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The free energy for compactified supermembranes at non-zero temperature
Volume 243, number 1,2
PHYSICS LETTERS B
21 June 1990
The free energy for compactified supermembranes at non-zero temperature A.A. B y t s e n k o ~ a n d S.D. O d i n t s o v 2 Department of Physics, The University of Newcastle upon Tyne, Newcastle upon Tyne NE I 7RU, UK
Received 7 December 1989; revised manuscript received 21 March 1990
The free energy for the closed supermembrane on a torus compactified background at non-zero temperature is calculated. The existence of the Hagedorn temperature and some cosmological applications are briefly discussed.
1. Introduction Recently, it has been suggested that s u p e r m e m b r a n e s can be candidates for a theory o f everything. The quant u m properties o f s u p e r m e m b r a n e s (which are s u p e r s y m m e t r i c extensions o f m e m b r a n e s [ 1 ] ) have been intensively investigated in recent papers [ 2-11 ]. S u p e r m e m b r a n e s have an intrinsic non-linear structure which is much more c o m p l i c a t e d than the superstring structure. However, for s u p e r m e m b r a n e s one can consider the same questions that one addresses in the case o f superstrings [ 12-14 ] ( t h e r m o d y n a m i c s o f supermembranes, cosmological applications, the calculation o f free energy) using the semiclassical a p p r o x i m a t i o n developed in refs. [2,4]. The t h e r m o d y n a m i c s o f s u p e r m e m b r a n e s on the eleven-dimensional Minkowski space has been investigated in ref. [ 10 ] in the f r a m e w o r k o f the semiclassical a p p r o x i m a t i o n . However, the question o f the existence o f massless states in the s u p e r m e m b r a n e remains open [ 6 - 8 ]. The spectrum o f the massless states o f the superm e m b r a n e d e p e n d s on the b a c k g r o u n d u n d e r consideration. (It seems [8 ] that this spectrum is continuous for the D = 11 s u p e r m e m b r a n e in M i n k o w s k i spacetime.) Probably, the torus c o m p a c t i f i e d b a c k g r o u n d is m o r e interesting as a ground state for the s u p e r m e m b r a n e . Moreover, the torus compactified b a c k g r o u n d is more useful for cosmological applications o f supermembranes. In the present letter we study a closed s u p e r m e m b r a n e in canonical equilibrium at high t e m p e r a t u r e on the c o m p a c t i f i e d b a c k g r o u n d ( S ~( t e m p e r a t u r e ) × R l ~- a - 1× Td ). We calculate the free energy o f the closed superm e m b r a n e on the b a c k g r o u n d S ~( t e m p e r a t u r e ) X R t ~_ d - ~X Td. Explicit examples are given for d = 2 and d = 7. It is shown that the H a g e d o r n t e m p e r a t u r e for the c o m p a c t i f i e d s u p e r m e m b r a n e at non-zero t e m p e r a t u r e is the same as for a s u p e r m e m b r a n e on a U i n k o w s k i space at non-zero t e m p e r a t u r e [ 10 ]. Some cosmological applications are also discussed.
2. Free energy for a compactified supermembrane at non-zero temperature Let us consider the calculation o f the free energy for the closed s u p e r m e m b r a n e c o m p a c t i f i e d on a torus at non-zero t e m p e r a t u r e (for the same calculation in the case o f the open s u p e r m e m b r a n e , see ref. [ 11 ] ). We begin Leningrad Polytechnical Institute, SU-195 251 Leningrad, USSR. 2 Permanent address: Department of Physics and Mathematics, Tomsk Pedagogical Institute, SU-634 041 Tomsk, USSR. 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
63
Volume 243, number 1,2
PHYSICS LETTERSB
21 June 1990
with a well-known expression, which is the sum of the free energies of supersymmetric quantum fields on the background $1 (temperature) × RD-a- 1XTa, 1
F(fl) = ? (det
a) '/2
r &-d-'k
.~aJ ~
{1--exp(-flWk))=Fb(fl)+Ff(fl)
(1)
l°g~l + e x p ( - f l W k )
Here fl is the inverse temperature, a = diag (R F 2, ..., R g-2), Rt are the radii of the toms, D--d--I
Wk=M2+
~
d
d
k] + Y~ Ry2(nj+gj),
j=l
Z R/-2(nj+gJ)z~t(n+g)a(n+g),
j=l
'g-(g, ..... gd),
j=l
gi = 0 or gi = ½ (twisted fields, see ref. [ 15 ] ), g (b)= g (f)_ g. After a little calculation, we get
F(fl)=--i dlTt-d(deta)l/2(2t)-(D-d+2)/2[O3(O~t ) --O4(Ol~t)] 0
×,,~aexp(-i-~tztt(n+g)a(n+g)-i-~nM2).
