- Email: [email protected]
A supertwistor description of the massless superparticle in ten-dimensional superspace
Volume 247, number 1
PHYSICS LETTERSB
6 September 1990
A supertwistor description of the massless superparticle in ten-dimensional superspace N a t h a n Berkovits Department of Physics, Rutgers University, Piscataway, NJ 08854, USA
Received 17 April 1990
Because of the mass-shell constraint PuPu= 0, the superparticle coordinates X~, PU, and 0a, can be replaced by supertwistor variables consisting of two bosonic SO( l, 9) Majorana-Weyl spinors and one fermionic SO( 1, 9) vector. In terms of these supertwistor variables, covariant quantization of the ten-dimensional superparticle is straightforward.
I. Introduction
The primary motivation for studying the ten-dimensional massless superparticle comes from it being the infinite-tension limit of the Green-Schwarz superstring [ 1 ]. Although the free Green-Schwarz superstring exists in three-, four-, six-, or ten-dimensional superspace, only the ten-dimensional superstring has been shown to allow interactions which are both Lorentz covariant and unitary (this is easily proven in the light-cone gauge by comparing with the Neveu-Schwarz-Ramond formalism for the superstring [2 ] ). Because the Green-Schwarz superstring is manifestly spacetime supersymmetric, calculations of scattering amplitudes using the GreenSchwarz formalism have the potential of being much simpler than analogous calculations using the NeveuSchwarz-Ramond formalism. Until now, however, all attempts to covariantly quantize the Green-Schwarz superstring have been plagued either by a large number of auxiliary fields [ 3 ] or by a complicated ghost structure [ 4 ]. As a result, there are no efficient methods presently available for the covariant calculation of superstring scattering amplitudes in the Green-Schwarz formalism. All previous attempts to covariantly quantize the ten-dimensional massless superparticle have been plagued by the same problems as the superstring [ 5,6 ]. It will be shown in this paper that by replacing the usual superspace coordinates of the ten-dimenElsevier Science Publishers B.V. (North-Holland)
sional superparticle with supertwistor variables, these covariant quantization problems can be avoided. It is possible that similar methods can be used to simplify the covariant quantization of the GreenSchwarz superstring. The massless superparticle in four-dimensional superspace was first described in a 1977 paper by Ferber [ 7 ]. He showed that by extending the SU (2, 2) conformal algebra to an SU (2, 2IN) superconformal algebra, the twistor theory ofPenrose [ 8 ] could be generalized to a supertwistor theory. In terms of the new supertwistor variables, it was straightforward to covariantly quantize the four-dimensional superparticle [9]. By replacing Ferber's complex-valued supertwistors with real-valued or quaterionic-valued supertwistors, Bengtsson and Cederwall were able to generalize these results to the massless superparticle in three- or six-dimensional superspace [ 10 ]. However, because of the non-associativity of octonions, they were unable to apply their supertwistor techniques to the ten-dimensional case. More recently, Sorokin, Tkach, Volkov, and Zheltukhin have found an equivalence at the classical level between the massless superparticle and the massless spinning particle (corresponding to the infinite-tension limit ofa Ramond string) in three, four, six, and ten dimensions [ 11 ]. This classical equivalence related the fermionic Siegel symmetry of the su45
Volume 247, number 1
PHYSICS LETTERS B
perparticle with the world-line supersymmetry of the spinning particle and identified 2aFu,bOb with q/u, where 0 b is the fermionic spinor of the superparticle, ~,u is the fermionic vector of the spinning particle, and 2~ is the bosonic spinor twistor variable. Although these theories are not equivalent at the quantum level, the relating of Siegel symmetry with worldline supersymmetry and the equation ~uu=2"F~bO b will turn out to be useful in defining a new set of supertwistor variables (i.e. different from those of Ferber, Bengtsson, and Cederwall) that can describe the massless superparticle in three-, four-, six-, and ten-dimensional superspace. In the first section of this paper, the massless superparticle action and super-Poincar6 generators will be described in terms of the usual superspace coordinates and the problems with covariant quantization will be reviewed. In the second section, a new definition of supertwistor variables will be introduced which can be used to describe the trajectory of a massless superparticle in three-, four-, six-, and tendimensional superspace. In the final section of this paper, the ten-dimensional massless superparticle action and superPoincar6 generators will be re-expressed in terms of these new supertwistor variables and covariant quantization will be performed in a straightforward manner.
2. The massless superparticle The ten-dimensional massless superparticle action can be written as [ 7 ]
pu=pu,
6 September 1990
q a = _ 2 i P u F ~ b O b,
Ml,.=½(Pl,X._P.XU
) + ~10 l" a F a~,b F .bc FcdO p dPp,
(2.2) where qa is the current of global spacetime supersymmetry transformations. In addition to world-line reparametrization invariance, the massless superparticle is also invariant under the following local fermionic Siegel transformation [12]: Oa--,O~ + P~'F~ub~cb, p~ ~ p ~ ' , " a X 1L --,X ,u -aO F aU b F ,b c x c P ,v
g~g--4iO~O~Xa.
