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Supersymmetric solitons
Volume 132B, number 1,2,3
PHYSICS LETTERS
24 November 1983
SUPERSYMMETRIC SOLITONS Wayne BOUCHER Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England Received 8 July 1983
We show that static bosonic supersymmetric configurations of a field theory are actually configurations that satisfy the field equations. This reduces the second order differential equations for bosons to first order ones. In particular, we show that the usual monopole solutions ofN = 2 super Yang-Mills are the only static bosonic supersymmetric solutions of the theory.
Introduction. To date, the most beautiful results to emerge from supersymmetric field theories have been those with a corresponding formal mathematical statement entailed by the super Lie algebraic structure underlying the theories. For example, Witten's analytic proof of the positive mass theorem for Einstein's theory of gravity [ 1 ], and the generalizations thereof for extended and gauged extended supergravity theories [2], have an obvious algebraic statement for the corresponding super-algebras which indeed was noted long before and which motivated, the analytic proof [3 ]. Similarly, one might think that some if not all of the quantum cancellations arising in quantized supersymmetric theories can be attributed to the manifest Bose-Fermi symmetry arising from the structure of the corresponding super-algebras. The starting point of this paper is the observation that many instantons and solitons that are known to occur in field theories in two, four, and other dimensions, are in fact supersymmetric, in a sense made precise below. This was (presumably) first noted by Witten and Olive [4] many years ago, although from a slightly different perspective than taken here. In this paper we will first show that there is a formal statement which says that, in any supersymmetric field theory that can be derived from a lagrangian and for which the central charges of static configurations are "topological" (i.e. they are conserved without use of the field equations), a static supersymmetric field 88
configuration with Fermi fields set equal to zero is in fact a configuraton that satisfies the Euler-Lagrange equations of motions. This is a useful statement in that imposing supersymmetry on a field configuration reduces the usual second-order differential equations of motion to first-order equations. Independently of this, one is interested in supersymmetric solutions of the equations of motion because one expects there to be a close relationship between stability of a given solution and whether or not it is supersymmetric. Indeed, when one comes to quantizing the theory one could conjecture that it is precisely the static supersymmetric solutions which give rise to extra states in the Hilbert space (cf. ref. [5] ). It should be emphasized that these results apply also to configurations of nonsupersymmetric field theories as long as these theories have supersymmetric extensions. Secondly, we will look at the specific example o f the N = 2 super Yang-Mills theory in four dimensions, with arbitrary (semisimple compact) gauge group G. We will see that the usual monopole solutions of Yang-Mills-Higgs type are supersymmetric, and that indeed these are the only finite energy bosonic supersymmetric static solutions of this theory. The N = 2 super Yang-Mills theory has one scalar field and one pseudoscalar field transforming under the adjoint representation of the Lie algebra of G and naively one might expect there to be supersymmetric solutions for which both fields enter nontrivially, that is, in such a
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PHYSICS LETTERS
Volume 132B, number 1,2,3
24 November 1983
way that one cannot use the global 0 ( 2 ) symmetry to rotate away one of the fields: we show that this cannot occur. Having understood the solitonic sector of the classical theory one should be able to understand at least the structure of the quantal Hilbert space.
(that is, M is the energy of the configuration, etc.: again we are using suggestive notation), then with an appropriate choice of gamma matrices, we have
Supersymmetric field configurations. In four dimensions the important part of the (extended Poincar6) supersymmetry algebra is
a 4N X 4N, where Z = V + iU is a complex antisymmetric N × N matrix. The following paragraph is a short review of work done by Ferrara et al. [7]. There exists a unitary matrix W such that B = W Z W T = - B T is real and block diagonal. Let
where
and
=
= zt
are hermitian central
charges [6] ' i . By a (partially) supersymmetric configuration, Is>, of a supersymmetric field theory, we will mean by definition a field configuration (not necessarily satisfying the equations of motion) for which there exists at least one Majorana spinor ecd such that ~9 Is> = 0 [in more traditional language, there is a supersymmetry transformation which leaves the background invariant: however we are using this notation to suggest that in the quantum theory, (static) supersymmetric configurations give rise to extra states in the quantal Hilbert space]. Assume now that (1) is valid as an operator equation on the configuration (of course, if the configuration does satisfy the equations of motion then (1) is valid automatically since our theory is supersymmetric). Then, in contrast to the case where there are no central charges, here partial supersymmetry does not imply total supersymmetry, and the concept of partial supersymmetry is the important one. From now on we drop the word "partial" and speak only of supersymmetric configurations. We will moreover be interested in particular in static configurations. Is>, which we take to imply that P m Is>=O, m = 1, 2,3. I f w e let Uij : , Vij : , M = ,1 Conventions: i, j go from 1 to N. ~ ¢ = diag (+ 1, - 1, - 1, - 1). {~,g, ~/u}= 2~/au, (3,5)2 = 1. The Q axe Majorana spinors, i.e ~= QC T. The charge conjugation matrix, C, satisfies C3,1sC-I = -3,gT, C T = -C. We set/~ = c = 1.
