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On the geometric meaning of the N = 1 Yang-Mills prepotential
Volume 117B, number 1, 2
PHYSICS LETTERS
4 November 1982
ON THE GEOMETRIC MEANING OF THE N = 1 Y A N G - M I L L S PREPOTENTIAL
E.A. IVANOV
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, USSR Received 28 June 1982
A deep similarity between intrinsic superspace geometries of the N = 1 Yang-Mills theory and minimal N = 1 supergravity is established. The N = 1 Yang-Mills prepotential is shown to take values in the coset space GC/G, Gc being the complexification of gauge group G.
1. The geometric structures inherent in the superspace description of supersymmetric gauge and supergravity theories radically differ from those in ordinary space-time. Gauge potentials in superspace have no chance to be the fundamental quantities as they carry too much degrees of freedom even in the fixed gauge. Therefore, in any self-contained superfield gauge theory they are expected to be composed of a lesser number of unconstrained superfields directly related to the physical field content of the theory, prepotentials. These are known for the N = 1 Yang-Mills [1] and supergravity [ 2 - 4 ] theories and, on the linearized level, for their N = 2 counterparts [5,6] * 1 The prepotentials provide an adequate realization of the minimal invariance group of a given superfield theory and hence can be regarded as natural carriers of its intrinsic geometry. But only in the case of N = 1 supergravity this role of them was exposed quite clearly. Ogievetsky and Sokatchev have shown [3] that the prepotential of minimal N = 1 supergravity Hm(x, O, "0) specifies the position of a real hypersurface R 4,4 = .~x m, 0 a, 0& } in the complex superspace C 4,2 = {x~a, 0~ }: Im x ~ = Hm(x, O, if), --
R -=
Re x~a = x m , :
(i)
, a The objects suggested in ref. [7] as prepotentials of full N = 2 supergravity seem not to be the true ones as they axe still subjected to certain constraints. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
The prepotentials of nonminimal N = 1 supergravity have a similar interpretation [7]. It is desirable to understand in analogous terms the other cases listed above and, before all, the text-book case o f N = 1 Yang-Mills. A deeper insight into its group and geometric structure may help in achieving a complete unconstrained superfield formulation of gauge theories and supergravities with higher N. Actually, the underlying geometry of the N = 1 Yang-Mills theory reveals a strong resemblance to that of minimal N = 1 supergravity and it is the aim of the present note to explicitly demonstrate this. The N = 1 Yang-Mills theory is shown to be associated with embedding of R 4,4 in the extended complex superspace C 4+M'2 = trXmL,tpL,i0 ~ ) (i = 1, ...,34) where ¢/L are local coordinates on the group G c, the complexification of the gauge group G, and M = dim G. The N = 1 Yang-Mills prepotential Vi(x, O, -0) is nothing but Im ¢~ restricted to the hypersurface R 4,4. It specifies the position of this hypersurface with respect to GC/G-directions in c4+M,2. 2. Let G be a compact group of internal symmetry with M hermitean generators T i. Our basic idea is to extend it to complex noncompact group G c with 2M. generators T[, T{~ -- (T[)? which satisfy the commutation relations: [ riL, Tk] --iciklrl,L [ viL, Tk] = 0 ,
[ R,T i T~] =iCiklTIR,
(2.1a) (2.1b) 59
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where C ikl are totally skew-symmetric structure constants of G. The group G c has the structure of direct product G L X G R with G L and G R generated, respectively, by T [ and T~ [e.g., if G = SU(n) then G c = SL(n, C)]. The initial group G appears as a diagonal in this product. Its generators are identified with the sum. T Li + .TRi = T i while the remaining generators A i = i ( r [ - T~) span the M-dimensional coset space GC/G. The latter is real and symmetric. We define the complex superspace C 4+M,2 = {x~, ¢~,, 0~) as a direct sum of ordinary chiral N = 1 superspace C4, 2 = {x~, 0~) and the group G L considered as a complex manifold with local coordinates ~o~: C4+M,2 = C 4,2 G G L.
