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Universality of the closed string phase transition
Nuclear Physics B334 (1990) 472-488 North-Holland
UNIVERSALITY OF THE CLOSED STRING PHASE TRANSITION* F. ENGLERT and J. ORLOFF** Unioersité Libre de Bruxelles, C. P. ?25, Boulevard du Triomphe, 1050 Brussels, Belgium Received 13 July 1989
The universality of the high-temperature phase for closed string theories in D > 4 flat space-time dimensions is established and extended to other theories involving random walk shaped structures .
1. Introduction
Theories of fundamental closed strings, exhibit, in the weak coupling limit, a phase transition when the dimensionality of space-time is at least equal to four [1] . In the semi-classical description, the high-temperature phase is characterized by the condensation, at a fixed temperature T = To , of an infinite string filling space with a finite constant average density . The condensate lives in thermal equilibrium with a scale invariant distribution of macroscopic closed strings cut off at low energies by a planckian spectrum of massless particles ; all macroscopic strings are random walk shaped [l, 2]. In this paper we shall show that the structure of this "stringy" phase is entirely determined by the random walk structure of the semi-classical closed strings and does not rely significantly on string theory itself. For instance, at high temperature, the scale invariant distribution is the same, not only for all fundamental closed string theories in a flat space-time background, but also for idealized thin cosmic strings although in the former case the random walk shape stems from the uncertainty principle while in the latter it is imposed as an external constraint . The only relevant model dependent parameter in the stringy phase is the temperature of the phase transition itself. In sect. 2 we study the character of the conformal group in the flat compact dimensions of a closed string theory. We derive its asymptotic behaviour for large bosonic and fermionic masses . The result depends only on the left- and right-sector central charges and on the number of space-time dimensions . * Supported in part by NATO C.R .G. 890404 . ** Chercheur IISN . 0550-3213/90/$03 .50 ~-= Elsevier Science Publishers B.V . (North-Holland)
F. Englert, J. Orloff / Closed string phase transition
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The asymptotic character is used in sect. 3 to derive the asymptotic distribution of states for a single closed string . The universality of the phase transition for all theories of fundamental closed strings in a flat space-time background follows as a corollary. In sect. 4, this universality is shown to be rooted in the random walk structure of semi-classical relativistic closed strings ; the description of the phase transition is then extended to include all such structures, without reference to an underlying fundamental scheme.
2. Universality of the asymptotic spectrum 2.1 . CLOSED BOSONIC STRING
Let us first consider a closed string in D space-time dimensions obtained by compactifying S = 26 - D dimensions of the usual bosonic string on a torus . The squared-mass partition function, which appears e.g. in the one-loop amplitude [3], is given by P(T; T) _ Y_ dML ; MR e2ariTML-2wi i:MR ML ; MR ( ; - ~~I(T)~1(T)~-241,5,S T
T)>
(2 .1)
where ML(R)
= 4 arm L(R)
= 1PL(R) + NL(R) - 1
(2 .2)
is the dimensionless squared mass of a left (right) string state with compact 8-dimensional momentum PL(R) and NL(R) oscillator excitations ; dmL ; MR is the number of different closed string states with fixed masses . Notice that such states are physical only if mL = mÁ= m 2 but this is not necessarily the case in the partition function appearing in eq. (2.1) . This function contains on the one hand the Dedekind 17-function, 00
,q(T)=e( iir/12)TFl
n=1
/ (1-e 2ariTn ) = (_1T) -1/2 n(-1 ,r) e
(2 .3)
which counts oscillator excitations and on the other hand the partition function of some lorentzian even self-dual lattice, 5;5( T ;
T
)_
ei p2-i~Tp2 (PL ;
PR)
(2 .4)
which represents both compact momenta and winding numbers [4]. For instance, for
F. Englert, J. Orloff / Closed string phase transition
47 4
8 = 1 compact dimension of radius R, this lattice would be given by (PLI PR) =
1 (
k
a' R
+1
R a'
;k
a' R
-1
R a'
(2 .5)
where k is some integer labelling the different possible momenta k/R and 1 is the number of times the string winds around the direction 8. Notice that since these integers have to be the same in PL and PR, T is not in general the direct product of independent left and right lattices, so that the compactification generally ruins the factorizability of P(T ; T) The modular properties of the rl-function (see eq. (2.3)), together with the evenness and the self-duality of the lattice l (see appendix A) imply that P(T+1 ;T+1)=P(T ;T),
P(-1/T ;-1/T)=(TT)(2-D)/2P(T ; r), (2 .6)
as required by the modular invariance of the full one-loop amplitude. This property turns out to be very powerful for obtaining estimates of the asymptotic spectrum in conformal theories. Indeed, all the information about the large-mass spectrum is encoded in the small-T partition function, P(,r ;T)
ó
(TT)(°-2)/2e2ffi/T-2~i/r[1+O(e-(2ff/ITD(R2 /« +a'/Rz))1, (2 .7)
which is dominated by the tachyon contribution . Unicity of this tachyonic ground state forbids a multiplicative constant in (2.7), a fact which will prove to be quite general . 2 .2 . FERMIONIC STRING
Let us now turn to fermionic closed strings, i.e. string theories which contain both two-dimensional bosons and fermions, and where conformal invariance is enlarged to a superconformal group. This extension can be made in the left or in the right sector (heterotic strings) or in both sectors (type-11 strings). We will first study extensively the heterotic strings, and then briefly sketch the computation for type-11 strings . Consider the supersymmetric right-moving sector of a heterotic theory in D = 10 - 2n space-time dimensions in the light-cone gauge. It consists of: (1) 8 - 2n transverse bosons ; (ii) Their 8 - 2n fermionic partners with + or - boundary conditions which can be bosonized into 4 - n compact bosons with momenta lying on the weight lattice of the transverse Lorentz group D4 - ; (iii) Extra degrees of freedom contributing 3n to the central charge in order to reach the light-cone gauge critical value ctot = 12; for these the simplest choice is either a set of 6n free fermions with non-trivial boundary conditions [5], or what
F. Englert, J. Orloff / Closed string phase transition
475
seems to be more general if not equivalent, a set of 3n free compact bosons . (See ref. [6] for a very clear review of bosonic compactification .) Although other choices are possible we will stick to this last one in this paper. Superconformal invariance of the resulting patchwork has of course to be checked but as we will see, this further constraint is not needed to determine uniquely the asymptotic spectrum ; (iv) Residual momenta left over from the superconformal ghosts which cannot be totally eliminated because of the picture-changing phenomenon [7] . This last point requires a "choice of picture" in order to make sure that every physical state appears only once. For our purposes, the most convenient way of achieving this is to start from the bosonic covariant formulation of the heterotic string [8] where the D,-,, weights are extended with the bosonized longitudinal fermions and bosonized superghost into D5 -,,, weights . We then use the "even lattice mapping" [6] between classes of DS-n,l and D,-,, which preserves modular invariance to all loops whereby the Dl, I ghost weights are mapped onto weights of a D4 ghosts" . We finally obtain for compact momenta a lorentzian lattice of the form F16+2n ;3n+8-n > which modular invariance forces to be even and self-dual . The choice of picture is now achieved by fixing the D4ghosu"-part of T to be the canonical representatives of the (v) and (s) classes *: (v) class : For every momentum of the form (
3n w 4-n 1 0 PO' 1'0'0 10) - ( PVL ; PVR,11 0'0'0) - ( PVL ; v
0
0)
(2 .8)
there is one pure momentum (i.e. without oscillator excitations) physical state. This state is a boson since a weight of D8 -n can take this particular form only if w4- n lies in the bosonic representations (0) or (v) of the transverse Lorentz group . All other bosonic physical states are obtained from these by action of the transverse oscillators . (s) class: For every momentum of the form 1 1 1, 1 1, 1 3n ( PSI21121-(PSL ;PsR12 2>2 z)-(PsL ;v 121 12)
,w 4-n ~21_ I
1 2
1
2
1
2)~
(2 .9)
there is one pure momentum physical fermionic state, with w4- n belonging to the classes (s) or (c). All other fermionic states are obtained from these by action of the transverse oscillators . The dimensionless squared-mass of these states is simply given again by eq. (2.2), but with NR counting oscillations in only 12 directions: 8 - 2n transverse spatial, * Along
with standard conventions, we shall take D roots to be (0, . . ., ± 1, 0, . . ., ± 1, 0 . . . . ), while the other classes of weights are : ( v ) = (1,0 . . . . ) + roots, ( s ) - ( i , i , z . . . . ) + roots, and (c) -t(-a " Z,Z, . . . +roots .
