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The zero tension limit of the superstring
Volume 253, number 3,4
PHYSICS LETTERS B
10 January 1991
The zero tension limit of the superstring U. L i n d s t r 6 m , B. Sundborg Institute of Theoretical Physics, University of Stockholm, Vanadisvgigen 9, S-113 46 Stockholm, Sweden
and G. T h e o d o r i d i s NORDITA, Blegdamsvej 17, Copenhagen O, Denmark Received 19 October 1990
We construct the limit T--,0 of the superstring starting from a reformulation of the same limit for the bosonic string. The resulting action is manifestly 2D-diffeomorphism invariant as well as Siegel-invariant. It is also invariant under superconformal transformations in the ambient superspace. The model describes a collection of superparticles moving on a supernull surface. We propose it as a high energy limit of the superstring.
The characteristic scale of string theory is given by the string tension T. At energies of the order of x / ~ or higher, string physics truly distinguishes itself from point particle physics. Consequently, various high energy limits [ 1-10] have been studied in order to gain insight into the elusive physical basis of the theory. In particular there have been speculations about an unbroken scale invariant phase. In such a framework the string tension (and thereby the masses of excited states) should be generated in some kind of spontaneous or dynamical symmetry breakdown. In any case, we expect some simplification to occur when a limit is taken, and indeed Gross [ 5 ] has found an infinite symmetry at high energy amplitudes. One may alternatively view a high energy limit as a zero tension limit, since only the energy measured in string units, E / x / ~ , is relevant. There are numerous ways of taking the T ~ 0 limit in the full theory, each one corresponding to the physical conditions in an equivalent high energy limit. When the external momenta are kept fixed (assuming massless external states), one gets the limit of refs. [4-6 ], though it is not clear to what extent interactions survive in this limit [amplitudes behave as e x p ( - c / T ) ]. A natural, closely related, choice is to take T--,0 in the string action in analogy to the m ~ 0 limit for the point par-
ticle. This can be done once the action is written in a suitable form [ 11,12 ]. One may then ask if the resuiting "null string" can be quantized [ 13,14 ] and if interactions can be introduced. At a fixed level of the quantum string the length approaches infinity as T - t / a , so that the "fundamental length" of string theory becomes infinite. Such null strings would then be reasonable candidates for the excitations in a scale invarianl phase. We expect the supersymmetric null string to make more sense than the bosonic one, just as in the tensionful case. A null superstring has been proposed previously in ref. [ 15 ], where the limit T--.0 is taken by first reformulating the Green-Schwarz superstring in terms of a phase space lagrangian. However, their action is only reparametrization invariant on shell. The same is true for the Siegel invariance [ 16,17 ], which in the ordinary superstring and superparticle allows unwanted (i.e. half the) spin degrees of freedom to be gauged away. A constraint on the parameters of the Siegel transformations is needed for the physical fermionic degrees of freedom to remain. Such a constraint seems to be missing in ref. [15]. In this paper we construct a null superstring action (in configuration space) without the drawbacks
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
3 19
Volume 253, number 3,4
PHYSICS LETTERS B
mentioned above. We first reexamine the bosonic null string in a geometric formulation, which is then easily taken over to the space-time supersymmetric case. Our discussion is classical in that we do not address the question of a critical dimension for our model. There is a standard way for putting an action of the type
S=m f L
(1)
into a form where it is possible to study the limit m-,0; introduce an auxiliary field (density) V and write the action in the (classically) equivalent form
S'=½ f V L 2 + V - l m 2.
(2)
10 January 1991
mation 8V= - 2 VO,A"L- ~will become meaningless on shell (where L = 0). Since local supersymmetries in general transform the lagrangian density into a total divergence, and we want to study the superstring whose Siegel symmetry is of this type, it is gratifying that there exists a different approach to the problem of taking the parameter to zero in actions of the form ( 1 ), which we now discuss. Let us first notice how we could have arrived at the "ein-bein" action (2) for the particle [ 18 ]. We start from the action (3) and derive the canonical momenta m2m Pm- N ~ 2 •
(9)
From this follows the constraint
The limit m--, 0 can then be straightforwardly reached. In this way the action for a massive particle
p2+m2=0,
Sm=-m~dr~
and the hamiltonian
2
(3)
H = 2 ( p 2 + m 2) ,
can be related to that o f a massless one So = - ½ f dr V~"2 ,
(4)
T~ d2xf~,
(5)
where
r~=a~x'"apx,.,
(6)
(7)
In both these cases the reparametrization invariance is preserved and the ambient space-time symmetry enlarged from Poincar6 to conformal symmetry. The approach just described has drawbacks, however. One obvious problem is to preserve a symmetry which is a symmetry of the action but under which the Lagrange density transforms into a total divergence. I f S L = 0 , A " then one has 8 S ' = ½f 2VLO,A"+ 8VL 2 .