(2.3)
Although this Siegel symmetry is essential for removing the unphysical fermionic degrees of freedom, it is also the source of covariant quantization problems. Because the fermionic variable K~ is an SO( 1, 9) Majorana-Weyl spinor, the Siegel symmetry has 16 off-shell generators. However, when the superparticle is on-shell (p2=0), P~F~bxb has only eight degrees of freedom and the number of independent Siegel symmetry generators is cut in half. Using the Batalin-Fradkin procedure of covariant quantization [ 13 ], this on-shell reducibility produces an infinite chain of ghosts [6 ]. Previously, the only way to avoid this complicated ghost structure was to introduce a large set of auxiliary fields [5 ]. Although these auxiliary fields bear a resemblance to the twistor variables that will be presented in this paper, the precise relationship between these two descriptions of the superparticle is unclear.
S = j dr [Pu(OTXU--io~oaffUabOb)+ ½gPUPu] , (2. 1 )
3. Definition of supertwistor variables
where X u and pu are bosonic SO ( 1, 9) vectors, 0 a is a 16-component fermionic SO ( 1, 9) Majorana-Weyl spinor, g is the world-line metric, FUab= Faub are real symmetric 16 × 16 matrices satisfying F g b P 'bc + Ft, ~,uuAc ab~F ' / ~ b c _ _ O~-,t ~,a , and
Twistor theory was first developed by Penrose as an alternative to the usual description of quantum fields. Instead of using spacetime points as the fundamental parameters on which the quantum fields depended, he used the light-like trajectories of massless particles. The spinor variables that labeled these light-like trajectories were called twistor variables. Penrose hypothesized that in a quantum theory of gravity, the classical description of a spacetime point would become ill-defined, but that a light-like trajectory would remain a meaningful concept. Although the application of twistor theory to quantum gravity
-
-
are the usual 32>(32 Majorana-Weyl ten-dimensional gamma-matrices. Using Noether's procedure, one finds the conserved superPoincar6 currents to be 46
Volume 247, number 1
PHYSICS LETTERSB
is still uncertain, twistors have proven useful in other areas of physics, including the geometrical interpretation of superspace constraints [ 14 ]. A light-like trajectory in d = 3, 4, 6, or 10 spacetime dimensions can be characterized by two real 2 d - 4 component SO ( 1, d - 1 ) spinors of opposite chirality, w ~ and 2b satisfying the 2 d - 4 constraints F ~ = (2bFg~2 ~) (Fauawa) --22a(2bWb) = 0 ,
(3.1)
where all repeated indices are summed over and FU~b is defined in eq. (2.1). Note that only d - 3 of these 2 d - 4 constraints are independent since 2aF~bF b and 2aF~ab(2CF~cd2d) are automatically zero by the F-matrix identity FUabFu~d+ FUcaFubd+ F~cFua d = 0 (valid in three, four, six, and ten dimensions). Using these same F-matrix identities, it can be shown that the constraints in (3.1) imply the existence of an SO( l, 9) vector X u satisfying w~ = XuFUab2b .
(3.2)
Since eq. (3.2) is also satisfied by replacing X u with X ~ + CA aFub2b, Wa and 2a define a light-like trajectory of points. Although the spinors w~ and 2a define a unique trajectory, there are d - 2 independent gauge transformations of Wa and 2a that leave this trajectory invariant. (This is easily seen by counting variables. A light-like line is parameterized by 2 d - 3 variables whereas the constrained spinors, w a and 2 ~, carry 2 ( 2 d - 4) - ( d - 3 ) degrees of freedom. ) These gauge transformations are [aw~, a2 b] = [Cw~, Ca b ] , [¢~Wa, ¢~2 b] --ed[2Fac2 -,u c ( F de u We)-- 2ad(2~Wc) _ 2 2 d W a It e ) + 2 2 d 2 b ] - F bd u (2 c Fce2
(3.3)
(exactly as in the case of the constraints, only d - 2 of these 2 d - 3 gauge transformations are independent). Just as a light-like line describes the trajectory of a massless particle in ordinary spacetime, a light-like superline can be used to describe the trajectory of a massless superparticle in superspace [14]. This superline is described by the following (I, d - 2 ) dimensional locus of points in d = 3-, 4-, 6-, or 10dimensional superspace:
6 September 1990
[ XU( t, x), oe( t, x) ] ,u bc KcP,v od+puFdelce], = [ X ~ + P ,ut - l O" a FabFv
(3.4) where 0 a and R7b are fermionic 2 d - 4 component spinors of opposite chirality and PuPu= 0 (implying that puFaub Xb has only 2-- d fermionic degrees of freedom). This superparticle trajectory can also be parametrized by two bosonic real 2 d - 4 component spinors of opposite chirality, wa and 2 b, and one fermionic real d-component vector, q/u, which satisfy the following 2 d - 3 constraints: g = (2aF~ab2b)~lU-~-O , G a = ( 2 b F~c 2 c) ( FaudWd) -- 2 2 a ( 2 bWb ) • Iz~vFuac Fcb2 u b +21~ =0 .
(3.5)
Note that as before, only d - 3 of the 2 d - 4 bosonic constraints are independent since 2bFUabG a - 4i~Ug is automatically zero. As a result of these constraints, there exists a locus of points, I X u, oa], in superspace satisfying the following equations: Wa=XuF~ab2b--i~luFUabOb ,
~u=2aFUabOb.
