deformed string amplitudes for closed and open stdngs and analize their properties. ... In the context of quantum gravity, string theory plays a privilegiated role: it.
STRING THEORY, QUANTUM GRAVITY AND QUANTUM GROUPS N. S~inchez DEMIRM,Observatoire de Paris, Section de Meudon 92195 Meudon Principal Cedex, France ABSTRACT:Within the perspective to understand physics at the Planck scale, we give a quantum group generalization of string theory.A deformation of the Helsenberg commutation relations defines a one-parameter family of string models.A hamiltonlan approach is developped.We find the effect of the quantum group deformation on the Virasoro algebra.We construct the N-point quantum group deformed string amplitudes for closed and open stdngs and analize their properties.We find the spectrum and high energy behaviour.The quantum group deformation removes the usual equal point singularities of propagators and shifts the intercept to s=0.1n spite of the fact that the theory exhlbtts Regge trajectories, the behavtour for s -~ == is of Born type. In the context of quantum gravity, string theory plays a privilegiated role: it gives a uv finite theory including the graviton.ln the absence of experimental guidance it is worthwhile to explore new mathematical structures connected with string theory, infinite dimensional algebras and beyond.The study of integrable theories and their associated Yang-Baxter (YB) algebras [1] suggested deformations of the universal envelopping Lie algebras [2] .This is now called a "quantum group".More recently,vertex representations of these algebras have been given [3,4].They involve operators fulfilling Heisenberg type relations.Here we present a quantum group generalization of bosonic string theory.Details are given in reference [5].Associating a SU(2)-quantum group for each space-time dimension yields [ ~m A , (~n B ] = 5n+ m TIAB cos'~n (sin ~n )2 n 3'2 Here y is the deformation parameter ( anisotropy parameter for statistical models).For 3,=0 we recover the usual Heisenberg commutation relations.~ AB stands for the Minkowski space-time metric.Hence,we construct a one-parameter continuos deformation of the usual string models.The scalar vertex operators for closed strings for a SU(2) quantum group reads V(z,~,p ) = VL(z ) VR(~ ) where z=exp 2i(~ +,c), VL(z)..
:expip. XL ( z ) :
XL A ( z ) = l q A 2
.i__pAlnz 2
,
VR(z)=
+ i~ ~
~
2
n,~O
"expip. X R ( ~ ) " z- n ~ n A sin n3,
We interpret here X A (~,~) = XLA(z ) + XRA(E) as the string coordinate.It obeys the usual wave equation on the world sheet (~) 2 _ ~)2 ) xA(~,~ )=0
but the
commutation relations depend on 3, even at equal times (~).Therefore we find
204
for the canonical conjugate momentum an expression world-sheet.The equations of motion derive from the action
non-local
in the
4
) S = 8=o~ o do d'c ,,-_o (.1)n )gx A ( On+ ' ,{: ) ~)I~ x A ( which is non-local on the world-sheet.The 1'-deformation has splitted the points in the kinetic energy by a distance 2c n = (n+1/2) with n ~ 7_. This action is translational invariant on ( ( > , ~ ) but lacks on manifest reparametrization invariance.The equations of motion are fully conformally invariant.The conserved Tp.vtensor is given by O0
T++ = 4 --
--
Z
(-1) n 2+ X A (On+).
'Y~ =,0
--
2+ X A (an-)
,
T+_ -- T_+ = 0
--
where x+--(G+~:). T++ can be expanded as T++ -- 8 , '~__, exp2in(l:+.o'),~Ln~ , Ln~ -- (=.-!-)t Z p(#.p)~/2 cos[ ( 2p- n ) (-~+ 1/2 ) 1'] (Xp (~n-p 2 p~Z sinyp sin-/(-I'-p) They fulfill Ln#= Ln "t'1 .These Ln~are deformations of the usual Virasoro operators.When 1' ~ 0 we recover the usual Ln and T~v operators [6].The Ln"(] satisfy the following algebra up to a central term [ Lnr,
LmS ]
8n,mS, r+l
+
= (n-m) Ln+mr+S
(4) -1
(Ln+m s'r 8m,n r,s+l
Bm,nr, -s + Ln+m r+s+l
(-Mn+m s'r Pm,nr, s+l + Mn+m r-s Pn,mS, r+l + + Mn+mr+S+l Pm,nr+l,-s-1) where
8mn rs = cos [l'(mr+ns)]
,
+ Ln+m r's
Bm,nr+l,'s-1 ) + (4)-1 Mn+mr+S Pm,n r,'s
+
Pmn rs = sin [1' (mr+ns) ]
and
Mn~-- (-1~ Z p(n-p) T?- O~pC(n.p (2p-n) sin [ 1'(2p-n)~+1/2 ) ] 2 pc z sin~,p sin ~, (n-p) The effect of the deformation on the Virasoro algebra transforms it in a three-index infinite algebra. The N-scalar tree amplitude for the present quantum group generalized strings are A M (k) -- .~ d}~M~' (y) IM ( k , y ) where k = ( k I ............ k M ) and
IM open (k~)
= R 1 ,~. For s -~ oo, t -~ =o with sit fixed, we find '~'4(s,t)open s,t ~
oo =
16 (1-cosy) log [ I,.~' ( 1 - c o s ® ) 1 + O( 1 ) s 2 cos e ( 1-cos • ) 8(1-cos~.) s3 cos ~ = l+2t/s.Once more this behaviour is different from the very soft behaviour of the usual strings. We have reported here a quantum group generalization of string amplitudes and studied their properties . These are the first steps to investigate the physical relevance of the models constructed in this note. This open several directions of research. REFERENCES [1]See for reviews: P.P. Kulish and E.K. Sklyanin, in Springer LectNotes in Physics (1981) vol 151. H.J. de Vega,lnt. Jour. Mod. Phys. A4, 735 (1990) and references therein. [2]V.G. Drinfeld, Dokl Akad Nauk 32 (1985) 25 M. Jimbo, LMP 11 (1986) 247 and LMP10 (1985) 63. [3]I.B. Frenkel and N.Jing, Yale preprint(1988),Mathematics. [4]V.G. Drinfeld, Dokl Akad Nauk 296 (1987) 135. [5]H.J.de Vega and N. S~nchez ,PAR-LPTHE 88-30 and Meudon-DEMIRM 88-101 preprint, Phys. Lett. 216B,97 (1989). [6]M. Green,J.H. Schwartz and E. Witten, Superstring Theory, vol 1 and 2(1987),Cambridge Univ. Press.
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