physique Thdorique de l'Eeole Normale Supdrieure 1, Paris, France
Received 20 February 1978
We present the supersymmetric version in 4 dimensions of the SU(N) non-linear o-model. Its coupling to supergravity is also discussed.
We wish to generalize to four dimensions the form of the supersymmetric non-linear a-model studied in two dimensions by Di Vecchia and Ferrara [1] and Witten [2], one of our aims being to elucidate some of the non-polynomial structure appearing in SO(N) (N > 4) supergravity theories [ 3 - 6 ] . In flat space, the lagrangian for N Wess-Zumino [7] scalar multiplets reads as follows:
~ × ~ = 0.
(z)
where the scalar ~bc~ fields are complex, and the X~ fields are Majorana spinors. The action is invariant under the supersymmetry transformations
Counting the remaining degrees of freedom for the bosons ( 2 N - 1) and for the fermions ( 2 N - 2) reveals a mismatch and therefore the theory cannot be supersymmetric as it stands. One more Bose degree of freedom has to be eliminated. The remaining global group of invariance is U(N) and one can eliminate one degree of freedom by promoting the U(1) subgroup to a local group of invariance, by introduction of an axial gauge field A u. This gauge field A u will have no kinetic term and thus can be eliminated from the action through its equations of motion. Thus one is led to the action
&ba = (1/X/5)~(1 - 7 5 ) X a, 6X~= -(i/v~-)Tg0u~ae,
L = ~ IDu~b~12 + } i 2 c~~ X a,
where
whe re
L = ½0uqb*~0u¢ ~ +-}i ~ y u 0 u × ~,
1
1
~ = v [ ~ + ~*~ + Vs(~ ~ -- ~*~)],
D.4~ '~ = ( 3 . - i A . ) ~ ,
as well as under a global U(N) ® U(N) group:
~bc~, Xc~being subject to the constraints (1, 2). The supersymmetry transformation laws are:
8q~~ = (CC~# + iAC~)~b#, (SXc~ = (C'C*# +iTsA'C~#)X# ' the C and C' matrices being real and antisymmetric, the A, A' matrices being real and symmetric. Inspired by refs. [ 1,2] we now impose the following conditions on the scalar and spinor fields:
~" I~al 2 = /22,
(1)
1 Laboratoire propre du C.N.R.S. associd ~ l'Ecole Normale supdrieure et ~ l'Universit6 de Paris-Sud.
~.X ~ = (3. + iA.~,s)×~,
&b~ = (l/x/2) g(1 - 75)X%
8X a = -(i/x/~)~'~Du~ae,
with D u'~ a = Ou "~ a _ i,y s "~ a A u .
The lagrangian and the constraints are globally SU(N),invariant and are locally invariant under U(1): A u -+A u + 0uA(x), ~ba --+ e i A ( x ) ~ a , X a -+ e - i A ( x ) q ' s
Xa .
The field A u can be eliminated through its equation 341
Volume 74B, number 4, 5
PHYSICS LETTERS
of motion:
17 April 1978
xN =
]1j21z-/h, \ ~N,I
N/d 1.1
after which the lagrangian reads:
The lagrangian is then written as
L =-~ 13uq~12 + { i ~ 7 " O u X
L = ~ ( a x b . a . z i a . z * J + ~i x;~x;
a
_ (1/8/.t2) [iq~*c~3.q~c~ + 2c~T.3,sXC~] 2, introducing quartic terms among fermions and bosons. Finally we show that the action, together with the constraints (1, 2) is invariant under supersymmetry. After a short calculation, one finds that:
+211. ~i~ij.~.hj._ _ (1/8/t2)(~,7.7 5 hi)2 ' where
I
Kff = ~
i j z k 0 2 Z*k
1 ¢ z, which vanishes because of constraint (2). Similarly, varying constraint (1) gives
" "~"
b,2
k.