[w A
®12 ,
0]
=
0
M
~
®12 ,
A A ?= 1
and let T =. Then as is obvious and is well-known, T is a positive semi-definite matrix, and so
A
M
Z
®121AI=[
M
B
must also be positive semidefinite, Letting {+bin, m = 1 to [N/2], bm ~>0} be of the set of non-trivial eigenvalues of B, this condition translates intoM ~>b m , for all m, with equality for some m if and only if Thas a zero eigenvalue. It is easy to see that the requirement that is) be supersymmetric implies that T has at least one zero eigenvalue, that is, the inequality must be saturated. Formal statement. Suppose we are given a supersymmetric field theory (in four dimensions) for which the central charges of static configurations are topological ("topological" in the sense that they are conserved without use of the equations of motion) and with physical Bose fields q5 and Fermi fields qz (q~ and are generic lables). Given a field configuration which: (i) has 'Is = 0. (ii) is static, and (iii) is supersymmetric in the sense given above. then this configuration is actually one which is a solution of the equations of motion of the theory. P r o o f o f formal statement. Before we give the proof of the formal statement we have to make a few 89
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explanatory comments. Firstly, we take our field theory to be derivable from a lagrangian. We assume that out lagrangian and (hence?) the supersymmetry transformations of the fields are analytic in the fields and the coupling constants at the origin, i.e. there is a Taylor expansion in terms of the fields and the coupling constants around the point where these are set to zero. This excludes supergravity theories from our considerations, although this does not mean that the conclusions do not hold for these theories [of course, eq. (1)has to be suitably interpreted for these theories, in addition], We also insist that our coupling constants be real, i.e. we do not allow any nilpotent Grassman-valued bit to be included in them, because only coupling constants which are real make sense when we go to the quantum field theory. We take physical Bose fields (respectively Fermi fields) to be fields which have dimension [m] (respectively [m 3/2] ) and which have second-order (respectively first-order) differential equations of motion. We take static to mean that extremizing the hamiltonian (which must necessarily exist) instead of the lagrangian for these configurations gives equivalent equations of motion (this definition is used because it copes best with the problems of gauge invariance). By a supersymmetric theory we mean field theory which has a (linear) realization of the supersymmetry algebra on the fields, assuming the equations of motion. In particular, (1) is satisfied. As is well-known, without use of "auxiliary" fields, which we are excluding as unphysical, in general (1) is not valid on a given configuration, i.e. one which may not satisfy the equations of motion. What we must first show is that conditions (i)-(iii) on a configuration are enough to imply that (1) is valid. Then we will use (1) to finish the proof. By the spin-statistics theorem, condition (i) ensures that the equations of motion are satisfied for the Fermi fields, since every term in the equations must contain at least one Fermi field, since we are assuming the coupling constants are real. Similarly, (1) must be valid on the Fermi fields. We must show that {Q, Qt }cI, contains no equations of motion, hence that (1) is valid on the Bose fields modulo gauge transformations and constraints, and thus that (1) is actually valid on the field configuration (which is only defined up to gauge equivalence and constraints). That {Q, Qt }~ contains no equations of mo90
24 November 1983
tion is a fact true independent of imposing conditions (i)-(iii) and follows from the following lemma. Lemma. For any supersymmetric field theory with physical fields only (in the sense given above) the supersymmetry algebra closes on the Bose fields without use of the equations of motion. Proofoflemma. We must make one more restriction on our theory to prove this lemma, namely that the number of Fermi fields in our theory be finite, perhaps too stringent a requirement for today's model builders. (We do not need this restriction if we assume condition (i), but we do not want to assume this for the lemma). If our lagrangian has no coupling constants with negative mass dimension then the result follows easily. Namely, Q has dimensions [m 1/2] so {Q, Q t } ~ has dimension [m2]. The equations of motion for qb have dimension [m 3 ] while the equations of motion for qs have dimension [m5/2]. Since the coupling constants and the fields have nonnegative mass dimension and since 2 < 5/2, 3 the result follows (we are using analyticity in the fields and coupling constants crucially here). If our lagrangian has coupling constants with negative mass dimension then we show that a simple field redefinition will give us a lagrangian equivalent to ours which has no coupling constants with negative mass dimension. We do not use the fact that our lagrangian is supersymmetric to prove this and so it is true for all lagrangians satisfying our other properties. To be specific, let L (~, qs g) be our lagrangian, with all coupling constants, g, taken to have mass dimension +1, O, or - 1 . Let
L'(e#', ~ ' , g ' )
=
i.tnL(la-ldp, I d - l ~ , Idg),
where ~ ' = # - l~b, etc., and where/~ is a constant with dimension of Ira]. The integer n is chosen as follows. Each term in the Taylor expansion of L is of the form f(qb, ~ , g ) = gk14pk2~k3~k4
,
where ~ represents a derivative, so that k 4 = O, 1, or 2. Each term in L must have dimension [m 4] so that we have 3
4 =ek 1 +k 2 + ~k 3 +k 4 ,
(2)
where g has dimension [me], e = +1, O, or - 1 . Then
24 November 1983
PHYSICS LETTERS
Volume 132B, number 1,2,3
= ~lko(~g)kl(I_1-l~)k,(C1-l~)k3ak4
)
version of N = 1 super Yang-Mills in six dimensions, and we shall for the most part be concerned with the latter theory, which is given by the lagrangian [8] *’
where k0=k2+k3
-kl=4-(l+e)kl
-;k3
-k,.