(2.2)
For convenience, we choose the exponential parametrization of group elements: gC(~L, ~°R) = gL(tPL)gR(tPR) = exp(i~°kLTk)exp( i~°kTk) --- exp [i(Re ~kLTk+ Im ~okLAk)l
[~0~t------(~o~)t I •
(2.3) Note that Re tpkLand Im tpkLprovide, respectively, a particular parametrization of G and GC/G. Since the coordinates x ~ , 0~ are related to their conjugates x~, O~ by P-parity, i.t is natural to accept the same l . convention for ~0lL, ~R" CkL P ~OR, k
i ~P Re ~PL, i Re tpL
Im~o~ e - I m ~ 0 } .
Correspondingly, if T i are scalars, A i should be pseudoscalars:
TiL P TiR,
T i P T i,
A i P - A i.
(2.4)
Clearly, (2.4) is the automorphism of the algebra (2.1) The group G c has a natural realization as the group of left multiplications of gC(tpL, ~R): t
gL ()kL)gL(q0L) = gL (tPL(~0t, XL)), gR(XR)gR(~PR) = gR(¢~(~0R, XR)),
(2.5)
thus inducing nonlinear transformations for ~0L, i ~PR" i To promote the global GC-transformations to the local ones, we assume that the group parameters X[ in (2.5) are arbitrary analytic functions of ordinary superspace coordinates: X}. = X~(XL, OL), 60
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X~t = (X~) t = X~,(XR,OR)-
(2.6)
The gauge group thus defined forms a semi-direct product with supersymmetry realized on x~, 0 [ * 2 This product is contained as a subgroup in the supergroup of general analytic coordinate transformations o f t 4+M,2 (to be more exact, in its "triangular" subgroup which leaves C 4,2 invariant). As implied by relation (2.1b), the left and right components of the gauge group G~oc = GLloc X GRloc commute with each other, so at the initial stage the "left" and "right" worlds are entirely disjoint (though conjugated). Now we wish to show that the N = 1 Yang-Mills theory has Gfoc as an invariance group and it naturally emerges upon extracting a special hypersurface in the superspace C 4+M,2 . This hypersurface is the real superspace R 4'4 = ( x m, 0 a, 0~}, just as in the case of N = 1 supergravity [3]. But it possesses now purely internal degrees of freedom besides those represented by Hm(x, O, "if), because of additional bosonic dimensions in c4+M, 2 . So, the embedding conditions (1.1) should be augmented with 23"/conditions Im ~0~ = Vi(x, O, #),
Re ~0~ =
Ui(x, O, 0),
(2.7)
where V i and U i are real pseudoscalar and scalar superfields. Their transformation propert!es in G~oc are uniquely determined by those of ~p].,~0}/(2.5). These superfields parametrize, respectively, the coset space GC/G and the subgroup G, hence they are of the Goldstone character with respect to corresponding transformations. We want G to be unbroken; then Ui(x, O, #) should be made to have no dynamical manifestations. To achieve this, one may proceed as in standard nonlinear o-models (see, e.g. ref. [8]) and arrange the theory to be invariant under the right gauge G-transformations: exp [i(UkT k + VkAk)] -+ exp [i(UkT k + VkAk)] exp(iXiTi),
(2.8)
where Xi = xi(x, O, -#) are M real superparameters. Then ui(x, O, -#) represent purely gauge degrees of freedom. From the geometric standpoint, the invariance under (2.8) means that different G-directions in C 4+M,2 are indistinguishable; the dynamics is required to depend only on the position of the hypersurface with respect to the directions spanning the coset space GC/G. • 2 Our consideration proceeds in the same way for rigid and local supersymmetries.
