SUHEP-4241-522

Finite Temperature Strings∗

arXiv:hep-th/9210016v1 2 Oct 1992

Mark J. Bowick1

Physics Department
Syracuse University
Syracuse, NY 13244-1130, USA

Abstract

In this talk a review of earlier work on finite temperature strings was presented.
Several topics were covered, including the canonical and microcanonical ensemble of strings,
the behavior of strings near the Hagedorn temperature as well as speculations on the
possible phases of high temperature strings. The connection of the string ensemble and,
more generally, statistical systems with an exponentially growing density of states with
number theory was also discussed.

September 29, 1992

∗
Invited talk delivered at the workshop “String Quantum Gravity and Physics at the Planck
Energy Scale,” Erice, June 21–28, 1992.
1
E-mail: Bowick@suhep.bitnet.
 1. Introduction

The theory of fundamental strings has attracted a lot of interest in the past decade as a
possible candidate for a unified theory of all the known interactions. Despite heroic efforts,
crucial questions still remain unanswered. For example, the basic degrees of freedom of
this theory are still not well understood. In particular near the Planck scale, where we
expect the stringy behavior to become more evident, it is very poorly understood what is
the correct configuration space and which states dominate the dynamics of string theory
at that scale.
One way to probe this region would be to study a very hot ensemble of strings. There
is, however, a difficulty as one tries to to heat the ensemble past a limiting temperature
−1
TH = βH [1,2], the famous Hagedorn temperature. The number of states of string theory
at a specific energy level increases exponentially with the energy. As a result, the parti-
tion function is infinite for β < βH due to the competition between the entropy and the
Boltzmann factor e−βE . What happens when we try to increase the temperature past this
point is, despite much effort, merely a speculation.
Since the effective string tension vanishes as one approaches TH one possibility is that
pumping more energy into the ensemble to increase its temperature merely results in the
formation of longer and “wigglier” strings by exciting higher oscillation modes [3]. This
might be related to a “string uncertainty principle”

h̄c GN E
∆x ≈ + 2 5 ,
E g c
where as we try to probe smaller distances by scattering more energetic strings, we obtain,
after a point, poorer resolution due to the formation of longer and longer strings.
Another suggestion is that the Hagedorn transition is really a first or second order
phase transition, where above TH we obtain a large genus zero contribution to the free
energy. In a string theory with smooth world sheets we expect the genus zero contribution
to the partition function to be zero. The reason is that the partition function is calculated
by toroidally compactifying the (euclidean) time direction with radius R = β/2π. Since
the sphere is simply connected, it cannot wrap around the time direction and it should give
a trivial β dependence. This suggests that smooth Riemann surfaces are not appropriate
for describing the high temperature phase of the string. The following picture is suggested
to hold [4–6]: The modes ϕ and ϕ∗ that wind once around S 1 become tachyonic at
TH and are supposed to acquire nonzero expectation values < ϕ > and < ϕ∗ > around

1
which we should look for stable solutions (in fact due to the dilaton the transition occurs
at Tcrit < TH ). Then the ϕ vertex operator creates tiny holes on the world sheet that
wind around the time direction tearing the world sheet. This suggests that the degrees of
freedom in the high-T phase are drastically reduced. In fact the large T behavior of the free
energy implies that there are fewer fundamental gauge invariant degrees of freedom than
any known field theory that lives in the embedding space (for example for the closed strings
one finds that the number of degrees of freedom is the same as that of a two dimensional
field theory!).
At this point it is useful to consider the parallel with low energy QCD. It has been
claimed long ago that large N QCD [7] or the strong coupling limit of QCD [8] can be
formulated as a string theory (for a recent review see [9] and references therein). In favor
of this point of view is the early success of the dual resonance models in describing the low
energy meson resonances. There are, however, difficulties with this picture. For example
the string in the relevant non-critical dimensions is not Lorentz invariant in the light
cone quantization and has one too many oscillators because of the Liouville mode in the
Polyakov quantization.
It is believed that SU (N ) QCD undergoes a first order deconfining transition at
some temperature Tdec . This can be the analogue of the Hagedorn transition of string
theory. We understand the high energy regime because of asymptotic freedom. The basic
degrees of freedom are weekly interacting quarks and gluons. In the strong coupling limit
perturbative QCD is no longer a good description anymore, since the perturbation series,
even as an asymptotic series, is not a good approximation. Hadrons, mesons and glueballs
are the basic degrees of freedom and an effective string theory description in terms of
thin chromo-electric flux tubes might be possible. The low temperature phase is described
by smooth world sheets and it is expected that such a description breaks down in the
high temperature regime; it is the underlying field theory that gives the basic degrees of
freedom in that case. In string theory, however, a 26 or 10 dimensional field theory would
be inappropriate for the high temperature phase, since such a theory would not have a
good ultraviolet behavior.
The study of finite temperature strings also suggests some perhaps deep connection
between multiplicative number theory and statistical systems with an exponentially grow-
ing density of states [10–13]. There exist several examples, like the bosonic, fermionic
or parafermionic Riemann gas and the known string and conformal field theories, where
the partition function is related to well studied number theoretic multiplicative arithmetic

2
functions and modular forms. Understanding such a connection for physically interesting
string theories could be of great importance for calculating and studying the behavior of
relevant physical quantities. The number theoretic methods developed are quite power-
ful and can be used to calculate the partition function or give a very good estimate of
the asymptotic behavior of the level densities of the above models. We should also not
underestimate the possible mathematical importance of these questions.
This presentation is organized as follows. In section two we introduce the main rele-
vant concepts. In section three we present the connection between string theory at finite
temperature and number theory. We stress the relation between the level density p(N )
of the bosonic string, the Dedekind eta function η(τ ) and Ramanujan’s τ -function τ (n).
We explain how one can use Hardy and Ramanujan’s results to compute p(N ). In section
four we make this connection deeper by discussing the Riemann gas of bosons, fermions
or parafermions. In section five we discuss the microcanonical and canonical ensemble
of strings emphasizing the limit of validity of the canonical ensemble near the Hagedorn
temperature. In section six we present the derivation of the asymptotic density of states
for the microcanonical ensemble and the effect of introducing chemical potentials in order
to impose conservation laws in the computation of βH . In section seven we describe how
we can obtain the partition function with the string path integral on the torus. In sec-
tion eight we discuss the possible physical behavior of the string ensemble near βH and in
section nine we discuss the results of Kogan and Sathiapalan and Atick and Witten.

