- Email: [email protected]
Supersymmetry and geometry in D < 4 nonlinear Σ-models
ANNALS
OF PHYSICS
168,
387403 (1986)
Supersymmetry and Geometry D < 4 Nonlinear E-Models GARY ATKINSON,* Department
UTPAL
of Physics and Astronomy,
CHATTOPADHYAY,~ University
of Maryland,
in
AND S. JAMES GATES, JR.: College
Park,
Maryland
20742
Received February 7, 1985; revised July 30, 1985
We discuss the augmentation of D < 4 supersymmetric u-models by the addition of higher derivative terms, such as the Skyrme and Wess-Zumino actions. These terms must satisfy differential constraints so that they preserve the symmetries of the usual kinetic action, In two dimensions, we explicitly construct the gauged WZ action for group manifolds. Some difficulties with the analogous four dimensional theories are noted. ‘C 1986 Academic press, IIIC.
I. INTRODUCTION
AND SUMMARY
Nonlinear sigma models provide an interesting arena where differential geometry, topology, and quantum field theory meet, sometimes with startling results [l-4]. For example, in four dimensions they act as effective field theories for low energy hadron physics and, if the Wess-Zumino chiral effective action [S] is included, these models incorporate all the information associated with the chiral anomalies [6]. Two dimensional sigma models in particular exhibit properties possessed by some four dimensional gauge theories such as instantons, renormalizability, and asymptotic freedom. These models also have the nice feature of being more amenable to rigorous study. Furthermore, it has been shown how nonlinear sigma models in two dimensions can be considered as a formulation of string theories (see, e.g., [7] and the references therein.), of which the more viable versions have supersymmetry as a necessary ingredient. These reasons and others indicate that work on D < 4 nonlinear sigma models can give valuable insight into effective and possibly realistic theories of nature. It is becoming increasingly clear that nonlinear sigma models augmented by Wess-Zumino and Skyrme [8] terms exhibit richer structures. Using the language of differential forms, we study the geometrical properties of the suitable extensions of these actions, which we call the generalized Wess-Zumino (WZ) and the generalized Skyrme action. A main result of this investigation is the formulation of a * Research supported in part by NSF under Grant PHY 82-18338. + Research supported in part by NSF under Grant PHY 83-08070. : Research supported in part by NSF under Grant PHY 84-16030. 387 0003-49 16/86 $7.50 Copyright B’I 1986 by Academx Press. Inc. All rlghts of reproduck,,, tn any lorm reserved
388
ATKINSON,
CHATTOPADHYAY,
AND GATES
specific geometric characterization of the generalized WZ action in terms of a differential equation that it must satisfy. If the bosonic fields of the theory are the coordinates of a group manifold, we can actually construct a solution to the above equation for even space-time dimensions. For odd dimensions, however, we show that a generalized WZ action cannot be constructed on a group manifold. Turning next to supersymmetry, we give the supersymmetric extensions of the generalized WZ and Skyrme terms for D = 2 and D = 3. In addition, for D = 2, we also construct the gauged version of the supersymmetric WZ term, both with and without central charges. The paper is organized as follows. In Section II, we define the generalized Skyrme (Sk) action and the generalized Wess-Zumino action in three space-time dimensions and describe the geometrical characterization of the latter. The whole discussion, however, can be readily generalized to arbitrary dimensions. In Section III, we specialize to the case where the bosonic fields of the theory coordinatize a group manifold and try to construct the WZ action using a technique due to Vainberg. A self contained discussion of the Vainberg technique is included. In Section IV, we discuss the supersymmetric extensions of the Skyrme and the WZ terms in 2 and 3 dimensions alluded to above. In the concluding section we make some comments on the recent attempts to supersymmetrize the Skyrme term [9] and the WZ action [lo] in four space-time dimensions.
II. COMPONENTS AND GEOMETRY We begin by considering within a geometrical context only component field theories in D = 3 space-time. Let 4”(x) (m = l,..., d) be a set of spin-zero fields defined over three dimensional space-time with coordinates x*, (a = 0, 1, 2). These fields may clearly be considered as a map 4: M3 --f 4 if M3 has coordinates x” and & has coordinates 4”. Writing dqP’= dx’(a,$“) we can consider the most general action that can be constructed from the d@” and possible tensors over the manifold ~8, assumed to be endowed with a metric g. If we impose the restriction that no more than quadratic temporal derivatives appear, this action must take the form S,,, = ; j [g,,,,,d@’ A *d@‘+ h,,(dqP
A dq5” A d&‘)
+kz,,(d@” * WV * * (d&’ * @“)I + LnpqrsWm A dq5” A dc,P) A * (d@ A dqY A d&). (We ignore the possibility of a term linear in dqP.) actions, S,, S,,, SrrC, and ,!& which we refer to action, the generalized Wess-Zumino action and (For simplicity, henceforth we will set S:, equal to
(2.1)
These terms precisely define four as the usual nonlinear a-model the generalized Skyrme action. zero.) In (2.1), the * denotes the
D-c
4
SUPERSYMMETRIC
389
C-MODELS
Hodge duality operation with respect to the Grassman algebra generated by d-x”. Although we are only considering D = 3, all terms in (2.1) are applicable to a spacetime of arbitrary dimension, except for the second term. Of course, it is possible to write for arbitrary D an appropriate form for this term. We need only introduce a D-form h = h,, .. . ..dqP A ... A d&“” in place of the 3-form. Choosing a specific x-coordinate frame, (2.1) implies
S,, = 1 d3x thmnpE‘“a14m) Ssk
=
f d3x
%n,,
(a,,&?
(a,@) @b,d”)
(aN)% (W”)
(abe.
In what follows, we will be concerned with the geometrical, as opposed to the topological, characterization of the action S,,. But it may be useful here to mention the distinction between these two characterizations. First, S,, is defined as the integral of a D-form h on JV. Next, we require that the isometries of M that are continuously connected to the identity (and, therefore, generated by the Killing vectors) are also symmetries of S,,. An action constructed from a D-form and satisfying this geometrical condition will be referred to as a generalized Wess-Zumino action. If J? is nontrival in the sense that there are symmetry transformations, i.e., isometries, which are not continuously connected to the identity, then we must consider the change of S,, under these transformations as well. Such considerations are intrinsically topological in nature and usually lead to quantization conditions, e.g., in four dimensional chiral theories. However, purely geometric aspects of the generalized Wess-Zumino action (as defined above) may be important and interesting even when d is trivial. An example of this is given in the o-model interpretation of the covariant string action [7]. Finally, even though the WZ term is related to chiral anomalies in even dimensions, it is impossible to interpret the odd dimensional WZ term in the same way, since there are no chiral fermions in odd dimensions. But this does not preclude the existence of the generalized Wess-Zumino action in odd dimensions. An example of this has been found for N= 2 supersymmetric a-models [ll] in D = 3. On the other hand we will show later that S,, cannot be constructed for group manifolds in odd dimensions. To study the symmetry properties of the action, we begin by varying the total action. In general, the variations Sdm may be functions of ii and possibly other fields f’(x). We may indicate this by writing Sbm = &zP(&‘,f’). Upon variation of the actions (2.2), we find
(2.3)
390
ATKINSON,
CHATTOPADHYAY,
SS,, =; j d3x C&,(k)] mnpq($a&? + 2
s
d3x kmnpq5
SK =; j d3x CUg)l,,
ta,af’)
(W’7
AND
GATES
(a,,@)
(av”)
@b4q)
@b,f)
(a’$“)
(ab’$‘),
(aa4’7 + j d3x g,, 5
(2.4) (W’)
(44’7,
(2.5)
where (with v” E &P),
C~“uam2p= ~qLzp.q+ Vqmhq, + ~q/4,,Anqp + Vq,pLq C~“k)L = VP&zn.p f VPdpn + VP9”&lp, CUW,,,
(2.6) (2.7)
= V’kmp,,,+ Vrmknpq+ KAnrpq + Kpkmrq + ~>qkmpr.
(2.8)
We recognize E,(T) as the Lie derivative of the tensor T taken along the vector field V= &Y”a,. In the special case when SqY’ is only a function of 4” and the f”s are constants, we see that for an arbitrary function F and tensors T we have
S= .[ F(T, dd, *d4),
(2.9)
d”S = 1 F(&,“( T), d#, *d/j).