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On imposing the natural gauge condition
Ui(x, 0 , 0 ) = 0,
(2.9)
we are left with M pseudoscalar superfields Vi(x, O, -0) which exist on the coset space GC/G and transform under G~oc according to the generic formula of nonlinear realizations [9] exp [i(Re XkT k + Im xkAk)] exp(iVkA k)
= exp(iVk'A k) exp [iKI(V, X)T/1
(2.10)
with )t~ given by (2.6). The transformation rule of matter superfields qb(x, 0, 0) is then defined following general prescriptions of ref. [9] :
c~'(x,O,-O)=exp[iKt(V,)t)~rt]O(x,O,-O),
(2.11)
~t being a proper matrix representation of T t. A simple consideration exploiting the automorphism (2.4) shows that the transformation law (2.10) admits the equivalent form: exp(iXk TLk) exp(2 v k ) exp(--ixkTk)
= exp(2Vk'T k)
(2.12)
(and the conjugated one, with T k instead of TLk). Since TLk satisfy the same commutation relations as T k, eq. (2.12) coincides with the standard transformation law o f N = 1 Yang-Mflls prepotential [1]. In fact, its conventional form is recovered when choosing a particular, non-self-conjugated representation for A i ~k = iTk
(~k = ~k, ~ k = 0)
(2.13)
(this choice is quite permissible because the structure of V i' , Ki(V, X) in (2.10), (2.12) is determined solely by the commutation relations of generators). Using the general connection between representations and nonlinear realizations [9] one may always pass from superflelds with the standard nonlinear transformation rule (2.11) to those transforming in G~oc linearly, according to the representation (2.13) or to the conjugated one:
~L = exp( i Vk ~k)op = exp(-- vk Tk)O~ = e x p ( - 2 V k Tk)qbR, qbL = exp(iXkTk)~bL'
(2.14)
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the transition from the real basis in the group space of G c to its complex left and right bases, in a perfect analogy with the connection between real and chiral bases in superspace. These formulas were known earlier [4,10], but our approach renders to them the clear group meaning. Note that the substitution of (2.13) into (2.10) yields the transformation o f N = 1 prepotential in the form given by Siegel and Gates [4]. Also, the invariance under the right gauge group (2.8) reduces to the well-known freedom of complexifying the prepotential. To summarize, we have derived the N = 1 YangMills prepotential from simple geometric and group principles similar to those constituting the basis of the Ogievetsky-Sokatchev formulation of minimal N = 1 supergravity [3]. 3. It follows from the above analysis that the N = 1 gauge theories can be thought of as a kind of generalized nonlinear o-models * 3 [exp(2iVkA k) is nothing but a "chiral field" on the coset space GC/G]. So, relevant invariants and other geometric characteristics should have an adequate expression in the universal language of Cartan differential forms which is of a common use in nonlinear realizations [9]. In the present case the basic forms are spinorial ones, they are introduced by the relation: exp(-i vkA k) [Da + I'clyL ]c~ exp(i VkA k) k k = l(coaA + ~ at T t ),
(3.1)
and by the conjugated one. Here Da is an ordinary covariant spinor derivative (it may correspond to the flat as well as curved geometries on I:14,4) and cl;L -=_ c~oLi,ri - Vc~ 1 L is the spinor connection on the group GL: L' L -1 q2~ = gL0tL)qYagL 0tL) + (1/i)gL(XL)D~g[1(XL). (3.2) One easily checks that under the nonlinear realization (2.10) the objects wa, k f2ka possess the standard transformation properties of Cartan forms. It is essential that the gauge superpotentials qyL, @R& = (QyL)t are purely subsidiary as they can be
qb~ = exp(iXkTk)qbR.
The relations (2.14) can be regarded as describing
3 An analogous fact for ordinary gauge theories has been established in ref. [ 11 ]. 61
Volume 117B, number 1, 2
PHYSICS LETTERS
covariantly expressed in terms of Vi(x, O, -if) by imposing the manifestly invariant conditions ~ ' = ~'.~ = 0,
(3.3)
that is a particular case of the inverse Higgs phenomenon [ 12]. The constraints (3.3) are algebraic with respect to c)y aL, ~.Rc~' so they can easily be solved to give q~L = (1/i) exp(--2 vkTkL)Dc~ exp(2 V k Tk), @.R = (c,yL)'~.
(3.4)
Oz
Thus, we are finally left with the forms ~ i , ~ . which are the connections on the coset space GC/G:
a i T i = (1/i) e x p ( - V k T k) D , exp(VkTk), ~ i T i = (1/i)exp(VkTk)Daexp(--VkTk).