47 6
F. Englert, J. Orloff / Closed string phase transition
their 4 - n bosonized fermionic partners and 3n compactified internal dimensions . Notice that such a theory can always be embedded in a consistent bosonic string theory of the type considered in the last section if we tensor F16+2n ;3n+s-n by the even, self-dual root lattice of Eg to obtain l's; s, i.e. in the same way as was originally done in refs. [9,10]. The fermionic string states are then the subset of bosonic states with the Eg root being identically 0, the D4gh°scs'' weight being as in (2 .8) or (2.9) and the oscillators in the corresponding 12 dimensions being frozen in their ground state . Let us call the bosonic and fermionic contributions to the partition function respectively B(T ; T) I`a(T ;
-
n
(T)-24
\
TI - Y,
(p.)
n (T)- 12 1-,
u(Te T),
F(T ;
T) =~1(T)-24
n (T)-12Fs(,r
2 2 ei"TP;L-i"TpáR .
We will now show that for
T
; Tl+
(2 .10)
small enou
B( r ; T) -~ F(T ; T) -> 2(TT)4-n e2artlT-ail-T T_o T_o so that (1) the spectrum is always asymptotically supersymmetric in these fermionic theories ; (ii) B + F is the same as (2.7) except for a factor in the exponential which reflects the difference in the right central charge. Why worry about B + F? Let us note that although what appears in the one-loop amplitudes like e.g. the cosmological constant is the difference B - F, what we need in the evaluation of a thermal ensemble of strings is rather the sum B + F, just like for an ordinary field theory containing both bosons and fermions in the Maxwell-Boltzmann limit (see subsect . 3.2) . We first show that although F16+2n :3n+4-n+4 is not necessarily a direct product, it always contains a direct product sublattice y = (po) X (0) where (po) is by definition the set of all (16 + 2n ; 4 + 2n)-dimensional vectors that can appear in F together with a D49'°sls ' root. Since by definition, all F vectors have a D4 weight for their last 4 components, any vector of the form (po = 0, (0)) will have integer scalar product with them and therefore belongs to the self-dual F. The closure of F under addition thus requires any vector of (po) X (0) to be in F. Having found such a sublattice, we can use it to decompose F into 4 cosets F16+2n ;3n+4-n+4= «po ), (O)}
ED
«P,), (C)}
C) «p,), (V)}
ED
«p,), (s)} ,
(2 .12)
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F. Englert, J. Orloff / Closed string phase transition
as was implicitly understood for physical states which belong to the last 2 classes . The four sets (pa) so defined have the same addition table and scalar products (modulo integers) as the four classes (a) of D4 weights [6], which implies that they are distinct . In order to get further information on (po ), we will need 2 basic properties of the unit cell volume in a lattice, vol[y*]=vol[y] -1
(=1
vol[ -y] = Ncoseu(r/y)vol[F]
if
y=y*),
for
y c r.
(2 .13)
that the unit cell volume of the These properties can for instance be used to check ,,,(weights/roots) = 4 and this is the Dn root lattice is 2: as we already saw, N,, square of the root volume since weights of a simply-laced Lie group form a lattice dual to the roots if these are normalized to 2. Using now the second property with (po ) X (0) and the fact that (po) and (0) are orthogonal, we get the volume, vol[( p.)] =
vol[(op[(0) ](0)]
=
4vol[ 1,]
= 2.
(2 .14)
Moreover, since I' and (0) are integer even lattices, (po) is integer even too so that it is contained in its dual (po)*. How many times? Using both properties, we see that (po )* must decompose into 22 = 4 cosets with respect to (po). What are those cosets? Since (po)* X (0)* contains T* = I', we are forced to identify (2.15)
(PO) *= (Po) ® (Pj ® (Ps) ® (Pj .
This is all modular invariance can tell us about (po) and its dual, but it is sufficient to derive the asymptotic lattice partition function . Indeed, the standard Poisson resummation (see appendix A) can now be expressed explicitly, Y- eimr(poL +PaL ) z- inT(PoR +P-R ) 2
l
( Po )
VOI[(po) ](
-1T)
_(8+n) (LT)
(2+n)
e
Tß(
2ni(pa pB)
,ß=0, v, s, c
- 1/T ;
(2 .16) For
T
small enough, the leading contribution always comes from
ro, because
p,,
PS
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F Fnglert, J. Orloff / Closed string phase transition
and p, being all odd, cannot be 0 so that 1.«( ,r
;T)
T_
~ 1(-1T)-(8+n)(1T)-(2+n)
for
a=0,v,s,c .
(2 .17)
The fact that all the classes yield the same result could have been anticipated since the only difference between classes is a little shift that is irrelevant at large distances . On the other hand, the numerical prefactor comes once more from the unicity of the (unphysical) ground state corrected by the volume of the unit cell of (po). Combining these results with the transformation properties of q (see eq. (2.3)), we get the announced result (2.11) and the fermionic analog of (2.7), Pthermal
- B ( T; T)
+ F(
T ; T ) -~ ( TT ) (D - 2)/2 e 2m/T - 7fi/T
(2.18)
At this level, it is an easy task to generalize the previous results for a type II string enjoying both left- and right-moving world-sheet supersymmetry : for chiral type II, we then just extend the above analysis in the left sector as well and find a lattice of the form r3n+8-n ;3n+8-n from which physical states are selected by only allowing the canonical vector or spinor weight for the left and right Däß°Sts" . We thus want to evaluate B (T ; T) _ (rv ; + rs ;s)(ni) -12 ' F(T ; T) = (F, ; s + Fs; U)(~~) -12 ' (2 .19) where, generalizing eq. (2.10), F.;,, is the lattice partition function of the vectors (pa; R) which can appear in r with a left Da weight of class a and a right weight of class ß . For the non-chiral type 11, it suffices to replace s by c in the left sector . Analyzing (po;o) the same way as we did for (po), we obtain the same expression as (2.11) with a changed exponent . Our results can finally be summarized very simply, Pthermal(T+TJ\ -T B( T, 7) + F( ;T)
TT
+(D-2)/2
ei~c/12r-imc/12T
2 .20
where c (c) is the left (right) light-cone central charge, i .e. 24 for a bosonic sector and 12 for a world-sheet supersymmetric sector. However, these results were concerned with a rather formal partition function of the rather immaterial parameter T . In sect. 3, we will show that universality at the level of this mathematically important partition function has physically observable consequences as well. 3. Universality of the critical string distribution 3 .1 . ASYMPTOTIC DEGENERACY BY STEEPEST DESCENT
The squared-mass partition functions P in sect. 2 were analytic functions of 2 independent variables T and j which were "dual" to the left and right squared
F. Englert, J. Orloff / Closed string phase transition
479
mass. However, these have to be equal to guarantee reparametrization invariance of physical states . This constraint is easily implemented by formally integrating over Re T=(T+T)/2, p(IMT)
= I z d(Re T) P(T ; T) _ 2 -
S(M) =
+ oc
M=ML=MR
dM
;Me-4a(Imr)M
dMs(M) e-4ar(ImT)M
r-
M'=ML =MR
dm,; m,8(M'-M),
(3 .2)
where we introduced the spectral density distribution s(M) which can be obtained by Laplace transforming p(ImT) * : f E+100
s (M) = - 2iJ
E -100
d(ImT) e 4f(ImT)Mp(ImT) .
(3 .3)
Our aim is now to obtain an asymptotic expression of s(M) for large M (i.e. small Im T) by saddle point integrations using the analyticity of P in both variables and its asymptotic form found in sect. 2, P(
T' T) --> ( Ti T-o
)(D-2)/2 e 2aribz/T-2sriP/T
,
(3 .4)
where b = c/24 is introduced to simplify later expressions . For Im T small, the first integral over Re T is dominated by an imaginary saddle at Re Tsaddle - i(IM T) x (b - b)/(b + b) giving p(ImT) 1mT~Ó
(4bb)(D-I)/2(b+b)-D(1MT)(2D-I)/2e~(h+b)2/Im, .
(3 .5)
Although this is not the only saddle point available (modular functions have singularities at every rational point of the Re T axis), it is easily seen to be the leading one since for any rational number p/q, p(_ p q
+
T; p +T I_ q
q2-D
1/2q
P(g2 Ti
q 2T) >
(3 .6)
so that the effective Im T in (3.5) would be larger by at least a factor of 4** . * Im T is defined here by Im T = (T - T)/2i, and is thus a complex variable as long as T is considered to be independent from T, like e.g . in eq. (2.1). ** For fermionic strings, where P = B + F is not strictly speaking a modular function by itself, an extra relative sign makes this statement even stronger.
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F. Englert, J. Orloff / Closed string phase transition
To obtain the density of states for large M, we now simply plug (3.5) into eq. (3.3) and evaluate the leading contribution around IM Tsaddle = (b + b)(4M)-1/2, s ( M)
i (bb)(D -1 )l2 M (D+1)/4 e 4r ~M (b+h) .
