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Nonlinear σ models with extended supersymmetry in four dimensions
Volume 90B, number 1,2
PHYSICS LETTERS
11 February 1980
N O N L I N E A R e MODELS WITH E X T E N D E D SUPERSYMMETRY IN FOUR DIMENSIONS Thomas L. CURTRIGHT The Enrico Fermi Institute, University of Chicago, Chicago, IL 6063 7, USA
and Daniel Z. FREEDMAN Institute for Theoretical Physics, State University o f New York, Stony Brook, N Y 11794, USA
Received 29 October 1979
Two nonlinear o models are discussed which have N = 2 supersymmetry in four dimensions. The f'trst is globally supersymmetric on a noncompact field manifold. The second model involves a compact manifold and requires coupling to N = 2 supergravity for consistency. Upon reduction to three and two dimensions, both models have N = 4 supersymmetry. In addition, the global model has vanishing (one-loop) charge renormalization in two dimensions.
Supersymmetric extensions o f the nonlinear o model are known in both two-dimensional [1,2] and four-dimensional [3] spacetime. These extensions have been studied considerably, especially in two dimensions. Interesting discussions o f their classical solutions [1,2], dynamical symmetry breaking mechanisms [2,4], S-matrices [5], nonlocal conservation laws [6], and their geometrical structures [7] have been given. In this note we discuss the construction o f new nonlinear models with higher supersymmetry, specifically N = 2 extended supersymmetry in four dimensions (and N = 4 supersymmetry after reduction to three or two dimensions). Two distinct types of N = 2 models appear to be possible. The first type has global N = 2 supersymmetry and scalar fields which span a noncompact manifold. A complete discussion o f this model is given here. The second type o f model involves a compact manifold o f scalar fields and requires local supersymmetry, i.e., coupling to extended supergravity is necessary for consistency. Partial results for t h i s model are given below. We start with n scalar multiplets o f N = 2 supersymmetry [8] consisting o f 2n complex scalar fields, ~b~(x) and q~(x), and n Dirac fields, ~ a ( x ) , where ot = 1 ..... n. The free lagrangian
"Q0 = i}a~/~ i}**~ + i ~ a ~
~ ,
(1)
is invariant under two independent commuting supersymmetry transformations, with Majorana spinor parameters 61 and 62. The transformation laws for the boson fields are (2)
8q~ = ~iL ~ ~ - eij~/R ~ c~ ,
where R , L = ½(1 + ")'5) are chiral projectors and eij = ~ - [ ( _ ) / _ ( _ ) i ] . The fermion fields transform as 8 ~a = - i ~ [ R ~ e i - Leii~b~6/] .
(3)
It is clear that the Bose fields initially span the manifold C 2n with linear internal symmetry U(2n). To simplify our notation, the t~ indices will usually be suppressed in subsequent formulae. Following the procedure o f ref. [3], we search for nonlinear constraints which are invariant under supersymmetry transformations. The obvious constraint ~i(9i = g2 is, surprisingly, not implementable without local supersymmetry and will be considered later. Instead, we first impose the isovector constraint ~b
= b,
(4)
where r a are the usual Pauli matrices and b is a fixed constant vector. Thus the SU(2) invariance on the " i " 71
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PHYSICS LETTERS
indices in (1) is explicitly broken down to a U(1) by eq. (4). Supersymmetry invariance of eq. (4) requires two independent Majorana spinor constraints, namely fi=R(~ifc-cij~jf)+Z(¢if-cij¢jfc)=O
,
(5)
where i = 1,2 and f e is the charge conjugate fermion field. At this point we count 4n - 3 real degrees of freedom for the bosons and 4n - 4 for the fermions. One more Bose degree of freedom must be removed to obtain a linear representation of supersymmetry. Following ref. [3] we eliminate this Fermi-Bose disparity by gauging a U(1) subgroup of the remaining global symmetry. To decide which U(1) is relevant, we examine the supersymmetry variation of eq, (5), which, after use of eq. (4) and Fierz rearrangement, takes the form
~ft' = l [ ( ~ f ) + (~75 f)')'sl
eijej
1
-
~(i~ au ~ + ffvuf) vu vs ei.