+½(v'a(qb'k)G(.,/a(al 1) J)
t z*J]
+ 2~t2 ~ - ~ - J "~-~a-al"
Varying the last constraint (2), one obtains:
The phase factor ~N/(~*N is irrelevant for writing the lagrangian because of the U(1) gauge invariance, but it is important for deriving the supersymmetric transformation laws of h i and zi:
(i/x/~) [(2ae)x a - 3'5Xa(~aT5 e) -i'~c~7"D.'~C~e];
6Z i = (1/X/2)(all)ij g(1 - 75)X/,
this expression vanishes indeed using constraint (1), replacingAu by its expression, and using a Fierz transformation. The U(1) invariance still present even after the elimination of A . allows us to fix the phase of one of the Bose fields. For instance one can choose ~N to be real and use the constraint (1) to eliminate it. This leaves us with 2N - 2 bosonic and fermionic degrees of freedom, the action still being supersymmetric and SU(N) invariant. We can rewrite the lagrangian using independent fields Z i (complex scalar fields) and h i (spinor field) (i = 1 ..... N - 1) defined by:
8 h i = - (i/v~-) (a'l- 1) ji T" 3 p ~Je + a ('al 1 )jk 8 ('a f i/a) h i
6(~*~)
= v'2 :~x
~ = o.
~)i = (oN/~*N)lJ2 zi/xil~,
It is easy to couple this model to supergravity. The corresponding lagrangian is in the 1.5 formalism [4,11] : L = - ( 1 / 4 K 2) VR 1
v'
-
--t
--
X i = ('~*N/'~N)l/2x/a('~ll)jihj,
a- ~
342
1
--p
I
~-~
+-~ i g [ x ~ @X '~ -{K2~'~7,75X~qSaDUq~ *~] + g rlD.qSal 2
z i z .1
Z*iaO "= aZ*l,
VK2(2~vsvux~)2,
where the gauge covariant derivatives are defined by
Du4) ~ = (3u - iAu)(p% c-/).Xa = (bu +i(1 + :K 1 2/a2)Au~,s)X% c-/)
u ¢ '~ = ( 3 . - : ~1"
2 2A u
'
uYs)¢~"
This lagrangian is invariant under the following local supersymmetry transformations:
8Va. = --it~g'-ya~' , so that
~
_ 2 e~.o. [~£75.}, q) v~o --~K 2 q;x'y. ~ t) (pa DvO *c~]
- (K V/2XQ) ~ ; D . ~ ' a T u T " X a - g '
where a = 1 + ziz*i/la2;(1/a)(a~)i]~. =.8i1 +ZiZ*i/la 2, a'l being defined with Z ~ instead of Z*; a 1 satisfies the relations
a O"Z ] = a Z i,
- g'ZJhjhi - ~"[5 "ZJhJvs hi].
dpU = (~)U/(p*N)l/2 id/N/~'
or
Z i = p~)i/~N,
_ (1/xQ-p2)Ig'~ih/hJ - g'ZivshJTshi .
Volume 74B, number 4, 5
PHYSICS LETTERS
t
84J. = (1/K)(~ue +-~t~2 ~baDu~b*a75e'), 6~b~ = (1/~/2) ~'(1 - ~/5)X~, 6Xa = --(i/x¢/2)7 ~ ¢-/)u(b~ e . As usual @ denotes the supersymmetric covariant derivatives. This lagrangian is also invariant under the following gauge transformations:
A u ~A~ + O~O, (~a ~
ei°~b~ ,
×~ --. e-i(1+½ ~:)°'~5 ×~, ¢~, --. e ½i~,~0v5 q/.u" t
The constraints on the fields are the same as in flat space. They still are invariant under supersymmetry transformations. Let us remark that (~aDuc~*~is a supercovariant object because o f the constraint (2). We can now compare our lagrangian with previous results. We note that we can take/a2 < 0 and the group SU(N - 1, 1) instead of SU(N). The reduction is made as previously for Ca to Zi; for ×~ only the phase factor is changed from (~*N/'~N)I/2 to ('~*N/'~N) (1+1K2~2)/2 and we redefine also ff'u as well as e' by t __
e,__
e.
