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Unidexterous superspace: The flax of (super)strings
Volume 194, number 1
PHYSICS LETTERS B
30 July 1987
UNIDEXTEROUS SUPERSPACE: THE FLAX OF (SUPER)STRINGS S. James GATES Jr. l Department of Physics and Astronomy, University of Maryland at CollegePark, CollegePark, MD 20742, USA
Roger BROOKS and Fuad M U H A M M A D 2 Centerfor Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 18 February 1987
We give the superspace formulation of the most general superconformalunidexterous theory. A proposal for the world-sheet superspace description of compactified strings is presented.
I. Introduction. In a previous paper [ 1 ], we gave the first and, to date, most complete discussion of the local superspace formulation of unidexterous (or one-handed) supersymmetrical theories. The global version of these N = 1/2 theories were first considered by Sakamoto [2]. Unidexterous superspace is the fundamental geometrical quantity in the construction of manifest world-sheet supersymmetry in NSR superstrings [ 3 ] and the heterotic string [4]. This superspace provides the basic building blocks (or flax) for the construction of all superstrings which involve world-sheet supersymmetry. Thus, a clear and concise way to probe for the existence of heretofore unknown strings is provided by a complete analysis of all superconformal theories in unidexterous superspace. It has been shown [ 5,6] that ordinary field theoretical techniques, combined with unidexterous superfields, can be used to deduce the existence of candidate string theories. In particular, the absence of scale and Lorentz anomalies enforces the condition that the effective action in unidexterous superspace possesses all of the symmetries which are present in the classical action. In the present paper, we will discuss how all of the known string theories which possess world-sheet supersymmetry can be formulated in unidexterous superspace. This is obvious for the heterotic string but, in principle, for the usual NSR string such a formulation should also exist. In fact, the matter gravitino linearized version of this theory was presented in ref. [ 6 ]. We will explicitly carry out the crucial steps for obtaining the complete theory and show that the previous result is exact. We also show how unidirectional supermultiplets provide a two-dimensional world-sheet description of string compactification. In addition, general unidexterous superconformal theories which include left- and right-moving multiplets will be discussed. 2. Unidexterous superconformal theories. In order to describe the unidexterous superspace formulation of all superconformal theories it is convenient to have a "spinning wheel", i.e., an action from which any superconformal theory arises as a special case. We should point out that such an action may have more accidental or non-manifest symmetries beyond those that are obvious. In particular, all of the spinning strings [ 7 ] and the heterotic string [4] must appear as special cases. In our first work [ 1 ], we showed that unidexterous superspace is essentially the same as all other superspaces. There are large numbers of multiplets which include scalar
t Researchsupported in part by NSF grant #PHY86-19077. 2 This work is supported in part by the US Department of Energy (DOE) under #DE-AC02-76ER03069. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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multiplets, vector multiplets, etc. going up to a supergravity multiplet. Not all of these multiplets are superconformal multiplets. O f the multiplets reviewed in ref. [ 1 ] only the scalar multiplet, X -~, and the minus spinor, q /, are superconformal multiplets. However, there are two additional superconformal multiplets which we overlooked. These are the N = 1/2 versions of unidirectional of chiral bosons. The right-moving boson or scalar "righton" multiplet has been given in the global context in ref. [ 5 ]. Minimal covariantization with respect to local supersymmetry yields an action for NR righton multiplets coupled to supergravity: S R = i ~ 1 f d2 o" d (
g -j (V + (~R ~V _ _ (ibR a q-A _ _ + + a/;V + (ibR aV ++ (~R /;), A _ _ + + t&~l =0,
where c/=l, ..., NR. After defining component fields by q}Ra--=~Ral, p + a - - = V + ~ R a l , iV +A _ + + aa I, 2_ _ + + ab-=A _ _ + + ea I and using the derrsity projection formula
f
d~- E-'~_
=e
(2.1) R--+ab--
' ( V + - i v / + + +)£e_ I,
(2.2)
we find the component result is given by SR------
d
+0R
--~Ra+iP+
+2~u__ + f f + a ~ + + ~ R
__p+
-t-2__ ++ gi/;(~+ + OR d ~ + + ~R /;"}-ip+ & ~ + + p +
-- _
a f i f f + a ~ + + ~ R b"
a+2V/++ +p+ gt~+ + ~R~)] ,
(2.3)
where the component supergravity quantities are defined by ~ + + ( e , ~,)=e_++ ' " 0 , , + a ) + + ( e , v/)d/,
ro++(e, v / ) = r o + + ( e ) = c + + . _ _ - - , [ea, eb] =ca.bCec.
o)__(e, v/)=c++ __ ++ +i2v/++ + V/ _ +,
(2.4)
In a similar manner, a unidexterous left-moving or scalar "lefton" multiplet has also been found in global
N = 1/2 superspace [ 8 ]. Once again minimal covariantization immediately yields the locally supersymmetric action of NL lefton multiplets coupled to supergravity: SL=i~
If
__ d 2 o ' d ~ - E _ I [V+~LaV
+ - - a9V__(JSLaV __ (J}Laq-A
(~Lfl], A+ - - [ a d l = 0 ,
(2.5)
where & = 1, ..., NL. Likewise, the component reduction of this action is obtained as S L = -- f d2o " e - I ½[ ~ + + ~ L a ~ _ _ --2+ + -- -- a ] ~ ( ~ / _ 0 L a ~
q~La + i f l +
a~__fl+ a + 2 V / _ _
+fl+ a ~ + + O L a
_ 0L/~ +2V/_ _ +fl+ a ~ _ _ OLd)
+i2+--ad(2~__p+a~__#ed+v/__+fl+a~ +iv/__ + cG/++q~La~ _ ~ L f l - - i v / + + + ~
fl+a
_~La~__~Lflq-iv/__
+V/++ +fl+ a~__~L]~)] ,
(2.6)
Here the component fields of the lefton multiplet are defined by OLa--=q~Lal, fl+a=V+gYLal, 2+ ad=A+--a~l,2++--a~=-iV+d+--a~l. The classical scale invariance of these actions is seen upon making the variations 8sV+=½SV++(V+S)J¢,
8sV++=SV++-i(V+S)V++(V++S)d/,
5sA__ ++at;=O, 8sA+ --ag=-½SA+ --a~. 36
8sV_
=SV__-(V__S)J/, (2.7)
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Now the most general unidexterous superspace action possessing local scale invariance takes the form Ssw = r d2 a d ( - E - l [ ~SM + L#MSM+ 5°r~ + L#L], d
~SM =i½~ + X - ~ __X~, L#R=i½[V
~MSM = -- ½r/_ :~ + q_ :,
+q~Ra~__ORe+A__
+ + abV + O R e ~
++O~6],
~L =i½ [V + q~L'~V - - ~ L dt + A + - - aBV __ q~L'~ __ q~Ld].