2. Strings as a Statistical Mechanical System

Consider a d dimensional string theory described by the embedding

X : Σ → Rd−1,1

of the Riemann surface (complex curve) Σ with coordinates (σ, τ ) into the d dimensional
Minkowski space Rd−1,1 with cartesian coordinates X µ , µ = 1, 2, . . . , d. The classical
free equations of motion
(∂σ2 − ∂τ2 )X µ = 0 (1)

and the boundary conditions, which for the open string are X ′µ (σ, τ ) = 0 for σ = 0 and
π, determine the mode expansion
X αµ
µ
x (σ, τ ) = xµ0 µ
+p τ +i √n e−inτ cos nσ . (2)
n
n6=0

3
Upon quantization, the constraint (L0 − a)|χ >= 0 on the physical states determines the
spectrum of the theory
1 2
M = N̂ − a . (3)
2
In the above formulas L0 is the zero mode of the stress energy tensor, a a normal ordering
P∞ P∞
constant and N̂ = n=1 nα−n · αn = n=1 nα†n · αn is the level number operator. The
number of states at a particular level increases very rapidly with N and as we will soon
see, it increases exponentially with N for large N . In order to compute the number of
states at level N , we consider the generating function for the level degeneracies

F (z) ≡ Tr z N̂
∞
X (4)
= d(N )z N
N=0

where d(N ) is the number of mass eigenstates at level N . We can compute F (z) using
P∞
N̂ = n=1 nα†n · αn
P
∞
nα+
n ·αn
∞
Y +
F (z) = Tr z 1 = Tr z nαn ·αn
n=1
( ∞  )d−2 (5)
Y 1
=
n=1
1 − zn
= |f (z)|d−2 ,

†
Q
∞
1
P
∞
where f (z) = 1−z n = p(n)z n is the classical partition function of Euler. It is the
n=1 n=0
generating function for the number of unrestricted partitions of N into positive integers.
Proof: Since
∞
Y
f (z) = (1 + z r + z 2r + z 3r + ...) , (6)
r=1

a typical term z k in f (z) is obtained by taking one contribution from each factor r =
1, ..., ∞ such that z k = (z)k1 (z 2 )k2 (z 3 )k3 ...(z n )kn with k = k1 + 2k2 + 3k3 + ... + nkn . This
yields a partition of k into sets {ki }.

3. Singularities of f (z) and Asymptotics of p(N )

The function f (z) has an essential singularity at all rational points of the unit circle
2πip
S 1 i.e. the set of points {e q : p, q ∈ Z} ≡ {zp,q }. The essential singularity arises
†
The tachyon may be accounted for by modifying this to F (z) = z1 |f (z)|d−2 .

4
 n
because zp,q = 1 for an infinite number of integers n = k · q ∀k ∈ Z. In order to estimate
the asymptotic behavior of the density of states we have to study the behavior of f (z) as
z → 1− . A crude estimate gives
∞ ∞
!
Y 1 X
lim f (z) = lim n
= lim exp − ln(1 − z n )
z→1− z→1−
n=1
1 − z z→1−
n=1
∞
! ∞
!
X n m
(z ) X zm
= lim exp = lim exp . (7)
z→1−
m,n=1
m z→1−
m=1
m(1 − z m )
∞
!
X 1 1
= lim exp
z→1−
m=1
m2 1 − z

This is the first appearance of exponential growth in the problem. One can improve the
above estimate by using the modular properties of f (z). Writing z as e2πiτ , the classical
partition function takes the form
∞
Y 1
f (τ ) = 2πiτ
, (8)
n=1
1 − e

which is the partition function of a single boson on the torus with modular parameter τ .
This is the first connection we see with additive number theory (combinatorics) and
conformal field theory. The function f (z) is closely related to an important modular form,
the Dedekind eta function η(τ ) via
∞
Y
iπτ
1
η(τ ) = e 12 1 − e2πinτ = z 24 /f (z) . (9)
n=1

This function has many fascinating number theoretic properties. For example
( ∞
)24 ∞
z Y X
24 n
η (τ ) = =z (1 − z ) ≡ τ (n)z n (10)
f (z)24 n=1 n=1

defines Ramanujan’s τ -function τ (n).

We list below some interesting properties of τ (n)
1) Ramanujan conjectured that τ (n) is a multiplicative function, i.e.

τ (mn) = τ (m)τ (n) if (m, n) = 1 ,

where (m, n) is the greatest common divisor of m and n. This was proved by Mordell
[14].

5
 P
2) τ (m)τ (n) = d11 τ ( mn
d2 ).
d|(m,n)
3) If τ (p) ≡ 0 (mod p) then τ (pn) ≡ 0 (mod p) ∀n.
4) Dyson [15] proved that

X (a − b)(a − c)(a − d)(a − e)(b − c)(b − d)(b − e)(c − d)(c − e)(d − e)

τ (n) =
1!2!3!4!
a,b,c,d,e

where a, b, c, d, e satisfy

a, b, c, d, e ≡ 1, 2, 3, 4, 5 (mod 5) ,

a + b+c+ d+ e = 0,

and
a2 + b2 + c2 + d2 + e2 = 10n .