(3 .7)
M -> oo Z
Of course, a saddle point evaluation is too crude to reveal the "spiky" nature of s(M), and the estimate (3.7) must really be understood as an average on domains ,AM > ~M _ >> 1. Finally, let us express this result in terms of the physical mass, p(m) dm=s(M=QaM 2 )dM ,
a
(1-D)/2
eRomdm,
m -D
m - 00 (4bb)
ßo =21r a'
(b+b),
(3 .8)
which has the familiar Hagedorn form studied previously in the context of bootstrap hadron models [11] and more recently rederived for particular fundamental closed string models [12]. Here, we showed that this density, including the numerical prefactor, is totally independent of the details of compactification such as the shape and size of the lattice F. This contrasts with common claims about this prefactor . For studying statistical ensembles of strings, it is also useful to re-express (3 .8) as the degeneracy at fixed energy by summing over the D - 1 uncompactified momenta. We then find in a box of length L, dpD
p(E)dE=
f (27r/L)
-1 D-1
fE+dEdm JE
E2 _P 2 ~
(D-1)/2
bbL2 E-x
p(
(1r 2 (b+b)a' 3 / Z E)
eR,E dE .
E
(3 .9)
In contrast to compact directions, the winding numbers in the D - 1 directions have here been kept to zero and a dependence on L and D is thus not surprising even at high energies. Had we included these windings, then for energies greater than the squared length of the box, there would be no difference left between compact and non-compact dimensions, and we could use (3 .8) with D = 1 and m = E, pwindings (
E) d E E» E Za,
eßoE dE . 3/2
E
(3 .10)
In this way we recovered in a general framework the asymptotic independence on compactification anticipated in ref. [13].
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F. Englert, J. Orloff / Closed string phase transition
In order to get a picture of the heavy strings appearing in (3.8), we would now like to compute the mean extension R(m) of a string of large mass m [1, 2]. Using the standard mode expansion of a coordinate transverse to the light-cone, x'(Q, t) = q' + 2a'p't
+ 1
ai 1 [ ak,Le -2ik(t+a) + _ e-2ik(t-a)] L ak,R ~ 2 k#0 k
(3 .11)
we first form a wave packet with q` - P, - 0 and then evaluate the fluctuation around this center-of-mass by averaging the mean squared distance between 2 points of the string over all states of the given mass m. For this, we note that the existence of good saddles in T can be interpreted as a kind of thermal equilibrium between the modes that contribute to the additive quantities ML(R) at effective "temperatures" X L1R), 2b ~L Tsaddle - Re Tsaddle + i Im Tsaddle - 1 m a' - I X _
Tsaddle
-
2b
-1
_ . '~R m a' - - 1 2 7r
(3 .12)
>
so that for instance, the excitation level of the oscillator ak,L follows a Bose-Einstein distribution, (nA,L)m-
1 k
(a k .La -k.L)m - (1 - e N Lk )
1 ~i,-0
(kí1L)-1 .
(3 .13)
We thus find the correlation length in the direction i to be, 1 ` i (dx )m- ~J() do((x'(a)-x,(O)/2)m=2a'
~L .R
2a'( À L 1 +ÀR l) -0
00
k-l
klz
+
k= 1
L
k=1
1
k
[ nk,L(ÄL) +n k,R( ÄR) +1 1
=a'(~L1+~R1)~z +O(logE), 3 (3 .14)
if we probe this correlation by a string of energy -- E to regulate the log divergence present even in the ground state [14,15]. Using now the fact that transverse lengths are not affected in a boost and isotropy to extrapolate this result for strings at rest, we finally find 7r
RZ(m)=(D-1)(dxi>2=(D-l) 12
b+b ' 3/2 M a bb
.
(3 .15)
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F Englert, J. Orloff / Closed string phase transition
Recalling the fixed energy degeneracy (3 .9), we recognize in its prefactor the same combination (b + b)/bb that appears in R2 (E) = R2(m = E) * , so that we can form a model independent ratio, D-1 (D-1)/2 L D-1 p(E)no windings (3 .16) R(E)>>L (
P(E)total
1277 )
( R(E) )
which indicates that R(E) is a good variable for characterizing the spatial extension of a string with energy E. In terms of R, (3 .9) takes the form D-1 L D - 1 (D-1)/2 dR eßoE(R) p(R) (3 .17) dR=2 (
127r )
IR(E))
R
3 .2. VALIDITY AND RESULTS OF A THERMAL ENSEMBLE
Having found the most general density of states in a single string, we now turn to the statistical ensemble of many strings. This problem can be addressed from two points of view: (i) One can view each state as a free point particle and sum up the contributions of all such particles that are possible in a given model . This procedure has the advantages of having a straightforward physical interpretation by mimicking a second quantized partition function as well as providing an easy bridge between the canonical and microcanonical ensembles [11]. On the other hand, interactions cannot easily be included in such an approach . (ii) A more recent approach rests on the usual imaginary time periodicity of thermal Green functions in field theory, which has been extended to strings by Polchinsky [3] . The partition function can then be formally expressed as a loop expansion in first quantization which coincides with the result of the previous approach in the zero loop approximation . Since loop corrections are hardly tractable anyway (see however ref. [16]), we will follow the first point of view, along the lines of refs. [1] and [2] . Let us consider a spatial box of size L sufficiently large so that boundary effects, including winding numbers for periodic boundary conditions can be neglected . The equilibrium partition function in the canonical ensemble at temperature /3 -1 is - PF(,8) = log Tre - PH = - J dEP B (E)log(1 - e - PE) - P F(E)log(1 + e-PE) dEN(E,ß), - PF(ß)tightest states + J F..
(3 .18)
* The sum over momenta simply brings factors of Im T that do not change the effective temperatures that the evaluation of R2(E) is the same as that of RZ(m). ~'L,R so
F. Englert, J. Orloff / Closed string phase transition
48 3
where Eo is some energy scale above which the asymptotic form (3.9) becomes ß-1, these strings are nearly at rest, and the mean reliable . Since Eo >> t~ 1 >_ occupation number of each second quantized string state with given energy is small enough to wash out any distinction between Bose and Fermi statistics. However, because of the large degeneracy of states (3.9), the total number of strings with energy E in equilibrium in our box can be large, N(E, ß) dE= e-OEp(E) dE D - 1 (D-1)/2
E - ( 127r ) Taking first the limit V = L D-1 strings at ß = ßo,
,
n(R)dR=2(
L
D-1
e(ßo -ß)E dE
f R(E))
(3 .19)
oo, we can compute the density n (R) of large D - 1 (D-1)/2 dR
127r
)
dR
-0 .0045 R4
RD
for
D= 4.
(3 .20)
We have thus derived the total model independence of the numerical prefactor in the scale invariant [1, 2] distribution (3.20)*. As we will see in the next section, the origin of this fact simply comes from the random walk structure of semi-classical heavy strings. As was pointed out in ref . [1], for D >, 4, the energy density is finite when ß approaches /3o from above and we can thus push more energy in our box. If L = oo, this energy will go into an "infinite string" which will then act like a thermostat with a fixed temperature ß6 1 . We thus see that the distribution (3.20) also applies beyond the critical point for closed strings of finite size. 4. Critical distribution of random walks To get further insight into the significance of the universal distribution (3.20), let us study the behaviour of discretized classical strings under similar conditions . Indeed, the uncertainty principle forbids the total localization of a quantized string and the best we can do is to localize small segments of energy -- a' -1 /2 inside a chain of patches with radius -- a' 1/2, thereby getting at each time a "thick" random walk with a step -a '1 /2 measuring a quantum spread. This problem is also * This result was used in ref. [17] . Note however that R is here the mean separation between 2 points on the string, whereas in refs . [1,17] R was the mean distance to the center-of-mass which is smaller by a factor of V2 .