(2.8)
In (2.8) the covariant derivative, ~ a , is defined by the expression
V A =fA BV B +fa.ll +gA" fg-- ~a + gA' f#,
(2.9)
where VA is the pure (1,0) supergravity covariant derivative [ 1,5,6]. The quantitiesfA B, fA, and gA are functions which may depend on simple unidexterous matter gravitino multiplets or vector multiplets. We note that superconformal invariance implies that this is the only way in which these multiplets can appear in Ssw. The quantities f¢ denote extra generators that may be present such as the generators of the extra non-manifest supersymmetries. The action in (2.8) is capable of describing all tachyon-free strings as special cases.
3. The complete (1,0) superspace geometry of the NSR string. It is well known that the N S R superstring can be described by the coupling of D = 2 , N = 1 (1,1) supergravity to scalar multiplets [9]. But from the point of view of (1,0) superspace this theory is just an extended supergravity theory. As supergravity theories can be expressed in terms of components, any extended supergravity theory can be expressed in terms of the simplest superfields. For D = 2, unidexterous superfields are these simplest superfields. In D = 4 , previous work has described how to project a larger superspace onto a smaller superspace [ 10]. In the following, these techniques will be applied to D = 2 superspaces. We begin with the (1,1) supergravity theory in (1,1) superspace. Using light-cone coordinates as opposed to "covariant" coordinates, the constraints on the supergravity theory take the forms [ 11 ] [V_,V_}=i2V__,
[V_,V+}=-i2Rd/,
[V+,V+}=i2V++,
[V+,V++}=O,
[v + +, v _ _ } = - i ( v + R ) V _ - i ( V R ) V
[V_,V++}=-RV++2(V+R).tl,
[V_,V__}=0,
[V+,V__}=RV_+2(V_R).It, + - 2 ( R 2 +iV +V _ R),//.
(3.1)
Application of the techniques of ref. [ 10] to the reduction of (1,1) to (1,0) superspace yields [ ~ + , 9 + } = i 2 ~ + + + i 2 ~ + - g-'+ - 9 _ _ +i4~v+ - S J/, [ ~+,
9 + + } = g"+ - S ~ +
[ 9 + , 9 _ _ }= - i 2 ~ + [~++, ~__ }=-(T__
- i2 g" + - ~v+ + - ~ _ _ - i 2 (
~v+ + - S + ~ +
- 7-'+ + _ _ - ) J / ,
- kg_ _ - ~ _ _ - i 2 ( hu_ _ - S+2~ + )~#, - S+27+) ~ + - i 2 g - ' + + - ~ v _ - 9 _ _ - 2 ( S 2 - i g - '
_ - ~v+ + __ - + V + 27+)J/. (3.2)
For the field strength of the matter gravitino multiplet ~'/A,B-- we find the constraints, g" +,+ - = 0, ~u +, + + - = 0, g J + , _ _ - =S. It is also useful to know that - i V +R I = g"+ + , _ _ - . The new (1,0) supergravity derivative ~A is related to the " p u r e " supergravity derivative VA and the sinistral matter gravitino ~UA- by the equations 37
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~+=V++i2~+-~
_-,///,
~++=V+++T+-~t_
30 July 1987
- V + ] + 2 T / + -(V
_]~+-)J//,
~
_=V
, (3.3)
[The details of obtaining the (1,0) results from (3.1) will be given in a longer version of this paper.] All of the quantities which appear above, in (3.2), are (1,0) superfelds. These expressions allow us to calculate the (1,1) superdeterminant ~ in terms of (1,0) superfields E and ~A and lead to the result ~ = E ( 1 +i~
T
-).
(3.4)
A locally supersymmetric (1,0) theory is obtained by noting that S=i
f d2a f d~-
f d~+~
~5a
= f d2o" f d~ E-I(V_LP-iT__
L~)I.
(3.5)
On the f r s t line of (3.5) all the superfields are (1,1) quantities. But on the second line we have integrated out the minus superspace coordinate so that all quantities are (1,0) superfields! The (1,1) form of the NSR lagrangian is very well known to be given by 5eNSR=--i½ (V +X-U)(V- X_a). Substitution of this into the expression on the first line of (3.5) implies SNSR= J
dZa d~- E-'
i½[(V
+Xa-)(V __X_o)+in_ -~v + r/__u+ 2 ~ +
-r/_ -a(V X_~)].
(3.6)
This is precisely the result that was given in ref. [ 6 ] where it was obtained by using a superfield Noether technique to lowest order. Our derivation above shows that the lowest-order Noether term yields the complete theory. At no point in our derivation have we neglected any terms. The procedure that we have outlined above may be applied to any type (p,q) superspace theory to derive a (1,0) form of the theory.
4. Heterotic versus heterodexterous strings. In the last section we saw how the NSR superstring can be described in terms of (1,0) superfields. Of course, the heterotic string [4] also has a (1,0) superfield form [1,12], Su = f d2a d~- E - ' i½ [(V + X-~)(V _ _ Xa) +i~/_ ¢V + q_ z].