(3.6)
Although this locus of points describes only a ( 1, d - 3 )-dimensional subset of a light-like superline, it can easily be enlarged to a complete (1, d - 2 ) - d i mensional superparticle trajectory by identifying the twistor variables [ wa, 2 b, ~uu ] with [ Wa + 4ia~u,F~c2 c, 2 b, ~uu+ ot2CFUcd2d ] ,
where o~ is a fermionic scalar (this identification resembles world-line supersymmetry where ~uu is identified with ~,U+otPU). Equivalently, one could say that [Wa, 2 b, ~1*u]
-+[Wa+4ioeg,,F~c2c, 2b,~u+oe2cFgd2d ]
(3.7)
is a fermionic gauge transformation that preserves the superparticle trajectory described by the supertwistor. Note that under the Siegel transformation [X ~, 0 a ] ---, [ XU--iObF~cFSd~cdP ~', 0 '~+ P "F~abXb ] , [Wa, 2 b, ~U] __+[W~ + (8ia29Ce) gGF~¢2~, 2 b , ~u + ( 2a2etCe)2CFUd2d ] , 47
Volume 247, number 1
PHYSICS LETTERSB
where pu = a2 ~FU~d2d. In addition to the fermionic gauge transformation, there are the following 2 d - 3 bosonic gauge transformations that also leave the light-like superline invariant: . c(F~d~w~) - 2C~a(2~Wc) [6wa, ~2 b, ~g/u ] _, ea[2Fac2 _2,~aWa +21~u • t,~,pF,,aeF~a p , ~ e )+22a2 b, - F ~bd (2 c F~2
~uv (F~de F euy - F ltde F~ef)2 f ] , [ ~wa, 62 b, &u ~'] --, [ Cwa, C2 b, C ~ ~'] .
(3.8)
For the same reasons as before, only d - 2 of these bosonic gauge transformations are independent. Since a light-like superline has ( 2 d - 3 , d - 2 ) degrees of freedom, the counting is correct [ ( 4 d - 8, d) - ( d - 3, 1)-(d-2, 1)= (2d-3, d-2)]. These new supertwistor variables differ in three major respects from the three-, four-, and six-dimensional supertwistor variables of Ferber [ 7 ], Bengtsson and Cederwall [ 10 ]. Firstly, the bosonic spinor 2 a is chosen to have the same, rather than the opposite, chirality of the fermionic spinor 0 b. Secondly, the fermionic twistor variable is a d-dimensional vector. And thirdly, the supertwistor variables are subject to fermionic, as well as bosonic, constraints.
6 September 1990
requirement that PuP ~' is zero off shell allows the Siegel symmetry to have the same number of on-shell and off-shell generators. (2) P°>~0 (since F°b is the unit matrix), i.e. the superparticle is moving forward in time. Bengtsson and Cederwall have argued that this restriction can be understood by observing that po is the hamiltonian of the second-quantized superparticle, which must be non-negative due to global spacetime supersymmetry [ 10 ]. Using the relations wa=XuF~b2b--igtuF~ab0 b, Pu=2bFgc2c and ~/uAaFU~bOb, the superparticle action in (2.1) can be rewritten as S = ~ dz { - 2 w a 0,2a-2i~uu0,~ ,~ + h a [ ( j.b l"~c); C)
( F~udWd) - 2/~a(,~ bWb )
• it q/~F ac v b] +21~t u Fcb2
+ a [ (2aFuab2b) ~tu] },
(4.1)
where ha and a are Lagrange multipliers which enforce the constraints in (3.5). In terms of the supertwistor variables, the superPoincar6 currents in (2.2) can be rewritten: pU=2bF~c2c,
qa=4iq/1,F~ab2 b ,
M l,~ = t 2 b ( F~F~Cd _ F~Fu~a) w a + i ( ~ " g t u _ ~uug/" ) .