(3.5)
All other quantities of the theory: the vector Cartan form, covariant derivatives and strengths are built up from ~2k, ~ k0~ following the standard procedure of ref. [13]. They have a familiar appearance, for this reason we do not present them here. The point to be emphasized is that our approach allows us to obtain all necessary quantities starting solely with the structure relations (2.1) and the standard nonlinear realization formula (3.1) supplemented by the constraints (3.3). Perhaps, it would be of interest to relate this formalism to the Levi" superform approach advocated by Schwartz [ 14] as the adequate geometric language for treating real hypersurfaces in complex superspaces. 4. The most interesting question following from the above consideration is how to generalize the proposed construction to higher N Yang-Mills theories, at least to the case N = 2. The necessity to complexify G in the N = 1 case be traced to the fact that the fundamental superspace o f N = 1 supersymmetry is complex superspace C 4,2 . Its true analog in the N = 2 case seems to be a superspace the bosonic coordinates of which form a quaternion [15]. So in the N = 2 case one may, instead of the extension T k --," { T k, i T k }, try the extension T k -+ { T k , qi ® T k } , where qi (i = 1,2, 3) are imaginary quaternion units. Then the corresponding prepotential should acquire an additional triplet index. That is just what occurs in the N = 2 electrodynamics [5]. Work along this line is now in progress. 62
4 November 1982
Other possible applications concern the N = 1 Yang-Mills theory itself. Once it is a kind of the nonlinear o-model on cosets GC]G, the idea arises that the corresponding linear o-models may exist, with G c as the vacuum symmetry group. In the conventional N = 1 Yang-Mills theory, the noncompactness of G c has no dynamical manifestations since the vacuum stability subgroup is compact and, besides, the Goldstone fields associated with spontaneous breaking GC/G are purely gauge degrees of freedom (they are contained in the superspin zero part of Vi(x, O, if). In the relevant linear o-model, the noncompactness would result in appearance of infinite-dimensional field multiplets, for any unitary representation of G c is infinite-dimensional. We also note that the geometric analogy between the N = 1 Yang-Mills and N = 1 supergravity naturally suggests their unification within a larger theory of the Kaluza-Klein type. One i = ggi as an independent coordinate may treat Re ~PL like x m in eq. (1.1), choose the base superspace to be FI4+M,4 = {x m, tfli, 0 ~ , 0&) rather than I:!4,4 , and construct a 4 + M-dimensional extension of minimal N = 1 supergravity through embedding R 4+M,4 into C 4+M,2 . The standard theory is expected to be recovered as the lowest order in a proper expansion in tpi . It is a pleasure for the author to thank A. Galperin, V. Ogievetsky, E. Sokatchev and K. Stelle for interest in the work and useful discussions.
References [1] S. Ferrara and B. Zumino, Nucl. Phys. B79 (1974) 413; A. Salam and J. Strathdee, Phys. Lett. 51B (1974) 35 3. [2] V. Ogievetsky and E. Sokatchev, Nucl. Phys. B124 (1977) 309. [3] V. Ogievetsky and E. Sokatchev, Phys. Lett. 79B (1978) 222; Yad. Fiz. 28 (1978) 1631 ; 31 (1980) 264,821. [4] W. Siegel and S.J. Gates, Nucl. Phys. B147 (1979) 77. [5] L. Mesincesku, Dubna preprint P2-12572 (1979). [6] S.J. Gates and W. Siegel, Nucl. Phys. B195 (1982) 39. [7] E. Sokatchev, in: Superspace and supergravity, eds. S. Hawking and M. Ro~ek (Cambridge U.P., London, 1981). [8] M.K. Gaillaxd and B. Zumino, Nucl. Phys. B193 (1981) 221. [9] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C.L. CaUan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247;
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D.V. Volkov, Sov. J. Part. and Nucl. 4 (1973) 3; V.I. Ogievetsky, in: Proc. tenth Winter School of Theoretical Physics in Karpacz, vol. I (1974) p. 117. [10] M.T. Grisaru, CALTECH preprint 68-894 (1981). [ 111 E.A. Ivanov and V.I. Ogievetsky, Pis'ma Zh. Eksp. Teor. Fiz. 23 (1976) 661; Lett. Math. Phys. 1 (1976) 309. [12] E.A. Ivanov and V.I. Ogievetsky, Teor. Mat. Fiz. 25 (1975) 164.
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[13] M.F. Sohnius, Nucl. Phys. B136 (1978) 461; J. Wess, in: Proc. VIII GIFT Seminar (Salamanca, 1981). [14] A.S. Schwarz, Yad. Fiz. 34 (1981) 1144; A.V. Gayduk, V.N. Romanov and A.S. Schwarz, Commun. Math. Phys. 79 (1981) 507. [15] A. Galperin, E. Ivanov and V. Ogievetsky, Dubna preprint E2-81-482 (1981); Yad. Fiz. 35 (1982) 790.