(6)
In the last terms in eq. (6) we find the current of the U(1) group which gauges the common phase of ~ and f a . This current term drops out of 6f i if we modify eq. (3) by making the derivative of the scalars U(1) covariant. The scalar and pseudoscalar spinor bilinears in eq. (6) also disappear if we add similar bilinear terms to eq. (3). The full theory is now specified. We have the constraints (4) and (5), and the lagrangian = D u ~ DUq~ + i ~ , E i f + ( ~ f ) M
+ (i 575 ~ ) N - ~¢(M 2 + N 2 ) .
(7)
In this expression, M, N, and Vu are real auxiliary fields, and Du~ i =(au+iVu)Oi,
D u f = ( a u +iVu) f .
(8)
The action is invariant under the boson variations (2) and the modified fermion variations 5 f = [ - i ~ + ( M - i 7 5 N ) ] [R4~iei - Lei]¢ie]] .
(9)
Variations of the auxiliary fields need not be specified since we work on-shell with respect to them, as in the 1.5 order formalism of supergravity [9]. The constraints (4) and (5) are now completely supersymmetric provided we use the on-shell values Vu = (i¢ a.~ + ~ 7 u f ) / ( 2 ~ ¢ ) ,
72
M = ff/(2¢¢),
11 February 1980 N =i~75f/(2¢¢).
(10)
Thus the lagrangian and the constraints are all N = 2 supersymmetric. If eqs. (10) are inserted in (7), we obtain the lagrangian with auxiliary fields eliminated, viz., ./2 ='~0 - [(i~-3~ ~ + ~ T u f ) 2 -- (t~f) 2 + (t~75 f)2]/(4~b) •
(11)
To define a perturbation expansion for the theory, we must resolve the constraints. The direction of the isovector b in eq. (4) is irrelevant, and we therefore choose the "north-pole" direction. The constraints become
qSl~b1 - ~2~b2 = b 3 > 0
,
~b1¢2= 0 ,
(12)
and they are easily solved. Clearly, the scalar fields lie in a noncompact manifold embedded in C 2n . Since the induced metric remains riemannian, however, there are no ghost-modes. If one takes the present theory and considers e 1 variations only, then ea2 and R f a can be discarded consistent with e 1 supersymmetry. This truncated theory is exactly the global model given in ref. [3]. Effects of the M and N fields cancel, and the coupling of Vu reduces to an axial coupling to a Majorana fermion. Evidently a superspace formulation of the N = 2 theory can be obtained without difficulty, since the lagrangian (7) is closely related to a known expression [10] involving jointly interacting N = 2 vector and scalar multiplets. It also seems that a self-coupling of N = 2 scalar multiplets has not previously been given, and that the nonlinear o-model constructed here provides the first example of such an interaction. Reduction of the model from four to three or two spacetime dimensions can be carried out according to well-known techniques [ 11 ]. Each four-component spinor of the four-dimensional theory becomes a pair of two-component spinors of the lower dimensional theory. Supersymmetry is preserved by the reduction. One therefore has invariance with respect to four distinct spinorial transformations, i.e., N = 4 supersymmetry. Various dynamical aspects of this new nonlinear theory are currently under investigation. We report here one result concerning the charge renormalization
Volume 90B, number 1,2
PHYSICS LETTERS
properties of the model in two dimensions. In the usual O(n) and CP(n) models in two dimensions, the boson contribution to the one-loop 3 function is negative [12], thereby imitating the asymptotic freedom of four-dimensional Yang-Mills theory. The supersymmetric addition of fermions does not change the value of the one-loop charge renormalization +1 because the fermion small-fluctuation determinant involves minimal coupling to vector potentials and is finite in two dimensions. We have calculated the oneloop 3 function in the present model both by ordinary perturbation theory, using the constraints (12), and by the moving frame method [13]. The boson contribution vanishes because the ~b~ and ~ba2 modes contribute with opposite signs. Again, fermions do not change this result. The determination of the/3 function in higher orders is of obvious interest in connection with a possible parallel with the N = 4 supersymmetric Y a n g Mills theory in four dimensions [14], for which the 3 function is known to vanish through two-loop order [15]. It is not clear to us, however, whether a twodimensional theory with vanishing charge renormalization would have pathological properties due to severe infrared divergences. We now consider the construction of a supersymmetric theory with the constraint ~i ~bi = g2 .
(13)
Proceeding as before; we see that eq. (13) is supersymmetric under the variations (2), if we impose the two Majorana spinor constraints
F i =R(c,bi 4 c + eij~l.4) +L(~i4 +eij~j4 c) = 0 .