The function η(τ ) is a modular form of weight + 12 since

η(−1/τ ) = (−iτ )1/2 η(τ ) . (11)

Using (11) we can map the region z → 1− , where f (z) has singularities, to the region
z ′ → 0+ , where f (z) is regular and close to 1. Under τ → −1/τ , z = e2πiτ → z ′ = e−2πi/τ
 2
and since z ′ = exp 4πln z , we obtain

 −1/2
′ (z ′ )1/24 − ln z
f (z ) = = z −1/24 f (z) (z ′ )1/24 .
η(−1/τ ) 2π

Thus
  12  
− ln z 1 −π 2
lim f (z) = lim z 24 exp (12)
z→1− z→1− 2π 6 ln z
Note that (12) is a slightly improved estimate of the behavior of f (z) as z → 1− as
compared to (7).
To extract p(N ) from the above results we follow Hardy and Ramanujan [16] who first
applied the techniques of complex analysis to the combinatoric problem of determining the
asymptotic behavior of the number of unrestricted partitions of an integer N .
I
1 f (z)
p(N ) = dz , (13)
2πi z N+1

6
where the contour is taken around the origin and within and close to the unit circle. Then
I  1/2  
1 − ln z −π 2 23
p(N ) = dz exp − (N + ) ln z . (14)
2πi 2π 6 ln z 24

A saddle point evaluation of p(N ) (see appendix) leads to the result

r !
1 2√
p(N ) ≈ √ exp π N . (15)
4 3N 3

A better result for p(N ) was derived by Hardy and Ramanujan. They showed that
 cλN  1
!
1 d e (−1)N d e 2 cλN
p(N ) = √ +
2π 2 dN λN 2π dN λN
√   1
! (16)
3 2 1 d e 3 cλN
+ √ cos Nπ − π +...
π 2 3 18 dN λN
q q
1 2
where λN = N− 24 and c = π 3. For N = 100 and 200, summing 6 terms of this
series gives p(100) = 190, 569, 291.996 and p(200) = 3, 972, 999, 029, 338.004, whereas the
exact results are p(100) = 190, 569, 292 and p(200) = 3, 972, 999, 029, 388 respectively [16].
Later an exact analytical result was found by Rademacher [17] using a slight variant of the
Hardy-Ramanujan analysis
  
cλN
d  sinh
∞
X
1 1 q
,
p(N ) = √ Aq (N )q 2 (17)
π 2 q=1
dN λN

where Aq (N ) is a certain sum of roots of unity.

4. Quantum Statistical Mechanics and Multiplicative Number Theory

Multiplicative number theory in the context of statistical systems with exponentially

growing number of states has been considered first by Julia and Spector[10–11]. Consider,
instead of the usual Fock space of a quantum field theory, a different labelling of the
quantum states. Place the creation operators {a†i } in cardinal order and associate the
Q
ith prime number with the ith creation operator. Then label the state i (a†i )ri |0 > by
Q
the integer N = i pri i . Since the factorization of N into prime numbers is unique, this
provides a basis for the Hilbert space of states. This is known as the Gödel numbering
[18]. We are arithmetizing this way a free quantum field theory the same way as the formal

7
calculus of propositions is arithmetized by associating the ith symbol in a proposition with
the ith prime number and determining the corresponding multiplicity by the symbol itself.
A simple example of a system with exponential growth of the number of states is the
Riemann gas. The partition function of such a system is defined as

Z(β) = Tr e−βH , (18)

with energies Ei = ln pi . Then

X X
Z(β) = exp(−β ri ln pi )
{ri } i
∞
X (19)
1
= = ζ(β)
Nβ
N=1

is the (ordinary) Riemann zeta function. The partition function therefore, has a simple
pole at β = 1. This can alternatively be seen by computing the density of states

1
ρ(E) = ρ(N )∆(N ) =
∆E
1 1
=
= N (1 + + . . .) = (20)
ln N+1
N
2N
1
= exp(E) + + O(e−E ) .
2
We see that βH = 1 corresponds to the simple pole of ζ(β) at β = 1.
Generally, multiplicative generating functions of the form (Dirichlet series)

X∞
f (n)
F (β) = (21)
n=1
nβ

are of interest, where f (n) is a general multiplicative arithmetic function on the natural
numbers valued in some field (usually the field of real or complex numbers). As in the case
of the ordinary Riemann zeta function, F (β) generally has a simple set of singularities but
a rich structure of complex zeroes.
The case f (n) = µ(n), where µ(n) is the Möbius function, gives the partition function
ζ(β)
of the fermionic analog of (18). This gives ZF (β) = ζ(2β) . Similarly the system of k-
parafermions is associated with [13]
n Q
f (n) ≡ µk (n) = 1 if n = i (pi )ri with 0 ≤ ri ≤ k − 1; (22)
0 otherwise.
8
It gives
ζ(β)
Zk (β) = . (23)
ζ(kβ)
Parafermions of order 2 and ∞ correspond to fermions and bosons respectively. Note that
a representation of the ordinary Riemann zeta function can be obtained by tensoring an
infinite set of parafermionic gases of order k at successively lower temperatures, such that
∞
Y
ζ(β) ≡ Z∞ (β) = Zk (k m β) . (24)
n=0

We hope that the above exposition has given a flavor of the important connection between
arithmetic gases and multiplicative number theory. Much remains to be understood for
the case of statistical systems that arise from string theories and conformal field theories.