(4.2)
4. Supertwistor description of the ten-dimensional superparticle In order to describe the dynamics o f a massless superparticle, it will be convenient to remove the gauge invariance [Wa, 2 b, g/u]~ [Cwa, C2 b, Cgt ~'] by identifying 2~F~'a2d with pu, the superparticle's momentum vector. The supertwistor [Wa, 2 b, ~Uu] will now describe not only the trajectory of the massless particle, but also its energy. This identification is possible if pu satisfies the following two conditions: (1) PuP~'=O, i.e. the superparticle is always on mass shell. Although the supertwistor description will have different off-shell degrees of freedom from the usual description in terms of superspace variables, this will only affect the field interpretation of the secondquantized superparticle, rather than the physical Hilbert space that comes from first quantization. The 48
Note that as a result of the supertwistor constraints, qF~UF"FPlq= 96iMtU"pPl. Canonical first quantization of the superparticle yields the following (anti-)commutation relations for the supertwistor variables: [Wa, 2b]
= - ~ 1" l ~ ab,
{~u",~u~}= ~i. u
(4.3 )
With these commutation relations, it is easy to show that the operators in (4.2) form a superPoincar6 algebra and that the supertwistor constraints in (3.5) generate the fermionic and bosonic gauge transformations (except for the scaling transformation) of (3.7) and (3.8). Because of their anti-commutation relations, the first-quantized gtu operators can be represented by SO ( 1, 9 ) 7-matrices, implying that the superparticle wavefunction transforms like a 32-component SO ( 1, 9) spinor containing both chiralities. In the "mo-
Volume 247, number 1
PHYSICS LETTERS B
m e n t u m " representation o f this wavefunction, 0A (2), the supertwistor constraints i m p l y that G~BOB= ~
(2bFgc2 c) F ~ d
-22"
2b
0A
v c ~x(F , u a b Fb~2 )(TUeAT,COc)--2i2aoA=O,
"JV 1"
g~08 = ~
1
(2bF~2c)Yu~0B = 0 ,
=AuFUd2d,
__ c ba " u f Ob(2)--Cuqc F,# d2 d =IX/~ C cu(FcfJ. ) (F~baF,d2" d)
- ~--i
(2eFufAy)(BcFub)+2ix/~2b(2CBc)
c o m m u t a t i o n relations explicitly d e p e n d on the supertwistor variables) and have only two levels o f reducibility [GaFu~b2b- 4i~'Ug= 0 and (2bFg¢2c) × (Fund2a) = (2bFgc2c) ( - - 4 i ~ ,u) = 0 where G" and g are the supertwistor constraints, (I~UabJ.b ) and --4i~ ,u are the ten first-level zero modes, and (2bFgc2c) is the only second-level zero m o d e ].
(4.4)
where the last term in G ] B comes from n o r m a l - o r d e r ing ambiguities, but can be fixed by requiring that 2 bFg, G~B - ix/~ y~g~ is automatically zero. If one splits the 32-component 0A into its two 16c o m p o n e n t spinors o f opposite chirality, 0a and 0 b, it is easy to check that in ten dimensions, the constraints are satisfied if a n d only if 0,(2)
6 September 1990
Acknowledgement I would like to thank Tom Banks, Michael Douglas, D a n Friedan, Yutaka Matsuo, Alexander Mikovic, Joel Shapiro, Steve Shenker, Sang Jin Sin, A n t o n Van de Ven, a n d John Y a m r o n for useful conversations, and the universities o f Rutgers a n d S U N Y at Stony Brook for their generosity and hospitality. I would especially like to thank Warren Siegel for m a n y helpful suggestions, including the suggestion to write the twistor formulas in covariant notation. This work was s u p p o r t e d by the N S F grant # 90-96107-01.
(4.5)
where q~a is the spacetime s u p e r s y m m e t r y generator, A u a n d Bc = CauFU~ad e p e n d on 2 only through the ten c o m b i n a t i o n s 2 bFgc2~, a n d A u satisfies the c o n d i t i o n Au(2bF~U)=O. Note that 0A is unchanged by the gauge t r a n s f o r m a t i o n s A U~ A u + CA bF~2 ~ or Ba-~ B~+ (2bFgc2 ~) (Fu~dFa) for a r b i t r a r y C a n d F a. By F o u r i e r transforming A u a n d B, into position space, where
7(x) = ~ d~°Xexp[i(AbF~2c)xu]f(2F2) , one sees that the physical Hilbert space o f the quantized superparticle can be constructed out o f a vector field, Au(X), a n d a spinor field, Ba(X), satisfying O ~ = O ~ 0 ~ u = 0"0~/~a=0. These can be interpreted as the on-shell N = 1 super-Yang-Mills fields in ten dimensions. An alternative a p p r o a c h to quantizing the ten-dimensional superparticle is to construct a BRST operator. Using the a p p r o a c h o f Batalin a n d F r a d k i n [ 13 ], this should be straightforward using supertwistor variables since the supertwistor constraints are all first class (although the structure "constants" in their
References [1]M.B. Green and J.H.Schwarz, Nucl. Phys. B 243 (1984) 285. [2 ] S. Mandelstam, Review of aspects of string theory, talk given at Niels Bohr Centennial Conf. (Copenhagen, Denmark, May 1985). [3] E. Nissimov, S. Pacheva and S. Solomon, Nucl. Phys. B 297 (1988) 349. [4] S.J. Gates Jr., M.T.Grisaru, U. Lindstrom, M. Ro~ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 225 (1989) 44. 15] Y. Eisenberg and S. Solomon, Nucl. Phys. B 309 (1988) 709. [ 6 ] U. Lindstrom, M. Ro6ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 224 (1989) 285. [7] A. Ferber, Nucl. Phys. B 132 (1978) 55. [8] R. Penrose and M.A.H. MacCallum, Phys. Rep. 6 (1972) 24. [9] T. Shirafuji, Prog. Theor. Phys. 70 (1983) 18. [ 10] I. Bengtsson and M. Cederwall, Nucl. Phys. B 302 (1988) 81. [ 11 ] D.P. Sorokin, V.I. Tkach, D.V. Volkov and A.A. Zheltukhin, Phys. Lett. B 216 (1989) 302. [ 12] W. Siegel, Phys. Len. B 128 (1983) 397. [ 13 ] I.A. Batalin and E.S. Fradkin, Phys. Lett. B 122 ( 1983 ) 157. [ 14] E. Witten, Nucl. Phys. B 266 (1986) 245.