(14)
These constraints leave 4n - I Bose and 4n - 4 Fermi degrees of freedom, and previous reasoning suggests that three additional boson modes should be eliminated by gauging an SU(2) subgroup of the global U(2n). To verify this we compute the transform of eq. (14) under the variations (2) and (3). Suppressing SU(2) doublet indices, we obtain 1
--
6 F = ~ [(43'53'u 4 ) 3'53'Ue + ½(~3'uv4) 3"uv3"sir2 e] + ~i(~" a au~) ta3"Ue,
(15)
where 3'uv = ½[3'u, %] and t 1 = 7-13,5, t 2 = 7-2, t 3 = r33' 5 . +1 Cf. Witten in ref. [1]. Also, see refs. [3,4].
11 February 1980
In eq. (15) we find the currents of an SU(2) gauge group which couples only to the Bose fields, together with axial and tensor spinor bilinears. The theory suggested by these results has lagrangian A? = c-/)ta~bc'/)uq~ + i t ~ 4 - a(~757u4A ta - ½g2AuA~ )
- t(i~Tuv4TUV - lg2TuvTUV ) ,
(16)
where a and t are parameters to be specified and Bu, Au, and Tuv are real vector, axial, and tensor auxiliary fields. The SU(2) covariant derivative is 1.
cOuqb = (a# + ~IB u " "c)q~ ,
(17)
and the solutions of the auxiliary field equations are B , = (i/g2) (~,~-ffu~b).
A u = (1/g2)(fj3"57u4),
(lS)
Tuv = (1/g2) i(~3"uv4).
The transformation rules are given by eq. (2) for bosons and by 84 = [-i3'u(c-/) u~b) - ~(Au3"53"u1 X [(R - ir2L ) e l i ,
llTuv3"Uv)(9] i (19)
for fermions. Eq. (19) has been chosen to cancel the previous terms in eq. (15) if the explicit solutions in eqs. (18) are used. Thus the constraints (13) and (14) are now supersymmetric under eqs. (2) and (19). We now discuss the variation of the lagrangian. Here one can anticipate difficulty because the SU(2) gauge group acts only on bosons and does not commute with supersymmetry. Hence local SU(2) invariance is not compatible with just global N = 2 supersymmetry and an invariant lagrangian cannot be obtained unless we include coupling to N = 2 supergravity [16]. Using e qs. (13) and (14) we actually find that many cancellations occur in 6Z?, but eventually we are left with two types of uncancelled terms. The first type consists of gauge non-invariant terms of generic form ~BvauCq/u3"v 4. This suggests that we consider an SU(2) doublet of spin-3/2 gauge fields 4 / and add Noether current terms 4vau4ry.~3'v4 to the lagrangian. The variation 5 4~ ~x (C-l)ue)t will then contain an SU(2) covariant derivative which cancels the B v terms above. The second type of uncancelled term is of the generic form ~Ov¢(A3" v + a3"V.,q) 4 involviug the axial auxiliary and the unspecified parameter a. There is an analogous term involving the tensor auxil73
Volume 90B, number 1,2
PHYSICS LETTERS
iary and the parameter t. One sees that no choice o f a can make this term vanish. We suggest instead that A u and Tuv be interpreted as auxiliary fields o f N = 2 supergravity [17] and that the additional terms in ~ i containing these auxiliaries will give the desired cancellation. It seems reasonable to conjecture that the nonlinear o model with constraints (13), (14) does exist as a locally supersymmetric theory, whose structure is possibly related to that theory discussed in ref. [18]. The full coupling to N = 2 supergravity will be pursued elsewhere. Here we simply note that a major question is whether the local theory will require a cosmological term [19] which can alter its physical interpretation. This is connected to the question o f whether the tr model coupling constant g becomes "locked" with Newton's constant K, in the form g ~ K - l , or whether g remains a free parameter, perhaps related to a cosmological constant. Finally we note the possibility that the constraints (4) and (13) can be simultaneously imposed on the locally supersymmetric theory. This would give 4n - 8 real spin-0 and spin-l/2 degrees o f freedom and a gauge group SU(2) × U(1). Evidence supporting this possibility is that the combined spinor constraints (5) and (14) are invariant under the variations (2) for bosons and a combination of (9) and (19) for fermions. We thank the Aspen Center for Physics for providing the pleasant surroundings in which this work was initiated. We also thank P. Fayet, J. Lukierski, A.M. Polyakov and E. Witten for useful discussions. T.L.C. was partially supported b y a Robert R. McCormick Fellowship. This research was also supported in part by the National Science Foundation under contracts PHY-7801224 and PHY-78-11969.