63
PHYSICS LETTERS
4 November 1982
ON THE GEOMETRIC MEANING OF THE N = 1 Y A N G - M I L L S PREPOTENTIAL
E.A. IVANOV
Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980 Dubna, USSR Received 28 June 1982
A deep similarity between intrinsic superspace geometries of the N = 1 Yang-Mills theory and minimal N = 1 supergravity is established. The N = 1 Yang-Mills prepotential is shown to take values in the coset space GC/G, Gc being the complexification of gauge group G.
1. The geometric structures inherent in the superspace description of supersymmetric gauge and supergravity theories radically differ from those in ordinary space-time. Gauge potentials in superspace have no chance to be the fundamental quantities as they carry too much degrees of freedom even in the fixed gauge. Therefore, in any self-contained superfield gauge theory they are expected to be composed of a lesser number of unconstrained superfields directly related to the physical field content of the theory, prepotentials. These are known for the N = 1 Yang-Mills [1] and supergravity [ 2 - 4 ] theories and, on the linearized level, for their N = 2 counterparts [5,6] * 1 The prepotentials provide an adequate realization of the minimal invariance group of a given superfield theory and hence can be regarded as natural carriers of its intrinsic geometry. But only in the case of N = 1 supergravity this role of them was exposed quite clearly. Ogievetsky and Sokatchev have shown [3] that the prepotential of minimal N = 1 supergravity Hm(x, O, "0) specifies the position of a real hypersurface R 4,4 = .~x m, 0 a, 0& } in the complex superspace C 4,2 = {x~a, 0~ }: Im x ~ = Hm(x, O, if), --
R -=
Re x~a = x m , :
(i)
, a The objects suggested in ref. [7] as prepotentials of full N = 2 supergravity seem not to be the true ones as they axe still subjected to certain constraints. 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
The prepotentials of nonminimal N = 1 supergravity have a similar interpretation [7]. It is desirable to understand in analogous terms the other cases listed above and, before all, the text-book case o f N = 1 Yang-Mills. A deeper insight into its group and geometric structure may help in achieving a complete unconstrained superfield formulation of gauge theories and supergravities with higher N. Actually, the underlying geometry of the N = 1 Yang-Mills theory reveals a strong resemblance to that of minimal N = 1 supergravity and it is the aim of the present note to explicitly demonstrate this. The N = 1 Yang-Mills theory is shown to be associated with embedding of R 4,4 in the extended complex superspace C 4+M'2 = trXmL,tpL,i0 ~ ) (i = 1, ...,34) where ¢/L are local coordinates on the group G c, the complexification of the gauge group G, and M = dim G. The N = 1 Yang-Mills prepotential Vi(x, O, -0) is nothing but Im ¢~ restricted to the hypersurface R 4,4. It specifies the position of this hypersurface with respect to GC/G-directions in c4+M,2. 2. Let G be a compact group of internal symmetry with M hermitean generators T i. Our basic idea is to extend it to complex noncompact group G c with 2M. generators T[, T{~ -- (T[)? which satisfy the commutation relations: [ riL, Tk] --iciklrl,L [ viL, Tk] = 0 ,
[ R,T i T~] =iCiklTIR,
(2.1a) (2.1b) 59
Volume 117B, number 1, 2
where C ikl are totally skew-symmetric structure constants of G. The group G c has the structure of direct product G L X G R with G L and G R generated, respectively, by T [ and T~ [e.g., if G = SU(n) then G c = SL(n, C)]. The initial group G appears as a diagonal in this product. Its generators are identified with the sum. T Li + .TRi = T i while the remaining generators A i = i ( r [ - T~) span the M-dimensional coset space GC/G. The latter is real and symmetric. We define the complex superspace C 4+M,2 = {x~, ¢~,, 0~) as a direct sum of ordinary chiral N = 1 superspace C4, 2 = {x~, 0~) and the group G L considered as a complex manifold with local coordinates ~o~: C4+M,2 = C 4,2 G G L.