(8)
In general the situation will be that the action is not invariant with 6V=0, and that the naive transfor320
where 2 is a Lagrange multiplier. The phase space lagrangian is
( 12 )
Integrating out the momenta, we pass to the configuration space lagrangian
L(x, 2) = ½(22/2)~ - 2)~m2 )
will give rise to that of the tensionless "null" string [ll,12l: S o = - ½ ~ d2~ Vy.
(11)
L = p 2 - 2 ( p 2 + m 2) .
and the bosonic string action
S-r=-T I d 2 ~ ~ - - -
(10)
.
( 13 )
We see that, putting V - ~= - 22, this agees with the action (2) with the particle lagrangian from (3). For the string this approach becomes more interesting. The momenta derived from the action ( 5 ) are T Pm-~'~~ ( X X ' )
2
['3('2~(m--(')(Y')Xtm]'
(14)
from which follow the constraints
p2+T2x'2=O,
px' = 0 .
(15)
This leads to a phase space lagrangian which contains two Lagrange multipliers 2 and p: L=p2-~,(p2
+ T i x '2 )
--ppx' .
(16)
Integrating out the momenta we arrive at the following configuration space lagrangian: 1
L(x, 2) = ~ [ 2 2 - 2 p 2 x ' + (p2-4T222)x'2] . (17)
Volume 253, number 3,4
PHYSICS LETTERS B
With the identification
.~ ( - 1 g
=~,
p
"~
P 422T2_p2)
(18)
the action (17) becomes the Weyl-invariant action: Sw = - ½T f d2~x/~gg'*ay,~e,
(19)
where g_--det g,a and 2 and p have been assigned the correct transformations for g-a to transform as a tensor under world sheet diffeomorphisms. Note that many first order actions, (actions involving an auxiliary metric), can be written down for the bosonic string [ 19 ]. The Weyl-invariant one, ( 19 ), is singled out by the procedure described above, i.e., by the quantization procedure, since what we have discussed is really setting up the path integral with the phase space as the starting point. Clearly, taking the limit T-+0 in this approach, e.g. in (16), gives something new; previously we had only one Lagrange multiplier field (see ( 7 ) ) , now we have two. It is in fact this extra field which allows us to overcome the difficulties related to the T ~ 0 limit for the superstring. Just as for the tensionful theory, the T ~ 0 limit of the action (16) allows for a geometrical interpretation of the Lagrange multipliers 2 and p. With the auxiliary vector density V" replacing 2 and p according to 1
2x/~ (I, - p ) - ~ V " ,
(20)
we obtain the following general form for the bosonic null string action: S,.= I d2~ V"V~Y'~P"
(21)
Clearly, V" can be taken to transform as a contravariant vector density (of weight - ~ ), and G " a = V " V p is then a tensor density with det G"P=0. The action (21 ) is the null string action that corresponds to the first order action (19) for the usual bosonic string. The V '~ field equations VPTt~ = 0
(22)
immediately show that ),,a has a null eigenvector, and we conclude that detT,p=0.
(23)
10 January 1991
which is the criterion for a null string. Instead of starting again from the tensionful theory for the superstring, we immediately generalize (21 ) to a globally supersymmetric theory. As usual, this is done simply by letting ~1 o, x m - - , H m ~ O , ~ x m - i 6 F " O . O ,
(24)
where 0(~) is a Majorana (Weyl) space-time spinor, world sheet scalar, and F m are Dirac matrices. Under global supersymmetry with (spinor) parameter e we have 6~0=e,
8~Xm=-itTFmS~o,
~8~H;7=0.