49
PHYSICS LETTERSB
6 September 1990
A supertwistor description of the massless superparticle in ten-dimensional superspace N a t h a n Berkovits Department of Physics, Rutgers University, Piscataway, NJ 08854, USA
Received 17 April 1990
Because of the mass-shell constraint PuPu= 0, the superparticle coordinates X~, PU, and 0a, can be replaced by supertwistor variables consisting of two bosonic SO( l, 9) Majorana-Weyl spinors and one fermionic SO( 1, 9) vector. In terms of these supertwistor variables, covariant quantization of the ten-dimensional superparticle is straightforward.
I. Introduction
The primary motivation for studying the ten-dimensional massless superparticle comes from it being the infinite-tension limit of the Green-Schwarz superstring [ 1 ]. Although the free Green-Schwarz superstring exists in three-, four-, six-, or ten-dimensional superspace, only the ten-dimensional superstring has been shown to allow interactions which are both Lorentz covariant and unitary (this is easily proven in the light-cone gauge by comparing with the Neveu-Schwarz-Ramond formalism for the superstring [2 ] ). Because the Green-Schwarz superstring is manifestly spacetime supersymmetric, calculations of scattering amplitudes using the GreenSchwarz formalism have the potential of being much simpler than analogous calculations using the NeveuSchwarz-Ramond formalism. Until now, however, all attempts to covariantly quantize the Green-Schwarz superstring have been plagued either by a large number of auxiliary fields [ 3 ] or by a complicated ghost structure [ 4 ]. As a result, there are no efficient methods presently available for the covariant calculation of superstring scattering amplitudes in the Green-Schwarz formalism. All previous attempts to covariantly quantize the ten-dimensional massless superparticle have been plagued by the same problems as the superstring [ 5,6 ]. It will be shown in this paper that by replacing the usual superspace coordinates of the ten-dimenElsevier Science Publishers B.V. (North-Holland)
sional superparticle with supertwistor variables, these covariant quantization problems can be avoided. It is possible that similar methods can be used to simplify the covariant quantization of the GreenSchwarz superstring. The massless superparticle in four-dimensional superspace was first described in a 1977 paper by Ferber [ 7 ]. He showed that by extending the SU (2, 2) conformal algebra to an SU (2, 2IN) superconformal algebra, the twistor theory ofPenrose [ 8 ] could be generalized to a supertwistor theory. In terms of the new supertwistor variables, it was straightforward to covariantly quantize the four-dimensional superparticle [9]. By replacing Ferber's complex-valued supertwistors with real-valued or quaterionic-valued supertwistors, Bengtsson and Cederwall were able to generalize these results to the massless superparticle in three- or six-dimensional superspace [ 10 ]. However, because of the non-associativity of octonions, they were unable to apply their supertwistor techniques to the ten-dimensional case. More recently, Sorokin, Tkach, Volkov, and Zheltukhin have found an equivalence at the classical level between the massless superparticle and the massless spinning particle (corresponding to the infinite-tension limit ofa Ramond string) in three, four, six, and ten dimensions [ 11 ]. This classical equivalence related the fermionic Siegel symmetry of the su45
Volume 247, number 1
PHYSICS LETTERS B
perparticle with the world-line supersymmetry of the spinning particle and identified 2aFu,bOb with q/u, where 0 b is the fermionic spinor of the superparticle, ~,u is the fermionic vector of the spinning particle, and 2~ is the bosonic spinor twistor variable. Although these theories are not equivalent at the quantum level, the relating of Siegel symmetry with worldline supersymmetry and the equation ~uu=2"F~bO b will turn out to be useful in defining a new set of supertwistor variables (i.e. different from those of Ferber, Bengtsson, and Cederwall) that can describe the massless superparticle in three-, four-, six-, and ten-dimensional superspace. In the first section of this paper, the massless superparticle action and super-Poincar6 generators will be described in terms of the usual superspace coordinates and the problems with covariant quantization will be reviewed. In the second section, a new definition of supertwistor variables will be introduced which can be used to describe the trajectory of a massless superparticle in three-, four-, six-, and tendimensional superspace. In the final section of this paper, the ten-dimensional massless superparticle action and superPoincar6 generators will be re-expressed in terms of these new supertwistor variables and covariant quantization will be performed in a straightforward manner.
2. The massless superparticle The ten-dimensional massless superparticle action can be written as [ 7 ]
pu=pu,
6 September 1990
q a = _ 2 i P u F ~ b O b,
Ml,.=½(Pl,X._P.XU
) + ~10 l" a F a~,b F .bc FcdO p dPp,
(2.2) where qa is the current of global spacetime supersymmetry transformations. In addition to world-line reparametrization invariance, the massless superparticle is also invariant under the following local fermionic Siegel transformation [12]: Oa--,O~ + P~'F~ub~cb, p~ ~ p ~ ' , " a X 1L --,X ,u -aO F aU b F ,b c x c P ,v
g~g--4iO~O~Xa.