(2.2)
For convenience, we choose the exponential parametrization of group elements: gC(~L, ~°R) = gL(tPL)gR(tPR) = exp(i~°kLTk)exp( i~°kTk) --- exp [i(Re ~kLTk+ Im ~okLAk)l
[~0~t------(~o~)t I •
(2.3) Note that Re tpkLand Im tpkLprovide, respectively, a particular parametrization of G and GC/G. Since the coordinates x ~ , 0~ are related to their conjugates x~, O~ by P-parity, i.t is natural to accept the same l . convention for ~0lL, ~R" CkL P ~OR, k
i ~P Re ~PL, i Re tpL
Im~o~ e - I m ~ 0 } .
Correspondingly, if T i are scalars, A i should be pseudoscalars:
TiL P TiR,
T i P T i,
A i P - A i.
(2.4)
Clearly, (2.4) is the automorphism of the algebra (2.1) The group G c has a natural realization as the group of left multiplications of gC(tpL, ~R): t
gL ()kL)gL(q0L) = gL (tPL(~0t, XL)), gR(XR)gR(~PR) = gR(¢~(~0R, XR)),
(2.5)
thus inducing nonlinear transformations for ~0L, i ~PR" i To promote the global GC-transformations to the local ones, we assume that the group parameters X[ in (2.5) are arbitrary analytic functions of ordinary superspace coordinates: X}. = X~(XL, OL), 60
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X~t = (X~) t = X~,(XR,OR)-
(2.6)
The gauge group thus defined forms a semi-direct product with supersymmetry realized on x~, 0 [ * 2 This product is contained as a subgroup in the supergroup of general analytic coordinate transformations o f t 4+M,2 (to be more exact, in its "triangular" subgroup which leaves C 4,2 invariant). As implied by relation (2.1b), the left and right components of the gauge group G~oc = GLloc X GRloc commute with each other, so at the initial stage the "left" and "right" worlds are entirely disjoint (though conjugated). Now we wish to show that the N = 1 Yang-Mills theory has Gfoc as an invariance group and it naturally emerges upon extracting a special hypersurface in the superspace C 4+M,2 . This hypersurface is the real superspace R 4'4 = ( x m, 0 a, 0~}, just as in the case of N = 1 supergravity [3]. But it possesses now purely internal degrees of freedom besides those represented by Hm(x, O, "if), because of additional bosonic dimensions in c4+M, 2 . So, the embedding conditions (1.1) should be augmented with 23"/conditions Im ~0~ = Vi(x, O, #),
Re ~0~ =
Ui(x, O, 0),
(2.7)
where V i and U i are real pseudoscalar and scalar superfields. Their transformation propert!es in G~oc are uniquely determined by those of ~p].,~0}/(2.5). These superfields parametrize, respectively, the coset space GC/G and the subgroup G, hence they are of the Goldstone character with respect to corresponding transformations. We want G to be unbroken; then Ui(x, O, #) should be made to have no dynamical manifestations. To achieve this, one may proceed as in standard nonlinear o-models (see, e.g. ref. [8]) and arrange the theory to be invariant under the right gauge G-transformations: exp [i(UkT k + VkAk)] -+ exp [i(UkT k + VkAk)] exp(iXiTi),
(2.8)
where Xi = xi(x, O, -#) are M real superparameters. Then ui(x, O, -#) represent purely gauge degrees of freedom. From the geometric standpoint, the invariance under (2.8) means that different G-directions in C 4+M,2 are indistinguishable; the dynamics is required to depend only on the position of the hypersurface with respect to the directions spanning the coset space GC/G. • 2 Our consideration proceeds in the same way for rigid and local supersymmetries.