We wish to show that k, is bounded below when we consider all terms in the lagrangian. If k, is unbounded below then either k, of k, must be unbounded above, in which case (2) tells us that E = -1 in the corres onding terms in the lagrangian, so that k, = 4 P - Sk3 - k4. But k3 cannot actually be greater than the number of Fermi fields in the theory, for otherwise qka = 0. Thus k, is bounded below, if we let -n be this lower bound then L ‘(Q’, \k’,g’) is a lagrangian with no coupling constants with negative mass dimension. The Bose and Fermi fields have a mass dimension one lower than their canonical values, but the argument given at the beginning of this proof still goes through. And L’ is as good as L for giving us our (classical) theory, having been obtained from L by a simple field redefinition and multiplicative constant. Then since the lemma is true for L ’ it is true for 15. We have now established that (1) is valid, given the hypotheses of our formal statement. We are basically done. Since the central charges are conserved without use of the equations of motion, the charges are welldefined and enter our variational problem as fixed boundary conditions. We noted previously that for fixed central charge, a supersymmetric configuration has minimal energy and so must be a solution of the variational problem for the hamiltonian, that is, the equations of motion. Notice that in order to use the formal statement one does not have to known what the central charge is as long as one knows it is “topological”. Also, it is easy to find counterexamples of the formal statement if any of conditions (i), (ii), or (iii) is dropped. For example, the simplest of supersymmetric lagrangians, the Wess-Zumino model, provides ample counterexamples. N= 2 super Yang-Mills. N= 2 super Yang-Mills in four dimensions is for algebraic purposes most conveniently thought of as the (naively) dimensionally reduced
FAB = aAAB ~ aBAA DAh=dAX-ie[AA,h]
-ie[AA,AB]
,
.
(3)
here h is a Weyl spinor, i.e. r7 h = h. Lagrangian (3) is transformed into a total derivative under the following supersymmetry transformations: &A= _‘~ABF 2
AB”
’
6AB = i&rgx - ixrg(Y,
where r70 = (Y. As was effectively shown by Witten and Olive [4], there is a central charge for this theory which is topological in origin. We are looking for static supersymmetric configurations, with Fermi fields set equal to zero. Static for Yang-Mills theories is taken to mean that there is a global gauge where 3, = A,, = 0. This implies that FuA = 0 for all A. And h = 0, SO 6AB = 0. Thus the only condition we need to satisfy is that 6X = 0 for some Weyl spinor cr. Certainly a necessary condition that 6X = 0 is that det (ii XABFJ,$B(x)) = 0, for all b and for all x, b being the Lie algebra index. We shall work with given fixed b and x and drop these labels for now. So we need det(ii CABFAB) = 0. Since FOA = 0 we are effectively working with SO(5) rather than with SO(l, 5) as a spacetime symmetry. We take the indices I, J to go from 1 to 5. FIJ is a real antisymmetric 5 X 5 matrix, and as such there exists a frame in which
(4)
*’ Conventions: A, B go from II to 5. rH are 8 x 8 Dirac matrices. {rA, rB}= 2rpB. tiB=diag(+l, -1, -1, -1, -1 -1). CM = _A [rA, FBI. (rTj2 = 1. AA and h are h are g-valued fieldsfg the Lie algebra of G. ( , ) is the usual inner product on g.
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PHYSICS LETTERS
Volume 132B, number 1,2,3
(remember we are working at a fixed point in spacetime and with fixed Lie algebra index, so such a frame does exist). We take as our basis of gamma matrices
O=i
24 November 1983 CABFA,~ @‘15 - $25)
0
r’) = (aI 0 1 631, ia 8 crl QD1 , io3 8 u2 0 1 ,
(r’, ia
8 a3 @ u1 , iu3 03u3 a u2, iu2 8 1 0 I),
Wl2
-F34)
+ b(F14 -F23
= In the particular we thus find that iA CABF 2
AB
=
-h,-A,,
and thus det (ii C ABFAB) = 0 is equivalent to A, = *hz. We take h, = h2 = :A, without loss of generality. Then i! XABF AB = diag(-h, 2
a1 =
(0, 0, 0, 13% 0, 0, 0)
and QI2=
I
(5)
b(F,5
+ $25)
+ a(F35 + $45) 0
J
Without loss of generality we may take a # 0. Let c = b/a= cl + ic2. Then setting the first and seventh elements of (5) equal to zero is equivalent to solving 1
-1, 0, 0, O,O, h, A).
We thus have four eigenvectors with eigenvalue zero. But in order to have 6h = 0 we not only need ii CAB X FAB to have a zero eigenvalue, we also need it to have a zero-eigenvector, 01,which satisfies the Weyl condition, r7a = 01.Using the explicit form for I77 we find that
+ $24)
b(F34 -Fl2)+a(F14-F23-*13-F24)
x, - A, )
•t h,, -A, + A,, x, + h,, A, + A,) 3
+ Fl3
0
frame in which FzJ is given by (4)
diag(-A,-h2,
h, - x2, -h,
0
Cl
-1
0 -cl c2
c2
c2
F25
0 I
c2 1 Cl
Fl5 ’
-cl
0
-1
=0
F35
>I F45 ,
But the determinant
of the above matrix is equal to
1 + 2(cf + c;> +
> 0
so we must have F,5=F25=F35=F45=0.