(4.1)
The differences between (3.6) and (4.1) are clear. In the NSR theory, a (1,0) sinistral matter gravitino multiplet is present and the minus spinor multiplets are equal in number to that of the scalar multiplets. Also the minus spinor multiplets transform in the same way as the scalar multiplets under spacetime Lorentz transformations. Yet both of these theories are known to be consistent strings. From the point of view of two-dimensional field theory this is easy to understand. As classical actions SNS~ and SH are scale and Lorentz invariant. In previous work [ 5,6 ], it was shown that upon quantization every (1,0) multiplet in the classical action contributes a potentially anomalous part to the effective action. This contribution takes the form
r ....
If962r
d2°"d~-
[VR[D+H+
]
( ~[[] n+__)_illL[D+n((9
)4
_
++] ( ( 0 ~-)- 3 H _ _ ++ ) ] , (4.2)
where VR and Ve are numbers which we shall refer to as the right and left degrees of freedom of the supermultiplet. On the basis of the previous work, we find the assignments given in table 1 for all of the multiplets which appear in the above actions. The origin of this table is simple to understand. If we began with nonsupersymmetric D = 2 ordinary gravity [(0,0) supergravity] then we know that the anomaly-freedom of the bosonic string requires 26 scalar fields. Thus VR= VL= -- 26 for the graviton. Next the calculations of refs. [ 5,6 ] 38
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Table 1 Superfield
UR
UL
x
1 ½
o
~R
1
0
11 -26
0 -15
~A E4~
give us the values of VR and //L for unidexterous supergravity. But the only way that (1,0) supergravity differs from (0,0) supergravity is the presence of a component dextral gravitino. This implies that a component dextral gravitino must have UR= 0 and UL= 11 (the difference between - 15 and - 2 6 ) . In the (1,0) sinistral gravitino multiplets [ 1 ] there appear the corresponding component gravitino fields. Thus, the sinistral gravitino multiplet must have UR= 11 and UL= 0 (the opposite of a component dextral gravitino). The internal degrees of the heterotic string can be described by sixteen unidirectional bosons or thirty-two unidirectional fermions. This implies that UR and UL for )7_ are one half the corresponding values for ~R- This observation about the relation between the degree of a unidirectional fermion versus that of a unidirectional boson also explains why UL= ~ for a scalar multiplet. At the component level, a massless scalar multiplet contains one right-moving boson, one left-moving boson, and one left-moving fermion. Summing over all of the fields in either the NSR or heterotic string actions shows that the total contribution to the anomalous part of the effective action is zero. Thus, the classical symmetries survive in the quantum theory. But comparing either actions in (3.5) and (4.1) with the most general unidexterous superconformal theory in (2.8), we note that no lefton multiplets are present! Classically, the lefton part of the action is also Lorentz and scale invariant. How do leftons change the considerations above? To rigorously answer this requires the quantization of the lefton multiplet as has been carried out recently for the non-supersymmetric case [ 13 ]. We are presently undertaking this step. But there is an intuitive way to argue the outcome of this. Note that F . . . . just counts the number of left degrees and right degrees in the classical action. It is known that the classical equations of motion for righton multiplets and lefton multiplets [ 5,6,8 ] take the respective forms D + • R= 0 and O__ ~L = 0. The scalar righton equation further implies O+ + ~ a ~ 0. Thus the spinor component of the scalar righton multiplet vanishes and the bosonic component is restricted to move to the right. Hence, we find /)R = l , /)L = 0. On the other hand, the equation of motion for the scalar lefton multiplet only restricts its component fields to move to the left. It then follows that the degrees of the scalar lefton multiplet are PR= 0 and //L-~-~" This assignment is also plausible when we recall that a component scalar field is equivalent to a scalar righton and a scalar lefton. So it is reasonable to expect a scalar multiplet to be equivalent to a scalar righton multiplet and a scalar lefton multiplet. Considering the action in (2.8) with only d scalar multiplets, ArE minus spinor multiplets, NR scalar righton multiplets, and ArE scalar lefton multiplets implies that when the conditions
VR=d+½NF+NR--26=O,
~ PL-----~d+~NL--15----0
(4.3)
are satisfied, the total contribution to F . . . . vanishes. This is highly suggestive of a "dimensional divesture" of the heterotic string theory to dimensions d = 1 0 - NL with an internal gauge group of the form GR × GL where the ranks of G R and GL satisfy the condition r k ( G R ) - r k ( G L ) - - 1 6 . Thus, upon quantization we expect the action S H ' = f d2trd( - E-~i½{(V+X~-)(V__X~_)+irl_ZV+q_ I "~ [(V + I~L a ) ( V _ _ ~ L a) -~-A+ - - a,/~(V _ _ ~ L a ) ( V _ _ ~L/~)] },
(4.4)
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[0~
fdZ~d( -
SH'=
E-'i½{(V+Xe)(V
_X_~)+[(V+q~Ra)(V_
+[(v+ClJLa)(V__tlJLa)-FA+ - c~(V
(~L &)(V
~R~)+A
++~(v+q~R~)(V++q~R~)]
~)L/~)] },
(4.5)
[0~0 the unidexterous superspace actions in (4.4) or (4.5) are very different from that in (4.1). Of course, unidirectional D = 2 fields are not particular to supersymmetry. For instance, adding these to the usual bosonic string yields a string propagating in d dimensions with an internal group GR × GL:
S = - f d2cre-'½{~++X~-~_ X~+ip+a~__p+~-ip_a~++p a},
(4.6)
where 0~
f d2a d2~ E - ' i ½ { ( 7 +X~-)(V _X~) + [(7 + ~R a)( 7 - ~R a) +A_ ++ a6(7 + ~ R a ) ( v ++ ~)R/5)]
"~- [(V + @L 6t)(V _ (~L ~) "~A+ - - 6t]~(V _ (~L &)(V __ ~)L/~)] },
(4.7)
where O<~a<~9-No, l<~&&<~No and rk(GL) =rk(GR) =No, d = 10--NG. Since the origin of this model is tachyon free, we expect it to remain so after divesture. Because of its right-left symmetry, this model is more economical in its internal group sector. For instance, in the case of d = 4, the rank of each factor is expected to be six. The form of this action in "covariant" notation is given by 1
Sr~sR= -- ~ f d2a d2~"E - ' -~- [(V Ot~)L a ) ( V
with
{(V~X~-)(V,X~) + [(V°t ~ R ~)(V¢~ ~ R d ) -FARO~aa6(P+)aB(VBglJR&)(Vat~JR 6)]
(~L a ) +ALC~aa~(P_)afl(V,O ~ L & ) ( V a~L/~)] },
(4.8)
(Y")"PAapaa6= (Ta)"aAeaaa~= 0 and P+ and P are chiral projection operators.