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F. Englert, J. Orloff / Closed string phase transition
interesting on its own since the classical "defect" strings occurring in the ordered phase of gauge theories, like the flux tubes in a type II supraconductor, possess a classical thickness which again plays the role of a short wavelength cut-off. Such string networks have also been studied numerically both in flat [18] and in curved space-time [19]. A discretized string state is specified by the position of N patches, xQ = (xi, . . .,xN },
D-1,
(4 .1)
together with their speeds zó. The segment inside each patch carries a fixed energy SE that contains a variable proportion of kinetic and potential stretching energy . What we want is to compute the number of different states with a fixed total energy E = NSE. To this aim, we will make two simplifying assumptions that will be easily relaxed later on : (i) We will disregard the kinetic energy of a segment, so that our counting becomes simply that of usual random walks ; (ii) We will restrict the space of the xQ to be a square lattice of spacing a. Counting walks is now an easy problem solved by the introduction of the "step" matrix S(x, y)- 1 if x and y are nearest neighbours but 0 otherwise . This is a step towards the solution because it is easily seen that S N(x,Y)
=
E
Z2, . . , ,N
S(x,z l )S(al,a3 ) . . .S(ZN ,Y)
(4 .2)
adds up 1N with every successful way of reaching y from x in N steps. Counting the number of homing walks p (N) is thus simply tracing, p( N) =
1
N
TrS N ,
(4 .3)
and correcting for the fact that each point of a given loop can in turn be the first in this way of counting. The Nth power of S is most conveniently taken after a Fourier transform whereupon it becomes diagonal by translational invariance, D-1
S( p) = r_ e ' °.(x-Y)S( x, Y) = Y_ 2cos( ap, ) Y
j=1
---~
2(D - 1) e-azpz/2(D-1),
p-0
SN(X
la r a
N~~
dD
(4.4)
1
(27r/a ) D-1
(2D - 2) Ne-(x-Y)
z
D-1 ~ (D-1)/2
(D-1)/2Na 2(
27rNa 2
a D -1
,
(4.5)
F. Englert, J. Orloff / Closed string phase transition
48 5
In the last equation, (2D - 2)N is the total number of walks of N steps, and the normalized spatial distribution that follows has the gaussian form required by the central limit theorem, all steps being independent . We can now examine what happens when we relax the two restrictions mentioned above, starting from the last one . Ad (ii) if the x-space is not discrete, S becomes a diffusion distribution S(x, y)dy and, resting on the central limit theorem, we get as before the total "number" of walks of N steps times a normed gaussian, SN(x>
y) dD-ly
N-oo
),
z z S Ne [(D - l)l21 N] (x -Y) (
D_1
l2aNe
2
(D_1)/2
dD-ly,
(4.6)
where s = jd y S(x, y) stands for the "number" of different single steps and ~2 = s - ljdy (x - y) 2S(x, y) is the mean distance travelled in one step. Ad (i) If we let each segment of the string carry a variable proportion of kinetic energy, ~2 changes from step to step and is reduced in the mean but this does not alter the generic form of S (4.6). For instance, in the discrete string model of ref . [18], the x-space is a face centered cubic lattice of spacing a (see fig . 1) and there are 3 types of segments in motion : 2(D - 1) segments of length a at rest on face joining the center edges (t = 0 on fig . 1); 4(D - 1)(D - 2) segments of length a/ speed of light (t = a/2 in fig. 1) ; and travelling at 1/ the of a face to a corner speed of light (t = a in fig . 1). and 2(D - 1) point-like segments travelling at the This model has therefore s = 4(D - 1) 2 and ~2 = a2/2 . We thus see that the central limit theorem makes (4.6) a truly generic formula . We can now evaluate the discretized closed string degeneracy in a box of length L
F
C
by (4.3),
L dD-1xSN(x, x) AN o
P(N) AN= J 0
( D - 1) LZ 2WZ2N
(D-1)~2
áN e Ntogs N (4 .7)
This should be compared with (3 .9), recalling that E --- N. If we include windings, i.e. sum on strings closing up to an integer number k of box lengths L, then for
Fig. 1 . Oscillation of the smallest planar loop showing the different types of segments in the model of ref . [18] .
486
F. Englert, J. Orloff / Closed string phase transition
sufficiently large N the gaussian sum
Y
p(N)3N=
kj= -oc
L d D-1XS N j, xi+ kjL) dNN AN e N J
(X
Nlogs
N >> L2/~2
(4 .8)
effectively removes the normalizing factor of (4.6) from (4.7) . This should be compared with (3 .10) . Since R2(N) is clearly - N~ z, we have understood (3 .16) and (3 .17), except for a numerical factor which we shall now check . The definition of R2(N) translated from sect . 3 is 1 Rz(N) = Nz
N
Y_
D-1
Y_ «xi - Xá-)zi
o,a'=1 j=1
= D- 1 N
/
(4 .9)
E\(Xa)2)~ a
with ~(
X
a)
2
_. -
fd D-1xSa(O,x)SN-a(X,0)XZ tdD-1xSQ(O X)SN-o(X 0)
~N-a),o-,oo
a(N -a) N
2.
4.10
For N large, the remaining sum over a can be approximated by an integral,
z
R (N)=
N2J N daa(N - a)=6N 2 , z
(4 .11)
which gives precisely the relation needed to recover (3 .16) and (3 .17), thereby explaining the universality of the density of states in high-energy strings. In terms of R, the only model dependent parameter is the Hagedorn temperature ßo 1. If in addition random walks are allowed to interact locally by cutting 2 strings meeting at a point and exchanging the partners, the energy is conserved in the interaction process . The system can then thermalize and, repeating the computation (3.18), we get a phase transition for those "thermal random walks" at the temperature ((N/E) logs) -1 which is the internal temperature of an infinite random walk. Finite closed strings are distributed again in a scale invariant way according to the universal distribution (3 .20). The occurrence of a phase transition and the universality of the "stringy phase" for D >_ 4 are thus a general property of "thermal random walks" which does not rely upon the detailed nature of the underlying string theory . The latter could be either the "fundamental" string theory considered in this paper or idealized thin cosmic strings or any linear structure. However, the critical hypothesis leading to universality is the type of interaction assumed here which amounts to both a weak
F. Englert, J. Orloff / Closed string phase transition
487
coupling limit and the absence of a relevant underlying local structure. The latter assumption would clearly be wrong for QCD flux tubes (where in fact the plasma phase takes over) and is also questionable for cosmic strings . In the case of fundamental strings which presently are assumed to be structureless, the only issue is the validity of weak coupling which anyhow limits our understanding of the theory. For instance, if the average mass shift for heavy states due to string loops would be growing faster than the bare mass, there would always be some energy scale above which perturbation theory would break down, no matter how small the coupling . Such an effect would undoubtedly alter the structure of the stringy phase. Otherwise, loops can only affect the value of the Hagedorn temperature. Keeping these problems in mind, one should nevertheless note that the existence of a stringy phase would have interesting consequences in cosmology . It provides indeed a natural explanation for inflation [1] and for large scale structure formation [17] and may therefore be of more general interest than the theories hitherto used to generate it. Appendix A For the sake of completeness, we give here a derivation of the Poisson resummation, which allows the transformation of a gaussian sum on a lorentzian (SL, 8R)dimensional lattice y into a gaussian sum on its dual y* . The partition function of this lattice shifted by an arbitrary vector p, Y(T ; T
P)
r, ei"(POL+PL)Z-i~T(POR+pR)Z = POEV
is by construction a periodic function. It can thus be decomposed into a Fourier series on the dual lattice, -y( T; T, P) = L_
9EY'
e2nip-9Y(T ; T;, q),
(A .1)
with Fourier components, Y(T ; T,q)
1 Vol
[Y I f
unit cell
1
_ dp e -2arip-g Y(T ; T, P)
00
Vol[Y] k-p+po=-oc lT
dke-2°rip
geivrvkL-iwfkR
R/2 e - (ie/T)4L+(iw1?)4n
-SL/2(-1T) -8
(A .2)
F. Englert, J. Orloff / Closed string phase transition
488
where we only used duality to set exp(27ripo q) = 1, so that this result holds for any lattice . If however y is an integral lattice, its dual can be decomposed into a certain number Nc of cosets, Y*= Yo®Y1® . . .
(A .3)
GYN-1
where Ya is y shifted by a certain vector pa representative of the coset and Yo - Y. Moreover, we can use eq. (2.13) to compute vol[Y] = vol[ y*] -1 = N, vol[y*] = N, .
(A .4)
If we finally define Y.(
T ; T) -
Y( T ;
(A .5)
T, p.),
we can express eq. (A.1) in closed form: YjT; T)
=N~ 1/Z(IT)-SL/z(-IT)-SR/2
N~-1 13=o
e 2aripa
-yp (_1/,r . -1/T) . (A .