(0, 0, 0, 0, 0, 1, 090)
are a basis for Weyl zero-eigenvectors over the complex numbers. We have until now work at a fixed spacetime point and with a fixed Lie algebra index. In order for the configuration to be supersymmetric, a necessary and sufficient condition is that we have at least one Weyl zero-eigenvector of ii 2ABF_&x) in common for all spacetime points and for all Lie algebra indices. That is, since oI and CY~are the Weyl zero-eigenvectors (in a given frame) at a given point and with given index, we need (IL= acuI + ba, to be a Weyl zero-eigenvector for all points and for all indices, for some constants a, b E C. This is an extremely powerful constraint. We have
Similarly, setting the fourth and sixth elements of (5) equal to zero is equivalent to solving x + c1y - c2z = cfl + ClZ = -c1x + y =
-9x
-
z = 0,
where x=F 12 -F347
Y =F,,
That is: Y = Cl&
z = --cp
which implies that x(1 +c; i.e. x=y=z=o.
92
+ b(F35 - $45)
t&=0,
)
-F23,
z=F13
+F24.
PHYSICS LETTERS
Volume 132B, number 1,2,3
1
Therefore we have proven that
=F14-F23
=F13 +F24=0.
= ~ (FjK' DI(nIFjK))
This can be restated as _ 1
FIj- ~elJKLMnKFLM,
i
(6)
1
-- ~ ( F j K , D I ( ~ e l J K L M F L M
njFKI - nKFij))
[using (17))] = 0
with
n K = (0, 0, 0, 0, 1).
(7)
In fact, if (6) is true for some constant (unit) vector nK, then by a (global) SO(5) rotation we can rotate n K into the form (7). An equivalent way of writing (6) is 1
(FIj" F i s ) so that
nlOlH = i n I ( F j K ' D I F j K )
F15 = F25 = F35 = F45 =F12_F34
24 November 1983
elJKLMFLM = n l F j K + n j F K I + n K F I j ,
niFij = 0.
(8)
We conclude that a necessary and sufficient condition for a static bosonic configuration for N = 1 super Yang-Mills in six dimensions to be supersymmetric is that the spatial part of FAB satisfies (6) for some constant (unit) vector n K. As is well-known, (6) implies the equations o f motion for FAB, which are D A F AB = 0. That is, FOA = 0 and 1
D I F I j = ~ e l J K L M n K D j F L M = O, by the Bianchi identities. Therefore our formal statement of the previous section has been upheld f o r N = 1 super Yang-Mills in six dimensions, and hence for N = 2 super Yang-Mills in four dimensions. To get to N = 2 super Yang-Mills in four dimensions one must dimensionally reduce the N = 1 super Yang-Mills in six dimensions. This just means that one assumes all fields are independent of the fourth and fifth coordinates o f spacetime, and the S O ( I , 5) spacetime symmetry is reduced to S O ( I , 3 ) × S O ( 2 ) s p a c e t i m e × chiral symmetry. By a spatial rotation and a chiral rotation there is a frame in four dimensions in which
n K = (sin 0, 0, 0, 0, cos 0 ) , for some constant 0. 0 = 0 corresponds to the usual Y a n g - M i l l s - H i g g s monopole solutions. If 0 @ 0, the configuration does not have finite energy: the energy density o f the configuration is H =
by the Bianchi identities and since n j F j K = 0 . I f 0 :/:0 then since 05 H = 0, we also have O1H= 0, that is, H is independent o f x 1 and M = f H d3x = oo (unless we have the trivial situation where F I j - 0). Thus we may restrict ourselves to the case 0 = 0. Defining A = A4, B = A 5 , we find that (6) is equivalent to
F 6 =--ei/kDk A ,
B =0
(i,j, k go from 1 to 3) which should be recognized as the Bogolmol'ny equations for Yang-Mills-Higgs monopoles. We have shown these to be the only bosonic supersymmetric solitons in N = 2 super Yang-Mills. We have also, incidently, shown that there are no bosonic supersymmetric solitons in N = 1 super Y a n g Mills in six dimensions. (Note: a previous attempt [9] to find supersymmetric monopoles found only these types of solutions and chiral rotations of them. We do not consider chiral rotations of a given solution to be new solutions, cf. ref. [9]. Indeed we do not consider a solution o f any set of field equations to be new if it is obtained from another solution by a symmetry operation of the theory, including if the symmetry happens to be supersymmetry). With a little more work one expects to extend this result to N = 4 super Yang-Mills in four dimensions, namely one expects that Yang-Mills Higgs monopoles will provide the only bosonic supersymmetric solitons. I wish to thank E. Corrigan, H. Osborn and especially G.W. Gibbons for help and discussions.
References [1] E. Witten, Commun. Math. Phys. 80 (1981) 381. [2] G.W. Gibbons and C.M. Hull, Phys. Lett. 109B (1982) 190;
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G.W. Gibbons, C.M. Hull and N.P. Warner, Nucl. Phys. B218 (1983) 173. [3] S. Deser and C. Teitelboim, Phys. Rev. Lett. 39 (1977) 249; M. Grisaru, Phys. Lett. 73B (19(78) 207. [4] E. Witten and D. Olive, Phys. Lett. 78B (1978) 97. [5] R. Jackiw, Rev. Mod. Phys. 49 (1977) 681. [6] R. Haag, J. Lopuszanski and M.F.Sohnius, Nucl. Phys. B88 (1975) 257.