(2.10)
The results in (2.3) and (2.6) imply that the variation of S,, can also be interpreted as the integral of the contraction of the exterior derivative of the 3-form h= hmnpd&” A dqS” A dtip with the vector field V. In coordinate free notation, (2.4) is expressed (after integrating by parts) as
W,,, = j- (4
(2.11)
V>,
or, in terms of coordinates,
cw,,,= Qi
d3X
EP”hCmnp,ql
(a,@?
tab&)
(wp)
64’.
(2.12)
This way of writing the variation is very interesting because dh is a 4-form. So when the map of T(M) -+ T(d), induced by 4, is used, dh is identically zero. However, (dh, V) is only a 3-form. So, (2.11) is nonzero, under the q&map, if h is not a closed form. But the action S,, has another property not shared by SO and S,,; under a
391
D-c 4 SUPERSYMMETRIC C-MODELS
“gauge” transformation of hmnp, S,, changes by a surface term. Replacing the “curl” of a two-form il,, we see
h,,
by
(2.13)
This brings us to addressing the geometrical characterization of the WZ term. To answer this we first discuss symmetries in Lagrangian field theories. Given a Lagrangian L(Y) depending on a set of fields Y’ we can define Lie differentiation E,(Y), denoted by f,(P) E 6Y’ where 6Y’ is a given variation that depends only on Y’ and some constants. We say that the Lie derivative generates a symmetry if E,(L) = 0, where S is the action (S = SdDx L(Y)). Often, however, we consider a theory described by two different Lagrangians L1 and L2 physically equivalent if L, and L, lead to the same equations of motion (i.e., if L, and L, differ by a total derivative). Because of this we can enlarge the definition of a symmetry. Instead of requiring f,(L) = 0, we can use the weaker definition f,(L) = a, r” for some quantity Y‘. Let us call the symmetries that satisfy f,(L) = 0 manifestly realized symmetries and those that satisfy f,(L) = 8, Y” nonmanifestly realized symmetries. When symmetries are gauged, this distinction is important. For manifestly realized symmetries, the Noether procedure leads to a “minimal” coupling of the gauge fields. For symmetries realized nonmanifestly, the Noether procedure leads to “nonminimal” coupling of the gauge fields. Some examples of this can be seen in gravitation, supergravity [12], gauging Killing vectors in N= 1 supersymmetric nonlinear o-models [13], and the gauging of the WZ term [63. Now it is simple to characterize the geometric meaning of the WZ term. Let g,,(d) be the metric of a nonlinear o-model and h,,(4) be a 3-form in the manifold of 4”‘. Now, the Killing vectors K,“’ generate a symmetry transformation of the action S, because fK(g) = 0. We want this to be a symmetry of the WZ action as well. However, from (2.13) it is clear that for this to be true it is sufficient to have the Lie derivative of h,, to be equal to exterior derivative of some twoform I,,. Therefore we say that the action contains a WZ term if (2.12) is satisfied and if for each Killing vector K,“’ we find the following condition to be true, d[f K(h)] = 0.
(2.14)
For the same reason, the Skyrme term obeys the stronger equation f K(k) = 0. III.
GROUP MANIFOLDS AND GEOMETRY
At this point, it is useful to consider the special case, where .X is a group manifold. In this case the @” fields can be mapped itit0 the elements U of a group, U= exp [&P(x)
Tm]
(3.1)
392
ATKINSON,
CHATTOPADHYAY,
AND
GATES
with T, being a matrix representation of the ith group generator. We define left and right Cartan-Maurer forms L,” and R,” by the equations U~‘iY,U=iL,“(qS) (a, U) U-’ = iR,Q)
T,(~?,cj”),
(3.2)
T,(Q”).
(3.3)
Throughout the remaining discussion we will adhere to the early-late convention for the A’ manifold. Thus a, b ,..., are anholonomic indices while m, n,..., are holonomic indices. These definitions permit L and R to be explicitly calculated L,“(ti)==(C,)-‘Tr[T(q)
T,,,],
R,“(m,=(C2)p’Tr[T”(~)
Tm],
(3.4) (3.5)
where AT, - [i&‘T,,, T,], A2T, = AAT,, etc., and the constant C, is determined so that L,“(O) = R,“(O) = 6,“. (Note that (3.2, 3.3) imply that L and R are regular as 4 approaches zero.) It follows that
Tr Cu-‘(a’u) u-‘(a, VI = -C2g,,(d)(a”~m)(a,~), k!m($) = kz&m”Lnb = k&,,a&b,
(3.6) (3.7)
and thus x7= -2c,
1 s
d3X Tr [Pu
a,,u-l J,
(3.8)
[a,,ua,,u~la[~uablu-II,
S,,=&jd’xTr
(3.9)
2
where we have already made a particular choice for kmnpq on group manifolds. Before we give the form of S,,, it is useful to note that a “spin-connection” w,,~ and “affine connection” r,,,,,p are defined by
a,y - W,ba~nb +
r,,p~~a = 0,
(3.10)
lam L,b + r,,PLpb = 0, and the anholonomy
coefficients Cobr by &,“a,,, Lb”a, - Lbma, &,“a,, = cabcL,“a,,
where L,”
is the inverse of L,“.
From
the definition
(3.11) of L,”
we find using
CT,, Tbl= -ifat,'Tc gwLno - gn L,” = fb/LmbL,‘.
(3.12)
D <
4
SUPERSYMMETRIC
393
C-MODELS
We now completely fix the geometry by assuming a gauge where o,,& = 0. This is consistent since a parallelizing contorsion always exists for group manifolds. This implies Cobc = fabc= Tabcr where Tabc is the torsion tensor for the manifold. Here Tabc is the “flattened” torsion tensor, Tabr = (rm, The equation of motion
- r,,,)
(3.13)
LamL,“L,p.
derived from (3.8) takes the form W(LmQ a,qq
(3.14)
= 0.
Following [6], we want to modify this equation by the addition tional to the D = 3 s-tensor, GB”(L,” 8,dm) T, = askd( Up’ d, U)( U-’
where a is a constant with the dimension ~a(L,“&,~m)
T,=
of a term propor-
8, U)( U- ’ ad U),
(3.15)
of mass. Using (3.2) this becomes
-iCITbT,T~EbedL~bL,cLpd(~,~m)(~,~fl)(~d~~)
(3.16)
or (3.17)
P( L,” a,@y = - OUbcd&~dL,bL,cL,d(a,~m)(a,Qln)(ad~P),
0 abed= id-
’ Tr [ T, Tb T, Td] .
(3.18)
Now we want to find an action that yields (3.15, 3.16) upon variation. Because we are in an odd dimensional space-time, a first guess might be Tr [U-‘dU U-*dU U-‘dU], since it is not identically zero. But this does not s contribute to an equation of motion! (This is also valid for any odd dimensional space-time; see Appendix.) So we must follow the construction of Witten, even in odd dimensions. To this end we note the following observation [14], (3.19)
Defining the operator K( (aF/&$)) so that (3.19) reads K((aF/&Y)) = F(d) - F(O), we note that K is a homotopy operator [ 151. In fact, the realization of K in (3.19) is such that applying K to the left hand side of (3.16) produces precisely the action S,. Explicitly we find -
s0
’ d~~“L,,(z~)[z~b(L,u(~~)
t&f=)]
= -&?-r
[abU c?~U~~‘].
(3.20)
2
Thus to find the action that gives (3.16) by variation, we make the replacement q5--+zqi in (3.16), multiply by -(l/C,) d”L,“(+) T,, trace and integrate. It is worthwile to note that this procedure is not guaranteed to produce an action whose
394
ATKINSON,
CHATTOPADHYAY,
AND
GATES
variation gives the equations of motion with which we started. We have to explicitly check that this is indeed the case. Following the prescription, we find that S,, = ;
j d3x 1; d&=
Tr [(0-l
0 z exp [iqPT,,,], Comparing
a,ir)(~-la,ir)(6-'a,6)~mZ,aT,],
zrno z L,“( 74)
(3.21)
this to (2.4) we see
The definition
of 0 implies that (3.21) can also be written as
(3.23)
~3=~a"(a,~i)(a,ir-*)(a,ir)
u-1.