(2)
Let us consider now the closed orientable supermembrane on the background $1 (temperature) × ~8 × T2. In the semiclassical approximation [ 4 ]:
M2(n)=(ntnzRtR2)2+2 ~ . . • (oG,,i+ a . ,i+
W.S,.A+S=). a .
meZ 2 m~O
n,k, +N~ b) + N I f) = 0 ,
nzk2 +N~ b) =N~ f) ---0.
(3)
Here W 2 = (nlmlRl)2+ (n2m2R2)2, in= (nl, r/2), ki=Ripi, i= 1, 2, Pi are discrete momenta, and ~r(b,
m,.2
1"1,2
'+a~
E --W--~m~ m m~Z2
,
Nl,f:~
=
~ mEZ2
~
cA+cA
Htl,2~Jm O m ,
a.,, S,,, are oscillators (see ref. [4] ). Using eq. (3) in eq. (2), we obtain (see also ref. [ 10 ] ) for the closed supermembrane:
Fc(fl)=-f dtlr-2(deta)l/2(2t)-'l/2[O3(O ~t)-04(O~t)] n
×
1/2
1/2
f
~ dZySTr ~ exp - ~ ' ( n + g ) a ( n + g ) -
~M2(n)+i2zry,(n,k,+N[ b ) + N t f))
- - I / 2 --1/2
+i2~y2(nzk2+ Ntb) ~ +N ( f ))) . Calculating the supertrace with the help of the technique shown in ref. [ 10 ], we obtain
64
Volume 243, number 1,2
PHYSICS LETTERS B
21 June 1990
E I/2
)
(
I/2
-1/2 -1/2
X[tanh(~nn,R,-iny,)tanh(~nER2-iny=)~(2ny.(t)=c)] -8
(4)
Here C=diag( n2 R~, n22 R 2~ 2J,
h(Z, I2) = ,~I~-I { 1 - 2 exp[ - (tmt2m) ~/2] cos(tin, Z) +exp[ -2(tmOm)~/2]}, q(Z, £2) = mini {1 + 2 exp[ - (trat2m) ~/2] cos(tin, z ) +exp[ -2(tm~2ra)~/2]} are the products over all d-tuples of natural numbers, ~ is a cut-off parameter (see the discussion in ref. [ 10] ), which is needed to make the ultraviolet behaviour regular. The asymptotics of the functions h, q are given in ref. [ 10 ]. Using these asymptotics one can show that the infrared behaviour ( t ~ ~ ) of the integrand (4) is regular. In the ultraviolet limit (t--, E~ 0) the behaviour of the integrand (4) is e x p ( - /rfl2~ / 8 X 7 n s ((3) - ~ - ] n~2 exp~ ~- nlRln2R2],
(5)
where the value of the Riemann zeta function ~(3) = 1.20205 .... Then with the help of the following asymptotics (t~,0): /567t3~(3)'~
{ 56/t3~(3) '~
,,
/56n3((3)'~
.~ e x p lnj~Rk, n2R2]'~ t i ; 4xdyexP~t2R,R2xyJ~Zexp~tER,R: J' --c~
--oo
it is possible to rewrite the expression (5) as follows: exp(-
7~']~2"~
/561t3~(3)'~
) exp k ~
)
Thus, we see that there is a limiting "Hagedorn temperature" in the theory. The Hagedorn temperature is fl~_ 16n 2 7~(3__)
R1R2 ' and if R t = n/a, R2 =/t/b, ~= b, fl~ is the same as in Minkowski space [ 10]. In this case, as in Minkowski space [ 10 ], tic depends on the cut-off parameter ~. (It is natural to consider t as the minimal length of the theory [ 10 ]. ) When fl < tic an ultraviolet divergence exists. Possibly, as in the case of superstrings, tic corresponds to a phase transition (it is essential to use the microcanonical ensemble to understand the behaviour of the system in this case). When T = 0 (fl = oo ) Fc = 0. However, ifg tb) ¢ g ~t) (in this case one must slightly modify eq. (4) ) then Fc (fl) ~ 0 at any T. (The analogous situation for the closed superstring has been discussed in ref. [ 14 ]. ) Using the results of the semiclassical quantization of the supermembrane one the background S~ (temperature) X ~ 3 X T 7 [2] in the same way as above we get 65