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[7] S. Ferrara, C.A. Savoy and B. Zumino, Phys. Lett. l l 0 B (1981) 393. [8] L. Brink, J.H. Schwarz and J. Scherk, Nucl. Phys. B121 (1977) 77; F. Gliozzi, D. Olive and J. Scherk, Nucl. Phys. B122 (1977) 253. [9] A. D'Adda, R. Horsley and P. DiVecchia, Phys. Lett. 76B (1978) 298.
PHYSICS LETTERS
24 November 1983
SUPERSYMMETRIC SOLITONS Wayne BOUCHER Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England Received 8 July 1983
We show that static bosonic supersymmetric configurations of a field theory are actually configurations that satisfy the field equations. This reduces the second order differential equations for bosons to first order ones. In particular, we show that the usual monopole solutions ofN = 2 super Yang-Mills are the only static bosonic supersymmetric solutions of the theory.
Introduction. To date, the most beautiful results to emerge from supersymmetric field theories have been those with a corresponding formal mathematical statement entailed by the super Lie algebraic structure underlying the theories. For example, Witten's analytic proof of the positive mass theorem for Einstein's theory of gravity [ 1 ], and the generalizations thereof for extended and gauged extended supergravity theories [2], have an obvious algebraic statement for the corresponding super-algebras which indeed was noted long before and which motivated, the analytic proof [3 ]. Similarly, one might think that some if not all of the quantum cancellations arising in quantized supersymmetric theories can be attributed to the manifest Bose-Fermi symmetry arising from the structure of the corresponding super-algebras. The starting point of this paper is the observation that many instantons and solitons that are known to occur in field theories in two, four, and other dimensions, are in fact supersymmetric, in a sense made precise below. This was (presumably) first noted by Witten and Olive [4] many years ago, although from a slightly different perspective than taken here. In this paper we will first show that there is a formal statement which says that, in any supersymmetric field theory that can be derived from a lagrangian and for which the central charges of static configurations are "topological" (i.e. they are conserved without use of the field equations), a static supersymmetric field 88
configuration with Fermi fields set equal to zero is in fact a configuraton that satisfies the Euler-Lagrange equations of motions. This is a useful statement in that imposing supersymmetry on a field configuration reduces the usual second-order differential equations of motion to first-order equations. Independently of this, one is interested in supersymmetric solutions of the equations of motion because one expects there to be a close relationship between stability of a given solution and whether or not it is supersymmetric. Indeed, when one comes to quantizing the theory one could conjecture that it is precisely the static supersymmetric solutions which give rise to extra states in the Hilbert space (cf. ref. [5] ). It should be emphasized that these results apply also to configurations of nonsupersymmetric field theories as long as these theories have supersymmetric extensions. Secondly, we will look at the specific example o f the N = 2 super Yang-Mills theory in four dimensions, with arbitrary (semisimple compact) gauge group G. We will see that the usual monopole solutions of Yang-Mills-Higgs type are supersymmetric, and that indeed these are the only finite energy bosonic supersymmetric static solutions of this theory. The N = 2 super Yang-Mills theory has one scalar field and one pseudoscalar field transforming under the adjoint representation of the Lie algebra of G and naively one might expect there to be supersymmetric solutions for which both fields enter nontrivially, that is, in such a
0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
PHYSICS LETTERS
Volume 132B, number 1,2,3
24 November 1983
way that one cannot use the global 0 ( 2 ) symmetry to rotate away one of the fields: we show that this cannot occur. Having understood the solitonic sector of the classical theory one should be able to understand at least the structure of the quantal Hilbert space.
(that is, M is the energy of the configuration, etc.: again we are using suggestive notation), then with an appropriate choice of gamma matrices, we have
Supersymmetric field configurations. In four dimensions the important part of the (extended Poincar6) supersymmetry algebra is
a 4N X 4N, where Z = V + iU is a complex antisymmetric N × N matrix. The following paragraph is a short review of work done by Ferrara et al. [7]. There exists a unitary matrix W such that B = W Z W T = - B T is real and block diagonal. Let
where
and
=
are hermitian central
charges [6] ' i . By a (partially) supersymmetric configuration, Is>, of a supersymmetric field theory, we will mean by definition a field configuration (not necessarily satisfying the equations of motion) for which there exists at least one Majorana spinor ecd such that ~9 Is> = 0 [in more traditional language, there is a supersymmetry transformation which leaves the background invariant: however we are using this notation to suggest that in the quantum theory, (static) supersymmetric configurations give rise to extra states in the quantal Hilbert space]. Assume now that (1) is valid as an operator equation on the configuration (of course, if the configuration does satisfy the equations of motion then (1) is valid automatically since our theory is supersymmetric). Then, in contrast to the case where there are no central charges, here partial supersymmetry does not imply total supersymmetry, and the concept of partial supersymmetry is the important one. From now on we drop the word "partial" and speak only of supersymmetric configurations. We will moreover be interested in particular in static configurations. Is>, which we take to imply that P m Is>=O, m = 1, 2,3. I f w e let Uij :