If we consider (x’, r) as the coordinates of a four dimensional manifold, then the three dimensional fields d”(x) are extended into four dimensional ones by (vyx, ?) = 7fyx). s ince all the four dimensional fields are defined in this way, we can visualize (x, z) space using Fig. 1. The boundaries of the (x, z) space are the origin, t = 0, and the three dimensional x-space at 7 = 1. The interior of the cone corresponds to (x, 7) for all other values of 7. It is interesting that such an interpretation is available since the cone construction is also used in singular homology
Cl0 The 6 operator also allows a simple rederivation -$-:Tr
[LJP~aa,~-‘] 2
= -&Tr
of (3.20) since
[(~-~-$$3”(&‘a~~)] 2
FIGURE
1
(3.24)
D
<
4
SUPERSYMMETRIC
395
C-MODELS
and the r.h.s. of (3.24) is equal to the integrand that enters (3.20). Additionally, Skyrme term can also be written extended into four dimensions since -&&Tr
[(a,.ir)(&,
the
irP’)(acaO)(ablO-‘)]
2
=&Tr[(
&I$)
iT,,((P’
a,,~)(ac.d-‘)(ablO))].
(3.25)
2
So we conclude that with this extension, apparently, the entire action in be written in the four dimensional manifold (x9, z). This surprising result an arbitrary dimensional x-space also. Given a group with U = exp [itimTm], with 4”(x) defined over a D-dimensional manifold, and of motion J(4) = 0, where
(2.1) can holds for elements equation
J(~)=d’(U-‘~,U)+a~,,[(~-‘~b,U)(U~’~r*~)(~--’~b1U)] + BE =l”‘yU-’
da,u)-‘(u-’
(3.26)
d,,,U);
the action for this equation is given by
With the exception of the Wess-Zumino term, each term is t-exact and thus depends only on the boundaries of the D + 1 dimensional manifold. However, the dimensionality of the manifold here plays a critical role. The proposed action for the WZ W term can be expressed as
(3.28) l$‘D=(~-ldfi)D=(ii-‘d~)
where v is the volume of the D-dimensional yields
A ..’
A (&‘dl?),
space-time. The variation
of (3.28)
The final four terms, for even D, add together to a total x-space divergence. But this is not true if D is odd. (See Appendix.) Thus the z-trick is only successful for even
396
ATKINSON,
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AND GATES
dimensional spacetime manifolds. (We note that failure of the r-trick is equivalent to the statement that the functional derivative curl of W, is nonzero for odd dimensions.) So, we conclude that a generalized WZ action does not exist in odd dimensions for a group manifold. In closing this section we should point out that the introduction of the r coordinate is not related necessarily to topological questions. This type of representation can occur whenever an equation of motion is used as a starting point to derive an action. For example, given the equation 04 = -cos (@), using the r-trick yields for an action
IV. SUPERSYMMETRIC EXTENSIONS In the previous sections we have discussed the relevant geometrical features of the Skyrme and WZ terms in ordinary space. Here, we will extend these to include supersymmetry. Additionally, we will consider D = 2 theories which are closely related to those in D = 3. In particular, we will give the D = 2 supersymmetry gauged version of the WZ action for a group manifold. For D = 3 the supersymmetric form of the Skyrme and WZ actions are Ssk= 1 d3x d% k ,,,(@)Dw”D%n,”
ay,waySw,
S,, = j d3x d28 h,, (~)(a,,~m)(D,~n)(DB~P),
(4.1) (4.2)
where CD” is a real “multiplet” of scalar supertields. Using the method of projection it can be verified that the purely bosonic terms here contain the appropriate component actions in (2.2). It is also interesting to note that for arbitrary variation of the superfield @” in (4.2) we find AS,., = f d3x d2tI h c,,,q,(aap~“)(D,~“)(Dp~P)
cm4
(4.3)
exactly like the change in the component action given in (2.12). Specializing to group manifolds, (4.1) becomes Ssk= f d3x d26’Tr [(D”U-‘)(Dfi’U)(iY’,
where U =exp [i@“T,,,].
U-‘)(a,,U)],
But due to the failure of the z-trick for constructing
(4.4)
the
D <
4
SUPERSYMMETRIC
397
C-MODELS
ordinary D = 3 WZW term, we have been unable to find an explicit realization (4.2) for group manifolds. The usual S, can be supersymmetrized as S, = j d3x d28 Tr [D’W’
D, U]
of
(4.5 1
on group manifolds. We note that the transformation U + g, ‘Ug, leaves (4.4) and (4.5) invariant for constant matrices g, and g,. We can gauge these symmetries by introducing gauge superconnections L, and R,4 and forming covariant derivatives V,U=D,U+iL,U-iUR,,
and using the minimal coupling prescription. Yang-Mills superconnection are [ 121
(4.6)
As usual the only constraints on D = 3
O=D,L,+D,L,+i{L,, 0 = D,R,
+ D,R,
L,)-2iL,,, + i{R,,
(4.7)
Ru} - 2iR,,.
It is instructive to consider the case of D = 2 where we can explicitly construct the WZ term and its gauged version on group manifolds. For the Skyrme term, the action can be obtained imply by replacing d3x by d2s in (4.1). But for D = 2, we replace (4.2) by
sWZW’ =sd2x d28(yS)“B h,,(@)(D,P’)(Dp@‘“).
(4.8)
(The D = 2 supersymmetric WZ action has been constructed by a number of authors [4, 173.) It is amusing to note that we can write this action also as an integral over an extra dimension z in the following way: s,, = [ d2x d28 [’ dz(y’)@ J
t2 H,,,,,(zQ)
Jo
@” D,@” D,rSp.
(4.9 1
where aEmh,, = H,, . Before we specialize to group manifolds, we want to point out that in two dimensions the N= 1 supersymmetry algebra can admit a central charge Z. For nonlinear supersymmetric a-models, we can in fact activate this central charge, which gets realized by means of Killing vectors G” of the underlying bosonic manifold in the following way [ 181 Z@” = moGm, where m, is some parameter with the dimension
(4.10) of mass. In the presence of the cen-
398
ATKINSON,
CHATTOPADHYAY,
tral charge the relevant supersymmetric are related by
AND
GATES
derivatives are fi, rather than D,; the two
ba = D, + i(y5),BeBZ, Ij,@”
= Da@“+
(4.11)
im,(y5),@‘?Gm,
Therefore, if the central charge is activated, the supersymmetric is given by
Wess-Zumino
Celltraly
S$=ged =
I
d2x d% (y5)@h,,(@)
We now return to (4.8). Specializing
b’,@”
to group manifolds
term
(4.12)
&F’.
this becomes
Under the global transformation U +g, -lUgR, (4.13) changes by a total spacetime derivative. Therefore, to gauge this symmetry we use the Noether method. We again introduce gauge superconnections L, and R, transforming as 6L,=DAA,-i[LA,
AL},
hR,=D,A.-i[R,,
AR},
(4.14)
where A, (AR) are infinitesimal parameters of the gauge transformations. all the independent constraints are such that [19]
[V,, VP} = 2zV,,+ 2i(y5),BPL
In D = 2 (4.15)
+ 2i(y5),,PR.
The Noether method yields Se@=
s,, + i -
s
d2x d20(y5)“B Tr [L, U DpV’
-R,
U-‘DBU]
(4.16)
d2x d20(ys)@’ Tr [ UR, U-IL,].
Actually, under a gauge transformation 6Sg;ged =
s
we find
d2x d26(y5)“8 Tr [A, D, L, - A,D,
RB].
(4.17)
This shows that only the diagonal subgroup (g, = g,, LA = RA) can be gauged.