Volume 243, number 1,2
PHYSICS LETTERS B
21 June 1990
F¢(fl)=- i dtn-7(deta)'/2(2t)-3[03(O i-a%o'(°l\ I i-a=ll L \ 12t ] 2t)3 1/2
X
f
d2y ~
--1/2
n,k~Z 7
('
,
exp-~-~(AB)2-~-~'(n+g)a(n+g)+i2~z[(n,k,+...+n6k6)y,+n7k7Y2]
X[tanh(~A_inyl)tanh(~B_iny2))~(2ny,
( ~t ) 2C , ) ]
)
-s .
(6)
Here
A=x/(nlRl)2+...+(n6R6) 2, B=n7R7, a=diag(RF2,...,R72), Cl=diag(A2, B2). For the case g = 0 eq. (6) has been obtained in another form in ref. [ 11 ]. One can also find fie as above. Two generalizations of the above calculation are immediately suggested. The first is the extension to other D (and d) [ 5 ]: D = 4, D = 5 or D = 7. Secondly, it is of interest to study the thermodynamics of supersymmetric pbranes, and also the thermodynamics of supermembranes on a curved background [ 9 ].
3. Cosmological applications Let the internal space be static. We assume the following form for the metric: gu, = diag( - 1, R2(t)gu,
A2(t)g,p),
where A = 0, i = 1..... 3, ol = 4, ..., 7. In the limit fl~ tic and assuming that ( 1 ) the Einstein equations are valid, and (2) the radiation-dominated epoch holds as for closed superstrings [ 12 ], one obtains
R(t)~t 2/3, p(t)~t -2,
(7)
as in the case of closed superstrings (Matsuo [ 12 ] ). Here T~,,= diag(p, Pgu,P 1gap).However, it is possible that the equations are not valid for superstrings (see Brandenberger and Vafa [ 12 ] ) and also for supermembranes. Thus, eqs. (7) are very speculative.
Acknowledgement S.D.O. thanks David Wiltshire for reading the manuscript and Paul Davies, David Toms and the Physics Department of the University of Newcastle upon Tyne for their kind hospitality. He would also like to thank the Royal Society for financial support.
References [ 1 ] P.A.M. Dirac, Proc. R. Soc. A 268 (1962) 57; P.A. Collins and R.W. Tucker, Nucl. Phys. B 112 ( 1976 ) 150; P.S.Howe and R.W. Tucker, J. Phys. A 10 ( 1977 ) L 155; J. Hughes, J. Liu and J. Polchinski, Phys. Lett. B 160 (1986) 370. [2 ] E. Bergshoeff, E. Sezgin and P.K. Townsend, Phys. Lett. B 189 ( 1987 ) 75; B 209 ( 1988 ) 451; Ann. Phys. 185 ( 1988 ) 330. [3] I. Bars, C.N. Pope and E. Sezgin, Phys. Lett. B 198 (1987) 455. [4 ] M.J. Duff, T. Inami, C.N. Pope, E. Sezgin and K.S. Stelle, Nucl. Phys. B 297 (1988) 515. [5 ] A. Achucarro, J.M. Evans, P.K. Townsend and D.L. Wiltshire, Phys. Lett. B 198 (1987) 75.