Note added. Generalizations o f the N = 2 supersymmetric model discussed in the text have been obtained based on the bose constraint ~.-aM°~:.¢.~ = b, where Ma# is an arbitrary hermitian n × n matrix and b is a fixed constant isovector. The fermi constraint, lagrangian and transformation rules are similar to (5), ( 7 ) - ( 9 ) o f the text with the matrix M inserted appropriately.
74
11 February 1980
References [1] P. Di Vecchia and S. Ferrara, Nucl. Phys. B130 (1977) 93; E. Witten, Phys. Rev. D16 (1977) 2991. [2] E. Witten, Nucl. Phys. B149 (1979) 285; A. D'Adda, P. Di Vecchia and M. Liischer, Nucl. Phys. B152 (1979) 125. [3] E. Cremmer and J. Scherk, Phys. Lett. 74B (1978) 341. [4] O. Alvarez, Phys. Rev. D17 (1978) 1123. [5] R. Shankar and E. Witten, Phys. Rev. D17 (1978) 2134. [6] E. Corrigan and C.K. Zachos, Phys. Lett. 88B (1979) 273; T.L. Curtright, Phys. Lett. 88B (1979) 276; T.L. Curtright and C.K. Zachos, Phys. Rev. D, to be published. [7] B. Zumino, CERN preprint, TH-2733. [8] A. Salam and J. Strathdee, Nucl. Phys. B97 (1975) 293; P. Fayet, Nucl. Phys. Bl13 (1976) 135. [9] E.S. Fradkin and M.A. Vasiliev, Lebedev Institute preprint (1976), unpublished; P. Townsend and P. van Nieuwenhuizen, Phys. Lett. 67B (1977) 439. [10] M. Sohnius, Nucl. Phys. B138 (1978) 109. [11] J. Scherk and J.H. Schwarz, Phys. Lett. 57B (1975) 463; L. Brink, J.H. Schwarz and J. Scherk, Nucl. Phys. B121 (1977) 77. [12] A.M. Polyakov, Phys. Lett. 59B (1975) 79; A. D'Adda, P. Di Vecchia and M. Liischer, Nucl. Phys. [13] A. D'Adda, P. Di Vecchia and M. Liischer, Phys. Rep. 49 (1979) 239; A.M. Polyakov, private communication. [14] F. Gliozzi, D. Olive and J. Scherk, Nucl. Phys. B122 (1977) 253. [15] D.R.T. Jones, Phys. Lett. 77B (1977) 199; E.C. Poggio and H.N. Pendleton, Phys. Lett. 77B (1977) 200. [16] S. Ferrara and P. van Nieuwenhuizen, Phys. Rev. Lett. 37 (1976) 1669. [17] P. Breitenlohner and A. Kabelschacht, Nucl. Phys. B148 (1979) 96; E.S. Fradkin and M.A. Vasiliev, Phys. Lett. 85B (1979) 47; B. De Wit and J.W. Van Holten, Nucl. Phys. B155 (1979) 530. [18] C.K. Zachos, Phys. Lett. 76B (1978) 329. [19] D.Z. Freedman, Phys. Rev. D15 (1977) 1173; D.Z. Freedman and A. Das, Nucl. Phys. B120 (1977) 221.
PHYSICS LETTERS
11 February 1980
N O N L I N E A R e MODELS WITH E X T E N D E D SUPERSYMMETRY IN FOUR DIMENSIONS Thomas L. CURTRIGHT The Enrico Fermi Institute, University of Chicago, Chicago, IL 6063 7, USA
and Daniel Z. FREEDMAN Institute for Theoretical Physics, State University o f New York, Stony Brook, N Y 11794, USA
Received 29 October 1979
Two nonlinear o models are discussed which have N = 2 supersymmetry in four dimensions. The f'trst is globally supersymmetric on a noncompact field manifold. The second model involves a compact manifold and requires coupling to N = 2 supergravity for consistency. Upon reduction to three and two dimensions, both models have N = 4 supersymmetry. In addition, the global model has vanishing (one-loop) charge renormalization in two dimensions.