5. (1,0) superspaceand the bosonic string. In closing this work, we make an amusing observation in this brief section. Even (0,0) supergravity can be described by unidexterous superspace. In four dimensions [ 16] it has been shown that non-supersymmetric theories can also be described by superfields if one is willing to introduce 40
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spurion superfields. With this in mind it should be possible to also describe the usual bosonic string in unidexterous superspace. To accomplish this we introduce a minus spinor superfield r/_ along with Lagrange multipliers Y+ +, Y - - , and r/_ ~-. The action which describes the bosonic string takes the form Sb.... ~c=
d 2 a d ( - E - ~ i ½ [ ( V + X -~) ( V
__Xa)+in_r/_
av + X_~+ Y++ (V + + r/_) + Y - - (V __ ~/_)],
(5.1)
where r/_ is a spurion superfield r / . = i(+. Reducing to components is straightforward and yields
if
S = - ~ d2ae-'[~++(e)X~-~__(e)X~_+2~__ +~+~-~++(e)X~_+ifl_~-~+~_+ig/+~-~__(e)g/+a +2+ --~,
+ + 2 + ++g/++ + + i 2 - - ~ , _ _
+~,++ +].
(5.2)
Notice that the unwanted spinors are set equal to zero by Lagrange multipliers. In D = 4 , N = 1 superspace it has been shown that consistent supergraph calculations can be carried out in the presence of spurions. It might be an interesting exercise to repeat such constructions here to derive the critical dimension o f the bosonic string by use o f (1,0) supergraphity techniques [ 5,6]. This would provide a derivation o f the extension o f the critical dimension formula d = 2 4 - ' n 6 to the case of m = - 1 [ for a theory with (0,0) local supersymmetry]. We should mention that a minus spinor ~/_ was also seen in ref. [ 1 ] to be necessary for the introduction o f a cosmological term in unidexterous superspace. One other intriguing possibility is to try to interpret the spurion superfield as being the v a c u u m value of a dynamical minus spinor. This might lead to a description where a compactified bosonic string model would arise as a spontaneously broken phase o f a supersymmetric model. If it is possible to carry out this scenario, then one could envision that a//string models might be spontaneously broken versions of the maximally supersymmetry (4,4) model.
Note added. After the completion o f this paper, W. Siegel brought the work of ref. [ 17 ] to our attention. There the possibility of four-dimensional superstrings is discussed. These constructions appear to correspond to the action in (4.4) for d = 4. In particular, the forty-four right-moving internal spinors are contained in the minus spinor superfields. The eighteen left-moving spinors arise by "fermionizing" the scalar components o f the lefton superfields. Thus, six lefton multiplets are equivalent to the appearance o f eighteen left-moving component fermions. These observations lend even stronger support for the interpretation o f (4.4) and (4.7) as manifestly world-sheet supersymmetric formulations of compactified string theories! The actions (4.7) and (4.8) provide the manifestly world-sheet supersymmetric formulation of the four-dimensional type-Ill superstrings [ 18,19 ]. In ref. [ 19] it is shown that the gauge groups o f these models are bounded in rank by 18. We wish to thank the referee for bringing this to our attention. We wish to thank T. Hfibsch, W. Siegel, and A.E. van de Ven for interesting discussions.
References [ 1] R. Brooks, F. Muhammad and S.J. Gates Jr., Nucl. Phys. B 268 (1986) 599. [2] M. Sakamoto, Phys. Lett. B 151 (1985) 115. [3] P. Ramond, Phys. Rev. D 3 (1971) 2415; A. Neveu and J. Schwarz, Nucl. Phys. B 31 (1971 ) 86. [4] D.J. Gross, J.A. Harvey, E. Martinec and R. Rohm, Phys. Rev. Left. 54 (1985) 502; Nucl. Phys. B 256 (1985) 253; (1986) 75. [5] M.T. Grisaru, L. Mezincescu and P.K. Townsend, Phys. Lett. B 179 (1986) 247. [ 6 ] S.J. Gates Jr., M.T. Grisaru, L. Mezincescu and P.K. Townsend, University of Texas at Austin preprint PPN UTTG-20-86. [ 7 ] M. AdemoUo,L. Brink, A. D'Adda, R. D'Auria, E. Napolitano, S. Sciuto, E. Del Giudice, P. Di Vecchia, S. Ferrara, F. Gliozzi, R. Musto, R. Pettorino and J. Schwarz, Nucl. Phys. B 111 (1976) 77; ,M. Ademollo, L. Brink, A. D'Adda, R. D'Auria, E. Napolitano, S. Sciuto, E. Del Giudice, P. Di Vecchia, S. Ferrara, F. Gliozzi, R. Musto and R. Pettorino, Nucl. Phys. B 114 (1976) 297; Phys. Left. B 62 (1976) 105. 41
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[8] w. Siegel, Nucl. Phys. B 238 (1984) 307. [9] L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. B 65 (1976) 471; S. Deser and B. Zumino, Phys. Lett. B 65 (1976) 369. [ 10] S.J. Gates Jr., A. Karlhede, U. Lindstr6m and M. Ro~ek, Class. Quant. Gray. 1 (1984) 277; Nucl. Phys. B 243 (1984) 221; J.M.F. Labastida, M. Ro~ek, E. S~inchez-Velasco and P. Wills, Phys. Lett. B 151 (1985) 111; J.M.F. Labastida, E. S~nchez-Velasco and P. Wills, Nucl. Phys. B 256 (1985) 394. [ 11 ] M. Ro~zek, P. van Nieuwenhuizen and S.C. Zbang, Ann. Phys. (NY) 172 (1986) 348. [ 12] P. Nelson and G. Moore, Nucl. Phys. B 274 (1986) 509; M. Evans and B. Ovrut, Phys. Lett. B 171 (1986) 177. [ 13 ] M. Bernstein and J. Sonnenschein, Quantization ofchiral bosons, Weizmann Institute preprint PPN WIS-86/47/Sept-PH. [14.] K.S. Narain, Phys. Lett. B 169 (1986) 41; K.S. Narain, M.H. Sarmadi and E. Witten, Nucl. Phys. B 279 (1987) 369. [.15] M. Mueller and E. Witten, Phys. Lett. B 182 (1986) 28. [16] L. Girardello and M.T. Grisaru, Nucl. Phys. B 194 (1982) 65. [ 17] H. Kawai, D.C. Lewellen and S.H.H. Tye, Phys. Rev. Lett. 57 (1986) 1832; Cornell preprint CLNS 68/732 (1986); I..Antoniadis, C. Bachas, C. Kounnas and P. Windey, Phys. Lett. B 171 (1986) 51; W. Lerche, D. L/ist and A.N. Schellekens, Nucl. Phys. B 287 (1987) 477. [ 18 ] R. Bluhm, L. Dolan and P. Goddard, Rockerfeller Preprint RU/B/187 ( December 1986); H. Kawai, D.C. Lewellen and S.H.H. Tye, Cornell Preprint CLNS 87/760 (January 1987). [ 19 ] L.J. Dixon, V. Kaplunovsky and C. Vafa, SLAC Preprint SLAC-PUB-4282 (March 1987 ).