66
Volume 243, number 1,2
PHYSICS LETTERS B
21 June 1990
[ 6] M.J. Duff, Proc. XVI Intern. School on Subnuclear physics (Erice, 1988 ), to appear; M.P. Blencowe and M.J. Duff, Nucl. Phys. B 310 (1988) 387. [ 7 ] K. Kikkawa and M. Yamasaki, Prog. Theor. Phys. 76 (1986) 1379; L. Mezincescu, R. Nepomechie and P. van Nieuwenhuizen, Nucl. Phys. B 309 (1988) 317; S. Ghandhi and K.S. Stelle, Class. Quantum Grav. 5 (1988) L127; I. Bars, Nucl. Phys. B 308 (1988) 462; I. Bars and C.N. Pope, Class. Quantum Gray. 5 (1988) 1; F. Paccanoni, P. Pasti and M. Tonin, Padova preprint DFPD/88/TH/9; U. Marquard, R. Kaizer and M. School, Karlsruhe preprint -THEP-4 ( 1989); C.N. Pope and K.S. Stelle, Imperial preprint TH. 1987-88/27. [8] B. de Wit, J. Hoppe and H. Nicolai, Nucl. Phys. B 305 (1988) 545; B. de Wit, M. Liischer and H. Nicolai, preprint DESY 88-162 (1988). [9] E. Bergshoeff, A. Salam, E. Sezgin and Y. Tanii, Phys. Lett. B 205 (1988) 237; H. Nicolai, E. Sezgin and Y. Tanii, Nucl. Phys. B 305 (1988) 483; M.J. Duff, C.N. Pope and E. Sezgin, Texas A & M preprint CTP-TAMU-01/89. [ 10] A.A. Bytsenko and S.A. Ktitorov, Phys. Lett. B 225 (1989) 325. [ 11 ] S.D. Odintsov, Newcastle preprint NCL-89-TP/18. [ 12] N. Matsuo, Z. Phys. C 36 (1987) 470; E. Alvarez and M.A.S. Osorio, Phys. Rev. D 36 ( 1987 ) 1175; Nucl. Phys. B 304 ( 1988 ) 327; M. McGuigan, Phys. Rev. D 38 (1988) 552; J. Atick and E. Witten, Nucl. Phys. B 310 (1988) 291; H. Brandenberger and C. Vafa, Nucl. Phys. B 316 (1989) 391. [ 13] S.D. Odintsov, Europhys. Lett. 8 (1989) 207; I.M. Lichtzier and S.D. Odintsov, Tomsk preprint No. 50 (1989). [ 14] R. Rohm, Nucl. Phys. B 237 (1984) 553; K. Kikkawa and M. Yamasaki, Phys. Lett. B 149 ( 1984 ) 357; I.L. Buchbinder and S.D. Odintsov, Intern. J. Mod. Phys. A 4 ( 1989 ) 4337. [ 15 ] C.J. Isham, Proc. R. Soc. London A 362 ( 1978 ) 383; A 364 ( 1978 ) 591; S.J. Avis and C.J. Isham, Nucl. Phys. B 156 (1979) 441.
67
PHYSICS LETTERS B
21 June 1990
The free energy for compactified supermembranes at non-zero temperature A.A. B y t s e n k o ~ a n d S.D. O d i n t s o v 2 Department of Physics, The University of Newcastle upon Tyne, Newcastle upon Tyne NE I 7RU, UK
Received 7 December 1989; revised manuscript received 21 March 1990
The free energy for the closed supermembrane on a torus compactified background at non-zero temperature is calculated. The existence of the Hagedorn temperature and some cosmological applications are briefly discussed.