Supersymmetric extensions o f the nonlinear o model are known in both two-dimensional [1,2] and four-dimensional [3] spacetime. These extensions have been studied considerably, especially in two dimensions. Interesting discussions o f their classical solutions [1,2], dynamical symmetry breaking mechanisms [2,4], S-matrices [5], nonlocal conservation laws [6], and their geometrical structures [7] have been given. In this note we discuss the construction o f new nonlinear models with higher supersymmetry, specifically N = 2 extended supersymmetry in four dimensions (and N = 4 supersymmetry after reduction to three or two dimensions). Two distinct types of N = 2 models appear to be possible. The first type has global N = 2 supersymmetry and scalar fields which span a noncompact manifold. A complete discussion o f this model is given here. The second type o f model involves a compact manifold o f scalar fields and requires local supersymmetry, i.e., coupling to extended supergravity is necessary for consistency. Partial results for t h i s model are given below. We start with n scalar multiplets o f N = 2 supersymmetry [8] consisting o f 2n complex scalar fields, ~b~(x) and q~(x), and n Dirac fields, ~ a ( x ) , where ot = 1 ..... n. The free lagrangian
"Q0 = i}a~/~ i}**~ + i ~ a ~
~ ,
(1)
is invariant under two independent commuting supersymmetry transformations, with Majorana spinor parameters 61 and 62. The transformation laws for the boson fields are (2)
8q~ = ~iL ~ ~ - eij~/R ~ c~ ,
where R , L = ½(1 + ")'5) are chiral projectors and eij = ~ - [ ( _ ) / _ ( _ ) i ] . The fermion fields transform as 8 ~a = - i ~ [ R ~ e i - Leii~b~6/] .
(3)
It is clear that the Bose fields initially span the manifold C 2n with linear internal symmetry U(2n). To simplify our notation, the t~ indices will usually be suppressed in subsequent formulae. Following the procedure o f ref. [3], we search for nonlinear constraints which are invariant under supersymmetry transformations. The obvious constraint ~i(9i = g2 is, surprisingly, not implementable without local supersymmetry and will be considered later. Instead, we first impose the isovector constraint ~b
= b,
(4)
where r a are the usual Pauli matrices and b is a fixed constant vector. Thus the SU(2) invariance on the " i " 71
Volume 90B, number 1,2
PHYSICS LETTERS
indices in (1) is explicitly broken down to a U(1) by eq. (4). Supersymmetry invariance of eq. (4) requires two independent Majorana spinor constraints, namely fi=R(~ifc-cij~jf)+Z(¢if-cij¢jfc)=O
,
(5)
where i = 1,2 and f e is the charge conjugate fermion field. At this point we count 4n - 3 real degrees of freedom for the bosons and 4n - 4 for the fermions. One more Bose degree of freedom must be removed to obtain a linear representation of supersymmetry. Following ref. [3] we eliminate this Fermi-Bose disparity by gauging a U(1) subgroup of the remaining global symmetry. To decide which U(1) is relevant, we examine the supersymmetry variation of eq, (5), which, after use of eq. (4) and Fierz rearrangement, takes the form
~ft' = l [ ( ~ f ) + (~75 f)')'sl
eijej
1
-
~(i~ au ~ + ffvuf) vu vs ei.
(6)
In the last terms in eq. (6) we find the current of the U(1) group which gauges the common phase of ~ and f a . This current term drops out of 6f i if we modify eq. (3) by making the derivative of the scalars U(1) covariant. The scalar and pseudoscalar spinor bilinears in eq. (6) also disappear if we add similar bilinear terms to eq. (3). The full theory is now specified. We have the constraints (4) and (5), and the lagrangian = D u ~ DUq~ + i ~ , E i f + ( ~ f ) M
+ (i 575 ~ ) N - ~¢(M 2 + N 2 ) .
(7)
In this expression, M, N, and Vu are real auxiliary fields, and Du~ i =(au+iVu)Oi,
D u f = ( a u +iVu) f .
(8)
The action is invariant under the boson variations (2) and the modified fermion variations 5 f = [ - i ~ + ( M - i 7 5 N ) ] [R4~iei - Lei]¢ie]] .