(2.3)
Although this Siegel symmetry is essential for removing the unphysical fermionic degrees of freedom, it is also the source of covariant quantization problems. Because the fermionic variable K~ is an SO( 1, 9) Majorana-Weyl spinor, the Siegel symmetry has 16 off-shell generators. However, when the superparticle is on-shell (p2=0), P~F~bxb has only eight degrees of freedom and the number of independent Siegel symmetry generators is cut in half. Using the Batalin-Fradkin procedure of covariant quantization [ 13 ], this on-shell reducibility produces an infinite chain of ghosts [6 ]. Previously, the only way to avoid this complicated ghost structure was to introduce a large set of auxiliary fields [5 ]. Although these auxiliary fields bear a resemblance to the twistor variables that will be presented in this paper, the precise relationship between these two descriptions of the superparticle is unclear.
S = j dr [Pu(OTXU--io~oaffUabOb)+ ½gPUPu] , (2. 1 )
3. Definition of supertwistor variables
where X u and pu are bosonic SO ( 1, 9) vectors, 0 a is a 16-component fermionic SO ( 1, 9) Majorana-Weyl spinor, g is the world-line metric, FUab= Faub are real symmetric 16 × 16 matrices satisfying F g b P 'bc + Ft, ~,uuAc ab~F ' / ~ b c _ _ O~-,t ~,a , and
Twistor theory was first developed by Penrose as an alternative to the usual description of quantum fields. Instead of using spacetime points as the fundamental parameters on which the quantum fields depended, he used the light-like trajectories of massless particles. The spinor variables that labeled these light-like trajectories were called twistor variables. Penrose hypothesized that in a quantum theory of gravity, the classical description of a spacetime point would become ill-defined, but that a light-like trajectory would remain a meaningful concept. Although the application of twistor theory to quantum gravity
-
-
are the usual 32>(32 Majorana-Weyl ten-dimensional gamma-matrices. Using Noether's procedure, one finds the conserved superPoincar6 currents to be 46
Volume 247, number 1
PHYSICS LETTERSB
is still uncertain, twistors have proven useful in other areas of physics, including the geometrical interpretation of superspace constraints [ 14 ]. A light-like trajectory in d = 3, 4, 6, or 10 spacetime dimensions can be characterized by two real 2 d - 4 component SO ( 1, d - 1 ) spinors of opposite chirality, w ~ and 2b satisfying the 2 d - 4 constraints F ~ = (2bFg~2 ~) (Fauawa) --22a(2bWb) = 0 ,
(3.1)
where all repeated indices are summed over and FU~b is defined in eq. (2.1). Note that only d - 3 of these 2 d - 4 constraints are independent since 2aF~bF b and 2aF~ab(2CF~cd2d) are automatically zero by the F-matrix identity FUabFu~d+ FUcaFubd+ F~cFua d = 0 (valid in three, four, six, and ten dimensions). Using these same F-matrix identities, it can be shown that the constraints in (3.1) imply the existence of an SO( l, 9) vector X u satisfying w~ = XuFUab2b .
(3.2)
Since eq. (3.2) is also satisfied by replacing X u with X ~ + CA aFub2b, Wa and 2a define a light-like trajectory of points. Although the spinors w~ and 2a define a unique trajectory, there are d - 2 independent gauge transformations of Wa and 2a that leave this trajectory invariant. (This is easily seen by counting variables. A light-like line is parameterized by 2 d - 3 variables whereas the constrained spinors, w a and 2 ~, carry 2 ( 2 d - 4) - ( d - 3 ) degrees of freedom. ) These gauge transformations are [aw~, a2 b] = [Cw~, Ca b ] , [¢~Wa, ¢~2 b] --ed[2Fac2 -,u c ( F de u We)-- 2ad(2~Wc) _ 2 2 d W a It e ) + 2 2 d 2 b ] - F bd u (2 c Fce2
(3.3)
(exactly as in the case of the constraints, only d - 2 of these 2 d - 3 gauge transformations are independent). Just as a light-like line describes the trajectory of a massless particle in ordinary spacetime, a light-like superline can be used to describe the trajectory of a massless superparticle in superspace [14]. This superline is described by the following (I, d - 2 ) dimensional locus of points in d = 3-, 4-, 6-, or 10dimensional superspace:
6 September 1990
[ XU( t, x), oe( t, x) ] ,u bc KcP,v od+puFdelce], = [ X ~ + P ,ut - l O" a FabFv
(3.4) where 0 a and R7b are fermionic 2 d - 4 component spinors of opposite chirality and PuPu= 0 (implying that puFaub Xb has only 2-- d fermionic degrees of freedom). This superparticle trajectory can also be parametrized by two bosonic real 2 d - 4 component spinors of opposite chirality, wa and 2 b, and one fermionic real d-component vector, q/u, which satisfy the following 2 d - 3 constraints: g = (2aF~ab2b)~lU-~-O , G a = ( 2 b F~c 2 c) ( FaudWd) -- 2 2 a ( 2 bWb ) • Iz~vFuac Fcb2 u b +21~ =0 .
(3.5)
Note that as before, only d - 3 of the 2 d - 4 bosonic constraints are independent since 2bFUabG a - 4i~Ug is automatically zero. As a result of these constraints, there exists a locus of points, I X u, oa], in superspace satisfying the following equations: Wa=XuF~ab2b--i~luFUabOb ,
~u=2aFUabOb.