In the case of SU(1, 1) writing 1//a 2 = - k K 2 we recover a class of coupling of the scalar multiplet to supergravity [8,9]. This explicitly shows the SU(1, 1) invariance discovered in this theory as well as in the extended 0 ( 4 ) supergravity [5]. In the case of SU(N, 1) if we take 1//~2 = _~2, we recover the coupling of N scalar multiplets to super-
17 April 1978
gravity as has been obtained after reduction of the coupling of N vector multiplet to the 0 ( 2 ) supergravity obtained by Luciani [ 10]. This study has shown that the appearance of the non-polynomial structure of the 0 ( 4 ) extended gravity is simply linked to the SU(1, 1) invariance of the theory. We cannot trivially extend this study when vector fields are involved since the SU(1, 1) is realized only on the equations of motion. But we can still show that the nonpolynomial structure of the interaction of the vector fields is simply obtained from the SU(N, 1) invariance. This will be described in a forthcoming paper. It is a pleasure to thank Dr. A. Neveu and B. Julia for useful discussions.
References [1] P. Di Vecchia and S. Ferrara, Nucl. Phys. B130 (1977) 93. [2] E. Witten, Phys. Rev. D16 (1977) 2991. [3] E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 68B (1977) 234. [4] E. Cremmer and J. Scherk, Nucl. Phys. B127 (1977) 259. [5] E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 74B (1978) 61. [6] B. De Wit and D.Z. Freedman, Nucl. Phys. B130 (1977) 105. [7] J. Wess and B. Zumino, Phys. Lett. 49B (1974) 52. [8] E. Cremmer and J. Scherk, Phys. Lett. 69B (1977) 97. [9] A. Das, M. Fischler and M. Ro~ek, preprint ITP-SB-77-38. [10] J.F. Luciani, preprint LPTENS 77/14, Nucl. Phys. B, to be published. [ 11] P. Van Nieuwenhuizen, ITP-SB-77-55 preprint.
343
PHYSICS LETTERS
17 April 1978
THE SUPERSYMMETRIC NON-LINEAR o-MODEL IN FOUR DIMENSIONS AND ITS COUPLING TO SUPERGRAVITY E. CREMMER and J. SCHERK Laboratoire de Physique Thdorique de l'Eeole Normale Supdrieure 1, Paris, France
Received 20 February 1978
We present the supersymmetric version in 4 dimensions of the SU(N) non-linear o-model. Its coupling to supergravity is also discussed.
We wish to generalize to four dimensions the form of the supersymmetric non-linear a-model studied in two dimensions by Di Vecchia and Ferrara [1] and Witten [2], one of our aims being to elucidate some of the non-polynomial structure appearing in SO(N) (N > 4) supergravity theories [ 3 - 6 ] . In flat space, the lagrangian for N Wess-Zumino [7] scalar multiplets reads as follows:
~ × ~ = 0.
(z)
where the scalar ~bc~ fields are complex, and the X~ fields are Majorana spinors. The action is invariant under the supersymmetry transformations
Counting the remaining degrees of freedom for the bosons ( 2 N - 1) and for the fermions ( 2 N - 2) reveals a mismatch and therefore the theory cannot be supersymmetric as it stands. One more Bose degree of freedom has to be eliminated. The remaining global group of invariance is U(N) and one can eliminate one degree of freedom by promoting the U(1) subgroup to a local group of invariance, by introduction of an axial gauge field A u. This gauge field A u will have no kinetic term and thus can be eliminated from the action through its equations of motion. Thus one is led to the action
&ba = (1/X/5)~(1 - 7 5 ) X a, 6X~= -(i/v~-)Tg0u~ae,
L = ~ IDu~b~12 + } i 2 c~~ X a,
where
whe re
L = ½0uqb*~0u¢ ~ +-}i ~ y u 0 u × ~,
1
1
~ = v [ ~ + ~*~ + Vs(~ ~ -- ~*~)],
D.4~ '~ = ( 3 . - i A . ) ~ ,
as well as under a global U(N) ® U(N) group:
~bc~, Xc~being subject to the constraints (1, 2). The supersymmetry transformation laws are:
8q~~ = (CC~# + iAC~)~b#, (SXc~ = (C'C*# +iTsA'C~#)X# ' the C and C' matrices being real and antisymmetric, the A, A' matrices being real and symmetric. Inspired by refs. [ 1,2] we now impose the following conditions on the scalar and spinor fields:
~" I~al 2 = /22,
(1)