D < 4 SUPERSYMMETRIC
C-MODELS
399
V. CONCLUSIONS We have shown that the class of actions, of which the Wess-Zumino action is an example, have straightforward supersymmetric extensions in two and three dimensions. Even without topological considerations, a number of interesting geometrical properties of the actions were found. The most important of these is the constraint on the tensor h given by (2.14). For group manifolds we can actually construct a solution of this equation only for even dimensional spacetimes. However, for some nongroup manifolds, specifically Kahler manifolds, nontrivial solutions of Eq. (2.14) exist within D = 3 supersymmetric theories [ 111. In four dimensions, there has been a recent attempt to supersymmetrize the WZ action [9]. There it is shown that no tensor j? (analogous to our h) exists for a compact Kahler manifolds. For noncompact Kahler manifolds, h is known to exist [20], but the appropriate four dimensional analog of (3.22) is not known. Such manifolds are important since they occur in phenomenologically interesting models. So, in our view, a more complete supersymmetric extension must include a construction of the tensor h in terms of the geometrical quantities that characterize noncompact Kahler manifolds. There are additional problems that plague both the four dimensional supersymmetric WZ and Skyrme actions [9, lo]. The first of these, the propagation of the “auxiliary” field components (F and G), renders somewhat problematic the usual geometric interpretation of the standard supersymmetric nonlinear o-model. Of greater concern is the presence of quartic temporal derivatives which obviate the raison detre for uniquely choosing the Skyrme term to stabilize the “Skyrmion” and raises questions about stability in general. In our opinion, more work is required to reconcile supersymmetry with higher derivative actions in four dimensions.
APPENDIX
In this Appendix we will show that the action given in (3.28) yields a local (local with respect to x0), manifestly z-independent equation of motion only in even dimension. To prove the assertion, let us rewrite the change in the action given in (3.29) in the following way:
where
400
ATKINSON,
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AND
GATES
From now on, we will omit the caret A since there is no possibility of confusion. We will also omit Tr while writing Ssl, and the wedge product between various forms will be implicit. We will show that m(D+‘)Tr
(UP’
6U
(W,-,-
1U-i g
W,,,
+P, jvj;d7Tr[( wDc’-$-T D-l -
1
(A.3)
(-1)-w,~,,ci’fgw,)o’au],
!?I=0
where Pi are defined by Pz = (1 + ( - 1)‘)/2. In (A.3), IV, is 1 by definition. Therefore we see that while in even dimensions, &!$ is zero modulo a surface integral, in odd dimensions &IS;, is given by the above nonvanishing expression modulo the same surface term. To derive (A.3), let us begin by noting some properties of WD which follow trivially from its definition. For D = 2n,
For D=2n+
1, dy-’
W *n+l=wY+l
d2U...dUdU-‘U, ‘.’
2n
2ni
I
Also note that dW,=PbW,+1,
UW2,+,=dUW2,,,
W2n+lU-‘=
-W2,dU-‘.
(‘4.4)
We will need an expression for 6 W,. It can be written as D-1
6W,=
-U-‘6UW,+
W,U-‘6U+
1 W,,,d(U-‘HJ)W,-,,-, m=O
(A.3
for D 2 1. To prove this let us first note that D-l
SW,=
c
w,,,6(u-‘dU)W,-,-I.
64.6)
Wl=O
Using the identity 6(U-‘dU)=-U-‘NJW,+
W,U-‘GU+d(U-‘6U),
(A.7)
D<
4 SUPERSYMMETRIC
401
C-MODELS
(A.6) becomes
D-l 6wD=
-
D-1
c
w,,,u-‘6uw,~,+
1 m=O
??I=0
w,,,+,u-‘6uw,~,~,
D-l
+ 1
(A.8 1
W,,, d(U-’ 6U)W&,,-,
m=O
from which (AS) follows immediately. Now note that if we plug (AS) into (A.2) the first two terms in 6 W, cancel precisely the first two terms on the right-hand side of (A.2) and we are left with D-l &d(U-‘6U)
1
(-l)“‘D+“wD-,-
,d$w,,,
PI=0 1
-
I?’ Y0
dT$-16U,
(A.9)
where we have used the cyclic property of trace and the anticommutativity Integrating by parts, we get
of forms.
as~=~~~~d~d[U-‘6UD~1(-l)“‘D+1)(W,,~,U~1~W~)]
m=O
1
+j- j; dzUp’ 6U F(W,)-2
(A.lO)
,
where F( W,) is given by D-l
F(W,)=
C (-l)m(D+“+ld
T?l=O
(
W,eJ1~
Wm).
(A.ll)
Since we can always write F( W,) as P,+ F( W,) + P; F( W,), it is sufficient to calculate F( W,) for even and odd D separately. The strategy in evaluating these will be to divide the sum over m in (A.1 1) into two groups corresponding to even and odd values of m. Thus, for even dimensions, (D = 2n), we have n-1
1
F(Wz,)= c d -W,,,~,U-1~w~m+w~n-2,-2u~~~w~~~I l?l=O
= n$ d[W,.-zm-z WI=0 =
du-‘!?,
aT
2m
-w
au-1 dU W,, Zn-2m--2 aT 1 -
dtT’&dU-&dUpldU m=O n-1
=m&, w2”-2m-2~
w,w,m=&
w2n
(A.12a)
402
ATKINSON,
while for odd dimensions
CHATTOPADHYAY,
AND GATES
(D = 2n + l), we have
(A.12b) In deriving (A.l2a), (A.12b) we have repeatedly made use of the identities given in (A.4). Combining these two we can write, F(W,)=P,t;y+P,
(A.13)
W~~-Dt;‘(-l)mWD_,~,U-L&dUW, t?l=O
Substituting
(A.13) in (A.lO) yields precisely (A.3). This completes our proof.
Next, we want to mention that one cannot take the expression J Tr W, (there is no integration over z here), as the WZ action even though one might be tempted to do so. Unlike in even dimensions, this term is not identically zero in odd dimensions. Nevertheless, it does not contribute to an equation of motion. This can be seen easily from Eq. (AS). Because of the cyclicity of trace, we find that when D is ..odd, 6[Tr
W,=D
jd(U-‘i%l)W,-,=D
(A.14)
jd(Up16uWD-,),
where in the last step we have used the fact that dW,- 1 is zero for odd dimensions. Thus the variation of the action is just a surface integral and therefore it does not give rise to any equation of motion. Finally, it is amusing to note that all Skyrme type terms on group manifolds can also be written in terms of W,,, as j Tr [W,* W,] with m = 2,..., D. Thus in D dimensions the most general group manifold theory that is no more than quadratic in temporal derivatives takes the form ^
w,*w,,,+c,
j’dT..,-I:
1 .
0
(A.15)
ACKNOWLEDGMENTS We gratefully Zachos.
acknowledge
useful
conversations
with
Darwin
Chang,
Kyriakos
Tsokos,
and Cosmas
D
<
4
SUPERSYMMETRIC
C-MODELS
403
REFERENCES
1. T. CURTRIGHT AND C. ZACHOS, in “Supergravity, Proceedings of the Supergravity Workshop in Stony Brook,” North-Holland, Amsterdam, 1979. 2. D. FRIEDAN, Phys. Rev. Lett. 45 (1980) 1057; Ph. D. dissertation, Univ. of California. Berkeley, 1980, unpublished. 3. L. ALVAREZ-GAUME, D. Z. FREEDMAN,AND S. MUKHI, Ann. Phys. (N. Y. ) 134 ( 1981). 85. 4. T. CURTRIGHT AND C. ZACHOS,Argonne preprint, ANL-HEP-PR-8450. 5. J. WESS AND B. ZWMINO, Phys. Lett. B 31 (1971), 95. 6. E. WITTEN, Nucl. Phys. B 223 (1983), 422. 7. M. HENNEAUX AND L. MEZINCESCU, Austin preprint, UTTG-2684 November 1984. 8. T. H. R. SKYRME. Proc. Roy. Sot. London Ser. A 260 (1961). 127; 262 (1961) 237. 9. E. A. BERGSHOEFF, R. I. NEPOMECHIE AND H. SCHNITZER, HUTP-84/AO48 (BRX-TH-170). 10. D. NEMESCHANSKYAND R. ROHM, Princeton preprint, 1984. 11. J. KOLLER, State Univ. of New York, preprint, ITP-SB-85-15. 12. S. J. GATES, JR., M. T. GRISARU, M. ROCEK, AND W. SIEGEL, Superspace. or One Thousand and One Lessons in Supersymmetry, Benjamin-Cummings, Reading, Mass., 1981. 13. J. BAGGER AND E. WITTEN, Nucl. Phys. B 222 (1983), 1. 14. M. M. VAINBERG, “Variational Methods for the Study of Nonlinear Operators,” Holden Day. San Francisco, 1964. 15. R. BOTT AND L. Tu, “Differential Forms in Algebraic Topology,” Graduate Texts in Math. Vol. 82, p. 34, Springer-Verlag, New York/Berlin, 1982; M. SPIVAK, “Calculus on Manifolds,” pp. 34, 93. Benjamin-Cummings, Menlo Park, Calif., 1965: H. CARTAN, “Differential Forms,” p. 36, Benjamin-Cummings, Menlo Park, Calif., 1970. 16. R. BOTT AND L. Tu, “Differential Forms in Algebraic Topology,” Graduate Texts in Math. Vol. 82. p. 185, Springer-Verlag, New York/Berlin, 1982. 17. S. J. GATES, JR., C. M. HULL. AND M. ROCEK, Stony Brook preprint, ITF-SB-84-53; P. S. HOWE AND G. SIERRA,Phys. Left. B 148 (1984) 451; R. ROHM, Princeton preprint, October 1984. 18. S. J. GATES, JR., Nucl. Phys. B 238 (1984), 349. 19. S. J. GATES, JR., “Supersymmetry and Yang-Mills Invariance in 1 + 1 dimensions,” MIT, Centre for Theoretical Physics Publication, No. 605, 1977, unpublished. 20. T. E. CLARK AND S. T. LOVE. Phys. Lett. B 138 (1984), 289.