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UNIDEXTEROUS SUPERSPACE: THE FLAX OF (SUPER)STRINGS S. James GATES Jr. l Department of Physics and Astronomy, University of Maryland at CollegePark, CollegePark, MD 20742, USA
Roger BROOKS and Fuad M U H A M M A D 2 Centerfor Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Received 18 February 1987
We give the superspace formulation of the most general superconformalunidexterous theory. A proposal for the world-sheet superspace description of compactified strings is presented.
I. Introduction. In a previous paper [ 1 ], we gave the first and, to date, most complete discussion of the local superspace formulation of unidexterous (or one-handed) supersymmetrical theories. The global version of these N = 1/2 theories were first considered by Sakamoto [2]. Unidexterous superspace is the fundamental geometrical quantity in the construction of manifest world-sheet supersymmetry in NSR superstrings [ 3 ] and the heterotic string [4]. This superspace provides the basic building blocks (or flax) for the construction of all superstrings which involve world-sheet supersymmetry. Thus, a clear and concise way to probe for the existence of heretofore unknown strings is provided by a complete analysis of all superconformal theories in unidexterous superspace. It has been shown [ 5,6] that ordinary field theoretical techniques, combined with unidexterous superfields, can be used to deduce the existence of candidate string theories. In particular, the absence of scale and Lorentz anomalies enforces the condition that the effective action in unidexterous superspace possesses all of the symmetries which are present in the classical action. In the present paper, we will discuss how all of the known string theories which possess world-sheet supersymmetry can be formulated in unidexterous superspace. This is obvious for the heterotic string but, in principle, for the usual NSR string such a formulation should also exist. In fact, the matter gravitino linearized version of this theory was presented in ref. [ 6 ]. We will explicitly carry out the crucial steps for obtaining the complete theory and show that the previous result is exact. We also show how unidirectional supermultiplets provide a two-dimensional world-sheet description of string compactification. In addition, general unidexterous superconformal theories which include left- and right-moving multiplets will be discussed. 2. Unidexterous superconformal theories. In order to describe the unidexterous superspace formulation of all superconformal theories it is convenient to have a "spinning wheel", i.e., an action from which any superconformal theory arises as a special case. We should point out that such an action may have more accidental or non-manifest symmetries beyond those that are obvious. In particular, all of the spinning strings [ 7 ] and the heterotic string [4] must appear as special cases. In our first work [ 1 ], we showed that unidexterous superspace is essentially the same as all other superspaces. There are large numbers of multiplets which include scalar
t Researchsupported in part by NSF grant #PHY86-19077. 2 This work is supported in part by the US Department of Energy (DOE) under #DE-AC02-76ER03069. 0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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multiplets, vector multiplets, etc. going up to a supergravity multiplet. Not all of these multiplets are superconformal multiplets. O f the multiplets reviewed in ref. [ 1 ] only the scalar multiplet, X -~, and the minus spinor, q /, are superconformal multiplets. However, there are two additional superconformal multiplets which we overlooked. These are the N = 1/2 versions of unidirectional of chiral bosons. The right-moving boson or scalar "righton" multiplet has been given in the global context in ref. [ 5 ]. Minimal covariantization with respect to local supersymmetry yields an action for NR righton multiplets coupled to supergravity: S R = i ~ 1 f d2 o" d (
g -j (V + (~R ~V _ _ (ibR a q-A _ _ + + a/;V + (ibR aV ++ (~R /;), A _ _ + + t&~l =0,
where c/=l, ..., NR. After defining component fields by q}Ra--=~Ral, p + a - - = V + ~ R a l , iV +A _ + + aa I, 2_ _ + + ab-=A _ _ + + ea I and using the derrsity projection formula
f
d~- E-'~_
=e
(2.1) R--+ab--
' ( V + - i v / + + +)£e_ I,
(2.2)
we find the component result is given by SR------
d
+0R
--~Ra+iP+
+2~u__ + f f + a ~ + + ~ R
__p+
-t-2__ ++ gi/;(~+ + OR d ~ + + ~R /;"}-ip+ & ~ + + p +
-- _
a f i f f + a ~ + + ~ R b"
a+2V/++ +p+ gt~+ + ~R~)] ,
(2.3)
where the component supergravity quantities are defined by ~ + + ( e , ~,)=e_++ ' " 0 , , + a ) + + ( e , v/)d/,
ro++(e, v / ) = r o + + ( e ) = c + + . _ _ - - , [ea, eb] =ca.bCec.