1 Laboratoire propre du C.N.R.S. associd ~ l'Ecole Normale supdrieure et ~ l'Universit6 de Paris-Sud.
~.X ~ = (3. + iA.~,s)×~,
&b~ = (l/x/2) g(1 - 75)X%
8X a = -(i/x/~)~'~Du~ae,
with D u'~ a = Ou "~ a _ i,y s "~ a A u .
The lagrangian and the constraints are globally SU(N),invariant and are locally invariant under U(1): A u -+A u + 0uA(x), ~ba --+ e i A ( x ) ~ a , X a -+ e - i A ( x ) q ' s
Xa .
The field A u can be eliminated through its equation 341
Volume 74B, number 4, 5
PHYSICS LETTERS
of motion:
17 April 1978
xN =
]1j21z-/h, \ ~N,I
N/d 1.1
after which the lagrangian reads:
The lagrangian is then written as
L =-~ 13uq~12 + { i ~ 7 " O u X
L = ~ ( a x b . a . z i a . z * J + ~i x;~x;
a
_ (1/8/.t2) [iq~*c~3.q~c~ + 2c~T.3,sXC~] 2, introducing quartic terms among fermions and bosons. Finally we show that the action, together with the constraints (1, 2) is invariant under supersymmetry. After a short calculation, one finds that:
+211. ~i~ij.~.hj._ _ (1/8/t2)(~,7.7 5 hi)2 ' where
I
Kff = ~
i j z k 0 2 Z*k
1 ¢ z, which vanishes because of constraint (2). Similarly, varying constraint (1) gives
" "~"
b,2
k.
+½(v'a(qb'k)G(.,/a(al 1) J)
t z*J]
+ 2~t2 ~ - ~ - J "~-~a-al"
Varying the last constraint (2), one obtains:
The phase factor ~N/(~*N is irrelevant for writing the lagrangian because of the U(1) gauge invariance, but it is important for deriving the supersymmetric transformation laws of h i and zi:
(i/x/~) [(2ae)x a - 3'5Xa(~aT5 e) -i'~c~7"D.'~C~e];
6Z i = (1/X/2)(all)ij g(1 - 75)X/,
this expression vanishes indeed using constraint (1), replacingAu by its expression, and using a Fierz transformation. The U(1) invariance still present even after the elimination of A . allows us to fix the phase of one of the Bose fields. For instance one can choose ~N to be real and use the constraint (1) to eliminate it. This leaves us with 2N - 2 bosonic and fermionic degrees of freedom, the action still being supersymmetric and SU(N) invariant. We can rewrite the lagrangian using independent fields Z i (complex scalar fields) and h i (spinor field) (i = 1 ..... N - 1) defined by:
8 h i = - (i/v~-) (a'l- 1) ji T" 3 p ~Je + a ('al 1 )jk 8 ('a f i/a) h i
6(~*~)
= v'2 :~x
~ = o.
~)i = (oN/~*N)lJ2 zi/xil~,
It is easy to couple this model to supergravity. The corresponding lagrangian is in the 1.5 formalism [4,11] : L = - ( 1 / 4 K 2) VR 1
v'
-
--t
--
X i = ('~*N/'~N)l/2x/a('~ll)jihj,
a- ~
342
1
--p
I
~-~
+-~ i g [ x ~ @X '~ -{K2~'~7,75X~qSaDUq~ *~] + g rlD.qSal 2
z i z .1
Z*iaO "= aZ*l,
VK2(2~vsvux~)2,
where the gauge covariant derivatives are defined by
Du4) ~ = (3u - iAu)(p% c-/).Xa = (bu +i(1 + :K 1 2/a2)Au~,s)X% c-/)
u ¢ '~ = ( 3 . - : ~1"
2 2A u
'
uYs)¢~"
This lagrangian is invariant under the following local supersymmetry transformations:
8Va. = --it~g'-ya~' , so that
~
_ 2 e~.o. [~£75.}, q) v~o --~K 2 q;x'y. ~ t) (pa DvO *c~]
- (K V/2XQ) ~ ; D . ~ ' a T u T " X a - g '
where a = 1 + ziz*i/la2;(1/a)(a~)i]~. =.8i1 +ZiZ*i/la 2, a'l being defined with Z ~ instead of Z*; a 1 satisfies the relations
a O"Z ] = a Z i,
- g'ZJhjhi - ~"[5 "ZJhJvs hi].