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On imposing the natural gauge condition
Ui(x, 0 , 0 ) = 0,
(2.9)
we are left with M pseudoscalar superfields Vi(x, O, -0) which exist on the coset space GC/G and transform under G~oc according to the generic formula of nonlinear realizations [9] exp [i(Re XkT k + Im xkAk)] exp(iVkA k)
= exp(iVk'A k) exp [iKI(V, X)T/1
(2.10)
with )t~ given by (2.6). The transformation rule of matter superfields qb(x, 0, 0) is then defined following general prescriptions of ref. [9] :
c~'(x,O,-O)=exp[iKt(V,)t)~rt]O(x,O,-O),
(2.11)
~t being a proper matrix representation of T t. A simple consideration exploiting the automorphism (2.4) shows that the transformation law (2.10) admits the equivalent form: exp(iXk TLk) exp(2 v k ) exp(--ixkTk)
= exp(2Vk'T k)
(2.12)
(and the conjugated one, with T k instead of TLk). Since TLk satisfy the same commutation relations as T k, eq. (2.12) coincides with the standard transformation law o f N = 1 Yang-Mflls prepotential [1]. In fact, its conventional form is recovered when choosing a particular, non-self-conjugated representation for A i ~k = iTk
(~k = ~k, ~ k = 0)
(2.13)
(this choice is quite permissible because the structure of V i' , Ki(V, X) in (2.10), (2.12) is determined solely by the commutation relations of generators). Using the general connection between representations and nonlinear realizations [9] one may always pass from superflelds with the standard nonlinear transformation rule (2.11) to those transforming in G~oc linearly, according to the representation (2.13) or to the conjugated one:
~L = exp( i Vk ~k)op = exp(-- vk Tk)O~ = e x p ( - 2 V k Tk)qbR, qbL = exp(iXkTk)~bL'
(2.14)
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the transition from the real basis in the group space of G c to its complex left and right bases, in a perfect analogy with the connection between real and chiral bases in superspace. These formulas were known earlier [4,10], but our approach renders to them the clear group meaning. Note that the substitution of (2.13) into (2.10) yields the transformation o f N = 1 prepotential in the form given by Siegel and Gates [4]. Also, the invariance under the right gauge group (2.8) reduces to the well-known freedom of complexifying the prepotential. To summarize, we have derived the N = 1 YangMills prepotential from simple geometric and group principles similar to those constituting the basis of the Ogievetsky-Sokatchev formulation of minimal N = 1 supergravity [3]. 3. It follows from the above analysis that the N = 1 gauge theories can be thought of as a kind of generalized nonlinear o-models * 3 [exp(2iVkA k) is nothing but a "chiral field" on the coset space GC/G]. So, relevant invariants and other geometric characteristics should have an adequate expression in the universal language of Cartan differential forms which is of a common use in nonlinear realizations [9]. In the present case the basic forms are spinorial ones, they are introduced by the relation: exp(-i vkA k) [Da + I'clyL ]c~ exp(i VkA k) k k = l(coaA + ~ at T t ),
(3.1)
and by the conjugated one. Here Da is an ordinary covariant spinor derivative (it may correspond to the flat as well as curved geometries on I:14,4) and cl;L -=_ c~oLi,ri - Vc~ 1 L is the spinor connection on the group GL: L' L -1 q2~ = gL0tL)qYagL 0tL) + (1/i)gL(XL)D~g[1(XL). (3.2) One easily checks that under the nonlinear realization (2.10) the objects wa, k f2ka possess the standard transformation properties of Cartan forms. It is essential that the gauge superpotentials qyL, @R& = (QyL)t are purely subsidiary as they can be
qb~ = exp(iXkTk)qbR.
The relations (2.14) can be regarded as describing
3 An analogous fact for ordinary gauge theories has been established in ref. [ 11 ]. 61
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covariantly expressed in terms of Vi(x, O, -if) by imposing the manifestly invariant conditions ~ ' = ~'.~ = 0,
(3.3)
that is a particular case of the inverse Higgs phenomenon [ 12]. The constraints (3.3) are algebraic with respect to c)y aL, ~.Rc~' so they can easily be solved to give q~L = (1/i) exp(--2 vkTkL)Dc~ exp(2 V k Tk), @.R = (c,yL)'~.
(3.4)
Oz
Thus, we are finally left with the forms ~ i , ~ . which are the connections on the coset space GC/G:
a i T i = (1/i) e x p ( - V k T k) D , exp(VkTk), ~ i T i = (1/i)exp(VkTk)Daexp(--VkTk).