OF PHYSICS
168,
387403 (1986)
Supersymmetry and Geometry D < 4 Nonlinear E-Models GARY ATKINSON,* Department
UTPAL
of Physics and Astronomy,
CHATTOPADHYAY,~ University
of Maryland,
in
AND S. JAMES GATES, JR.: College
Park,
Maryland
20742
Received February 7, 1985; revised July 30, 1985
We discuss the augmentation of D < 4 supersymmetric u-models by the addition of higher derivative terms, such as the Skyrme and Wess-Zumino actions. These terms must satisfy differential constraints so that they preserve the symmetries of the usual kinetic action, In two dimensions, we explicitly construct the gauged WZ action for group manifolds. Some difficulties with the analogous four dimensional theories are noted. ‘C 1986 Academic press, IIIC.
I. INTRODUCTION
AND SUMMARY
Nonlinear sigma models provide an interesting arena where differential geometry, topology, and quantum field theory meet, sometimes with startling results [l-4]. For example, in four dimensions they act as effective field theories for low energy hadron physics and, if the Wess-Zumino chiral effective action [S] is included, these models incorporate all the information associated with the chiral anomalies [6]. Two dimensional sigma models in particular exhibit properties possessed by some four dimensional gauge theories such as instantons, renormalizability, and asymptotic freedom. These models also have the nice feature of being more amenable to rigorous study. Furthermore, it has been shown how nonlinear sigma models in two dimensions can be considered as a formulation of string theories (see, e.g., [7] and the references therein.), of which the more viable versions have supersymmetry as a necessary ingredient. These reasons and others indicate that work on D < 4 nonlinear sigma models can give valuable insight into effective and possibly realistic theories of nature. It is becoming increasingly clear that nonlinear sigma models augmented by Wess-Zumino and Skyrme [8] terms exhibit richer structures. Using the language of differential forms, we study the geometrical properties of the suitable extensions of these actions, which we call the generalized Wess-Zumino (WZ) and the generalized Skyrme action. A main result of this investigation is the formulation of a * Research supported in part by NSF under Grant PHY 82-18338. + Research supported in part by NSF under Grant PHY 83-08070. : Research supported in part by NSF under Grant PHY 84-16030. 387 0003-49 16/86 $7.50 Copyright B’I 1986 by Academx Press. Inc. All rlghts of reproduck,,, tn any lorm reserved
388
ATKINSON,
CHATTOPADHYAY,
AND GATES
specific geometric characterization of the generalized WZ action in terms of a differential equation that it must satisfy. If the bosonic fields of the theory are the coordinates of a group manifold, we can actually construct a solution to the above equation for even space-time dimensions. For odd dimensions, however, we show that a generalized WZ action cannot be constructed on a group manifold. Turning next to supersymmetry, we give the supersymmetric extensions of the generalized WZ and Skyrme terms for D = 2 and D = 3. In addition, for D = 2, we also construct the gauged version of the supersymmetric WZ term, both with and without central charges. The paper is organized as follows. In Section II, we define the generalized Skyrme (Sk) action and the generalized Wess-Zumino action in three space-time dimensions and describe the geometrical characterization of the latter. The whole discussion, however, can be readily generalized to arbitrary dimensions. In Section III, we specialize to the case where the bosonic fields of the theory coordinatize a group manifold and try to construct the WZ action using a technique due to Vainberg. A self contained discussion of the Vainberg technique is included. In Section IV, we discuss the supersymmetric extensions of the Skyrme and the WZ terms in 2 and 3 dimensions alluded to above. In the concluding section we make some comments on the recent attempts to supersymmetrize the Skyrme term [9] and the WZ action [lo] in four space-time dimensions.
II. COMPONENTS AND GEOMETRY We begin by considering within a geometrical context only component field theories in D = 3 space-time. Let 4”(x) (m = l,..., d) be a set of spin-zero fields defined over three dimensional space-time with coordinates x*, (a = 0, 1, 2). These fields may clearly be considered as a map 4: M3 --f 4 if M3 has coordinates x” and & has coordinates 4”. Writing dqP’= dx’(a,$“) we can consider the most general action that can be constructed from the d@” and possible tensors over the manifold ~8, assumed to be endowed with a metric g. If we impose the restriction that no more than quadratic temporal derivatives appear, this action must take the form S,,, = ; j [g,,,,,d@’ A *d@‘+ h,,(dqP
A dq5” A d&‘)
+kz,,(d@” * WV * * (d&’ * @“)I + LnpqrsWm A dq5” A dc,P) A * (d@ A dqY A d&). (We ignore the possibility of a term linear in dqP.) actions, S,, S,,, SrrC, and ,!& which we refer to action, the generalized Wess-Zumino action and (For simplicity, henceforth we will set S:, equal to
(2.1)
These terms precisely define four as the usual nonlinear a-model the generalized Skyrme action. zero.) In (2.1), the * denotes the
D-c
4
SUPERSYMMETRIC
389
C-MODELS
Hodge duality operation with respect to the Grassman algebra generated by d-x”. Although we are only considering D = 3, all terms in (2.1) are applicable to a spacetime of arbitrary dimension, except for the second term. Of course, it is possible to write for arbitrary D an appropriate form for this term. We need only introduce a D-form h = h,, .. . ..dqP A ... A d&“” in place of the 3-form. Choosing a specific x-coordinate frame, (2.1) implies
S,, = 1 d3x thmnpE‘“a14m) Ssk
=
f d3x
%n,,
(a,,&?
(a,@) @b,d”)
(aN)% (W”)
(abe.