(9)
Variations of the auxiliary fields need not be specified since we work on-shell with respect to them, as in the 1.5 order formalism of supergravity [9]. The constraints (4) and (5) are now completely supersymmetric provided we use the on-shell values Vu = (i¢ a.~ + ~ 7 u f ) / ( 2 ~ ¢ ) ,
72
M = ff/(2¢¢),
11 February 1980 N =i~75f/(2¢¢).
(10)
Thus the lagrangian and the constraints are all N = 2 supersymmetric. If eqs. (10) are inserted in (7), we obtain the lagrangian with auxiliary fields eliminated, viz., ./2 ='~0 - [(i~-3~ ~ + ~ T u f ) 2 -- (t~f) 2 + (t~75 f)2]/(4~b) •
(11)
To define a perturbation expansion for the theory, we must resolve the constraints. The direction of the isovector b in eq. (4) is irrelevant, and we therefore choose the "north-pole" direction. The constraints become
qSl~b1 - ~2~b2 = b 3 > 0
,
~b1¢2= 0 ,
(12)
and they are easily solved. Clearly, the scalar fields lie in a noncompact manifold embedded in C 2n . Since the induced metric remains riemannian, however, there are no ghost-modes. If one takes the present theory and considers e 1 variations only, then ea2 and R f a can be discarded consistent with e 1 supersymmetry. This truncated theory is exactly the global model given in ref. [3]. Effects of the M and N fields cancel, and the coupling of Vu reduces to an axial coupling to a Majorana fermion. Evidently a superspace formulation of the N = 2 theory can be obtained without difficulty, since the lagrangian (7) is closely related to a known expression [10] involving jointly interacting N = 2 vector and scalar multiplets. It also seems that a self-coupling of N = 2 scalar multiplets has not previously been given, and that the nonlinear o-model constructed here provides the first example of such an interaction. Reduction of the model from four to three or two spacetime dimensions can be carried out according to well-known techniques [ 11 ]. Each four-component spinor of the four-dimensional theory becomes a pair of two-component spinors of the lower dimensional theory. Supersymmetry is preserved by the reduction. One therefore has invariance with respect to four distinct spinorial transformations, i.e., N = 4 supersymmetry. Various dynamical aspects of this new nonlinear theory are currently under investigation. We report here one result concerning the charge renormalization
Volume 90B, number 1,2
PHYSICS LETTERS
properties of the model in two dimensions. In the usual O(n) and CP(n) models in two dimensions, the boson contribution to the one-loop 3 function is negative [12], thereby imitating the asymptotic freedom of four-dimensional Yang-Mills theory. The supersymmetric addition of fermions does not change the value of the one-loop charge renormalization +1 because the fermion small-fluctuation determinant involves minimal coupling to vector potentials and is finite in two dimensions. We have calculated the oneloop 3 function in the present model both by ordinary perturbation theory, using the constraints (12), and by the moving frame method [13]. The boson contribution vanishes because the ~b~ and ~ba2 modes contribute with opposite signs. Again, fermions do not change this result. The determination of the/3 function in higher orders is of obvious interest in connection with a possible parallel with the N = 4 supersymmetric Y a n g Mills theory in four dimensions [14], for which the 3 function is known to vanish through two-loop order [15]. It is not clear to us, however, whether a twodimensional theory with vanishing charge renormalization would have pathological properties due to severe infrared divergences. We now consider the construction of a supersymmetric theory with the constraint ~i ~bi = g2 .
(13)
Proceeding as before; we see that eq. (13) is supersymmetric under the variations (2), if we impose the two Majorana spinor constraints
F i =R(c,bi 4 c + eij~l.4) +L(~i4 +eij~j4 c) = 0 .
(14)
These constraints leave 4n - I Bose and 4n - 4 Fermi degrees of freedom, and previous reasoning suggests that three additional boson modes should be eliminated by gauging an SU(2) subgroup of the global U(2n). To verify this we compute the transform of eq. (14) under the variations (2) and (3). Suppressing SU(2) doublet indices, we obtain 1
--
6 F = ~ [(43'53'u 4 ) 3'53'Ue + ½(~3'uv4) 3"uv3"sir2 e] + ~i(~" a au~) ta3"Ue,
(15)
where 3'uv = ½[3'u, %] and t 1 = 7-13,5, t 2 = 7-2, t 3 = r33' 5 . +1 Cf. Witten in ref. [1]. Also, see refs. [3,4].