(3.5)
All other quantities of the theory: the vector Cartan form, covariant derivatives and strengths are built up from ~2k, ~ k0~ following the standard procedure of ref. [13]. They have a familiar appearance, for this reason we do not present them here. The point to be emphasized is that our approach allows us to obtain all necessary quantities starting solely with the structure relations (2.1) and the standard nonlinear realization formula (3.1) supplemented by the constraints (3.3). Perhaps, it would be of interest to relate this formalism to the Levi" superform approach advocated by Schwartz [ 14] as the adequate geometric language for treating real hypersurfaces in complex superspaces. 4. The most interesting question following from the above consideration is how to generalize the proposed construction to higher N Yang-Mills theories, at least to the case N = 2. The necessity to complexify G in the N = 1 case be traced to the fact that the fundamental superspace o f N = 1 supersymmetry is complex superspace C 4,2 . Its true analog in the N = 2 case seems to be a superspace the bosonic coordinates of which form a quaternion [15]. So in the N = 2 case one may, instead of the extension T k --," { T k, i T k }, try the extension T k -+ { T k , qi ® T k } , where qi (i = 1,2, 3) are imaginary quaternion units. Then the corresponding prepotential should acquire an additional triplet index. That is just what occurs in the N = 2 electrodynamics [5]. Work along this line is now in progress. 62
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Other possible applications concern the N = 1 Yang-Mills theory itself. Once it is a kind of the nonlinear o-model on cosets GC]G, the idea arises that the corresponding linear o-models may exist, with G c as the vacuum symmetry group. In the conventional N = 1 Yang-Mills theory, the noncompactness of G c has no dynamical manifestations since the vacuum stability subgroup is compact and, besides, the Goldstone fields associated with spontaneous breaking GC/G are purely gauge degrees of freedom (they are contained in the superspin zero part of Vi(x, O, if). In the relevant linear o-model, the noncompactness would result in appearance of infinite-dimensional field multiplets, for any unitary representation of G c is infinite-dimensional. We also note that the geometric analogy between the N = 1 Yang-Mills and N = 1 supergravity naturally suggests their unification within a larger theory of the Kaluza-Klein type. One i = ggi as an independent coordinate may treat Re ~PL like x m in eq. (1.1), choose the base superspace to be FI4+M,4 = {x m, tfli, 0 ~ , 0&) rather than I:!4,4 , and construct a 4 + M-dimensional extension of minimal N = 1 supergravity through embedding R 4+M,4 into C 4+M,2 . The standard theory is expected to be recovered as the lowest order in a proper expansion in tpi . It is a pleasure for the author to thank A. Galperin, V. Ogievetsky, E. Sokatchev and K. Stelle for interest in the work and useful discussions.
References [1] S. Ferrara and B. Zumino, Nucl. Phys. B79 (1974) 413; A. Salam and J. Strathdee, Phys. Lett. 51B (1974) 35 3. [2] V. Ogievetsky and E. Sokatchev, Nucl. Phys. B124 (1977) 309. [3] V. Ogievetsky and E. Sokatchev, Phys. Lett. 79B (1978) 222; Yad. Fiz. 28 (1978) 1631 ; 31 (1980) 264,821. [4] W. Siegel and S.J. Gates, Nucl. Phys. B147 (1979) 77. [5] L. Mesincesku, Dubna preprint P2-12572 (1979). [6] S.J. Gates and W. Siegel, Nucl. Phys. B195 (1982) 39. [7] E. Sokatchev, in: Superspace and supergravity, eds. S. Hawking and M. Ro~ek (Cambridge U.P., London, 1981). [8] M.K. Gaillaxd and B. Zumino, Nucl. Phys. B193 (1981) 221. [9] S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2239; C.L. CaUan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247;
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D.V. Volkov, Sov. J. Part. and Nucl. 4 (1973) 3; V.I. Ogievetsky, in: Proc. tenth Winter School of Theoretical Physics in Karpacz, vol. I (1974) p. 117. [10] M.T. Grisaru, CALTECH preprint 68-894 (1981). [ 111 E.A. Ivanov and V.I. Ogievetsky, Pis'ma Zh. Eksp. Teor. Fiz. 23 (1976) 661; Lett. Math. Phys. 1 (1976) 309. [12] E.A. Ivanov and V.I. Ogievetsky, Teor. Mat. Fiz. 25 (1975) 164.
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[13] M.F. Sohnius, Nucl. Phys. B136 (1978) 461; J. Wess, in: Proc. VIII GIFT Seminar (Salamanca, 1981). [14] A.S. Schwarz, Yad. Fiz. 34 (1981) 1144; A.V. Gayduk, V.N. Romanov and A.S. Schwarz, Commun. Math. Phys. 79 (1981) 507. [15] A. Galperin, E. Ivanov and V. Ogievetsky, Dubna preprint E2-81-482 (1981); Yad. Fiz. 35 (1982) 790.
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