(25)
Hence, the action S v - f d2~ V"V~HmHp,,,
(26)
is globally supersymmetric if we take 5~V~= 0. Next we turn to the Siegel symmetry. This is a local fermionic symmetry of the Green-Schwarz superstring which halves the fermionic degrees of freedom in 0: 5,~O=iI~,x ~ ,
&~X'"=iOF'"6,,O,
= ~ 8 ~ I ~ = - 2iS~0P~O, 0 ,
(27)
with x" satisfying a certain constraint [17,20]. To have invariance in the Green-Schwarz action one adds a Wess-Zumino term to the part of the action which corresponds to our (26). We shall see that this will not be necessary in our case. In fact, if we demand x" = V"x ,
(28)
with x(~) a space-time spinor and world sheet scalar density of weight ½, we have for the variation of the action (26): 8 S v = 2 f d2~ V"7~/~(8~ V P - 2 V P V P ~ p t f x ) .
(29)
Thus, the action is Siegel-invariant if we put 8 V '~ = 2 V '~ VPOp6x,
(30)
which is in fact highly reminiscent of the transformation of the metric in the tensionful theory. Note that the V '~ field equations ~ We use conventions adopted to D= 10 and Majorana-Weyl spinors. In other dimensions where one uses Weylspinors the transformations must be modified, e.g. OF'~O,~O --* ½( Or~a.o-o.~rmo). 321
Volume 253, number 3,4 (i)
8V""
PHYSICS LETTERSB
V"HS'H~,,,=0,
~l~,V"l~pVa=O
(31) show that in the combination that enters the x variation, ~'x=/~,,V"tc, the operator ~ is nilpotent (on shell) and hence x only carries half the degrees of freedom o f a spinor, in complete analogy to the T4:0 case. Just as for the Green-Schwarz superstring there is also a local bosonic symmetry of the action (26):
10 January 1991
pect that the original super-Poincar6 group, the symmetry group of the Green-Schwarz superstring, gets replaced by the super-conformal group. This is indeed the case in the dimensions D = 2 - 6 [21 ] where the superconformal group has been shown to exist, and possibly also in D = 10, where it may exist [22]. In fact, the superconformal group transforms the super line element ds 2 = H mH i nd~"d~ p
( 36 )
This symmetry is required for closing the algebra of the ~c-transformations. In fact, we find
into g2(X, 0)ds 2, hence the invariance of the action (26) under this group is almost immediate. In D = 4 the superconformal transformations o f X m and 0 can be found in ref. [23] and they lead to the following transformations of V":
[8~2, 8~, ] = G ( x = - V~(O~OKt2 )1<,1)
8, v ~ - i [~S, V ~ ] = - ½ q O V ~ ,
8~ V " = 0 ,
8~0=2V"0~0 ,
8~X'"=iGF"8.~O :
(32)
+ 8~(2= -- 4i(~2 Va/~/~xl ) )
+ 8~(U = 4i (~72 w v P / ~
8 h V " - i [ b , n K ",, V , ] = - b X V
))
+ terms zc equations of motion.
8DV'~=-i[DA, V " ] = - D V
( 33 )
Here ¢" is the parameter of 2D diffeomorphisms. The bosonic null string is easily seen to consist of massless particles moving on a null surface• For the present model, described by the action (26), we have the equations of motion (31) and (ii) (iii)
8X m" OB(V'~VPHm)=o, 80:
V'~VP(O,~O)I~p+O,,(V"V~O~Ip)=O.
(34) In the diffeomorphism gauge V " = ( 1 , 0) we find from (31) and (34) (i)~(Ho)2=0,
HoH~ = 0 ,
(ii) ~ H ~ ' = 0 , (iii) ~G/~o = 0 ,
(35)
and we recognize the equation of motion for a massless superparticle at each a. Also, the motion is confined to the superspace equivalent of a null hypersurface. Masslessness is usually connected with the conformal group. The massless particle has the conformal group replacing the Poincar6 group as symmetry group in the ambient space-time. Likewise, for the tensionless bosonic string the same extension of the symmetry group occurs. It is therefore natural to ex322
~.