In what follows, we will be concerned with the geometrical, as opposed to the topological, characterization of the action S,,. But it may be useful here to mention the distinction between these two characterizations. First, S,, is defined as the integral of a D-form h on JV. Next, we require that the isometries of M that are continuously connected to the identity (and, therefore, generated by the Killing vectors) are also symmetries of S,,. An action constructed from a D-form and satisfying this geometrical condition will be referred to as a generalized Wess-Zumino action. If J? is nontrival in the sense that there are symmetry transformations, i.e., isometries, which are not continuously connected to the identity, then we must consider the change of S,, under these transformations as well. Such considerations are intrinsically topological in nature and usually lead to quantization conditions, e.g., in four dimensional chiral theories. However, purely geometric aspects of the generalized Wess-Zumino action (as defined above) may be important and interesting even when d is trivial. An example of this is given in the o-model interpretation of the covariant string action [7]. Finally, even though the WZ term is related to chiral anomalies in even dimensions, it is impossible to interpret the odd dimensional WZ term in the same way, since there are no chiral fermions in odd dimensions. But this does not preclude the existence of the generalized Wess-Zumino action in odd dimensions. An example of this has been found for N= 2 supersymmetric a-models [ll] in D = 3. On the other hand we will show later that S,, cannot be constructed for group manifolds in odd dimensions. To study the symmetry properties of the action, we begin by varying the total action. In general, the variations Sdm may be functions of ii and possibly other fields f’(x). We may indicate this by writing Sbm = &zP(&‘,f’). Upon variation of the actions (2.2), we find
(2.3)
390
ATKINSON,
CHATTOPADHYAY,
SS,, =; j d3x C&,(k)] mnpq($a&? + 2
s
d3x kmnpq5
SK =; j d3x CUg)l,,
ta,af’)
(W’7
AND
GATES
(a,,@)
(av”)
@b4q)
@b,f)
(a’$“)
(ab’$‘),
(aa4’7 + j d3x g,, 5
(2.4) (W’)
(44’7,
(2.5)
where (with v” E &P),
C~“uam2p= ~qLzp.q+ Vqmhq, + ~q/4,,Anqp + Vq,pLq C~“k)L = VP&zn.p f VPdpn + VP9”&lp, CUW,,,
(2.6) (2.7)
= V’kmp,,,+ Vrmknpq+ KAnrpq + Kpkmrq + ~>qkmpr.
(2.8)
We recognize E,(T) as the Lie derivative of the tensor T taken along the vector field V= &Y”a,. In the special case when SqY’ is only a function of 4” and the f”s are constants, we see that for an arbitrary function F and tensors T we have
S= .[ F(T, dd, *d4),
(2.9)
d”S = 1 F(&,“( T), d#, *d/j).
(2.10)
The results in (2.3) and (2.6) imply that the variation of S,, can also be interpreted as the integral of the contraction of the exterior derivative of the 3-form h= hmnpd&” A dqS” A dtip with the vector field V. In coordinate free notation, (2.4) is expressed (after integrating by parts) as
W,,, = j- (4
(2.11)
V>,
or, in terms of coordinates,
cw,,,= Qi
d3X
EP”hCmnp,ql
(a,@?
tab&)
(wp)
64’.
(2.12)
This way of writing the variation is very interesting because dh is a 4-form. So when the map of T(M) -+ T(d), induced by 4, is used, dh is identically zero. However, (dh, V) is only a 3-form. So, (2.11) is nonzero, under the q&map, if h is not a closed form. But the action S,, has another property not shared by SO and S,,; under a
391
D-c 4 SUPERSYMMETRIC C-MODELS
“gauge” transformation of hmnp, S,, changes by a surface term. Replacing the “curl” of a two-form il,, we see
h,,
by
(2.13)
This brings us to addressing the geometrical characterization of the WZ term. To answer this we first discuss symmetries in Lagrangian field theories. Given a Lagrangian L(Y) depending on a set of fields Y’ we can define Lie differentiation E,(Y), denoted by f,(P) E 6Y’ where 6Y’ is a given variation that depends only on Y’ and some constants. We say that the Lie derivative generates a symmetry if E,(L) = 0, where S is the action (S = SdDx L(Y)). Often, however, we consider a theory described by two different Lagrangians L1 and L2 physically equivalent if L, and L, lead to the same equations of motion (i.e., if L, and L, differ by a total derivative). Because of this we can enlarge the definition of a symmetry. Instead of requiring f,(L) = 0, we can use the weaker definition f,(L) = a, r” for some quantity Y‘. Let us call the symmetries that satisfy f,(L) = 0 manifestly realized symmetries and those that satisfy f,(L) = 8, Y” nonmanifestly realized symmetries. When symmetries are gauged, this distinction is important. For manifestly realized symmetries, the Noether procedure leads to a “minimal” coupling of the gauge fields. For symmetries realized nonmanifestly, the Noether procedure leads to “nonminimal” coupling of the gauge fields. Some examples of this can be seen in gravitation, supergravity [12], gauging Killing vectors in N= 1 supersymmetric nonlinear o-models [13], and the gauging of the WZ term [63. Now it is simple to characterize the geometric meaning of the WZ term. Let g,,(d) be the metric of a nonlinear o-model and h,,(4) be a 3-form in the manifold of 4”‘. Now, the Killing vectors K,“’ generate a symmetry transformation of the action S, because fK(g) = 0. We want this to be a symmetry of the WZ action as well. However, from (2.13) it is clear that for this to be true it is sufficient to have the Lie derivative of h,, to be equal to exterior derivative of some twoform I,,. Therefore we say that the action contains a WZ term if (2.12) is satisfied and if for each Killing vector K,“’ we find the following condition to be true, d[f K(h)] = 0.
(2.14)
For the same reason, the Skyrme term obeys the stronger equation f K(k) = 0. III.
GROUP MANIFOLDS AND GEOMETRY
At this point, it is useful to consider the special case, where .X is a group manifold. In this case the @” fields can be mapped itit0 the elements U of a group, U= exp [&P(x)
Tm]
(3.1)
392
ATKINSON,
CHATTOPADHYAY,
AND
GATES
with T, being a matrix representation of the ith group generator. We define left and right Cartan-Maurer forms L,” and R,” by the equations U~‘iY,U=iL,“(qS) (a, U) U-’ = iR,Q)
T,(~?,cj”),
(3.2)
T,(Q”).
(3.3)
Throughout the remaining discussion we will adhere to the early-late convention for the A’ manifold. Thus a, b ,..., are anholonomic indices while m, n,..., are holonomic indices. These definitions permit L and R to be explicitly calculated L,“(ti)==(C,)-‘Tr[T(q)
T,,,],
R,“(m,=(C2)p’Tr[T”(~)
Tm],
(3.4) (3.5)
where AT, - [i&‘T,,, T,], A2T, = AAT,, etc., and the constant C, is determined so that L,“(O) = R,“(O) = 6,“. (Note that (3.2, 3.3) imply that L and R are regular as 4 approaches zero.) It follows that
Tr Cu-‘(a’u) u-‘(a, VI = -C2g,,(d)(a”~m)(a,~), k!m($) = kz&m”Lnb = k&,,a&b,
(3.6) (3.7)
and thus x7= -2c,
1 s
d3X Tr [Pu
a,,u-l J,
(3.8)
[a,,ua,,u~la[~uablu-II,
S,,=&jd’xTr
(3.9)
2
where we have already made a particular choice for kmnpq on group manifolds. Before we give the form of S,,, it is useful to note that a “spin-connection” w,,~ and “affine connection” r,,,,,p are defined by
a,y - W,ba~nb +
r,,p~~a = 0,
(3.10)
lam L,b + r,,PLpb = 0, and the anholonomy
coefficients Cobr by &,“a,,, Lb”a, - Lbma, &,“a,, = cabcL,“a,,
where L,”
is the inverse of L,“.
From
the definition
(3.11) of L,”
we find using
CT,, Tbl= -ifat,'Tc gwLno - gn L,” = fb/LmbL,‘.
(3.12)
D <
4
SUPERSYMMETRIC
393
C-MODELS
We now completely fix the geometry by assuming a gauge where o,,& = 0. This is consistent since a parallelizing contorsion always exists for group manifolds. This implies Cobc = fabc= Tabcr where Tabc is the torsion tensor for the manifold. Here Tabc is the “flattened” torsion tensor, Tabr = (rm, The equation of motion
- r,,,)
(3.13)
LamL,“L,p.
derived from (3.8) takes the form W(LmQ a,qq
(3.14)
= 0.
Following [6], we want to modify this equation by the addition tional to the D = 3 s-tensor, GB”(L,” 8,dm) T, = askd( Up’ d, U)( U-’
where a is a constant with the dimension ~a(L,“&,~m)
T,=
of a term propor-
8, U)( U- ’ ad U),
(3.15)
of mass. Using (3.2) this becomes
-iCITbT,T~EbedL~bL,cLpd(~,~m)(~,~fl)(~d~~)
(3.16)
or (3.17)
P( L,” a,@y = - OUbcd&~dL,bL,cL,d(a,~m)(a,Qln)(ad~P),
0 abed= id-
’ Tr [ T, Tb T, Td] .