11 February 1980
In eq. (15) we find the currents of an SU(2) gauge group which couples only to the Bose fields, together with axial and tensor spinor bilinears. The theory suggested by these results has lagrangian A? = c-/)ta~bc'/)uq~ + i t ~ 4 - a(~757u4A ta - ½g2AuA~ )
- t(i~Tuv4TUV - lg2TuvTUV ) ,
(16)
where a and t are parameters to be specified and Bu, Au, and Tuv are real vector, axial, and tensor auxiliary fields. The SU(2) covariant derivative is 1.
cOuqb = (a# + ~IB u " "c)q~ ,
(17)
and the solutions of the auxiliary field equations are B , = (i/g2) (~,~-ffu~b).
A u = (1/g2)(fj3"57u4),
(lS)
Tuv = (1/g2) i(~3"uv4).
The transformation rules are given by eq. (2) for bosons and by 84 = [-i3'u(c-/) u~b) - ~(Au3"53"u1 X [(R - ir2L ) e l i ,
llTuv3"Uv)(9] i (19)
for fermions. Eq. (19) has been chosen to cancel the previous terms in eq. (15) if the explicit solutions in eqs. (18) are used. Thus the constraints (13) and (14) are now supersymmetric under eqs. (2) and (19). We now discuss the variation of the lagrangian. Here one can anticipate difficulty because the SU(2) gauge group acts only on bosons and does not commute with supersymmetry. Hence local SU(2) invariance is not compatible with just global N = 2 supersymmetry and an invariant lagrangian cannot be obtained unless we include coupling to N = 2 supergravity [16]. Using e qs. (13) and (14) we actually find that many cancellations occur in 6Z?, but eventually we are left with two types of uncancelled terms. The first type consists of gauge non-invariant terms of generic form ~BvauCq/u3"v 4. This suggests that we consider an SU(2) doublet of spin-3/2 gauge fields 4 / and add Noether current terms 4vau4ry.~3'v4 to the lagrangian. The variation 5 4~ ~x (C-l)ue)t will then contain an SU(2) covariant derivative which cancels the B v terms above. The second type of uncancelled term is of the generic form ~Ov¢(A3" v + a3"V.,q) 4 involviug the axial auxiliary and the unspecified parameter a. There is an analogous term involving the tensor auxil73
Volume 90B, number 1,2
PHYSICS LETTERS
iary and the parameter t. One sees that no choice o f a can make this term vanish. We suggest instead that A u and Tuv be interpreted as auxiliary fields o f N = 2 supergravity [17] and that the additional terms in ~ i containing these auxiliaries will give the desired cancellation. It seems reasonable to conjecture that the nonlinear o model with constraints (13), (14) does exist as a locally supersymmetric theory, whose structure is possibly related to that theory discussed in ref. [18]. The full coupling to N = 2 supergravity will be pursued elsewhere. Here we simply note that a major question is whether the local theory will require a cosmological term [19] which can alter its physical interpretation. This is connected to the question o f whether the tr model coupling constant g becomes "locked" with Newton's constant K, in the form g ~ K - l , or whether g remains a free parameter, perhaps related to a cosmological constant. Finally we note the possibility that the constraints (4) and (13) can be simultaneously imposed on the locally supersymmetric theory. This would give 4n - 8 real spin-0 and spin-l/2 degrees o f freedom and a gauge group SU(2) × U(1). Evidence supporting this possibility is that the combined spinor constraints (5) and (14) are invariant under the variations (2) for bosons and a combination of (9) and (19) for fermions. We thank the Aspen Center for Physics for providing the pleasant surroundings in which this work was initiated. We also thank P. Fayet, J. Lukierski, A.M. Polyakov and E. Witten for useful discussions. T.L.C. was partially supported b y a Robert R. McCormick Fellowship. This research was also supported in part by the National Science Foundation under contracts PHY-7801224 and PHY-78-11969.
Note added. Generalizations o f the N = 2 supersymmetric model discussed in the text have been obtained based on the bose constraint ~.-aM°~:.¢.~ = b, where Ma# is an arbitrary hermitian n × n matrix and b is a fixed constant isovector. The fermi constraint, lagrangian and transformation rules are similar to (5), ( 7 ) - ( 9 ) o f the text with the matrix M inserted appropriately.
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