'~ ,
(37)
Here S, K'" and/I are the generators of S-supersymmerry, special conformal and dilation transformations, respectively. V" is inert under the rest of the superconformal group. We have derived a null superstring which is manifestly reparametrization and Siegel invariant (in contrast to the model in ref. [ 15 ] ). Our action is a straightforward generalization of the bosonic null string in a novel formulation. The "classical" restrictions on the space-time dimension where our action can be formulated are less severe than for the tensionfull superstring, unless we also demand superconformal invariance in the ambient superspace, or maybe that the theory should be equivalent (in a lightcone gauge) to a (sector of ) a spinning null string, in analogy to the T:/: 0 case. We conclude by some more comments on our action. For the usual Green-Schwarz action there are restrictions from the gamma matrix algebra on the dimensions D in which it is invariant, namely D = 3, 4, 6 and 10. To show invariance of our action under Siegel symmetry, all we need is the Majorana properties of the spinors. Hence invariance results in all dimensions which support Majorana (or Weyl) spinors. Finally, extensions to higher N (more supersymmetries) of our action is straightforward. We thank P. van Nieuwenhuizen, M. Ro6ek and
Volume 253, number 3,4
PHYSICS LETTERS B
W. Siegel for p r o v i d i n g useful i n f o r m a t i o n .
References [ 1 ] D+ Amati, M. Ciafaloni and G. Veneziano, Phys. tett. B 197 (1987) 81; Intern. J. Mod. Phys. A 3 (1988) 1615; Phys. Lett. B 216 (1989) 41. [2] l.J. Muzinich and M. Soldate, Phys. Rev. D 37 (1988) 359. [3] B. Sundborg, Nucl. Phys. B 306 (1988) 545. [4] D.J. Gross and P.E. Mende, Phys. Lett. B 197 (1987) 129; Nucl. Phys. B 303 (1988) 407. [5] D.J. Gross, Phys. Rev. Lett. 60 (1988) 1229. [6] D.J. Gross and J. Manes, Nucl. Phys. B 326 (1989) 73. [7] E. Alvarez, Phys. Rev. D 31 (1985) 418. [8] B. Sundborg, Nucl. Phys. B 254 (1985) 583. [9] M.J. Bowick and L.C.R. Wijewardhana, Phys. Rev. Lett. 54 (1985) 2485. [ 10] J.J. Atick and E. Witten, Nucl. Phys. B 310 (1988) 291.
10 January 1991
[ 11 ] A. Schild, Phys. Rev. D 16 (1977) 1722. [12]A. Karlhede and U. Lindstrfm, Class. Quantum Grav. 3 (1986) L73. [ 13 ] F. Lizzi, B. Rai, G. Sparano and A. Srivasta, Phys. Lett. B 182 (1986) 326. [ 14 ] J. Gamboa, C. Ramirez and M. Ruiz-Altaba, Phys..Lett. B 225 (1989) 335. [ 15 ] A. Barcelos-Neto and M. Ruiz-Altaba, Phys. Lett. B 228 (1989) 193. [16] W. Siegel, Phys. Lett. B 128 (1983) 397. [ 17 ] M.J. Green and J.H. Schwarz, Phys. Lett. B 136 (1984) 367. [ 18 ] R. Marnelius, Acta Phys. Pol. B 13 ( 1982 ) 669. [ 19] U. Lindstr6m, Intern. J. Mod. Phys. 3 (1988) 2401. [20] M.J. Green and J.H. Schwarz, Nucl. Phys. B 243 (1984) 285. [21 ] B.S. DeWitt and P. van Nieuwenhuizen, J. Math. Phys. 23 (1982) 1953. [22]W. Siegel, Introduction to string field theory (World Scientific, Singapore, 1988). [23]J. Gates Jr, M.T. Grisaru, M. Ro~ek and W. Siegel, Superspace (Benjamin/Cummings, Menlo Park, CA, 1983 ).
323
PHYSICS LETTERS B
10 January 1991
The zero tension limit of the superstring U. L i n d s t r 6 m , B. Sundborg Institute of Theoretical Physics, University of Stockholm, Vanadisvgigen 9, S-113 46 Stockholm, Sweden
and G. T h e o d o r i d i s NORDITA, Blegdamsvej 17, Copenhagen O, Denmark Received 19 October 1990
We construct the limit T--,0 of the superstring starting from a reformulation of the same limit for the bosonic string. The resulting action is manifestly 2D-diffeomorphism invariant as well as Siegel-invariant. It is also invariant under superconformal transformations in the ambient superspace. The model describes a collection of superparticles moving on a supernull surface. We propose it as a high energy limit of the superstring.