(3.6)
Although this locus of points describes only a ( 1, d - 3 )-dimensional subset of a light-like superline, it can easily be enlarged to a complete (1, d - 2 ) - d i mensional superparticle trajectory by identifying the twistor variables [ wa, 2 b, ~uu ] with [ Wa + 4ia~u,F~c2 c, 2 b, ~uu+ ot2CFUcd2d ] ,
where o~ is a fermionic scalar (this identification resembles world-line supersymmetry where ~uu is identified with ~,U+otPU). Equivalently, one could say that [Wa, 2 b, ~1*u]
-+[Wa+4ioeg,,F~c2c, 2b,~u+oe2cFgd2d ]
(3.7)
is a fermionic gauge transformation that preserves the superparticle trajectory described by the supertwistor. Note that under the Siegel transformation [X ~, 0 a ] ---, [ XU--iObF~cFSd~cdP ~', 0 '~+ P "F~abXb ] , [Wa, 2 b, ~U] __+[W~ + (8ia29Ce) gGF~¢2~, 2 b , ~u + ( 2a2etCe)2CFUd2d ] , 47
Volume 247, number 1
PHYSICS LETTERSB
where pu = a2 ~FU~d2d. In addition to the fermionic gauge transformation, there are the following 2 d - 3 bosonic gauge transformations that also leave the light-like superline invariant: . c(F~d~w~) - 2C~a(2~Wc) [6wa, ~2 b, ~g/u ] _, ea[2Fac2 _2,~aWa +21~u • t,~,pF,,aeF~a p , ~ e )+22a2 b, - F ~bd (2 c F~2
~uv (F~de F euy - F ltde F~ef)2 f ] , [ ~wa, 62 b, &u ~'] --, [ Cwa, C2 b, C ~ ~'] .
(3.8)
For the same reasons as before, only d - 2 of these bosonic gauge transformations are independent. Since a light-like superline has ( 2 d - 3 , d - 2 ) degrees of freedom, the counting is correct [ ( 4 d - 8, d) - ( d - 3, 1)-(d-2, 1)= (2d-3, d-2)]. These new supertwistor variables differ in three major respects from the three-, four-, and six-dimensional supertwistor variables of Ferber [ 7 ], Bengtsson and Cederwall [ 10 ]. Firstly, the bosonic spinor 2 a is chosen to have the same, rather than the opposite, chirality of the fermionic spinor 0 b. Secondly, the fermionic twistor variable is a d-dimensional vector. And thirdly, the supertwistor variables are subject to fermionic, as well as bosonic, constraints.
6 September 1990
requirement that PuP ~' is zero off shell allows the Siegel symmetry to have the same number of on-shell and off-shell generators. (2) P°>~0 (since F°b is the unit matrix), i.e. the superparticle is moving forward in time. Bengtsson and Cederwall have argued that this restriction can be understood by observing that po is the hamiltonian of the second-quantized superparticle, which must be non-negative due to global spacetime supersymmetry [ 10 ]. Using the relations wa=XuF~b2b--igtuF~ab0 b, Pu=2bFgc2c and ~/uAaFU~bOb, the superparticle action in (2.1) can be rewritten as S = ~ dz { - 2 w a 0,2a-2i~uu0,~ ,~ + h a [ ( j.b l"~c); C)
( F~udWd) - 2/~a(,~ bWb )
• it q/~F ac v b] +21~t u Fcb2
+ a [ (2aFuab2b) ~tu] },
(4.1)
where ha and a are Lagrange multipliers which enforce the constraints in (3.5). In terms of the supertwistor variables, the superPoincar6 currents in (2.2) can be rewritten: pU=2bF~c2c,
qa=4iq/1,F~ab2 b ,
M l,~ = t 2 b ( F~F~Cd _ F~Fu~a) w a + i ( ~ " g t u _ ~uug/" ) .
(4.2)
4. Supertwistor description of the ten-dimensional superparticle In order to describe the dynamics o f a massless superparticle, it will be convenient to remove the gauge invariance [Wa, 2 b, g/u]~ [Cwa, C2 b, Cgt ~'] by identifying 2~F~'a2d with pu, the superparticle's momentum vector. The supertwistor [Wa, 2 b, ~Uu] will now describe not only the trajectory of the massless particle, but also its energy. This identification is possible if pu satisfies the following two conditions: (1) PuP~'=O, i.e. the superparticle is always on mass shell. Although the supertwistor description will have different off-shell degrees of freedom from the usual description in terms of superspace variables, this will only affect the field interpretation of the secondquantized superparticle, rather than the physical Hilbert space that comes from first quantization. The 48
Note that as a result of the supertwistor constraints, qF~UF"FPlq= 96iMtU"pPl. Canonical first quantization of the superparticle yields the following (anti-)commutation relations for the supertwistor variables: [Wa, 2b]
= - ~ 1" l ~ ab,
{~u",~u~}= ~i. u
(4.3 )
With these commutation relations, it is easy to show that the operators in (4.2) form a superPoincar6 algebra and that the supertwistor constraints in (3.5) generate the fermionic and bosonic gauge transformations (except for the scaling transformation) of (3.7) and (3.8). Because of their anti-commutation relations, the first-quantized gtu operators can be represented by SO ( 1, 9 ) 7-matrices, implying that the superparticle wavefunction transforms like a 32-component SO ( 1, 9) spinor containing both chiralities. In the "mo-
Volume 247, number 1
PHYSICS LETTERS B
m e n t u m " representation o f this wavefunction, 0A (2), the supertwistor constraints i m p l y that G~BOB= ~
(2bFgc2 c) F ~ d
-22"
2b
0A
v c ~x(F , u a b Fb~2 )(TUeAT,COc)--2i2aoA=O,
"JV 1"
g~08 = ~
1
(2bF~2c)Yu~0B = 0 ,
=AuFUd2d,
__ c ba " u f Ob(2)--Cuqc F,# d2 d =IX/~ C cu(FcfJ. ) (F~baF,d2" d)
- ~--i
(2eFufAy)(BcFub)+2ix/~2b(2CBc)
c o m m u t a t i o n relations explicitly d e p e n d on the supertwistor variables) and have only two levels o f reducibility [GaFu~b2b- 4i~'Ug= 0 and (2bFg¢2c) × (Fund2a) = (2bFgc2c) ( - - 4 i ~ ,u) = 0 where G" and g are the supertwistor constraints, (I~UabJ.b ) and --4i~ ,u are the ten first-level zero modes, and (2bFgc2c) is the only second-level zero m o d e ].