(3.18)
Now we want to find an action that yields (3.15, 3.16) upon variation. Because we are in an odd dimensional space-time, a first guess might be Tr [U-‘dU U-*dU U-‘dU], since it is not identically zero. But this does not s contribute to an equation of motion! (This is also valid for any odd dimensional space-time; see Appendix.) So we must follow the construction of Witten, even in odd dimensions. To this end we note the following observation [14], (3.19)
Defining the operator K( (aF/&$)) so that (3.19) reads K((aF/&Y)) = F(d) - F(O), we note that K is a homotopy operator [ 151. In fact, the realization of K in (3.19) is such that applying K to the left hand side of (3.16) produces precisely the action S,. Explicitly we find -
s0
’ d~~“L,,(z~)[z~b(L,u(~~)
t&f=)]
= -&?-r
[abU c?~U~~‘].
(3.20)
2
Thus to find the action that gives (3.16) by variation, we make the replacement q5--+zqi in (3.16), multiply by -(l/C,) d”L,“(+) T,, trace and integrate. It is worthwile to note that this procedure is not guaranteed to produce an action whose
394
ATKINSON,
CHATTOPADHYAY,
AND
GATES
variation gives the equations of motion with which we started. We have to explicitly check that this is indeed the case. Following the prescription, we find that S,, = ;
j d3x 1; d&=
Tr [(0-l
0 z exp [iqPT,,,], Comparing
a,ir)(~-la,ir)(6-'a,6)~mZ,aT,],
zrno z L,“( 74)
(3.21)
this to (2.4) we see
The definition
of 0 implies that (3.21) can also be written as
(3.23)
~3=~a"(a,~i)(a,ir-*)(a,ir)
u-1.
If we consider (x’, r) as the coordinates of a four dimensional manifold, then the three dimensional fields d”(x) are extended into four dimensional ones by (vyx, ?) = 7fyx). s ince all the four dimensional fields are defined in this way, we can visualize (x, z) space using Fig. 1. The boundaries of the (x, z) space are the origin, t = 0, and the three dimensional x-space at 7 = 1. The interior of the cone corresponds to (x, 7) for all other values of 7. It is interesting that such an interpretation is available since the cone construction is also used in singular homology
Cl0 The 6 operator also allows a simple rederivation -$-:Tr
[LJP~aa,~-‘] 2
= -&Tr
of (3.20) since
[(~-~-$$3”(&‘a~~)] 2
FIGURE
1
(3.24)
D
<
4
SUPERSYMMETRIC
395
C-MODELS
and the r.h.s. of (3.24) is equal to the integrand that enters (3.20). Additionally, Skyrme term can also be written extended into four dimensions since -&&Tr
[(a,.ir)(&,
the
irP’)(acaO)(ablO-‘)]
2
=&Tr[(
&I$)
iT,,((P’
a,,~)(ac.d-‘)(ablO))].
(3.25)
2
So we conclude that with this extension, apparently, the entire action in be written in the four dimensional manifold (x9, z). This surprising result an arbitrary dimensional x-space also. Given a group with U = exp [itimTm], with 4”(x) defined over a D-dimensional manifold, and of motion J(4) = 0, where
(2.1) can holds for elements equation
J(~)=d’(U-‘~,U)+a~,,[(~-‘~b,U)(U~’~r*~)(~--’~b1U)] + BE =l”‘yU-’
da,u)-‘(u-’
(3.26)
d,,,U);
the action for this equation is given by
With the exception of the Wess-Zumino term, each term is t-exact and thus depends only on the boundaries of the D + 1 dimensional manifold. However, the dimensionality of the manifold here plays a critical role. The proposed action for the WZ W term can be expressed as
(3.28) l$‘D=(~-ldfi)D=(ii-‘d~)
where v is the volume of the D-dimensional yields
A ..’
A (&‘dl?),
space-time. The variation
of (3.28)
The final four terms, for even D, add together to a total x-space divergence. But this is not true if D is odd. (See Appendix.) Thus the z-trick is only successful for even
396
ATKINSON,
CHATTOPADHYAY,
AND GATES
dimensional spacetime manifolds. (We note that failure of the r-trick is equivalent to the statement that the functional derivative curl of W, is nonzero for odd dimensions.) So, we conclude that a generalized WZ action does not exist in odd dimensions for a group manifold. In closing this section we should point out that the introduction of the r coordinate is not related necessarily to topological questions. This type of representation can occur whenever an equation of motion is used as a starting point to derive an action. For example, given the equation 04 = -cos (@), using the r-trick yields for an action
IV. SUPERSYMMETRIC EXTENSIONS In the previous sections we have discussed the relevant geometrical features of the Skyrme and WZ terms in ordinary space. Here, we will extend these to include supersymmetry. Additionally, we will consider D = 2 theories which are closely related to those in D = 3. In particular, we will give the D = 2 supersymmetry gauged version of the WZ action for a group manifold. For D = 3 the supersymmetric form of the Skyrme and WZ actions are Ssk= 1 d3x d% k ,,,(@)Dw”D%n,”
ay,waySw,
S,, = j d3x d28 h,, (~)(a,,~m)(D,~n)(DB~P),
(4.1) (4.2)
where CD” is a real “multiplet” of scalar supertields. Using the method of projection it can be verified that the purely bosonic terms here contain the appropriate component actions in (2.2). It is also interesting to note that for arbitrary variation of the superfield @” in (4.2) we find AS,., = f d3x d2tI h c,,,q,(aap~“)(D,~“)(Dp~P)
cm4
(4.3)
exactly like the change in the component action given in (2.12). Specializing to group manifolds, (4.1) becomes Ssk= f d3x d26’Tr [(D”U-‘)(Dfi’U)(iY’,
where U =exp [i@“T,,,].
U-‘)(a,,U)],
But due to the failure of the z-trick for constructing
(4.4)
the
D <
4
SUPERSYMMETRIC
397
C-MODELS
ordinary D = 3 WZW term, we have been unable to find an explicit realization (4.2) for group manifolds. The usual S, can be supersymmetrized as S, = j d3x d28 Tr [D’W’
D, U]
of
(4.5 1
on group manifolds. We note that the transformation U + g, ‘Ug, leaves (4.4) and (4.5) invariant for constant matrices g, and g,. We can gauge these symmetries by introducing gauge superconnections L, and R,4 and forming covariant derivatives V,U=D,U+iL,U-iUR,,
and using the minimal coupling prescription. Yang-Mills superconnection are [ 121
(4.6)
As usual the only constraints on D = 3
O=D,L,+D,L,+i{L,, 0 = D,R,
+ D,R,
L,)-2iL,,, + i{R,,
(4.7)
Ru} - 2iR,,.
It is instructive to consider the case of D = 2 where we can explicitly construct the WZ term and its gauged version on group manifolds. For the Skyrme term, the action can be obtained imply by replacing d3x by d2s in (4.1). But for D = 2, we replace (4.2) by
sWZW’ =sd2x d28(yS)“B h,,(@)(D,P’)(Dp@‘“).
(4.8)
(The D = 2 supersymmetric WZ action has been constructed by a number of authors [4, 173.) It is amusing to note that we can write this action also as an integral over an extra dimension z in the following way: s,, = [ d2x d28 [’ dz(y’)@ J
t2 H,,,,,(zQ)
Jo
@” D,@” D,rSp.
(4.9 1
where aEmh,, = H,, . Before we specialize to group manifolds, we want to point out that in two dimensions the N= 1 supersymmetry algebra can admit a central charge Z. For nonlinear supersymmetric a-models, we can in fact activate this central charge, which gets realized by means of Killing vectors G” of the underlying bosonic manifold in the following way [ 181 Z@” = moGm, where m, is some parameter with the dimension
(4.10) of mass. In the presence of the cen-
398
ATKINSON,
CHATTOPADHYAY,
tral charge the relevant supersymmetric are related by
AND
GATES
derivatives are fi, rather than D,; the two
ba = D, + i(y5),BeBZ, Ij,@”
= Da@“+
(4.11)
im,(y5),@‘?Gm,
Therefore, if the central charge is activated, the supersymmetric is given by
Wess-Zumino
Celltraly
S$=ged =
I
d2x d% (y5)@h,,(@)
We now return to (4.8). Specializing
b’,@”
to group manifolds
term
(4.12)
&F’.
this becomes
Under the global transformation U +g, -lUgR, (4.13) changes by a total spacetime derivative. Therefore, to gauge this symmetry we use the Noether method. We again introduce gauge superconnections L, and R, transforming as 6L,=DAA,-i[LA,
AL},
hR,=D,A.-i[R,,
AR},
(4.14)
where A, (AR) are infinitesimal parameters of the gauge transformations. all the independent constraints are such that [19]
[V,, VP} = 2zV,,+ 2i(y5),BPL
In D = 2 (4.15)
+ 2i(y5),,PR.