[w A
®12 ,
0]
=
0
M
~
®12 ,
A A ?= 1
and let T =
A
M
Z
®121AI=[
M
B
must also be positive semidefinite, Letting {+bin, m = 1 to [N/2], bm ~>0} be of the set of non-trivial eigenvalues of B, this condition translates intoM ~>b m , for all m, with equality for some m if and only if Thas a zero eigenvalue. It is easy to see that the requirement that is) be supersymmetric implies that T has at least one zero eigenvalue, that is, the inequality must be saturated. Formal statement. Suppose we are given a supersymmetric field theory (in four dimensions) for which the central charges of static configurations are topological ("topological" in the sense that they are conserved without use of the equations of motion) and with physical Bose fields q5 and Fermi fields qz (q~ and are generic lables). Given a field configuration which: (i) has 'Is = 0. (ii) is static, and (iii) is supersymmetric in the sense given above. then this configuration is actually one which is a solution of the equations of motion of the theory. P r o o f o f formal statement. Before we give the proof of the formal statement we have to make a few 89
Volume 132B, number 1,2,3
PHYSICS LETTERS
explanatory comments. Firstly, we take our field theory to be derivable from a lagrangian. We assume that out lagrangian and (hence?) the supersymmetry transformations of the fields are analytic in the fields and the coupling constants at the origin, i.e. there is a Taylor expansion in terms of the fields and the coupling constants around the point where these are set to zero. This excludes supergravity theories from our considerations, although this does not mean that the conclusions do not hold for these theories [of course, eq. (1)has to be suitably interpreted for these theories, in addition], We also insist that our coupling constants be real, i.e. we do not allow any nilpotent Grassman-valued bit to be included in them, because only coupling constants which are real make sense when we go to the quantum field theory. We take physical Bose fields (respectively Fermi fields) to be fields which have dimension [m] (respectively [m 3/2] ) and which have second-order (respectively first-order) differential equations of motion. We take static to mean that extremizing the hamiltonian (which must necessarily exist) instead of the lagrangian for these configurations gives equivalent equations of motion (this definition is used because it copes best with the problems of gauge invariance). By a supersymmetric theory we mean field theory which has a (linear) realization of the supersymmetry algebra on the fields, assuming the equations of motion. In particular, (1) is satisfied. As is well-known, without use of "auxiliary" fields, which we are excluding as unphysical, in general (1) is not valid on a given configuration, i.e. one which may not satisfy the equations of motion. What we must first show is that conditions (i)-(iii) on a configuration are enough to imply that (1) is valid. Then we will use (1) to finish the proof. By the spin-statistics theorem, condition (i) ensures that the equations of motion are satisfied for the Fermi fields, since every term in the equations must contain at least one Fermi field, since we are assuming the coupling constants are real. Similarly, (1) must be valid on the Fermi fields. We must show that {Q, Qt }cI, contains no equations of motion, hence that (1) is valid on the Bose fields modulo gauge transformations and constraints, and thus that (1) is actually valid on the field configuration (which is only defined up to gauge equivalence and constraints). That {Q, Qt }~ contains no equations of mo90
24 November 1983
tion is a fact true independent of imposing conditions (i)-(iii) and follows from the following lemma. Lemma. For any supersymmetric field theory with physical fields only (in the sense given above) the supersymmetry algebra closes on the Bose fields without use of the equations of motion. Proofoflemma. We must make one more restriction on our theory to prove this lemma, namely that the number of Fermi fields in our theory be finite, perhaps too stringent a requirement for today's model builders. (We do not need this restriction if we assume condition (i), but we do not want to assume this for the lemma). If our lagrangian has no coupling constants with negative mass dimension then the result follows easily. Namely, Q has dimensions [m 1/2] so {Q, Q t } ~ has dimension [m2]. The equations of motion for qb have dimension [m 3 ] while the equations of motion for qs have dimension [m5/2]. Since the coupling constants and the fields have nonnegative mass dimension and since 2 < 5/2, 3 the result follows (we are using analyticity in the fields and coupling constants crucially here). If our lagrangian has coupling constants with negative mass dimension then we show that a simple field redefinition will give us a lagrangian equivalent to ours which has no coupling constants with negative mass dimension. We do not use the fact that our lagrangian is supersymmetric to prove this and so it is true for all lagrangians satisfying our other properties. To be specific, let L (~, qs g) be our lagrangian, with all coupling constants, g, taken to have mass dimension +1, O, or - 1 . Let
L'(e#', ~ ' , g ' )
=
i.tnL(la-ldp, I d - l ~ , Idg),
where ~ ' = # - l~b, etc., and where/~ is a constant with dimension of Ira]. The integer n is chosen as follows. Each term in the Taylor expansion of L is of the form f(qb, ~ , g ) = gk14pk2~k3~k4
,
where ~ represents a derivative, so that k 4 = O, 1, or 2. Each term in L must have dimension [m 4] so that we have 3
4 =ek 1 +k 2 + ~k 3 +k 4 ,
(2)
where g has dimension [me], e = +1, O, or - 1 . Then
24 November 1983
PHYSICS LETTERS
Volume 132B, number 1,2,3
= ~lko(~g)kl(I_1-l~)k,(C1-l~)k3ak4
)
version of N = 1 super Yang-Mills in six dimensions, and we shall for the most part be concerned with the latter theory, which is given by the lagrangian [8] *’
where k0=k2+k3
-kl=4-(l+e)kl
-;k3
-k,.