References [1]
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
[13] [14] [l5] [16] [17] [18] [19]
Y . Aharonov, F. Englert and J. Orloff, Phys . Lett . B199 (1987) 366 D. Mitchell and N . Turok, Phys . Rev . Lett . 58 (1987) 1577 ; Nucl . Phys . B294 (1987) 1138 J . Polchinsky, Commun . Math . Phys . 104 (1986) 93 K . Narain, Phys . Lett. B169 (1986) 41 1 . Antoniadis, C. Bachas and C . Kounnas, Nucl . Phys . B2ß9 (1987) 87 R . Bluhm, L . Dolan and P . Goddard, Nucl . Phys . B309 (1988) 330 W. Lerche and A .N . Schellekens, CERN -TH .4925 (1987) D. Friedan, E . Martinec and S.H . Shenker, Nucl . Phys . B272 (1986) 93 W . Lerche, D . L6st and A.N . Schellekens, Nucl . Phys . B2ß7 (1987) 477 A . Casher, F . Englert, H. Nicolai and A . Taormina, Phys. Lett . B162 (1985) 121 F . Englert, H . Nicolai and A . Schellekens, Nucl . Phys . B274 (1986) 315 S . Frautschi, Phys . Rev. D3 (1971) 2821 ; R .D . Carlitz, Phys . Rev . D5 (1972) 3231 B . Sundborg, Nucl . Phys. B254 (1985) 583 ; M.J . Bowick and L .C .R . Wijewardhana, Phys. Rev. Lett . 54 (1985) 2485 ; M. Gleiser and J .G . Taylor, Phys . Lett. B164 (1985) 36 ; S .-H . Tye, Phys . Lett . B15ß (1985) 388 ; E . Alvarez, Nucl . Phys. B269 (1986) 596 R . Brandenberger and C . Vafa, BROWN-HET-6703, HUTP-88/A035 A. Casher, E.G . Floratos and N .C . Tsamis, Phys. Lett . B199 (1987) 377 M . Karliner, I. Klebanov and L . Susskind, Int. J . Mod . Phys . A 3 (1988) 1981 J .J . Attick and E . Witten, Nucl . Phys . B310 (1988) 291 F . Englert, J . Orloff and T. Piran, Phys . Lett. B212 (1988) 423 A .G . Smith and A . Vilenkin, Phys . Rev . D36 (1987) 990 A. Albrecht and N . Turok, Phys. Rev. Lett . 54 (1985) 1868 ; D .P . Benett and F .R . Bouchet, Phys . Rev . Lett. 60 (1988) 257 ; T . Vachaspati and A . Vilenkin, Phys . Rev . D30 (1984) 2036
UNIVERSALITY OF THE CLOSED STRING PHASE TRANSITION* F. ENGLERT and J. ORLOFF** Unioersité Libre de Bruxelles, C. P. ?25, Boulevard du Triomphe, 1050 Brussels, Belgium Received 13 July 1989
The universality of the high-temperature phase for closed string theories in D > 4 flat space-time dimensions is established and extended to other theories involving random walk shaped structures .
1. Introduction
Theories of fundamental closed strings, exhibit, in the weak coupling limit, a phase transition when the dimensionality of space-time is at least equal to four [1] . In the semi-classical description, the high-temperature phase is characterized by the condensation, at a fixed temperature T = To , of an infinite string filling space with a finite constant average density . The condensate lives in thermal equilibrium with a scale invariant distribution of macroscopic closed strings cut off at low energies by a planckian spectrum of massless particles ; all macroscopic strings are random walk shaped [l, 2]. In this paper we shall show that the structure of this "stringy" phase is entirely determined by the random walk structure of the semi-classical closed strings and does not rely significantly on string theory itself. For instance, at high temperature, the scale invariant distribution is the same, not only for all fundamental closed string theories in a flat space-time background, but also for idealized thin cosmic strings although in the former case the random walk shape stems from the uncertainty principle while in the latter it is imposed as an external constraint . The only relevant model dependent parameter in the stringy phase is the temperature of the phase transition itself. In sect. 2 we study the character of the conformal group in the flat compact dimensions of a closed string theory. We derive its asymptotic behaviour for large bosonic and fermionic masses . The result depends only on the left- and right-sector central charges and on the number of space-time dimensions . * Supported in part by NATO C.R .G. 890404 . ** Chercheur IISN . 0550-3213/90/$03 .50 ~-= Elsevier Science Publishers B.V . (North-Holland)
F. Englert, J. Orloff / Closed string phase transition
473
The asymptotic character is used in sect. 3 to derive the asymptotic distribution of states for a single closed string . The universality of the phase transition for all theories of fundamental closed strings in a flat space-time background follows as a corollary. In sect. 4, this universality is shown to be rooted in the random walk structure of semi-classical relativistic closed strings ; the description of the phase transition is then extended to include all such structures, without reference to an underlying fundamental scheme.
2. Universality of the asymptotic spectrum 2.1 . CLOSED BOSONIC STRING
Let us first consider a closed string in D space-time dimensions obtained by compactifying S = 26 - D dimensions of the usual bosonic string on a torus . The squared-mass partition function, which appears e.g. in the one-loop amplitude [3], is given by P(T; T) _ Y_ dML ; MR e2ariTML-2wi i:MR ML ; MR ( ; - ~~I(T)~1(T)~-241,5,S T
T)>
(2 .1)
where ML(R)
= 4 arm L(R)
= 1PL(R) + NL(R) - 1
(2 .2)
is the dimensionless squared mass of a left (right) string state with compact 8-dimensional momentum PL(R) and NL(R) oscillator excitations ; dmL ; MR is the number of different closed string states with fixed masses . Notice that such states are physical only if mL = mÁ= m 2 but this is not necessarily the case in the partition function appearing in eq. (2.1) . This function contains on the one hand the Dedekind 17-function, 00
,q(T)=e( iir/12)TFl
n=1
/ (1-e 2ariTn ) = (_1T) -1/2 n(-1 ,r) e
(2 .3)
which counts oscillator excitations and on the other hand the partition function of some lorentzian even self-dual lattice, 5;5( T ;
T
)_
ei p2-i~Tp2 (PL ;
PR)
(2 .4)
which represents both compact momenta and winding numbers [4]. For instance, for
F. Englert, J. Orloff / Closed string phase transition
47 4
8 = 1 compact dimension of radius R, this lattice would be given by (PLI PR) =
1 (
k
a' R
+1
R a'
;k
a' R
-1
R a'
(2 .5)
where k is some integer labelling the different possible momenta k/R and 1 is the number of times the string winds around the direction 8. Notice that since these integers have to be the same in PL and PR, T is not in general the direct product of independent left and right lattices, so that the compactification generally ruins the factorizability of P(T ; T) The modular properties of the rl-function (see eq. (2.3)), together with the evenness and the self-duality of the lattice l (see appendix A) imply that P(T+1 ;T+1)=P(T ;T),
P(-1/T ;-1/T)=(TT)(2-D)/2P(T ; r), (2 .6)
as required by the modular invariance of the full one-loop amplitude. This property turns out to be very powerful for obtaining estimates of the asymptotic spectrum in conformal theories. Indeed, all the information about the large-mass spectrum is encoded in the small-T partition function, P(,r ;T)
ó
(TT)(°-2)/2e2ffi/T-2~i/r[1+O(e-(2ff/ITD(R2 /« +a'/Rz))1, (2 .7)
which is dominated by the tachyon contribution . Unicity of this tachyonic ground state forbids a multiplicative constant in (2.7), a fact which will prove to be quite general . 2 .2 . FERMIONIC STRING
Let us now turn to fermionic closed strings, i.e. string theories which contain both two-dimensional bosons and fermions, and where conformal invariance is enlarged to a superconformal group. This extension can be made in the left or in the right sector (heterotic strings) or in both sectors (type-11 strings). We will first study extensively the heterotic strings, and then briefly sketch the computation for type-11 strings . Consider the supersymmetric right-moving sector of a heterotic theory in D = 10 - 2n space-time dimensions in the light-cone gauge. It consists of: (1) 8 - 2n transverse bosons ; (ii) Their 8 - 2n fermionic partners with + or - boundary conditions which can be bosonized into 4 - n compact bosons with momenta lying on the weight lattice of the transverse Lorentz group D4 - ; (iii) Extra degrees of freedom contributing 3n to the central charge in order to reach the light-cone gauge critical value ctot = 12; for these the simplest choice is either a set of 6n free fermions with non-trivial boundary conditions [5], or what
F. Englert, J. Orloff / Closed string phase transition
475
seems to be more general if not equivalent, a set of 3n free compact bosons . (See ref. [6] for a very clear review of bosonic compactification .) Although other choices are possible we will stick to this last one in this paper. Superconformal invariance of the resulting patchwork has of course to be checked but as we will see, this further constraint is not needed to determine uniquely the asymptotic spectrum ; (iv) Residual momenta left over from the superconformal ghosts which cannot be totally eliminated because of the picture-changing phenomenon [7] . This last point requires a "choice of picture" in order to make sure that every physical state appears only once. For our purposes, the most convenient way of achieving this is to start from the bosonic covariant formulation of the heterotic string [8] where the D,-,, weights are extended with the bosonized longitudinal fermions and bosonized superghost into D5 -,,, weights . We then use the "even lattice mapping" [6] between classes of DS-n,l and D,-,, which preserves modular invariance to all loops whereby the Dl, I ghost weights are mapped onto weights of a D4 ghosts" . We finally obtain for compact momenta a lorentzian lattice of the form F16+2n ;3n+8-n > which modular invariance forces to be even and self-dual . The choice of picture is now achieved by fixing the D4ghosu"-part of T to be the canonical representatives of the (v) and (s) classes *: (v) class : For every momentum of the form (
3n w 4-n 1 0 PO' 1'0'0 10) - ( PVL ; PVR,11 0'0'0) - ( PVL ; v
0
0)
(2 .8)
there is one pure momentum (i.e. without oscillator excitations) physical state. This state is a boson since a weight of D8 -n can take this particular form only if w4- n lies in the bosonic representations (0) or (v) of the transverse Lorentz group . All other bosonic physical states are obtained from these by action of the transverse oscillators . (s) class: For every momentum of the form 1 1 1, 1 1, 1 3n ( PSI21121-(PSL ;PsR12 2>2 z)-(PsL ;v 121 12)
,w 4-n ~21_ I
1 2
1
2
1
2)~
(2 .9)
there is one pure momentum physical fermionic state, with w4- n belonging to the classes (s) or (c). All other fermionic states are obtained from these by action of the transverse oscillators . The dimensionless squared-mass of these states is simply given again by eq. (2.2), but with NR counting oscillations in only 12 directions: 8 - 2n transverse spatial, * Along
with standard conventions, we shall take D roots to be (0, . . ., ± 1, 0, . . ., ± 1, 0 . . . . ), while the other classes of weights are : ( v ) = (1,0 . . . . ) + roots, ( s ) - ( i , i , z . . . . ) + roots, and (c) -t(-a " Z,Z, . . . +roots .