5. The String Microcanonical and Canonical Ensemble

We are now ready to compute the microcanonical density of states for the string
ensemble. Since the mass spectrum is given by
1 2
M = N − const , (25)
4
the asymptotic form for p(N ) translates into the string density of states

Ω(M ) ∼ M −α exp (βM ) . (26)

The microcanonical density of states Ω(E) is defined by

Ω(E) = Tr δ(E − H̃)

Z
dβ̃
= Tr eiβ̃(E−H̃)
2π
β0Z+i∞
dβ βE
= e Tr e−β H̃ (27)
2πi
β0 −i∞
β0Z+i∞
dβ βE
= e Z(β) ,
2πi
β0 −i∞

where β = iβ̃ and β0 is chosen for the convergence of Z(β) ≡ Tr e−β H̃ , the canonical
partition function. Inverting the Laplace transform we obtain
Z ∞
Z(β) = dE Ω(E)e−βE (28)
0

9
In a typical statistical mechanical system with N degrees of freedom Ω(E) ∼ E N for large
E. Thus the entropy S = ln Ω(E) ∼ N ln E is extensive. For large N , the integrand is
sharply peaked and the canonical partition function can be evaluated by a saddle point
approximation. Since Z ∞
Z(β) = dE exp (ln Ω(E) − βE) (29)
0

the saddle point occurs at E0 such that

d ln Ω(E) N
=β= , (30)
dE E=E0 E0

N
i.e. E0 = β
= N T . Thus fixing the temperature T (canonical ensemble) is equivalent
in the thermodynamic limit to fixing E in the microcanonical ensemble at N T and T is
really the average energy per degree of freedom. The fluctuations are given by

Z∞  2

1 2 ∂
Z(β) = Z0 dE exp (E − E0 ) (S(E) − βE)
2 ∂E 2
0
(31)
Z∞  
1 ∂ 2 S(E)
= Z0 dE exp (E − E0 )2 .
2 ∂E 2
0

This may be rewritten in terms of the microcanonical (formal) specific heat CV . Since

−1
SM = ln Ω(E) and TM = ∂E ∂S , we have

∂ 2S 1 dT 1 1
2
=− 2 =− 2
, (32)
∂E T dE V T (CV )M

dE
where CV = dT . Then
 −1
1 ∂ 2S
(CV )M =− 2 . (33)
T ∂E 2
Thus (31) becomes
 
1 1 −1
Z(β) = Z0 exp − (E − E0 )2 2 (CV )M . (34)
2 T

Convergence requires that CM > 0 in order that saddle point approximation is valid.
For strings Ω(E) ∼ E −α exp βM E and as one approaches TH from below the integrand
Ω(E)E −βM E flattens out. Typically α > 1 and Z(β) is well defined for β = βH , but it
+
diverges for β > βH . Fixing T as β → βH no longer corresponds to fixing a precise E and

10
the canonical and microcanonical ensemble need not agree. The entropy S = ln Ω(E) ∼
βH E − α ln E is no longer extensive as well. For further discussion see section 7.
For free particles
Y
−1 Y

Z(β) = 1 − e−βEk1 b 1 + e−βEk1 f .
k1 b k1 f

Therefore
∞ 
X 
fB (βr) r fF (βr)
ln Z(β) = − (−1) , (35)
r=1
r r

where the single-string partition functions fB and fF are

X Z∞
−βEk,b
fB = e = dE e−βE ωB (E)
k,b 0
(36)
X Z∞
fF = e−βEk,f = dE e−βE ωF (E)
k,f 0

Substituting (35) into (27) we obtain the multi-string density of states from the single
string density of states

β0Z+i∞ (∞  )
dβ βE X fB (βr) f F (βr)
Ω(E) = e exp − (−1)r . (37)
2πi r=1
r r
β0 −i∞

The r = 1 term gives the familiar Maxwell-Boltzmann expression

X∞ Z Y
n X
1
Ω(E) = dEi ω(Ei )δ(E − Ei ) (38)
n=1
n!
i=1 i

where ω(E) = ωB (E) + ωF (E)

6. From the One-String Density of States to the Multi-String

Density of States for Toroidal Compactification

Consider d-dimensional strings compactified on an internal manifold of dimension c

that is for simplicity taken to be a torus with radii Ri with i = 1, 2, . . . , c. In computing the
full density of states, it is important to incorporate the effects of conservation laws [19] such
as conservation of momentum, winding number and, in the case of the heterotic string, of

11
U (1) charges or windings/momenta in the internal directions. For every conserved charge
QA we introduce a chemical potential µA and compute
 
Ω(E, µA ) = tr e2πiµA QA δ(E − H)
  (39)
Z(β, µA ) = tr e2πiµA QA e−βH .
The condition that the total conserved charge be some fixed value qA (typically qA = 0)
is then enforced by multiplying Ω(E, µA ) or Z(β, µA ) by exp{−2πiµA qA } and integrating
µA over the range (− 21 , 12 )
Z 1
2
Ω(E, qA ) = dµA Ω(E, µA )e−2πiµA qA . (40)
− 12

The full density of states will be derived from the single-string density of states as discussed
in the previous section. In order to compute the single string density of states, consider
the energy of a general string state
c  2
X 
2 2 n i 2
E =k + + (2Ri ) m2i + 4nL + 4nR . (41)
i=1
Ri2