1. Introduction Recently, it has been suggested that s u p e r m e m b r a n e s can be candidates for a theory o f everything. The quant u m properties o f s u p e r m e m b r a n e s (which are s u p e r s y m m e t r i c extensions o f m e m b r a n e s [ 1 ] ) have been intensively investigated in recent papers [ 2-11 ]. S u p e r m e m b r a n e s have an intrinsic non-linear structure which is much more c o m p l i c a t e d than the superstring structure. However, for s u p e r m e m b r a n e s one can consider the same questions that one addresses in the case o f superstrings [ 12-14 ] ( t h e r m o d y n a m i c s o f supermembranes, cosmological applications, the calculation o f free energy) using the semiclassical a p p r o x i m a t i o n developed in refs. [2,4]. The t h e r m o d y n a m i c s o f s u p e r m e m b r a n e s on the eleven-dimensional Minkowski space has been investigated in ref. [ 10 ] in the f r a m e w o r k o f the semiclassical a p p r o x i m a t i o n . However, the question o f the existence o f massless states in the s u p e r m e m b r a n e remains open [ 6 - 8 ]. The spectrum o f the massless states o f the superm e m b r a n e d e p e n d s on the b a c k g r o u n d u n d e r consideration. (It seems [8 ] that this spectrum is continuous for the D = 11 s u p e r m e m b r a n e in M i n k o w s k i spacetime.) Probably, the torus c o m p a c t i f i e d b a c k g r o u n d is m o r e interesting as a ground state for the s u p e r m e m b r a n e . Moreover, the torus compactified b a c k g r o u n d is more useful for cosmological applications o f supermembranes. In the present letter we study a closed s u p e r m e m b r a n e in canonical equilibrium at high t e m p e r a t u r e on the c o m p a c t i f i e d b a c k g r o u n d ( S ~( t e m p e r a t u r e ) × R l ~- a - 1× Td ). We calculate the free energy o f the closed superm e m b r a n e on the b a c k g r o u n d S ~( t e m p e r a t u r e ) X R t ~_ d - ~X Td. Explicit examples are given for d = 2 and d = 7. It is shown that the H a g e d o r n t e m p e r a t u r e for the c o m p a c t i f i e d s u p e r m e m b r a n e at non-zero t e m p e r a t u r e is the same as for a s u p e r m e m b r a n e on a U i n k o w s k i space at non-zero t e m p e r a t u r e [ 10 ]. Some cosmological applications are also discussed.
2. Free energy for a compactified supermembrane at non-zero temperature Let us consider the calculation o f the free energy for the closed s u p e r m e m b r a n e c o m p a c t i f i e d on a torus at non-zero t e m p e r a t u r e (for the same calculation in the case o f the open s u p e r m e m b r a n e , see ref. [ 11 ] ). We begin Leningrad Polytechnical Institute, SU-195 251 Leningrad, USSR. 2 Permanent address: Department of Physics and Mathematics, Tomsk Pedagogical Institute, SU-634 041 Tomsk, USSR. 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )
63
Volume 243, number 1,2
PHYSICS LETTERSB
21 June 1990
with a well-known expression, which is the sum of the free energies of supersymmetric quantum fields on the background $1 (temperature) × RD-a- 1XTa, 1
F(fl) = ? (det
a) '/2
r &-d-'k
.~aJ ~
{1--exp(-flWk))=Fb(fl)+Ff(fl)
(1)
l°g~l + e x p ( - f l W k )
Here fl is the inverse temperature, a = diag (R F 2, ..., R g-2), Rt are the radii of the toms, D--d--I
Wk=M2+
~
d
d
k] + Y~ Ry2(nj+gj),
j=l
Z R/-2(nj+gJ)z~t(n+g)a(n+g),
j=l
'g-(g, ..... gd),
j=l
gi = 0 or gi = ½ (twisted fields, see ref. [ 15 ] ), g (b)= g (f)_ g. After a little calculation, we get
F(fl)=--i dlTt-d(deta)l/2(2t)-(D-d+2)/2[O3(O~t ) --O4(Ol~t)] 0
×,,~aexp(-i-~tztt(n+g)a(n+g)-i-~nM2).