dpU = (~)U/(p*N)l/2 id/N/~'
or
Z i = p~)i/~N,
_ (1/xQ-p2)Ig'~ih/hJ - g'ZivshJTshi .
Volume 74B, number 4, 5
PHYSICS LETTERS
t
84J. = (1/K)(~ue +-~t~2 ~baDu~b*a75e'), 6~b~ = (1/~/2) ~'(1 - ~/5)X~, 6Xa = --(i/x¢/2)7 ~ ¢-/)u(b~ e . As usual @ denotes the supersymmetric covariant derivatives. This lagrangian is also invariant under the following gauge transformations:
A u ~A~ + O~O, (~a ~
ei°~b~ ,
×~ --. e-i(1+½ ~:)°'~5 ×~, ¢~, --. e ½i~,~0v5 q/.u" t
The constraints on the fields are the same as in flat space. They still are invariant under supersymmetry transformations. Let us remark that (~aDuc~*~is a supercovariant object because o f the constraint (2). We can now compare our lagrangian with previous results. We note that we can take/a2 < 0 and the group SU(N - 1, 1) instead of SU(N). The reduction is made as previously for Ca to Zi; for ×~ only the phase factor is changed from (~*N/'~N)I/2 to ('~*N/'~N) (1+1K2~2)/2 and we redefine also ff'u as well as e' by t __
e,__
e.
In the case of SU(1, 1) writing 1//a 2 = - k K 2 we recover a class of coupling of the scalar multiplet to supergravity [8,9]. This explicitly shows the SU(1, 1) invariance discovered in this theory as well as in the extended 0 ( 4 ) supergravity [5]. In the case of SU(N, 1) if we take 1//~2 = _~2, we recover the coupling of N scalar multiplets to super-
17 April 1978
gravity as has been obtained after reduction of the coupling of N vector multiplet to the 0 ( 2 ) supergravity obtained by Luciani [ 10]. This study has shown that the appearance of the non-polynomial structure of the 0 ( 4 ) extended gravity is simply linked to the SU(1, 1) invariance of the theory. We cannot trivially extend this study when vector fields are involved since the SU(1, 1) is realized only on the equations of motion. But we can still show that the nonpolynomial structure of the interaction of the vector fields is simply obtained from the SU(N, 1) invariance. This will be described in a forthcoming paper. It is a pleasure to thank Dr. A. Neveu and B. Julia for useful discussions.
References [1] P. Di Vecchia and S. Ferrara, Nucl. Phys. B130 (1977) 93. [2] E. Witten, Phys. Rev. D16 (1977) 2991. [3] E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 68B (1977) 234. [4] E. Cremmer and J. Scherk, Nucl. Phys. B127 (1977) 259. [5] E. Cremmer, J. Scherk and S. Ferrara, Phys. Lett. 74B (1978) 61. [6] B. De Wit and D.Z. Freedman, Nucl. Phys. B130 (1977) 105. [7] J. Wess and B. Zumino, Phys. Lett. 49B (1974) 52. [8] E. Cremmer and J. Scherk, Phys. Lett. 69B (1977) 97. [9] A. Das, M. Fischler and M. Ro~ek, preprint ITP-SB-77-38. [10] J.F. Luciani, preprint LPTENS 77/14, Nucl. Phys. B, to be published. [ 11] P. Van Nieuwenhuizen, ITP-SB-77-55 preprint.
343