We wish to show that k, is bounded below when we consider all terms in the lagrangian. If k, is unbounded below then either k, of k, must be unbounded above, in which case (2) tells us that E = -1 in the corres onding terms in the lagrangian, so that k, = 4 P - Sk3 - k4. But k3 cannot actually be greater than the number of Fermi fields in the theory, for otherwise qka = 0. Thus k, is bounded below, if we let -n be this lower bound then L ‘(Q’, \k’,g’) is a lagrangian with no coupling constants with negative mass dimension. The Bose and Fermi fields have a mass dimension one lower than their canonical values, but the argument given at the beginning of this proof still goes through. And L’ is as good as L for giving us our (classical) theory, having been obtained from L by a simple field redefinition and multiplicative constant. Then since the lemma is true for L ’ it is true for 15. We have now established that (1) is valid, given the hypotheses of our formal statement. We are basically done. Since the central charges are conserved without use of the equations of motion, the charges are welldefined and enter our variational problem as fixed boundary conditions. We noted previously that for fixed central charge, a supersymmetric configuration has minimal energy and so must be a solution of the variational problem for the hamiltonian, that is, the equations of motion. Notice that in order to use the formal statement one does not have to known what the central charge is as long as one knows it is “topological”. Also, it is easy to find counterexamples of the formal statement if any of conditions (i), (ii), or (iii) is dropped. For example, the simplest of supersymmetric lagrangians, the Wess-Zumino model, provides ample counterexamples. N= 2 super Yang-Mills. N= 2 super Yang-Mills in four dimensions is for algebraic purposes most conveniently thought of as the (naively) dimensionally reduced
FAB = aAAB ~ aBAA DAh=dAX-ie[AA,h]
-ie[AA,AB]
,
.
(3)
here h is a Weyl spinor, i.e. r7 h = h. Lagrangian (3) is transformed into a total derivative under the following supersymmetry transformations: &A= _‘~ABF 2
AB”
’
6AB = i&rgx - ixrg(Y,
where r70 = (Y. As was effectively shown by Witten and Olive [4], there is a central charge for this theory which is topological in origin. We are looking for static supersymmetric configurations, with Fermi fields set equal to zero. Static for Yang-Mills theories is taken to mean that there is a global gauge where 3, = A,, = 0. This implies that FuA = 0 for all A. And h = 0, SO 6AB = 0. Thus the only condition we need to satisfy is that 6X = 0 for some Weyl spinor cr. Certainly a necessary condition that 6X = 0 is that det (ii XABFJ,$B(x)) = 0, for all b and for all x, b being the Lie algebra index. We shall work with given fixed b and x and drop these labels for now. So we need det(ii CABFAB) = 0. Since FOA = 0 we are effectively working with SO(5) rather than with SO(l, 5) as a spacetime symmetry. We take the indices I, J to go from 1 to 5. FIJ is a real antisymmetric 5 X 5 matrix, and as such there exists a frame in which
(4)
*’ Conventions: A, B go from II to 5. rH are 8 x 8 Dirac matrices. {rA, rB}= 2rpB. tiB=diag(+l, -1, -1, -1, -1 -1). CM = _A [rA, FBI. (rTj2 = 1. AA and h are h are g-valued fieldsfg the Lie algebra of G. ( , ) is the usual inner product on g.
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Volume 132B, number 1,2,3
(remember we are working at a fixed point in spacetime and with fixed Lie algebra index, so such a frame does exist). We take as our basis of gamma matrices
O=i
24 November 1983 CABFA,~ @‘15 - $25)
0
r’) = (aI 0 1 631, ia 8 crl QD1 , io3 8 u2 0 1 ,
(r’, ia
8 a3 @ u1 , iu3 03u3 a u2, iu2 8 1 0 I),
Wl2
-F34)
+ b(F14 -F23
= In the particular we thus find that iA CABF 2
AB
=
-h,-A,,
and thus det (ii C ABFAB) = 0 is equivalent to A, = *hz. We take h, = h2 = :A, without loss of generality. Then i! XABF AB = diag(-h, 2
a1 =
(0, 0, 0, 13% 0, 0, 0)
and QI2=
I
(5)
b(F,5
+ $25)
+ a(F35 + $45) 0
J
Without loss of generality we may take a # 0. Let c = b/a= cl + ic2. Then setting the first and seventh elements of (5) equal to zero is equivalent to solving 1
-1, 0, 0, O,O, h, A).
We thus have four eigenvectors with eigenvalue zero. But in order to have 6h = 0 we not only need ii CAB X FAB to have a zero eigenvalue, we also need it to have a zero-eigenvector, 01,which satisfies the Weyl condition, r7a = 01.Using the explicit form for I77 we find that
+ $24)
b(F34 -Fl2)+a(F14-F23-*13-F24)
x, - A, )
•t h,, -A, + A,, x, + h,, A, + A,) 3
+ Fl3
0
frame in which FzJ is given by (4)
diag(-A,-h2,
h, - x2, -h,
0
Cl
-1
0 -cl c2
c2
c2
F25
0 I
c2 1 Cl
Fl5 ’
-cl
0
-1
=0
F35
>I F45 ,
But the determinant
of the above matrix is equal to
1 + 2(cf + c;> +
> 0
so we must have F,5=F25=F35=F45=0.
(0, 0, 0, 0, 0, 1, 090)
are a basis for Weyl zero-eigenvectors over the complex numbers. We have until now work at a fixed spacetime point and with a fixed Lie algebra index. In order for the configuration to be supersymmetric, a necessary and sufficient condition is that we have at least one Weyl zero-eigenvector of ii 2ABF_&x) in common for all spacetime points and for all Lie algebra indices. That is, since oI and CY~are the Weyl zero-eigenvectors (in a given frame) at a given point and with given index, we need (IL= acuI + ba, to be a Weyl zero-eigenvector for all points and for all indices, for some constants a, b E C. This is an extremely powerful constraint. We have
Similarly, setting the fourth and sixth elements of (5) equal to zero is equivalent to solving x + c1y - c2z = cfl + ClZ = -c1x + y =
-9x
-
z = 0,
where x=F 12 -F347
Y =F,,
That is: Y = Cl&
z = --cp
which implies that x(1 +c; i.e. x=y=z=o.