The Noether method yields Se@=
s,, + i -
s
d2x d20(y5)“B Tr [L, U DpV’
-R,
U-‘DBU]
(4.16)
d2x d20(ys)@’ Tr [ UR, U-IL,].
Actually, under a gauge transformation 6Sg;ged =
s
we find
d2x d26(y5)“8 Tr [A, D, L, - A,D,
RB].
(4.17)
This shows that only the diagonal subgroup (g, = g,, LA = RA) can be gauged.
D < 4 SUPERSYMMETRIC
C-MODELS
399
V. CONCLUSIONS We have shown that the class of actions, of which the Wess-Zumino action is an example, have straightforward supersymmetric extensions in two and three dimensions. Even without topological considerations, a number of interesting geometrical properties of the actions were found. The most important of these is the constraint on the tensor h given by (2.14). For group manifolds we can actually construct a solution of this equation only for even dimensional spacetimes. However, for some nongroup manifolds, specifically Kahler manifolds, nontrivial solutions of Eq. (2.14) exist within D = 3 supersymmetric theories [ 111. In four dimensions, there has been a recent attempt to supersymmetrize the WZ action [9]. There it is shown that no tensor j? (analogous to our h) exists for a compact Kahler manifolds. For noncompact Kahler manifolds, h is known to exist [20], but the appropriate four dimensional analog of (3.22) is not known. Such manifolds are important since they occur in phenomenologically interesting models. So, in our view, a more complete supersymmetric extension must include a construction of the tensor h in terms of the geometrical quantities that characterize noncompact Kahler manifolds. There are additional problems that plague both the four dimensional supersymmetric WZ and Skyrme actions [9, lo]. The first of these, the propagation of the “auxiliary” field components (F and G), renders somewhat problematic the usual geometric interpretation of the standard supersymmetric nonlinear o-model. Of greater concern is the presence of quartic temporal derivatives which obviate the raison detre for uniquely choosing the Skyrme term to stabilize the “Skyrmion” and raises questions about stability in general. In our opinion, more work is required to reconcile supersymmetry with higher derivative actions in four dimensions.
APPENDIX
In this Appendix we will show that the action given in (3.28) yields a local (local with respect to x0), manifestly z-independent equation of motion only in even dimension. To prove the assertion, let us rewrite the change in the action given in (3.29) in the following way:
where
400
ATKINSON,
CHATTOPADHYAY,
AND
GATES
From now on, we will omit the caret A since there is no possibility of confusion. We will also omit Tr while writing Ssl, and the wedge product between various forms will be implicit. We will show that m(D+‘)Tr
(UP’
6U
(W,-,-
1U-i g
W,,,
+P, jvj;d7Tr[( wDc’-$-T D-l -
1
(A.3)
(-1)-w,~,,ci’fgw,)o’au],
!?I=0
where Pi are defined by Pz = (1 + ( - 1)‘)/2. In (A.3), IV, is 1 by definition. Therefore we see that while in even dimensions, &!$ is zero modulo a surface integral, in odd dimensions &IS;, is given by the above nonvanishing expression modulo the same surface term. To derive (A.3), let us begin by noting some properties of WD which follow trivially from its definition. For D = 2n,
For D=2n+
1, dy-’
W *n+l=wY+l
d2U...dUdU-‘U, ‘.’
2n
2ni
I
Also note that dW,=PbW,+1,
UW2,+,=dUW2,,,
W2n+lU-‘=
-W2,dU-‘.
(‘4.4)
We will need an expression for 6 W,. It can be written as D-1
6W,=
-U-‘6UW,+
W,U-‘6U+
1 W,,,d(U-‘HJ)W,-,,-, m=O
(A.3
for D 2 1. To prove this let us first note that D-l
SW,=
c
w,,,6(u-‘dU)W,-,-I.
64.6)
Wl=O
Using the identity 6(U-‘dU)=-U-‘NJW,+
W,U-‘GU+d(U-‘6U),
(A.7)
D<
4 SUPERSYMMETRIC
401
C-MODELS
(A.6) becomes
D-l 6wD=
-
D-1
c
w,,,u-‘6uw,~,+
1 m=O
??I=0
w,,,+,u-‘6uw,~,~,
D-l
+ 1
(A.8 1
W,,, d(U-’ 6U)W&,,-,
m=O
from which (AS) follows immediately. Now note that if we plug (AS) into (A.2) the first two terms in 6 W, cancel precisely the first two terms on the right-hand side of (A.2) and we are left with D-l &d(U-‘6U)
1
(-l)“‘D+“wD-,-
,d$w,,,
PI=0 1
-
I?’ Y0
dT$-16U,
(A.9)
where we have used the cyclic property of trace and the anticommutativity Integrating by parts, we get
of forms.
as~=~~~~d~d[U-‘6UD~1(-l)“‘D+1)(W,,~,U~1~W~)]
m=O
1
+j- j; dzUp’ 6U F(W,)-2
(A.lO)
,
where F( W,) is given by D-l
F(W,)=
C (-l)m(D+“+ld
T?l=O
(
W,eJ1~
Wm).
(A.ll)
Since we can always write F( W,) as P,+ F( W,) + P; F( W,), it is sufficient to calculate F( W,) for even and odd D separately. The strategy in evaluating these will be to divide the sum over m in (A.1 1) into two groups corresponding to even and odd values of m. Thus, for even dimensions, (D = 2n), we have n-1
1
F(Wz,)= c d -W,,,~,U-1~w~m+w~n-2,-2u~~~w~~~I l?l=O
= n$ d[W,.-zm-z WI=0 =
du-‘!?,
aT
2m
-w
au-1 dU W,, Zn-2m--2 aT 1 -
dtT’&dU-&dUpldU m=O n-1
=m&, w2”-2m-2~
w,w,m=&
w2n
(A.12a)
402
ATKINSON,
while for odd dimensions
CHATTOPADHYAY,
AND GATES
(D = 2n + l), we have
(A.12b) In deriving (A.l2a), (A.12b) we have repeatedly made use of the identities given in (A.4). Combining these two we can write, F(W,)=P,t;y+P,
(A.13)
W~~-Dt;‘(-l)mWD_,~,U-L&dUW, t?l=O
Substituting
(A.13) in (A.lO) yields precisely (A.3). This completes our proof.
Next, we want to mention that one cannot take the expression J Tr W, (there is no integration over z here), as the WZ action even though one might be tempted to do so. Unlike in even dimensions, this term is not identically zero in odd dimensions. Nevertheless, it does not contribute to an equation of motion. This can be seen easily from Eq. (AS). Because of the cyclicity of trace, we find that when D is ..odd, 6[Tr
W,=D
jd(U-‘i%l)W,-,=D
(A.14)
jd(Up16uWD-,),
where in the last step we have used the fact that dW,- 1 is zero for odd dimensions. Thus the variation of the action is just a surface integral and therefore it does not give rise to any equation of motion. Finally, it is amusing to note that all Skyrme type terms on group manifolds can also be written in terms of W,,, as j Tr [W,* W,] with m = 2,..., D. Thus in D dimensions the most general group manifold theory that is no more than quadratic in temporal derivatives takes the form ^
w,*w,,,+c,
j’dT..,-I:
1 .
0
(A.15)
ACKNOWLEDGMENTS We gratefully Zachos.
acknowledge
useful
conversations
with
Darwin
Chang,
Kyriakos
Tsokos,
and Cosmas
D
<
4
SUPERSYMMETRIC
C-MODELS
403
REFERENCES
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