Here k is the momentum in the non-compact directions and ni and mi are the momentum
and winding quantum numbers, respectively. The integers nL and nR label the internal
level numbers describing oscillatory mode excitations of the string.
As described in section 1, the number of left-moving (right-moving) excitations at a
P P
given level is given by the generating function F (z) = α z nLα (F (z̄) = α z̄ nRα ) where
the sum runs over all possible states α. For a bosonic sector f (z) = η −24 (z) = 1z f (z)24 (the
1 ϑ42 (z)
extra factor z corresponds to the tachyon), while for a superstring sector f (z) = η 12 (z) .
As before we compute the asymptotic level densities using a saddle point evaluation
I
1 dz
d(n) = n+1
f (z) . (42)
2πi z
The resulting level densities for left and right sectors are
√ √
−(d+1)/4 4π aL nL −(d+1)/4 4π aR nR
d(nL ) ∼ nL e , d(nR ) ∼ nR e (43)
1
where aL,R = 2 for a superstring sector and aL,R = 1 for a bosonic string sector (so for the
heterotic string aL = 1 and aR = 21 ). Physical states must also obey the level-matching
P
condition L0 − L̄0 = 0, or nL − nR = − i mi ni . This and (41) give
 2
2 2 ni
8nL = E − k − + 2Ri mi
Ri
 2
2 2 ni
8nR = E − k − − 2Ri mi .
Ri

12
Therefore the single-string density of states is (including the chemical potentials κ, µ and
ν)
Z
dD k X EdE 2πi(κ·k+µ·m+ν·n)
ω(E, κ, µ, ν)dE ∼ VD e
(2π)D m ,n 4
i i
 √ √ (44)
4π aL nL (E,k,ni ,mi )+4π aR nR (E,k,ni ,mi )
e
.
nL (E, k, ni, mi )(d+1)/4 nR (E, k, ni, mi )(d+1)/4
Here D = d − c − 1 is the number of non-compact space directions, and VD is the volume
of the non-compact directions.)
Note that for each Ri there are regions with different behaviour. If Ri ≫ E ≫ 1/Ri we
ignore the winding quantum numbers mi and the sum over the closely spaced momentum
levels is well approximated by an integral over a continuous spectrum of momentum; thus
the string behaves as if the direction i were noncompact. Likewise, if 1/Ri ≫ E ≫ Ri the
sum over momenta drops out and the sum over winding becomes continuous; this situation
is dual to the previous one under R → 1/2R. Finally we have the truly high-energy regime,
E ≫ Ri , 1/Ri . Here we can closely approximate the sum over mi , ni by an integral dmi dni .
Furthermore, for large E we can evaluate the integral (44) by saddle point methods. If
we expand the arguments of the square roots to first order in k 2 , (ni /Ri ± 2Rmi )2 (valid
for large E in the saddle point approximation since the region (ni /Ri ± 2Rmi )2 /E 2 ≪ 1,
k 2 /E 2 ≪ 1 dominates), the integral becomes quadratic and yields

1
ω(E, κ, µ, ν) ∼ eβH (κ,µ,ν)E (45)
E D/2+1

where
(
2
2 )
µ µ
√ κ2 4R + νR − νR
βH (κ, µ, ν) = βH − 2π √ √ + √ 2 + 4R
√ 2 (46)
aL + aR aL aR

and
√ √ √
βH = 2π ( aL + aR )

is the usual Hagedorn temperature [3]. Thus in effect the Hagedorn temperature depends
on the chemical potentials.
This result can now be used to derive the multi-string density of states as sketched in
the previous section. First we will compute the Maxwell-Boltzmann contribution (38) to
the total density of states, and then we will argue that the corrections due to Bose-Einstein

13
or Fermi-Dirac statistics do not substantially alter the asymptotic form of the density of
states.
The MB expression (38) becomes a sum of terms of the form

Z Y
n
!
1 dEi βH (κ,µ,ν)Ei X
D/2+1
e δ E− Ei . (47)
n! i=1 E i
i

The integrals over Ei are divergent at the lower end and must be cut off at some energy
m0 >
∼Ms where the asymptotic expression (45) is no longer valid. It is clear that the
dominant contribution to (47) is that in which most of the energy is in one string, Ei ≈ E.
Thus the integral is approximately
!n−1
1 eβH (κ,µ,ν)E 2
Ωn (E, κ, µ, ν) ∼ for D > 0;
(n − 1)! E D/2+1 Dm0
D/2
(48)
βH (κ,µ,ν)E
1 e n−1
∼ [ln(E/m0 )] for D = 0.
(n − 1)! E

Summing over n then gives

eβH (κ,µ,ν)E 2/ DmD/2
Ω(E, κ, µ, ν) ∼ e 0
for D > 0;
E D/2+1 (49)
1 βH (κ,µ,ν)E
∼ e for D = 0.
m0

The density of states without the conservation constraints is given by setting κ = µ =

ν = 0; this expression has been found in the literature. To obtain the density of states
including the constraint of zero total momentum and winding, we use (46) and integrate
over κ, µ, ν to find
eβH E
Ω(E) ∼ (50)
E d−δD
where we define δD = 1 if D = 0 and δD = 0 otherwise. Imposing the constraints changes
the power of E; this is in contrast to other thermodynamic situations where conservation
laws are not so important essentially because one string carries most of the energy and
bears the burden of the conservation laws.
We can also compute corrections to (50) due to Bose-Einstein and Fermi-Dirac statis-
tics. One finds that the effect of these corrections is to multiply (50) by a function that
depends on low-energy physics but to leading order not on E.