47 6
F. Englert, J. Orloff / Closed string phase transition
their 4 - n bosonized fermionic partners and 3n compactified internal dimensions . Notice that such a theory can always be embedded in a consistent bosonic string theory of the type considered in the last section if we tensor F16+2n ;3n+s-n by the even, self-dual root lattice of Eg to obtain l's; s, i.e. in the same way as was originally done in refs. [9,10]. The fermionic string states are then the subset of bosonic states with the Eg root being identically 0, the D4gh°scs'' weight being as in (2 .8) or (2.9) and the oscillators in the corresponding 12 dimensions being frozen in their ground state . Let us call the bosonic and fermionic contributions to the partition function respectively B(T ; T) I`a(T ;
-
n
(T)-24
\
TI - Y,
(p.)
n (T)- 12 1-,
u(Te T),
F(T ;
T) =~1(T)-24
n (T)-12Fs(,r
2 2 ei"TP;L-i"TpáR .
We will now show that for
T
; Tl+
(2 .10)
small enou
B( r ; T) -~ F(T ; T) -> 2(TT)4-n e2artlT-ail-T T_o T_o so that (1) the spectrum is always asymptotically supersymmetric in these fermionic theories ; (ii) B + F is the same as (2.7) except for a factor in the exponential which reflects the difference in the right central charge. Why worry about B + F? Let us note that although what appears in the one-loop amplitudes like e.g. the cosmological constant is the difference B - F, what we need in the evaluation of a thermal ensemble of strings is rather the sum B + F, just like for an ordinary field theory containing both bosons and fermions in the Maxwell-Boltzmann limit (see subsect . 3.2) . We first show that although F16+2n :3n+4-n+4 is not necessarily a direct product, it always contains a direct product sublattice y = (po) X (0) where (po) is by definition the set of all (16 + 2n ; 4 + 2n)-dimensional vectors that can appear in F together with a D49'°sls ' root. Since by definition, all F vectors have a D4 weight for their last 4 components, any vector of the form (po = 0, (0)) will have integer scalar product with them and therefore belongs to the self-dual F. The closure of F under addition thus requires any vector of (po) X (0) to be in F. Having found such a sublattice, we can use it to decompose F into 4 cosets F16+2n ;3n+4-n+4= «po ), (O)}
ED
«P,), (C)}
C) «p,), (V)}
ED
«p,), (s)} ,
(2 .12)
477
F. Englert, J. Orloff / Closed string phase transition
as was implicitly understood for physical states which belong to the last 2 classes . The four sets (pa) so defined have the same addition table and scalar products (modulo integers) as the four classes (a) of D4 weights [6], which implies that they are distinct . In order to get further information on (po ), we will need 2 basic properties of the unit cell volume in a lattice, vol[y*]=vol[y] -1
(=1
vol[ -y] = Ncoseu(r/y)vol[F]
if
y=y*),
for
y c r.
(2 .13)
that the unit cell volume of the These properties can for instance be used to check ,,,(weights/roots) = 4 and this is the Dn root lattice is 2: as we already saw, N,, square of the root volume since weights of a simply-laced Lie group form a lattice dual to the roots if these are normalized to 2. Using now the second property with (po ) X (0) and the fact that (po) and (0) are orthogonal, we get the volume, vol[( p.)] =
vol[(op[(0) ](0)]
=
4vol[ 1,]
= 2.
(2 .14)
Moreover, since I' and (0) are integer even lattices, (po) is integer even too so that it is contained in its dual (po)*. How many times? Using both properties, we see that (po )* must decompose into 22 = 4 cosets with respect to (po). What are those cosets? Since (po)* X (0)* contains T* = I', we are forced to identify (2.15)
(PO) *= (Po) ® (Pj ® (Ps) ® (Pj .
This is all modular invariance can tell us about (po) and its dual, but it is sufficient to derive the asymptotic lattice partition function . Indeed, the standard Poisson resummation (see appendix A) can now be expressed explicitly, Y- eimr(poL +PaL ) z- inT(PoR +P-R ) 2
l
( Po )
VOI[(po) ](
-1T)
_(8+n) (LT)
(2+n)
e
Tß(
2ni(pa pB)
,ß=0, v, s, c
- 1/T ;
(2 .16) For
T
small enough, the leading contribution always comes from
ro, because
p,,
PS
478
F Fnglert, J. Orloff / Closed string phase transition
and p, being all odd, cannot be 0 so that 1.«( ,r
;T)
T_
~ 1(-1T)-(8+n)(1T)-(2+n)
for
a=0,v,s,c .
(2 .17)
The fact that all the classes yield the same result could have been anticipated since the only difference between classes is a little shift that is irrelevant at large distances . On the other hand, the numerical prefactor comes once more from the unicity of the (unphysical) ground state corrected by the volume of the unit cell of (po). Combining these results with the transformation properties of q (see eq. (2.3)), we get the announced result (2.11) and the fermionic analog of (2.7), Pthermal
- B ( T; T)
+ F(
T ; T ) -~ ( TT ) (D - 2)/2 e 2m/T - 7fi/T
(2.18)
At this level, it is an easy task to generalize the previous results for a type II string enjoying both left- and right-moving world-sheet supersymmetry : for chiral type II, we then just extend the above analysis in the left sector as well and find a lattice of the form r3n+8-n ;3n+8-n from which physical states are selected by only allowing the canonical vector or spinor weight for the left and right Däß°Sts" . We thus want to evaluate B (T ; T) _ (rv ; + rs ;s)(ni) -12 ' F(T ; T) = (F, ; s + Fs; U)(~~) -12 ' (2 .19) where, generalizing eq. (2.10), F.;,, is the lattice partition function of the vectors (pa; R) which can appear in r with a left Da weight of class a and a right weight of class ß . For the non-chiral type 11, it suffices to replace s by c in the left sector . Analyzing (po;o) the same way as we did for (po), we obtain the same expression as (2.11) with a changed exponent . Our results can finally be summarized very simply, Pthermal(T+TJ\ -T B( T, 7) + F( ;T)
TT
+(D-2)/2
ei~c/12r-imc/12T
2 .20
where c (c) is the left (right) light-cone central charge, i .e. 24 for a bosonic sector and 12 for a world-sheet supersymmetric sector. However, these results were concerned with a rather formal partition function of the rather immaterial parameter T . In sect. 3, we will show that universality at the level of this mathematically important partition function has physically observable consequences as well. 3. Universality of the critical string distribution 3 .1 . ASYMPTOTIC DEGENERACY BY STEEPEST DESCENT
The squared-mass partition functions P in sect. 2 were analytic functions of 2 independent variables T and j which were "dual" to the left and right squared
F. Englert, J. Orloff / Closed string phase transition
479
mass. However, these have to be equal to guarantee reparametrization invariance of physical states . This constraint is easily implemented by formally integrating over Re T=(T+T)/2, p(IMT)
= I z d(Re T) P(T ; T) _ 2 -
S(M) =
+ oc
M=ML=MR
dM
;Me-4a(Imr)M
dMs(M) e-4ar(ImT)M
r-
M'=ML =MR
dm,; m,8(M'-M),
(3 .2)
where we introduced the spectral density distribution s(M) which can be obtained by Laplace transforming p(ImT) * : f E+100
s (M) = - 2iJ
E -100
d(ImT) e 4f(ImT)Mp(ImT) .
(3 .3)
Our aim is now to obtain an asymptotic expression of s(M) for large M (i.e. small Im T) by saddle point integrations using the analyticity of P in both variables and its asymptotic form found in sect. 2, P(
T' T) --> ( Ti T-o
)(D-2)/2 e 2aribz/T-2sriP/T
,
(3 .4)
where b = c/24 is introduced to simplify later expressions . For Im T small, the first integral over Re T is dominated by an imaginary saddle at Re Tsaddle - i(IM T) x (b - b)/(b + b) giving p(ImT) 1mT~Ó
(4bb)(D-I)/2(b+b)-D(1MT)(2D-I)/2e~(h+b)2/Im, .