(2)
Let us consider now the closed orientable supermembrane on the background $1 (temperature) × ~8 × T2. In the semiclassical approximation [ 4 ]:
M2(n)=(ntnzRtR2)2+2 ~ . . • (oG,,i+ a . ,i+
W.S,.A+S=). a .
meZ 2 m~O
n,k, +N~ b) + N I f) = 0 ,
nzk2 +N~ b) =N~ f) ---0.
(3)
Here W 2 = (nlmlRl)2+ (n2m2R2)2, in= (nl, r/2), ki=Ripi, i= 1, 2, Pi are discrete momenta, and ~r(b,
m,.2
1"1,2
'+a~
E --W--~m~ m m~Z2
,
Nl,f:~
=
~ mEZ2
~
cA+cA
Htl,2~Jm O m ,
a.,, S,,, are oscillators (see ref. [4] ). Using eq. (3) in eq. (2), we obtain (see also ref. [ 10 ] ) for the closed supermembrane:
Fc(fl)=-f dtlr-2(deta)l/2(2t)-'l/2[O3(O ~t)-04(O~t)] n
×
1/2
1/2
f
~ dZySTr ~ exp - ~ ' ( n + g ) a ( n + g ) -
~M2(n)+i2zry,(n,k,+N[ b ) + N t f))
- - I / 2 --1/2
+i2~y2(nzk2+ Ntb) ~ +N ( f ))) . Calculating the supertrace with the help of the technique shown in ref. [ 10 ], we obtain
64
Volume 243, number 1,2
PHYSICS LETTERS B
21 June 1990
E I/2
)
(
I/2
-1/2 -1/2
X[tanh(~nn,R,-iny,)tanh(~nER2-iny=)~(2ny.(t)=c)] -8
(4)
Here C=diag( n2 R~, n22 R 2~ 2J,
h(Z, I2) = ,~I~-I { 1 - 2 exp[ - (tmt2m) ~/2] cos(tin, Z) +exp[ -2(tmOm)~/2]}, q(Z, £2) = mini {1 + 2 exp[ - (trat2m) ~/2] cos(tin, z ) +exp[ -2(tm~2ra)~/2]} are the products over all d-tuples of natural numbers, ~ is a cut-off parameter (see the discussion in ref. [ 10] ), which is needed to make the ultraviolet behaviour regular. The asymptotics of the functions h, q are given in ref. [ 10 ]. Using these asymptotics one can show that the infrared behaviour ( t ~ ~ ) of the integrand (4) is regular. In the ultraviolet limit (t--, E~ 0) the behaviour of the integrand (4) is e x p ( - /rfl2~ / 8 X 7 n s ((3) - ~ - ] n~2 exp~ ~- nlRln2R2],
(5)
where the value of the Riemann zeta function ~(3) = 1.20205 .... Then with the help of the following asymptotics (t~,0): /567t3~(3)'~
{ 56/t3~(3) '~
,,
/56n3((3)'~
.~ e x p lnj~Rk, n2R2]'~ t i ; 4xdyexP~t2R,R2xyJ~Zexp~tER,R: J' --c~
--oo
it is possible to rewrite the expression (5) as follows: exp(-
7~']~2"~
/561t3~(3)'~
) exp k ~
)
Thus, we see that there is a limiting "Hagedorn temperature" in the theory. The Hagedorn temperature is fl~_ 16n 2 7~(3__)
R1R2 ' and if R t = n/a, R2 =/t/b, ~= b, fl~ is the same as in Minkowski space [ 10]. In this case, as in Minkowski space [ 10 ], tic depends on the cut-off parameter ~. (It is natural to consider t as the minimal length of the theory [ 10 ]. ) When fl < tic an ultraviolet divergence exists. Possibly, as in the case of superstrings, tic corresponds to a phase transition (it is essential to use the microcanonical ensemble to understand the behaviour of the system in this case). When T = 0 (fl = oo ) Fc = 0. However, ifg tb) ¢ g ~t) (in this case one must slightly modify eq. (4) ) then Fc (fl) ~ 0 at any T. (The analogous situation for the closed superstring has been discussed in ref. [ 14 ]. ) Using the results of the semiclassical quantization of the supermembrane one the background S~ (temperature) X ~ 3 X T 7 [2] in the same way as above we get 65