14
 Thus for the type II string the asymptotic density of states is

eβH E
Ω(E) ∼ . (51)
E 10−δD
If we consider the heterotic E8 × E8 or Spin(32)/Z2 string we see that conservation of
the 16 U(1) charges should also be imposed. In the bosonic formulation this is equivalent
to momentum/winding conservation for the internal degrees of freedom, and, as one can
easily convince oneself by a slight modification of the above argument, gives an extra factor
of 1/E 8 . Thus the asymptotic density of states for the heterotic string is

eβH E
Ω(E) ∼ . (52)
E 18−δD
It will be useful for us to deduce which string configurations contribute most signifi-
cantly to the total string density of states; this can be done by inspection of (47) or the
corresponding expression including quantum statistics. As we have already noted, the
dominant configuration for the integral for a fixed number n of strings is when Ei ≈ E,
i.e. when most of the energy is in a single string. For D > 0 it also appears that string
configurations with a small total number of strings dominate the sum, but for D = 0
configurations with a large total number of strings dominate the sum. (These qualitative
features also hold true when corrections due to quantum statistics are included.) We will
return to a discussion of this single string dominance of the energy in section 7

7. Path-Integral Derivation

In the meantime, we turn to the second derivation of the string density of states
[20–23] and [3]. Once again (27) is used to relate Ω(E) to the complex function Z(β).
Next recall that the free energy F = − β1 ln Z for a single free particle of mass M can be
computed from a first-quantized one-loop path integral on a space with compactified time
R1
direction of circumference β. The action is Sp = 21 0 dτ [e−1 (dX/dτ )2 + eM 2 ], and in
the path integral we integrate over all maps X µ (τ ) of the circle into the target spacetime
S 1 × Rd−1 and over all one-metrics g = e2 on the circle. Explicitly,
∞ Z
X Z
Dg
2 ln Z = DXe−Sp (−1)nF̂ (53)
n=−∞
Vol(Diff) n

where in the path integral over metrics we must eliminate the volume of the diffeomorphism
group, and where the boundary condition on the X path integral is X 0 (1) = X 0 (0) +

15
nβ, X i (1) = X i (0); n is the number of times that the circle winds around the compact
time direction. We also define F̂ to be zero for bosons and one for fermions; the extra
factor (−1)nF̂ corresponds to anti-periodic boundary conditions for fermions in the second-
quantized formalism. One easily shows (taking into account the Killing vector on the circle)
that Z Z ∞
Dg ds
=
Vol(Diff) 0 s
where s is the proper length of the circle. One can also show
Z
n2 β 2 M2s
DXe−Sp = (V β)(2πs)−d/2 e− 2s − 2
n

where V is the spatial volume. These give the free energy

Z ∞ X∞
V ds −d/2 n2 β 2 M2s
F =− (2πs) e− 2s − 2 (−1)nF̂ . (54)
2 0 s n=−∞

It is fairly simple to demonstrate that this is equivalent to the standard expressions for
the free energy of a single free particle.
Likewise, the free energy for string is obtained by doing the functional integral over
maps of the torus into S 1 × Rd−1 , with the (conformal gauge-fixed) action which we may
take to be that of the heterotic string (our units are such that α′ = 21 )
Z
1

S= d2 σ ∂z X µ ∂z̄ Xµ + ψ̄ µ ∂z ψ̄µ + ψ µ ∂z̄ ψµ + λi ∂z̄ λi .
2π

(The λi provide a fermionic representation of the gauge degrees of freedom.) The maps
are allowed to wind in the time direction on the torus, but not in the space direction, as
in the case of the particle. The result is
Z ∞ Z 1
V dτ2 2 1 b
F = − π −5 dτ1 (2πτ2 )−5 12 ch[G](q)
4 0 τ2 − 21 η(q)8 η̄(q̄)
X X  2 2 (55)
β n
C̄π1 π2 (q̄) exp − (−1)nπ1 .
π1 ,π =0,1 n
2πτ2
2

Here q = exp{2πi(τ1 + iτ2 )}, η is the Dedekind function; ch[G]b is the character function of
the gauge group G = E8 ×E8 or SO(32), ch[E b8 × E d =
b8 ] = (ϑ82 +ϑ83 +ϑ84 )2 /4η 16 , ch[SO(32)]
(ϑ16 16 16 16 4 4 4
2 +ϑ3 +ϑ4 )/2η ; and C̄00 = ϑ̄3 , C̄01 = −ϑ̄4 , C̄10 = −ϑ̄2 , and C̄11 = 0. This expression
is analogous to (54); the integral over πτ2 corresponds to that over s, the sum over n is the

16
sum over sectors where the time direction on the torus wraps n times around the target
time dimension, and the rest of the expression integrated over τ1 gives the sum over all
single string states of (−1)nF̂ exp{−M 2 πτ2 /2}. We could therefore alternatively derive
this expression directly from (54) by thinking of the string as a collection of particles
corresponding to its various modes.
To make this expression look more familiar from the string point of view, we recall
1
that the region S of the τ = τ1 + iτ2 plane |τ1 | < 2, 0 < τ2 < ∞ corresponds to an
1
infinite number of copies of the fundamental domain F defined by |τ1 | < 2
, |τ | > 1; S
is tiled by images of F under maps τ ′ = (pτ + q)/(rτ + s), p, q, r, s ∈ Z, ps − qr = 1.
The free energy for the heterotic string can therefore be rewritten as an integral over the
fundamental domain
Z X
V d2 τ 1 b
F = − π −5 (2πτ2 )−5 12 ch[ G](q) C̄π1 π2 (q̄)
4 F τ2 η(q)8 η̄(q̄) π1 ,π2 =0,1
  (56)
X β 2

2 2 2
exp − m τ2 + (n − mτ1 ) (−1)mπ2 +nπ1 +mn
m,n
2πτ 2

and likewise for the type II string

Z X X
V d2 τ 1
F = − π −5 (2πτ2 )−5 8 Cπ1 π2 (q)C̄π̄1 π̄2 (q̄)
8 F τ2 η(q)8 η̄(q̄) π1 ,π2 =0,1 π̄1 ,π̄2 =0,1
  (57)
X β 2