The characteristic scale of string theory is given by the string tension T. At energies of the order of x / ~ or higher, string physics truly distinguishes itself from point particle physics. Consequently, various high energy limits [ 1-10] have been studied in order to gain insight into the elusive physical basis of the theory. In particular there have been speculations about an unbroken scale invariant phase. In such a framework the string tension (and thereby the masses of excited states) should be generated in some kind of spontaneous or dynamical symmetry breakdown. In any case, we expect some simplification to occur when a limit is taken, and indeed Gross [ 5 ] has found an infinite symmetry at high energy amplitudes. One may alternatively view a high energy limit as a zero tension limit, since only the energy measured in string units, E / x / ~ , is relevant. There are numerous ways of taking the T ~ 0 limit in the full theory, each one corresponding to the physical conditions in an equivalent high energy limit. When the external momenta are kept fixed (assuming massless external states), one gets the limit of refs. [4-6 ], though it is not clear to what extent interactions survive in this limit [amplitudes behave as e x p ( - c / T ) ]. A natural, closely related, choice is to take T--,0 in the string action in analogy to the m ~ 0 limit for the point par-
ticle. This can be done once the action is written in a suitable form [ 11,12 ]. One may then ask if the resuiting "null string" can be quantized [ 13,14 ] and if interactions can be introduced. At a fixed level of the quantum string the length approaches infinity as T - t / a , so that the "fundamental length" of string theory becomes infinite. Such null strings would then be reasonable candidates for the excitations in a scale invarianl phase. We expect the supersymmetric null string to make more sense than the bosonic one, just as in the tensionful case. A null superstring has been proposed previously in ref. [ 15 ], where the limit T--.0 is taken by first reformulating the Green-Schwarz superstring in terms of a phase space lagrangian. However, their action is only reparametrization invariant on shell. The same is true for the Siegel invariance [ 16,17 ], which in the ordinary superstring and superparticle allows unwanted (i.e. half the) spin degrees of freedom to be gauged away. A constraint on the parameters of the Siegel transformations is needed for the physical fermionic degrees of freedom to remain. Such a constraint seems to be missing in ref. [15]. In this paper we construct a null superstring action (in configuration space) without the drawbacks
0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )
3 19
Volume 253, number 3,4
PHYSICS LETTERS B
mentioned above. We first reexamine the bosonic null string in a geometric formulation, which is then easily taken over to the space-time supersymmetric case. Our discussion is classical in that we do not address the question of a critical dimension for our model. There is a standard way for putting an action of the type
S=m f L
(1)
into a form where it is possible to study the limit m-,0; introduce an auxiliary field (density) V and write the action in the (classically) equivalent form
S'=½ f V L 2 + V - l m 2.
(2)
10 January 1991
mation 8V= - 2 VO,A"L- ~will become meaningless on shell (where L = 0). Since local supersymmetries in general transform the lagrangian density into a total divergence, and we want to study the superstring whose Siegel symmetry is of this type, it is gratifying that there exists a different approach to the problem of taking the parameter to zero in actions of the form ( 1 ), which we now discuss. Let us first notice how we could have arrived at the "ein-bein" action (2) for the particle [ 18 ]. We start from the action (3) and derive the canonical momenta m2m Pm- N ~ 2 •
(9)
From this follows the constraint
The limit m--, 0 can then be straightforwardly reached. In this way the action for a massive particle
p2+m2=0,
Sm=-m~dr~
and the hamiltonian
2
(3)
H = 2 ( p 2 + m 2) ,
can be related to that o f a massless one So = - ½ f dr V~"2 ,
(4)
T~ d2xf~,
(5)
where
r~=a~x'"apx,.,
(6)
(7)
In both these cases the reparametrization invariance is preserved and the ambient space-time symmetry enlarged from Poincar6 to conformal symmetry. The approach just described has drawbacks, however. One obvious problem is to preserve a symmetry which is a symmetry of the action but under which the Lagrange density transforms into a total divergence. I f S L = 0 , A " then one has 8 S ' = ½f 2VLO,A"+ 8VL 2 .
(8)
In general the situation will be that the action is not invariant with 6V=0, and that the naive transfor320
where 2 is a Lagrange multiplier. The phase space lagrangian is
( 12 )
Integrating out the momenta, we pass to the configuration space lagrangian
L(x, 2) = ½(22/2)~ - 2)~m2 )
will give rise to that of the tensionless "null" string [ll,12l: S o = - ½ ~ d2~ Vy.