o)__(e, v/)=c++ __ ++ +i2v/++ + V/ _ +,
(2.4)
In a similar manner, a unidexterous left-moving or scalar "lefton" multiplet has also been found in global
N = 1/2 superspace [ 8 ]. Once again minimal covariantization immediately yields the locally supersymmetric action of NL lefton multiplets coupled to supergravity: SL=i~
If
__ d 2 o ' d ~ - E _ I [V+~LaV
+ - - a9V__(JSLaV __ (J}Laq-A
(~Lfl], A+ - - [ a d l = 0 ,
(2.5)
where & = 1, ..., NL. Likewise, the component reduction of this action is obtained as S L = -- f d2o " e - I ½[ ~ + + ~ L a ~ _ _ --2+ + -- -- a ] ~ ( ~ / _ 0 L a ~
q~La + i f l +
a~__fl+ a + 2 V / _ _
+fl+ a ~ + + O L a
_ 0L/~ +2V/_ _ +fl+ a ~ _ _ OLd)
+i2+--ad(2~__p+a~__#ed+v/__+fl+a~ +iv/__ + cG/++q~La~ _ ~ L f l - - i v / + + + ~
fl+a
_~La~__~Lflq-iv/__
+V/++ +fl+ a~__~L]~)] ,
(2.6)
Here the component fields of the lefton multiplet are defined by OLa--=q~Lal, fl+a=V+gYLal, 2+ ad=A+--a~l,2++--a~=-iV+d+--a~l. The classical scale invariance of these actions is seen upon making the variations 8sV+=½SV++(V+S)J¢,
8sV++=SV++-i(V+S)V++(V++S)d/,
5sA__ ++at;=O, 8sA+ --ag=-½SA+ --a~. 36
8sV_
=SV__-(V__S)J/, (2.7)
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Now the most general unidexterous superspace action possessing local scale invariance takes the form Ssw = r d2 a d ( - E - l [ ~SM + L#MSM+ 5°r~ + L#L], d
~SM =i½~ + X - ~ __X~, L#R=i½[V
~MSM = -- ½r/_ :~ + q_ :,
+q~Ra~__ORe+A__
+ + abV + O R e ~
++O~6],
~L =i½ [V + q~L'~V - - ~ L dt + A + - - aBV __ q~L'~ __ q~Ld].
(2.8)
In (2.8) the covariant derivative, ~ a , is defined by the expression
V A =fA BV B +fa.ll +gA" fg-- ~a + gA' f#,
(2.9)
where VA is the pure (1,0) supergravity covariant derivative [ 1,5,6]. The quantitiesfA B, fA, and gA are functions which may depend on simple unidexterous matter gravitino multiplets or vector multiplets. We note that superconformal invariance implies that this is the only way in which these multiplets can appear in Ssw. The quantities f¢ denote extra generators that may be present such as the generators of the extra non-manifest supersymmetries. The action in (2.8) is capable of describing all tachyon-free strings as special cases.
3. The complete (1,0) superspace geometry of the NSR string. It is well known that the N S R superstring can be described by the coupling of D = 2 , N = 1 (1,1) supergravity to scalar multiplets [9]. But from the point of view of (1,0) superspace this theory is just an extended supergravity theory. As supergravity theories can be expressed in terms of components, any extended supergravity theory can be expressed in terms of the simplest superfields. For D = 2, unidexterous superfields are these simplest superfields. In D = 4 , previous work has described how to project a larger superspace onto a smaller superspace [ 10]. In the following, these techniques will be applied to D = 2 superspaces. We begin with the (1,1) supergravity theory in (1,1) superspace. Using light-cone coordinates as opposed to "covariant" coordinates, the constraints on the supergravity theory take the forms [ 11 ] [V_,V_}=i2V__,
[V_,V+}=-i2Rd/,
[V+,V+}=i2V++,
[V+,V++}=O,
[v + +, v _ _ } = - i ( v + R ) V _ - i ( V R ) V
[V_,V++}=-RV++2(V+R).tl,
[V_,V__}=0,
[V+,V__}=RV_+2(V_R).It, + - 2 ( R 2 +iV +V _ R),//.
(3.1)
Application of the techniques of ref. [ 10] to the reduction of (1,1) to (1,0) superspace yields [ ~ + , 9 + } = i 2 ~ + + + i 2 ~ + - g-'+ - 9 _ _ +i4~v+ - S J/, [ ~+,
9 + + } = g"+ - S ~ +
[ 9 + , 9 _ _ }= - i 2 ~ + [~++, ~__ }=-(T__
- i2 g" + - ~v+ + - ~ _ _ - i 2 (
~v+ + - S + ~ +
- 7-'+ + _ _ - ) J / ,
- kg_ _ - ~ _ _ - i 2 ( hu_ _ - S+2~ + )~#, - S+27+) ~ + - i 2 g - ' + + - ~ v _ - 9 _ _ - 2 ( S 2 - i g - '
_ - ~v+ + __ - + V + 27+)J/. (3.2)
For the field strength of the matter gravitino multiplet ~'/A,B-- we find the constraints, g" +,+ - = 0, ~u +, + + - = 0, g J + , _ _ - =S. It is also useful to know that - i V +R I = g"+ + , _ _ - . The new (1,0) supergravity derivative ~A is related to the " p u r e " supergravity derivative VA and the sinistral matter gravitino ~UA- by the equations 37
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~+=V++i2~+-~
_-,///,
~++=V+++T+-~t_
30 July 1987
- V + ] + 2 T / + -(V
_]~+-)J//,
~
_=V
, (3.3)
[The details of obtaining the (1,0) results from (3.1) will be given in a longer version of this paper.] All of the quantities which appear above, in (3.2), are (1,0) superfelds. These expressions allow us to calculate the (1,1) superdeterminant ~ in terms of (1,0) superfields E and ~A and lead to the result ~ = E ( 1 +i~
T
-).