(3 .5)
Although this is not the only saddle point available (modular functions have singularities at every rational point of the Re T axis), it is easily seen to be the leading one since for any rational number p/q, p(_ p q
+
T; p +T I_ q
q2-D
1/2q
P(g2 Ti
q 2T) >
(3 .6)
so that the effective Im T in (3.5) would be larger by at least a factor of 4** . * Im T is defined here by Im T = (T - T)/2i, and is thus a complex variable as long as T is considered to be independent from T, like e.g . in eq. (2.1). ** For fermionic strings, where P = B + F is not strictly speaking a modular function by itself, an extra relative sign makes this statement even stronger.
480
F. Englert, J. Orloff / Closed string phase transition
To obtain the density of states for large M, we now simply plug (3.5) into eq. (3.3) and evaluate the leading contribution around IM Tsaddle = (b + b)(4M)-1/2, s ( M)
i (bb)(D -1 )l2 M (D+1)/4 e 4r ~M (b+h) .
(3 .7)
M -> oo Z
Of course, a saddle point evaluation is too crude to reveal the "spiky" nature of s(M), and the estimate (3.7) must really be understood as an average on domains ,AM > ~M _ >> 1. Finally, let us express this result in terms of the physical mass, p(m) dm=s(M=QaM 2 )dM ,
a
(1-D)/2
eRomdm,
m -D
m - 00 (4bb)
ßo =21r a'
(b+b),
(3 .8)
which has the familiar Hagedorn form studied previously in the context of bootstrap hadron models [11] and more recently rederived for particular fundamental closed string models [12]. Here, we showed that this density, including the numerical prefactor, is totally independent of the details of compactification such as the shape and size of the lattice F. This contrasts with common claims about this prefactor . For studying statistical ensembles of strings, it is also useful to re-express (3 .8) as the degeneracy at fixed energy by summing over the D - 1 uncompactified momenta. We then find in a box of length L, dpD
p(E)dE=
f (27r/L)
-1 D-1
fE+dEdm JE
E2 _P 2 ~
(D-1)/2
bbL2 E-x
p(
(1r 2 (b+b)a' 3 / Z E)
eR,E dE .
E
(3 .9)
In contrast to compact directions, the winding numbers in the D - 1 directions have here been kept to zero and a dependence on L and D is thus not surprising even at high energies. Had we included these windings, then for energies greater than the squared length of the box, there would be no difference left between compact and non-compact dimensions, and we could use (3 .8) with D = 1 and m = E, pwindings (
E) d E E» E Za,
eßoE dE . 3/2
E
(3 .10)
In this way we recovered in a general framework the asymptotic independence on compactification anticipated in ref. [13].
481
F. Englert, J. Orloff / Closed string phase transition
In order to get a picture of the heavy strings appearing in (3.8), we would now like to compute the mean extension R(m) of a string of large mass m [1, 2]. Using the standard mode expansion of a coordinate transverse to the light-cone, x'(Q, t) = q' + 2a'p't
+ 1
ai 1 [ ak,Le -2ik(t+a) + _ e-2ik(t-a)] L ak,R ~ 2 k#0 k
(3 .11)
we first form a wave packet with q` - P, - 0 and then evaluate the fluctuation around this center-of-mass by averaging the mean squared distance between 2 points of the string over all states of the given mass m. For this, we note that the existence of good saddles in T can be interpreted as a kind of thermal equilibrium between the modes that contribute to the additive quantities ML(R) at effective "temperatures" X L1R), 2b ~L Tsaddle - Re Tsaddle + i Im Tsaddle - 1 m a' - I X _
Tsaddle
-
2b
-1
_ . '~R m a' - - 1 2 7r
(3 .12)
>
so that for instance, the excitation level of the oscillator ak,L follows a Bose-Einstein distribution, (nA,L)m-
1 k
(a k .La -k.L)m - (1 - e N Lk )
1 ~i,-0
(kí1L)-1 .
(3 .13)
We thus find the correlation length in the direction i to be, 1 ` i (dx )m- ~J() do((x'(a)-x,(O)/2)m=2a'
~L .R
2a'( À L 1 +ÀR l) -0
00
k-l
klz
+
k= 1
L
k=1
1
k
[ nk,L(ÄL) +n k,R( ÄR) +1 1
=a'(~L1+~R1)~z +O(logE), 3 (3 .14)
if we probe this correlation by a string of energy -- E to regulate the log divergence present even in the ground state [14,15]. Using now the fact that transverse lengths are not affected in a boost and isotropy to extrapolate this result for strings at rest, we finally find 7r
RZ(m)=(D-1)(dxi>2=(D-l) 12
b+b ' 3/2 M a bb
.
(3 .15)
48 2
F Englert, J. Orloff / Closed string phase transition
Recalling the fixed energy degeneracy (3 .9), we recognize in its prefactor the same combination (b + b)/bb that appears in R2 (E) = R2(m = E) * , so that we can form a model independent ratio, D-1 (D-1)/2 L D-1 p(E)no windings (3 .16) R(E)>>L (
P(E)total
1277 )
( R(E) )
which indicates that R(E) is a good variable for characterizing the spatial extension of a string with energy E. In terms of R, (3 .9) takes the form D-1 L D - 1 (D-1)/2 dR eßoE(R) p(R) (3 .17) dR=2 (
127r )
IR(E))
R
3 .2. VALIDITY AND RESULTS OF A THERMAL ENSEMBLE
Having found the most general density of states in a single string, we now turn to the statistical ensemble of many strings. This problem can be addressed from two points of view: (i) One can view each state as a free point particle and sum up the contributions of all such particles that are possible in a given model . This procedure has the advantages of having a straightforward physical interpretation by mimicking a second quantized partition function as well as providing an easy bridge between the canonical and microcanonical ensembles [11]. On the other hand, interactions cannot easily be included in such an approach . (ii) A more recent approach rests on the usual imaginary time periodicity of thermal Green functions in field theory, which has been extended to strings by Polchinsky [3] . The partition function can then be formally expressed as a loop expansion in first quantization which coincides with the result of the previous approach in the zero loop approximation . Since loop corrections are hardly tractable anyway (see however ref. [16]), we will follow the first point of view, along the lines of refs. [1] and [2] . Let us consider a spatial box of size L sufficiently large so that boundary effects, including winding numbers for periodic boundary conditions can be neglected . The equilibrium partition function in the canonical ensemble at temperature /3 -1 is - PF(,8) = log Tre - PH = - J dEP B (E)log(1 - e - PE) - P F(E)log(1 + e-PE) dEN(E,ß), - PF(ß)tightest states + J F..
(3 .18)
* The sum over momenta simply brings factors of Im T that do not change the effective temperatures that the evaluation of R2(E) is the same as that of RZ(m). ~'L,R so
F. Englert, J. Orloff / Closed string phase transition
48 3
where Eo is some energy scale above which the asymptotic form (3.9) becomes ß-1, these strings are nearly at rest, and the mean reliable . Since Eo >> t~ 1 >_ occupation number of each second quantized string state with given energy is small enough to wash out any distinction between Bose and Fermi statistics. However, because of the large degeneracy of states (3.9), the total number of strings with energy E in equilibrium in our box can be large, N(E, ß) dE= e-OEp(E) dE D - 1 (D-1)/2
E - ( 127r ) Taking first the limit V = L D-1 strings at ß = ßo,
,
n(R)dR=2(
L
D-1
e(ßo -ß)E dE
f R(E))
(3 .19)
oo, we can compute the density n (R) of large D - 1 (D-1)/2 dR
127r
)
dR
-0 .0045 R4
RD
for
D= 4.
(3 .20)
We have thus derived the total model independence of the numerical prefactor in the scale invariant [1, 2] distribution (3.20)*. As we will see in the next section, the origin of this fact simply comes from the random walk structure of semi-classical heavy strings. As was pointed out in ref . [1], for D >, 4, the energy density is finite when ß approaches /3o from above and we can thus push more energy in our box. If L = oo, this energy will go into an "infinite string" which will then act like a thermostat with a fixed temperature ß6 1 . We thus see that the distribution (3.20) also applies beyond the critical point for closed strings of finite size. 4. Critical distribution of random walks To get further insight into the significance of the universal distribution (3.20), let us study the behaviour of discretized classical strings under similar conditions . Indeed, the uncertainty principle forbids the total localization of a quantized string and the best we can do is to localize small segments of energy -- a' -1 /2 inside a chain of patches with radius -- a' 1/2, thereby getting at each time a "thick" random walk with a step -a '1 /2 measuring a quantum spread. This problem is also * This result was used in ref. [17] . Note however that R is here the mean separation between 2 points on the string, whereas in refs . [1,17] R was the mean distance to the center-of-mass which is smaller by a factor of V2 .