Volume 243, number 1,2
PHYSICS LETTERS B
21 June 1990
F¢(fl)=- i dtn-7(deta)'/2(2t)-3[03(O i-a%o'(°l\ I i-a=ll L \ 12t ] 2t)3 1/2
X
f
d2y ~
--1/2
n,k~Z 7
('
,
exp-~-~(AB)2-~-~'(n+g)a(n+g)+i2~z[(n,k,+...+n6k6)y,+n7k7Y2]
X[tanh(~A_inyl)tanh(~B_iny2))~(2ny,
( ~t ) 2C , ) ]
)
-s .
(6)
Here
A=x/(nlRl)2+...+(n6R6) 2, B=n7R7, a=diag(RF2,...,R72), Cl=diag(A2, B2). For the case g = 0 eq. (6) has been obtained in another form in ref. [ 11 ]. One can also find fie as above. Two generalizations of the above calculation are immediately suggested. The first is the extension to other D (and d) [ 5 ]: D = 4, D = 5 or D = 7. Secondly, it is of interest to study the thermodynamics of supersymmetric pbranes, and also the thermodynamics of supermembranes on a curved background [ 9 ].
3. Cosmological applications Let the internal space be static. We assume the following form for the metric: gu, = diag( - 1, R2(t)gu,
A2(t)g,p),
where A = 0, i = 1..... 3, ol = 4, ..., 7. In the limit fl~ tic and assuming that ( 1 ) the Einstein equations are valid, and (2) the radiation-dominated epoch holds as for closed superstrings [ 12 ], one obtains
R(t)~t 2/3, p(t)~t -2,
(7)
as in the case of closed superstrings (Matsuo [ 12 ] ). Here T~,,= diag(p, Pgu,P 1gap).However, it is possible that the equations are not valid for superstrings (see Brandenberger and Vafa [ 12 ] ) and also for supermembranes. Thus, eqs. (7) are very speculative.
Acknowledgement S.D.O. thanks David Wiltshire for reading the manuscript and Paul Davies, David Toms and the Physics Department of the University of Newcastle upon Tyne for their kind hospitality. He would also like to thank the Royal Society for financial support.
References [ 1 ] P.A.M. Dirac, Proc. R. Soc. A 268 (1962) 57; P.A. Collins and R.W. Tucker, Nucl. Phys. B 112 ( 1976 ) 150; P.S.Howe and R.W. Tucker, J. Phys. A 10 ( 1977 ) L 155; J. Hughes, J. Liu and J. Polchinski, Phys. Lett. B 160 (1986) 370. [2 ] E. Bergshoeff, E. Sezgin and P.K. Townsend, Phys. Lett. B 189 ( 1987 ) 75; B 209 ( 1988 ) 451; Ann. Phys. 185 ( 1988 ) 330. [3] I. Bars, C.N. Pope and E. Sezgin, Phys. Lett. B 198 (1987) 455. [4 ] M.J. Duff, T. Inami, C.N. Pope, E. Sezgin and K.S. Stelle, Nucl. Phys. B 297 (1988) 515. [5 ] A. Achucarro, J.M. Evans, P.K. Townsend and D.L. Wiltshire, Phys. Lett. B 198 (1987) 75.
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21 June 1990
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