(3.4)
A locally supersymmetric (1,0) theory is obtained by noting that S=i
f d2a f d~-
f d~+~
~5a
= f d2o" f d~ E-I(V_LP-iT__
L~)I.
(3.5)
On the f r s t line of (3.5) all the superfields are (1,1) quantities. But on the second line we have integrated out the minus superspace coordinate so that all quantities are (1,0) superfields! The (1,1) form of the NSR lagrangian is very well known to be given by 5eNSR=--i½ (V +X-U)(V- X_a). Substitution of this into the expression on the first line of (3.5) implies SNSR= J
dZa d~- E-'
i½[(V
+Xa-)(V __X_o)+in_ -~v + r/__u+ 2 ~ +
-r/_ -a(V X_~)].
(3.6)
This is precisely the result that was given in ref. [ 6 ] where it was obtained by using a superfield Noether technique to lowest order. Our derivation above shows that the lowest-order Noether term yields the complete theory. At no point in our derivation have we neglected any terms. The procedure that we have outlined above may be applied to any type (p,q) superspace theory to derive a (1,0) form of the theory.
4. Heterotic versus heterodexterous strings. In the last section we saw how the NSR superstring can be described in terms of (1,0) superfields. Of course, the heterotic string [4] also has a (1,0) superfield form [1,12], Su = f d2a d~- E - ' i½ [(V + X-~)(V _ _ Xa) +i~/_ ¢V + q_ z].
(4.1)
The differences between (3.6) and (4.1) are clear. In the NSR theory, a (1,0) sinistral matter gravitino multiplet is present and the minus spinor multiplets are equal in number to that of the scalar multiplets. Also the minus spinor multiplets transform in the same way as the scalar multiplets under spacetime Lorentz transformations. Yet both of these theories are known to be consistent strings. From the point of view of two-dimensional field theory this is easy to understand. As classical actions SNS~ and SH are scale and Lorentz invariant. In previous work [ 5,6 ], it was shown that upon quantization every (1,0) multiplet in the classical action contributes a potentially anomalous part to the effective action. This contribution takes the form
r ....
If962r
d2°"d~-
[VR[D+H+
]
( ~[[] n+__)_illL[D+n((9
)4
_
++] ( ( 0 ~-)- 3 H _ _ ++ ) ] , (4.2)
where VR and Ve are numbers which we shall refer to as the right and left degrees of freedom of the supermultiplet. On the basis of the previous work, we find the assignments given in table 1 for all of the multiplets which appear in the above actions. The origin of this table is simple to understand. If we began with nonsupersymmetric D = 2 ordinary gravity [(0,0) supergravity] then we know that the anomaly-freedom of the bosonic string requires 26 scalar fields. Thus VR= VL= -- 26 for the graviton. Next the calculations of refs. [ 5,6 ] 38
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Table 1 Superfield
UR
UL
x
1 ½
o
~R
1
0
11 -26
0 -15
~A E4~
give us the values of VR and //L for unidexterous supergravity. But the only way that (1,0) supergravity differs from (0,0) supergravity is the presence of a component dextral gravitino. This implies that a component dextral gravitino must have UR= 0 and UL= 11 (the difference between - 15 and - 2 6 ) . In the (1,0) sinistral gravitino multiplets [ 1 ] there appear the corresponding component gravitino fields. Thus, the sinistral gravitino multiplet must have UR= 11 and UL= 0 (the opposite of a component dextral gravitino). The internal degrees of the heterotic string can be described by sixteen unidirectional bosons or thirty-two unidirectional fermions. This implies that UR and UL for )7_ are one half the corresponding values for ~R- This observation about the relation between the degree of a unidirectional fermion versus that of a unidirectional boson also explains why UL= ~ for a scalar multiplet. At the component level, a massless scalar multiplet contains one right-moving boson, one left-moving boson, and one left-moving fermion. Summing over all of the fields in either the NSR or heterotic string actions shows that the total contribution to the anomalous part of the effective action is zero. Thus, the classical symmetries survive in the quantum theory. But comparing either actions in (3.5) and (4.1) with the most general unidexterous superconformal theory in (2.8), we note that no lefton multiplets are present! Classically, the lefton part of the action is also Lorentz and scale invariant. How do leftons change the considerations above? To rigorously answer this requires the quantization of the lefton multiplet as has been carried out recently for the non-supersymmetric case [ 13 ]. We are presently undertaking this step. But there is an intuitive way to argue the outcome of this. Note that F . . . . just counts the number of left degrees and right degrees in the classical action. It is known that the classical equations of motion for righton multiplets and lefton multiplets [ 5,6,8 ] take the respective forms D + • R= 0 and O__ ~L = 0. The scalar righton equation further implies O+ + ~ a ~ 0. Thus the spinor component of the scalar righton multiplet vanishes and the bosonic component is restricted to move to the right. Hence, we find /)R = l , /)L = 0. On the other hand, the equation of motion for the scalar lefton multiplet only restricts its component fields to move to the left. It then follows that the degrees of the scalar lefton multiplet are PR= 0 and //L-~-~" This assignment is also plausible when we recall that a component scalar field is equivalent to a scalar righton and a scalar lefton. So it is reasonable to expect a scalar multiplet to be equivalent to a scalar righton multiplet and a scalar lefton multiplet. Considering the action in (2.8) with only d scalar multiplets, ArE minus spinor multiplets, NR scalar righton multiplets, and ArE scalar lefton multiplets implies that when the conditions