2 2 2
exp − m τ2 + (n − mτ1 ) (−1)m(π2 +π̄2 )+n(π1 +π̄1 ) .
m,n
2πτ 2

From (56) one can proceed to calculate the asymptotic density of states .
In these expressions the sum over m is the sum over maps of the torus in which the
space direction of the torus is wound m times about the target S 1 . While one might think
of these as corresponding to strange states of the string in which it wraps around the time
direction, this interpretation is not necessarily correct if the fundamental expression for
the free energy is (55) which directly corresponds to Z(β) = tr e−βH . This trace contains
only ordinary string states and not the strange ‘winding states;’ the latter appear to arise
only as mathematical artifacts when we transform the expression for ln Z as an integral
over S into an integral over F .
Note the following interesting point. (55) is equivalent to the modular invariant (56)
the cosmological constant Λ = 0. However, if Λ 6= 0, (55) gives ∞ · Λ + F ′ and (56) gives
Λ + F ′ where Λ is cosmological constant as it is usually defined in string theory (as an

17
integral over F ) and F ′ is the temperature dependent part of F . This is also a difficulty
for string field theory -i.e. to naturally produce the fundamental domain.

8. Physical Behaviour of Strings at High Temperature

We now turn to an attempt to extract a better understanding of the physics of strings

at high temperature from the above calculations [3,23]. To start with, assume that we
have at our disposal a thermal bath that can interact with the string ensemble to heat it
up. Within the context of fundamental strings this assumption is not necessarily valid as
all matter is presumably made out of strings, but we make this initial assumption to aid
us in isolating the source of any strange behavior that we may encounter; we will see that
the assumption of the existence of such a reservoir doesn’t obviously appear to be related
to any such behavior. Therefore suppose that we have a heat reservoir with a density of
states Ωr (E).
The total density of states for the string ensemble in contact with this reservoir is
Z E
ΩT (E) = dEs Ω(Es )Ωr (E − Es ) .
0

If we assume that the reservoir is large (in a sense to be made precise shortly), then most
of the energy will be in the reservoir, i.e. E ≫ Es . In this case we can expand

∂ 1 ∂2
ln Ωr (E − Es ) ∼
= ln Ωr (E) − Es ln Ωr (E) + Es2 ln Ωr (E) + − · · · .
∂E 2 ∂E 2
∂
We identify ∂E
ln Ωr (E) = β as the inverse temperature of the reservoir. We can drop the
2
∂
higher terms when the reservoir is sufficiently large; precisely, Es2 ∂E 2 Sr (E) ≪ 1 or

1 ∂E
2
CV r = − ≫ Es2 . (58)
β ∂β

Thus we can use the ensemble

Z ∞
ΩT (E) = Ωr (E) dEs e−βEs Ω(Es ) (59)
0

as long as we ensure that the heat capacity of the reservoir is large enough in the sense of
(58); replacing the upper limit by ∞ has no significant effect.
Now we are prepared to study the properties of the string gas. As we raise the
temperature, (59) stays well defined all the way to βH . Even at βH (59) is well defined

18
because of the power law tail ∼ 1/E (d−δD ) . (Of course Ω(E) takes a different, convergent,
form for low energy;nsince the string is a gas of
o massless particles at very low energies, we
1/(D+1)
expect Ω(E) ∼ exp a D+1 D E
D/(D+1)
VD there for some constant a). Furthermore,
the mean energy density R∞
dEe−βE EΩ(E)
hEi = R0 ∞ (60)
0
dEe−βE Ω(E)
is finite for β = βH ; e−βH E Ω(E) will presumably look something like Fig. 1 where the
maximum E0 is somewhere between the low energy and high energy domains where Ω(E)
is known, i.e. E0 will be of order one in string scale units.
What happens when we attempt to raise the temperature further? It is apparent
from (59) that for β < βH this ensemble is not defined; its mean energy becomes infinite.
Therefore it seems that we cannot pass βH . But in principle we could imagine adding more
energy to the string system in an attempt to raise its temperature; where does this energy
go? A clue was pointed out at the end of section five: at large energies the density of
states is dominated by one (or a small number of) highly energetic, and therefore long and
wiggly, strings. Thus it seems quite reasonable that any energy that we add to the system
in an attempt to raise the temperature goes into making strings in very high oscillation
modes. The low mass modes are populated with a thermal distribution at βH , and the rest
of the energy is in oscillation modes. It may be useful to think of this transition to long-
string dominance as a higher-order phase transition with an infinite latent heat. The low
temperature phase is what we know, and the ‘high temperature phase’ is the configuration
where the fractional energy in long string is one, but this phase is inaccessible because it
corresponds to infinite energy. (This is assuming that the fundamental degrees of freedom
are not drastically modified at some finite energy density, as could well be the case.)

9. X-Y Model and the Berezinskiĭ-Kosterlitz-Thouless Transition

Consider the string action in the conformal gauge

Z
1
S= dσdτ (∂α X µ ∂ α X µ ) , (61)
2π
where we take α′ = 1/2. Compactifying in the time direction, in order to account for the
finite temperature, and using periodic boundary conditions for closed string X µ (σ +π, τ ) =
X µ (σ, τ ) and X µ (σ, τ + π) = X µ (σ, τ ) + nβδ0µ , we obtain the mode expansion for the time
coordinate
2πnτ mβσ
X 0 (σ, τ ) = X̃ 0 + + + (oscillators) (62)
β π