(11)
L = p 2 - 2 ( p 2 + m 2) .
and the bosonic string action
S-r=-T I d 2 ~ ~ - - -
(10)
.
( 13 )
We see that, putting V - ~= - 22, this agees with the action (2) with the particle lagrangian from (3). For the string this approach becomes more interesting. The momenta derived from the action ( 5 ) are T Pm-~'~~ ( X X ' )
2
['3('2~(m--(')(Y')Xtm]'
(14)
from which follow the constraints
p2+T2x'2=O,
px' = 0 .
(15)
This leads to a phase space lagrangian which contains two Lagrange multipliers 2 and p: L=p2-~,(p2
+ T i x '2 )
--ppx' .
(16)
Integrating out the momenta we arrive at the following configuration space lagrangian: 1
L(x, 2) = ~ [ 2 2 - 2 p 2 x ' + (p2-4T222)x'2] . (17)
Volume 253, number 3,4
PHYSICS LETTERS B
With the identification
.~ ( - 1 g
=~,
p
"~
P 422T2_p2)
(18)
the action (17) becomes the Weyl-invariant action: Sw = - ½T f d2~x/~gg'*ay,~e,
(19)
where g_--det g,a and 2 and p have been assigned the correct transformations for g-a to transform as a tensor under world sheet diffeomorphisms. Note that many first order actions, (actions involving an auxiliary metric), can be written down for the bosonic string [ 19 ]. The Weyl-invariant one, ( 19 ), is singled out by the procedure described above, i.e., by the quantization procedure, since what we have discussed is really setting up the path integral with the phase space as the starting point. Clearly, taking the limit T-+0 in this approach, e.g. in (16), gives something new; previously we had only one Lagrange multiplier field (see ( 7 ) ) , now we have two. It is in fact this extra field which allows us to overcome the difficulties related to the T ~ 0 limit for the superstring. Just as for the tensionful theory, the T ~ 0 limit of the action (16) allows for a geometrical interpretation of the Lagrange multipliers 2 and p. With the auxiliary vector density V" replacing 2 and p according to 1
2x/~ (I, - p ) - ~ V " ,
(20)
we obtain the following general form for the bosonic null string action: S,.= I d2~ V"V~Y'~P"
(21)
Clearly, V" can be taken to transform as a contravariant vector density (of weight - ~ ), and G " a = V " V p is then a tensor density with det G"P=0. The action (21 ) is the null string action that corresponds to the first order action (19) for the usual bosonic string. The V '~ field equations VPTt~ = 0
(22)
immediately show that ),,a has a null eigenvector, and we conclude that detT,p=0.
(23)
10 January 1991
which is the criterion for a null string. Instead of starting again from the tensionful theory for the superstring, we immediately generalize (21 ) to a globally supersymmetric theory. As usual, this is done simply by letting ~1 o, x m - - , H m ~ O , ~ x m - i 6 F " O . O ,
(24)
where 0(~) is a Majorana (Weyl) space-time spinor, world sheet scalar, and F m are Dirac matrices. Under global supersymmetry with (spinor) parameter e we have 6~0=e,
8~Xm=-itTFmS~o,
~8~H;7=0.
(25)
Hence, the action S v - f d2~ V"V~HmHp,,,
(26)
is globally supersymmetric if we take 5~V~= 0. Next we turn to the Siegel symmetry. This is a local fermionic symmetry of the Green-Schwarz superstring which halves the fermionic degrees of freedom in 0: 5,~O=iI~,x ~ ,
&~X'"=iOF'"6,,O,
= ~ 8 ~ I ~ = - 2iS~0P~O, 0 ,
(27)
with x" satisfying a certain constraint [17,20]. To have invariance in the Green-Schwarz action one adds a Wess-Zumino term to the part of the action which corresponds to our (26). We shall see that this will not be necessary in our case. In fact, if we demand x" = V"x ,
(28)
with x(~) a space-time spinor and world sheet scalar density of weight ½, we have for the variation of the action (26): 8 S v = 2 f d2~ V"7~/~(8~ V P - 2 V P V P ~ p t f x ) .