VR=d+½NF+NR--26=O,
~ PL-----~d+~NL--15----0
(4.3)
are satisfied, the total contribution to F . . . . vanishes. This is highly suggestive of a "dimensional divesture" of the heterotic string theory to dimensions d = 1 0 - NL with an internal gauge group of the form GR × GL where the ranks of G R and GL satisfy the condition r k ( G R ) - r k ( G L ) - - 1 6 . Thus, upon quantization we expect the action S H ' = f d2trd( - E-~i½{(V+X~-)(V__X~_)+irl_ZV+q_ I "~ [(V + I~L a ) ( V _ _ ~ L a) -~-A+ - - a,/~(V _ _ ~ L a ) ( V _ _ ~L/~)] },
(4.4)
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[0~
fdZ~d( -
SH'=
E-'i½{(V+Xe)(V
_X_~)+[(V+q~Ra)(V_
+[(v+ClJLa)(V__tlJLa)-FA+ - c~(V
(~L &)(V
~R~)+A
++~(v+q~R~)(V++q~R~)]
~)L/~)] },
(4.5)
[0~
S = - f d2cre-'½{~++X~-~_ X~+ip+a~__p+~-ip_a~++p a},
(4.6)
where 0~
f d2a d2~ E - ' i ½ { ( 7 +X~-)(V _X~) + [(7 + ~R a)( 7 - ~R a) +A_ ++ a6(7 + ~ R a ) ( v ++ ~)R/5)]
"~- [(V + @L 6t)(V _ (~L ~) "~A+ - - 6t]~(V _ (~L &)(V __ ~)L/~)] },
(4.7)
where O<~a<~9-No, l<~&&<~No and rk(GL) =rk(GR) =No, d = 10--NG. Since the origin of this model is tachyon free, we expect it to remain so after divesture. Because of its right-left symmetry, this model is more economical in its internal group sector. For instance, in the case of d = 4, the rank of each factor is expected to be six. The form of this action in "covariant" notation is given by 1
Sr~sR= -- ~ f d2a d2~"E - ' -~- [(V Ot~)L a ) ( V
with
{(V~X~-)(V,X~) + [(V°t ~ R ~)(V¢~ ~ R d ) -FARO~aa6(P+)aB(VBglJR&)(Vat~JR 6)]
(~L a ) +ALC~aa~(P_)afl(V,O ~ L & ) ( V a~L/~)] },
(4.8)
(Y")"PAapaa6= (Ta)"aAeaaa~= 0 and P+ and P are chiral projection operators.
5. (1,0) superspaceand the bosonic string. In closing this work, we make an amusing observation in this brief section. Even (0,0) supergravity can be described by unidexterous superspace. In four dimensions [ 16] it has been shown that non-supersymmetric theories can also be described by superfields if one is willing to introduce 40
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spurion superfields. With this in mind it should be possible to also describe the usual bosonic string in unidexterous superspace. To accomplish this we introduce a minus spinor superfield r/_ along with Lagrange multipliers Y+ +, Y - - , and r/_ ~-. The action which describes the bosonic string takes the form Sb.... ~c=
d 2 a d ( - E - ~ i ½ [ ( V + X -~) ( V
__Xa)+in_r/_
av + X_~+ Y++ (V + + r/_) + Y - - (V __ ~/_)],
(5.1)
where r/_ is a spurion superfield r / . = i(+. Reducing to components is straightforward and yields
if
S = - ~ d2ae-'[~++(e)X~-~__(e)X~_+2~__ +~+~-~++(e)X~_+ifl_~-~+~_+ig/+~-~__(e)g/+a +2+ --~,
+ + 2 + ++g/++ + + i 2 - - ~ , _ _
+~,++ +].
(5.2)
Notice that the unwanted spinors are set equal to zero by Lagrange multipliers. In D = 4 , N = 1 superspace it has been shown that consistent supergraph calculations can be carried out in the presence of spurions. It might be an interesting exercise to repeat such constructions here to derive the critical dimension o f the bosonic string by use o f (1,0) supergraphity techniques [ 5,6]. This would provide a derivation o f the extension o f the critical dimension formula d = 2 4 - ' n 6 to the case of m = - 1 [ for a theory with (0,0) local supersymmetry]. We should mention that a minus spinor ~/_ was also seen in ref. [ 1 ] to be necessary for the introduction o f a cosmological term in unidexterous superspace. One other intriguing possibility is to try to interpret the spurion superfield as being the v a c u u m value of a dynamical minus spinor. This might lead to a description where a compactified bosonic string model would arise as a spontaneously broken phase o f a supersymmetric model. If it is possible to carry out this scenario, then one could envision that a//string models might be spontaneously broken versions of the maximally supersymmetry (4,4) model.
Note added. After the completion o f this paper, W. Siegel brought the work of ref. [ 17 ] to our attention. There the possibility of four-dimensional superstrings is discussed. These constructions appear to correspond to the action in (4.4) for d = 4. In particular, the forty-four right-moving internal spinors are contained in the minus spinor superfields. The eighteen left-moving spinors arise by "fermionizing" the scalar components o f the lefton superfields. Thus, six lefton multiplets are equivalent to the appearance o f eighteen left-moving component fermions. These observations lend even stronger support for the interpretation o f (4.4) and (4.7) as manifestly world-sheet supersymmetric formulations of compactified string theories! The actions (4.7) and (4.8) provide the manifestly world-sheet supersymmetric formulation of the four-dimensional type-Ill superstrings [ 18,19 ]. In ref. [ 19] it is shown that the gauge groups o f these models are bounded in rank by 18. We wish to thank the referee for bringing this to our attention. We wish to thank T. Hfibsch, W. Siegel, and A.E. van de Ven for interesting discussions.
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