19
and  2  2
1 2 1 nπ mβ 1 nπ mβ
E = NL + + − 1 + NR + − −1 (63)
4 2 β 2π 2 β 2π
with the constraint NL − NR = m · n. The state NL = NR = 0 with n = 0 and m = ±1
has
1
E(T )2 = M (T )2 = −8 + (64)
π2T 2
√
For T < TH = 1/2π 2 its mass-squared is positive but at T = TH it becomes massless and
for T > TH tachyonic. Call the m = ±1 winding states ϕ and ϕ∗ respectively. Thus the
Hagedorn divergence (T > TH ) corresponds to the existence of a tachyon and a divergence
at the boundary τ2 → ∞ of the moduli space in the path-integral language [4–5]

Z∞
2
F (β) ∼ (const.) dτ2 e−M (β)τ2
→∞ for M 2 (β) < 0 . (65)

β
The action for the ϕ mode given by X 0 = 2π ϕ with ϕ(σ, τ + π) = ϕ(σ, τ ) + 2π is
Z
β2
Sϕ = (∂ϕ)2
8π 3
Z (66)
g β2
= (∂ϕ)2 , g≡ .
2 4π 3

The action Sϕ is that of the Villain model which is the continuum limit of that of the X-Y
model (planar magnet)
g X
S=− cos(φi − φi+δ ) . (67)
2
<i,δ>

The vortex solutions of the model are given by

 
i z − z0
ϕ(z) = − m ln , (68)
2 z̄ − z̄0

where z0 is the position of the vortex and m is the winding number. The corresponding
free energy is  
πg A
Fm = ( m2 − 1) ln , (69)
2 a2
where a is the lattice spacing cutoff. The above model has an infinite order Berezin-
√
skiĭ-Kosterlitz-Thouless phase transition at gc = π2 or βc = 2 2π = βH ! In the low
temperature regime (g > gc ) the system is dominated by spin wave excitations and is in an
unscreened scale invariant phase with power law spin-spin correlations and tightly bound

20
vortex-antivortex pairs. In the high temperature regime (g < gc ) the topological order is
destroyed, the vortices unbind and are free (vortex condensation). The phase is screened
with exponentially decaying spin-spin correlations characterized by a coherence length ξ
which breaks the scale invariance.
Consider a chemical potential µ for vortices, where µ corresponds to the energy of
dissociation of a vortex-antivortex bound state. The RG flows for the system are in the
fugacity y (y = e−µ/kT ) and T plane as shown in Fig.1.
For T < Tc and y small the system flows to y = 0 or to µ = ∞ and vortices are bound
(region I). For T > Tc and y small the system flows off to y large out of the domain of
perturbation theory but µ is effectively small and vortices can unbind.
To describe string theory in this phase one would have to look at the coupled β-
functions for all the massless moduli at β = βH which include gµν , Bµν , the dilaton σ
and ϕ. This might have solutions corresponding to a conformal fixed point for T > TH .
Atick and Witten [6] suggested that ϕ gets a v.e.v. < ϕ > before TH thus obviating
the difficulties of the Hagedorn divergence. But they found the effective potential for ϕ
obtained from its interactions with the dilaton σ is of the type

Vef f (ϕ) ∼ m2 (T )ϕ∗ ϕ − λ̂(ϕ∗ ϕ)2

with
λ̂ > 0 and m2 (T ) = −8 + 1/π 2 T 2 .

This implies a first-order phase transition. Vef f (ϕ) must, however, be stabilized by higher-
order terms and < ϕ > is not calculable in perturbation theory.
Atick and Witten argued that, had λ̂ been negative, there would be a second order
phase transition at T = TH . For T just greater than TH then

2 m2
|< ϕ >| =
2λ̂

and the free energy is

−m2 −m2
F = V (< ϕ >) = =
4λ̂ 4λg 2 T
This is a genus zero (sphere) contribution to the free energy and implies the world-sheet is
spontaneously tearing to become non-simply-connected. They then argue that this picture

21
is valid even for the first-order transition case λ̂ > 0. This is not, so far, justified by any
calculation and fails in the weak coupling limit g → 0.

Appendix

In this appendix we compute the asymptotic form of the level density p(N ) which led
to (15). From (14) we have that
I  1/2
1 − ln z
p(N ) = dz exp[g(z)] , (70)
2πi 2π
−π 2 π2 1 N+c
with g(z) = 6ℓnz − (N + c) ln z and c = 23/24. Then g ′ (z) = 6(ln z)2 · z − z , which has
a maximum at ln z0 = − √π6 · √ 1
N+c
. Then
2π √
g(z0 ) = √ N + c . (71)
6
Thus the leading behavior of p(N ) is
r r !
2 23
p(N ) ∼ exp π N+
3 24 (72)
 √ 
∼ exp βc N
q
with βc = π 23 . Corrections are obtained by looking at the quadratic fluctuations
1
g(z) = g(z0 ) + (z − z0 )2 g ′′ (z0 ) + . . . , (73)
2
√  q 
1
where g ′′ (z0 ) = 2 π 6 (N + c)3/2 exp π 23 √N+c . Then (70) is a Gaussian integral which
gives r !
1 1 2√
p(N ) = √ exp π N . (74)
4 3N 3
For the string in d dimensions (74) easily generalizes to
r !
2 √
p(N ) = (const.)N −(d+1)/4 exp π (d − 2) N . (75)
3

Acknowledgements

This was supported by the Outstanding Junior Investigator Grant DOE DE-FG02-
85ER40231. I also wish to thank Steve Giddings for collaboration on which part of this
review is based and Konstantinos Anagnostopoulos for much assistance in the preparation
of this manuscript.

22
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23
 Figure Captions

Fig. 1. R.G. flows for the Villain model. The x axis is Tc /T − 1 and the y axis the
fugacity e−µ/kT . The x axis is a line of trivial fixed points that can be reached
from region I. Points from region II flow to large y and out of the domain of
validity of perturbation theory.

24