(29)
Thus, the action is Siegel-invariant if we put 8 V '~ = 2 V '~ VPOp6x,
(30)
which is in fact highly reminiscent of the transformation of the metric in the tensionful theory. Note that the V '~ field equations ~ We use conventions adopted to D= 10 and Majorana-Weyl spinors. In other dimensions where one uses Weylspinors the transformations must be modified, e.g. OF'~O,~O --* ½( Or~a.o-o.~rmo). 321
Volume 253, number 3,4 (i)
8V""
PHYSICS LETTERSB
V"HS'H~,,,=0,
~l~,V"l~pVa=O
(31) show that in the combination that enters the x variation, ~'x=/~,,V"tc, the operator ~ is nilpotent (on shell) and hence x only carries half the degrees of freedom o f a spinor, in complete analogy to the T4:0 case. Just as for the Green-Schwarz superstring there is also a local bosonic symmetry of the action (26):
10 January 1991
pect that the original super-Poincar6 group, the symmetry group of the Green-Schwarz superstring, gets replaced by the super-conformal group. This is indeed the case in the dimensions D = 2 - 6 [21 ] where the superconformal group has been shown to exist, and possibly also in D = 10, where it may exist [22]. In fact, the superconformal group transforms the super line element ds 2 = H mH i nd~"d~ p
( 36 )
This symmetry is required for closing the algebra of the ~c-transformations. In fact, we find
into g2(X, 0)ds 2, hence the invariance of the action (26) under this group is almost immediate. In D = 4 the superconformal transformations o f X m and 0 can be found in ref. [23] and they lead to the following transformations of V":
[8~2, 8~, ] = G ( x = - V~(O~OKt2 )1<,1)
8, v ~ - i [~S, V ~ ] = - ½ q O V ~ ,
8~ V " = 0 ,
8~0=2V"0~0 ,
8~X'"=iGF"8.~O :
(32)
+ 8~(2= -- 4i(~2 Va/~/~xl ) )
+ 8~(U = 4i (~72 w v P / ~
8 h V " - i [ b , n K ",, V , ] = - b X V
))
+ terms zc equations of motion.
8DV'~=-i[DA, V " ] = - D V
( 33 )
Here ¢" is the parameter of 2D diffeomorphisms. The bosonic null string is easily seen to consist of massless particles moving on a null surface• For the present model, described by the action (26), we have the equations of motion (31) and (ii) (iii)
8X m" OB(V'~VPHm)=o, 80:
V'~VP(O,~O)I~p+O,,(V"V~O~Ip)=O.
(34) In the diffeomorphism gauge V " = ( 1 , 0) we find from (31) and (34) (i)~(Ho)2=0,
HoH~ = 0 ,
(ii) ~ H ~ ' = 0 , (iii) ~G/~o = 0 ,
(35)
and we recognize the equation of motion for a massless superparticle at each a. Also, the motion is confined to the superspace equivalent of a null hypersurface. Masslessness is usually connected with the conformal group. The massless particle has the conformal group replacing the Poincar6 group as symmetry group in the ambient space-time. Likewise, for the tensionless bosonic string the same extension of the symmetry group occurs. It is therefore natural to ex322
~.
'~ ,
(37)
Here S, K'" and/I are the generators of S-supersymmerry, special conformal and dilation transformations, respectively. V" is inert under the rest of the superconformal group. We have derived a null superstring which is manifestly reparametrization and Siegel invariant (in contrast to the model in ref. [ 15 ] ). Our action is a straightforward generalization of the bosonic null string in a novel formulation. The "classical" restrictions on the space-time dimension where our action can be formulated are less severe than for the tensionfull superstring, unless we also demand superconformal invariance in the ambient superspace, or maybe that the theory should be equivalent (in a lightcone gauge) to a (sector of ) a spinning null string, in analogy to the T:/: 0 case. We conclude by some more comments on our action. For the usual Green-Schwarz action there are restrictions from the gamma matrix algebra on the dimensions D in which it is invariant, namely D = 3, 4, 6 and 10. To show invariance of our action under Siegel symmetry, all we need is the Majorana properties of the spinors. Hence invariance results in all dimensions which support Majorana (or Weyl) spinors. Finally, extensions to higher N (more supersymmetries) of our action is straightforward. We thank P. van Nieuwenhuizen, M. Ro6ek and
Volume 253, number 3,4
PHYSICS LETTERS B
W. Siegel for p r o v i d i n g useful i n f o r m a t i o n .
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