92
+ b(F35 - $45)
t&=0,
)
-F23,
z=F13
+F24.
PHYSICS LETTERS
Volume 132B, number 1,2,3
1
Therefore we have proven that
=F14-F23
=F13 +F24=0.
= ~ (FjK' DI(nIFjK))
This can be restated as _ 1
FIj- ~elJKLMnKFLM,
i
(6)
1
-- ~ ( F j K , D I ( ~ e l J K L M F L M
njFKI - nKFij))
[using (17))] = 0
with
n K = (0, 0, 0, 0, 1).
(7)
In fact, if (6) is true for some constant (unit) vector nK, then by a (global) SO(5) rotation we can rotate n K into the form (7). An equivalent way of writing (6) is 1
(FIj" F i s ) so that
nlOlH = i n I ( F j K ' D I F j K )
F15 = F25 = F35 = F45 =F12_F34
24 November 1983
elJKLMFLM = n l F j K + n j F K I + n K F I j ,
niFij = 0.
(8)
We conclude that a necessary and sufficient condition for a static bosonic configuration for N = 1 super Yang-Mills in six dimensions to be supersymmetric is that the spatial part of FAB satisfies (6) for some constant (unit) vector n K. As is well-known, (6) implies the equations o f motion for FAB, which are D A F AB = 0. That is, FOA = 0 and 1
D I F I j = ~ e l J K L M n K D j F L M = O, by the Bianchi identities. Therefore our formal statement of the previous section has been upheld f o r N = 1 super Yang-Mills in six dimensions, and hence for N = 2 super Yang-Mills in four dimensions. To get to N = 2 super Yang-Mills in four dimensions one must dimensionally reduce the N = 1 super Yang-Mills in six dimensions. This just means that one assumes all fields are independent of the fourth and fifth coordinates o f spacetime, and the S O ( I , 5) spacetime symmetry is reduced to S O ( I , 3 ) × S O ( 2 ) s p a c e t i m e × chiral symmetry. By a spatial rotation and a chiral rotation there is a frame in four dimensions in which
n K = (sin 0, 0, 0, 0, cos 0 ) , for some constant 0. 0 = 0 corresponds to the usual Y a n g - M i l l s - H i g g s monopole solutions. If 0 @ 0, the configuration does not have finite energy: the energy density o f the configuration is H =
by the Bianchi identities and since n j F j K = 0 . I f 0 :/:0 then since 05 H = 0, we also have O1H= 0, that is, H is independent o f x 1 and M = f H d3x = oo (unless we have the trivial situation where F I j - 0). Thus we may restrict ourselves to the case 0 = 0. Defining A = A4, B = A 5 , we find that (6) is equivalent to
F 6 =--ei/kDk A ,
B =0
(i,j, k go from 1 to 3) which should be recognized as the Bogolmol'ny equations for Yang-Mills-Higgs monopoles. We have shown these to be the only bosonic supersymmetric solitons in N = 2 super Yang-Mills. We have also, incidently, shown that there are no bosonic supersymmetric solitons in N = 1 super Y a n g Mills in six dimensions. (Note: a previous attempt [9] to find supersymmetric monopoles found only these types of solutions and chiral rotations of them. We do not consider chiral rotations of a given solution to be new solutions, cf. ref. [9]. Indeed we do not consider a solution o f any set of field equations to be new if it is obtained from another solution by a symmetry operation of the theory, including if the symmetry happens to be supersymmetry). With a little more work one expects to extend this result to N = 4 super Yang-Mills in four dimensions, namely one expects that Yang-Mills Higgs monopoles will provide the only bosonic supersymmetric solitons. I wish to thank E. Corrigan, H. Osborn and especially G.W. Gibbons for help and discussions.
References [1] E. Witten, Commun. Math. Phys. 80 (1981) 381. [2] G.W. Gibbons and C.M. Hull, Phys. Lett. 109B (1982) 190;
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G.W. Gibbons, C.M. Hull and N.P. Warner, Nucl. Phys. B218 (1983) 173. [3] S. Deser and C. Teitelboim, Phys. Rev. Lett. 39 (1977) 249; M. Grisaru, Phys. Lett. 73B (19(78) 207. [4] E. Witten and D. Olive, Phys. Lett. 78B (1978) 97. [5] R. Jackiw, Rev. Mod. Phys. 49 (1977) 681. [6] R. Haag, J. Lopuszanski and M.F.Sohnius, Nucl. Phys. B88 (1975) 257.
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[7] S. Ferrara, C.A. Savoy and B. Zumino, Phys. Lett. l l 0 B (1981) 393. [8] L. Brink, J.H. Schwarz and J. Scherk, Nucl. Phys. B121 (1977) 77; F. Gliozzi, D. Olive and J. Scherk, Nucl. Phys. B122 (1977) 253. [9] A. D'Adda, R. Horsley and P. DiVecchia, Phys. Lett. 76B (1978) 298.