(4.4)
where the last term in G ] B comes from n o r m a l - o r d e r ing ambiguities, but can be fixed by requiring that 2 bFg, G~B - ix/~ y~g~ is automatically zero. If one splits the 32-component 0A into its two 16c o m p o n e n t spinors o f opposite chirality, 0a and 0 b, it is easy to check that in ten dimensions, the constraints are satisfied if a n d only if 0,(2)
6 September 1990
Acknowledgement I would like to thank Tom Banks, Michael Douglas, D a n Friedan, Yutaka Matsuo, Alexander Mikovic, Joel Shapiro, Steve Shenker, Sang Jin Sin, A n t o n Van de Ven, a n d John Y a m r o n for useful conversations, and the universities o f Rutgers a n d S U N Y at Stony Brook for their generosity and hospitality. I would especially like to thank Warren Siegel for m a n y helpful suggestions, including the suggestion to write the twistor formulas in covariant notation. This work was s u p p o r t e d by the N S F grant # 90-96107-01.
(4.5)
where q~a is the spacetime s u p e r s y m m e t r y generator, A u a n d Bc = CauFU~ad e p e n d on 2 only through the ten c o m b i n a t i o n s 2 bFgc2~, a n d A u satisfies the c o n d i t i o n Au(2bF~U)=O. Note that 0A is unchanged by the gauge t r a n s f o r m a t i o n s A U~ A u + CA bF~2 ~ or Ba-~ B~+ (2bFgc2 ~) (Fu~dFa) for a r b i t r a r y C a n d F a. By F o u r i e r transforming A u a n d B, into position space, where
7(x) = ~ d~°Xexp[i(AbF~2c)xu]f(2F2) , one sees that the physical Hilbert space o f the quantized superparticle can be constructed out o f a vector field, Au(X), a n d a spinor field, Ba(X), satisfying O ~ = O ~ 0 ~ u = 0"0~/~a=0. These can be interpreted as the on-shell N = 1 super-Yang-Mills fields in ten dimensions. An alternative a p p r o a c h to quantizing the ten-dimensional superparticle is to construct a BRST operator. Using the a p p r o a c h o f Batalin a n d F r a d k i n [ 13 ], this should be straightforward using supertwistor variables since the supertwistor constraints are all first class (although the structure "constants" in their
References [1]M.B. Green and J.H.Schwarz, Nucl. Phys. B 243 (1984) 285. [2 ] S. Mandelstam, Review of aspects of string theory, talk given at Niels Bohr Centennial Conf. (Copenhagen, Denmark, May 1985). [3] E. Nissimov, S. Pacheva and S. Solomon, Nucl. Phys. B 297 (1988) 349. [4] S.J. Gates Jr., M.T.Grisaru, U. Lindstrom, M. Ro~ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 225 (1989) 44. 15] Y. Eisenberg and S. Solomon, Nucl. Phys. B 309 (1988) 709. [ 6 ] U. Lindstrom, M. Ro6ek, W. Siegel, P. van Nieuwenhuizen and A.E. van de Ven, Phys. Lett. B 224 (1989) 285. [7] A. Ferber, Nucl. Phys. B 132 (1978) 55. [8] R. Penrose and M.A.H. MacCallum, Phys. Rep. 6 (1972) 24. [9] T. Shirafuji, Prog. Theor. Phys. 70 (1983) 18. [ 10] I. Bengtsson and M. Cederwall, Nucl. Phys. B 302 (1988) 81. [ 11 ] D.P. Sorokin, V.I. Tkach, D.V. Volkov and A.A. Zheltukhin, Phys. Lett. B 216 (1989) 302. [ 12] W. Siegel, Phys. Len. B 128 (1983) 397. [ 13 ] I.A. Batalin and E.S. Fradkin, Phys. Lett. B 122 ( 1983 ) 157. [ 14] E. Witten, Nucl. Phys. B 266 (1986) 245.
49