484
F. Englert, J. Orloff / Closed string phase transition
interesting on its own since the classical "defect" strings occurring in the ordered phase of gauge theories, like the flux tubes in a type II supraconductor, possess a classical thickness which again plays the role of a short wavelength cut-off. Such string networks have also been studied numerically both in flat [18] and in curved space-time [19]. A discretized string state is specified by the position of N patches, xQ = (xi, . . .,xN },
D-1,
(4 .1)
together with their speeds zó. The segment inside each patch carries a fixed energy SE that contains a variable proportion of kinetic and potential stretching energy . What we want is to compute the number of different states with a fixed total energy E = NSE. To this aim, we will make two simplifying assumptions that will be easily relaxed later on : (i) We will disregard the kinetic energy of a segment, so that our counting becomes simply that of usual random walks ; (ii) We will restrict the space of the xQ to be a square lattice of spacing a. Counting walks is now an easy problem solved by the introduction of the "step" matrix S(x, y)- 1 if x and y are nearest neighbours but 0 otherwise . This is a step towards the solution because it is easily seen that S N(x,Y)
=
E
Z2, . . , ,N
S(x,z l )S(al,a3 ) . . .S(ZN ,Y)
(4 .2)
adds up 1N with every successful way of reaching y from x in N steps. Counting the number of homing walks p (N) is thus simply tracing, p( N) =
1
N
TrS N ,
(4 .3)
and correcting for the fact that each point of a given loop can in turn be the first in this way of counting. The Nth power of S is most conveniently taken after a Fourier transform whereupon it becomes diagonal by translational invariance, D-1
S( p) = r_ e ' °.(x-Y)S( x, Y) = Y_ 2cos( ap, ) Y
j=1
---~
2(D - 1) e-azpz/2(D-1),
p-0
SN(X
la r a
N~~
dD
(4.4)
1
(27r/a ) D-1
(2D - 2) Ne-(x-Y)
z
D-1 ~ (D-1)/2
(D-1)/2Na 2(
27rNa 2
a D -1
,
(4.5)
F. Englert, J. Orloff / Closed string phase transition
48 5
In the last equation, (2D - 2)N is the total number of walks of N steps, and the normalized spatial distribution that follows has the gaussian form required by the central limit theorem, all steps being independent . We can now examine what happens when we relax the two restrictions mentioned above, starting from the last one . Ad (ii) if the x-space is not discrete, S becomes a diffusion distribution S(x, y)dy and, resting on the central limit theorem, we get as before the total "number" of walks of N steps times a normed gaussian, SN(x>
y) dD-ly
N-oo
),
z z S Ne [(D - l)l21 N] (x -Y) (
D_1
l2aNe
2
(D_1)/2
dD-ly,
(4.6)
where s = jd y S(x, y) stands for the "number" of different single steps and ~2 = s - ljdy (x - y) 2S(x, y) is the mean distance travelled in one step. Ad (i) If we let each segment of the string carry a variable proportion of kinetic energy, ~2 changes from step to step and is reduced in the mean but this does not alter the generic form of S (4.6). For instance, in the discrete string model of ref . [18], the x-space is a face centered cubic lattice of spacing a (see fig . 1) and there are 3 types of segments in motion : 2(D - 1) segments of length a at rest on face joining the center edges (t = 0 on fig . 1); 4(D - 1)(D - 2) segments of length a/ speed of light (t = a/2 in fig. 1) ; and travelling at 1/ the of a face to a corner speed of light (t = a in fig . 1). and 2(D - 1) point-like segments travelling at the This model has therefore s = 4(D - 1) 2 and ~2 = a2/2 . We thus see that the central limit theorem makes (4.6) a truly generic formula . We can now evaluate the discretized closed string degeneracy in a box of length L
F
C
by (4.3),
L dD-1xSN(x, x) AN o
P(N) AN= J 0
( D - 1) LZ 2WZ2N
(D-1)~2
áN e Ntogs N (4 .7)
This should be compared with (3 .9), recalling that E --- N. If we include windings, i.e. sum on strings closing up to an integer number k of box lengths L, then for
Fig. 1 . Oscillation of the smallest planar loop showing the different types of segments in the model of ref . [18] .
486
F. Englert, J. Orloff / Closed string phase transition
sufficiently large N the gaussian sum
Y
p(N)3N=
kj= -oc
L d D-1XS N j, xi+ kjL) dNN AN e N J
(X
Nlogs
N >> L2/~2
(4 .8)
effectively removes the normalizing factor of (4.6) from (4.7) . This should be compared with (3 .10) . Since R2(N) is clearly - N~ z, we have understood (3 .16) and (3 .17), except for a numerical factor which we shall now check . The definition of R2(N) translated from sect . 3 is 1 Rz(N) = Nz
N
Y_
D-1
Y_ «xi - Xá-)zi
o,a'=1 j=1
= D- 1 N
/
(4 .9)
E\(Xa)2)~ a
with ~(
X
a)
2
_. -
fd D-1xSa(O,x)SN-a(X,0)XZ tdD-1xSQ(O X)SN-o(X 0)
~N-a),o-,oo
a(N -a) N
2.
4.10
For N large, the remaining sum over a can be approximated by an integral,
z
R (N)=
N2J N daa(N - a)=6N 2 , z
(4 .11)
which gives precisely the relation needed to recover (3 .16) and (3 .17), thereby explaining the universality of the density of states in high-energy strings. In terms of R, the only model dependent parameter is the Hagedorn temperature ßo 1. If in addition random walks are allowed to interact locally by cutting 2 strings meeting at a point and exchanging the partners, the energy is conserved in the interaction process . The system can then thermalize and, repeating the computation (3.18), we get a phase transition for those "thermal random walks" at the temperature ((N/E) logs) -1 which is the internal temperature of an infinite random walk. Finite closed strings are distributed again in a scale invariant way according to the universal distribution (3 .20). The occurrence of a phase transition and the universality of the "stringy phase" for D >_ 4 are thus a general property of "thermal random walks" which does not rely upon the detailed nature of the underlying string theory . The latter could be either the "fundamental" string theory considered in this paper or idealized thin cosmic strings or any linear structure. However, the critical hypothesis leading to universality is the type of interaction assumed here which amounts to both a weak
F. Englert, J. Orloff / Closed string phase transition
487
coupling limit and the absence of a relevant underlying local structure. The latter assumption would clearly be wrong for QCD flux tubes (where in fact the plasma phase takes over) and is also questionable for cosmic strings . In the case of fundamental strings which presently are assumed to be structureless, the only issue is the validity of weak coupling which anyhow limits our understanding of the theory. For instance, if the average mass shift for heavy states due to string loops would be growing faster than the bare mass, there would always be some energy scale above which perturbation theory would break down, no matter how small the coupling . Such an effect would undoubtedly alter the structure of the stringy phase. Otherwise, loops can only affect the value of the Hagedorn temperature. Keeping these problems in mind, one should nevertheless note that the existence of a stringy phase would have interesting consequences in cosmology . It provides indeed a natural explanation for inflation [1] and for large scale structure formation [17] and may therefore be of more general interest than the theories hitherto used to generate it. Appendix A For the sake of completeness, we give here a derivation of the Poisson resummation, which allows the transformation of a gaussian sum on a lorentzian (SL, 8R)dimensional lattice y into a gaussian sum on its dual y* . The partition function of this lattice shifted by an arbitrary vector p, Y(T ; T
P)
r, ei"(POL+PL)Z-i~T(POR+pR)Z = POEV
is by construction a periodic function. It can thus be decomposed into a Fourier series on the dual lattice, -y( T; T, P) = L_
9EY'
e2nip-9Y(T ; T;, q),
(A .1)
with Fourier components, Y(T ; T,q)
1 Vol
[Y I f
unit cell
1
_ dp e -2arip-g Y(T ; T, P)
00
Vol[Y] k-p+po=-oc lT
dke-2°rip
geivrvkL-iwfkR
R/2 e - (ie/T)4L+(iw1?)4n
-SL/2(-1T) -8
(A .2)
F. Englert, J. Orloff / Closed string phase transition
488
where we only used duality to set exp(27ripo q) = 1, so that this result holds for any lattice . If however y is an integral lattice, its dual can be decomposed into a certain number Nc of cosets, Y*= Yo®Y1® . . .
(A .3)
GYN-1
where Ya is y shifted by a certain vector pa representative of the coset and Yo - Y. Moreover, we can use eq. (2.13) to compute vol[Y] = vol[ y*] -1 = N, vol[y*] = N, .
(A .4)
If we finally define Y.(
T ; T) -
Y( T ;
(A .5)
T, p.),
we can express eq. (A.1) in closed form: YjT; T)
=N~ 1/Z(IT)-SL/z(-IT)-SR/2
N~-1 13=o
e 2aripa
-yp (_1/,r . -1/T) . (A .
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