Hagedorn Strings
Feng-Li Lin
師大物理系
Outline
High Temperature strings
String/Black Hole Correspondence
Hagedorn strings on AdS
A puzzle
• If black hole can form at the LHC, then can field theory explain the BekensteinHawking entropy?
• For field theory, the entropy associated with the high energy states at energy E
is proportional to log E.
• However, the black hole entropy is proportional to E(d-1)/(d-2).
• Therefore, there is a mis-match in entropy for the process
a + b −→ virtual particles −→ black hole
High Temperature Strings
Peculiarity of string spectrum
• finite set of massless particles, i.e. gauge bosons, graviton
a
Aµ
,
gµν
• exponential degeneracy of states at high energy (Hagedorn Spectrum)
eβs E
with
βs ∼ l s
• Hagedorn entropy
Sstring ∼ βs E
• At high energy, the string behavior is very different from particle theory
Random Walk of a String
• The above Hagedorn spectrum can be understood as random walk in target
space since the energy of a string is proportional to the length of random walk:
1 eβs "
ω(") ∼ V · ·
" W (")
• in d non-compact dimensions
ω(") ∼ V ·
eβs "
"1+d/2
• all space are compact
eβs "
ω(") ∼
"
Thermal Partition Function of String
• Counting the physical string states, the thermal partition function of a single
string is
Z1 (β) =
!
F0
d2 τ
T r(e−βHw.s. )
τ2
• Or using the path integral formulation
Z1 (β)
#
$
∞
2
2
"
β
|m
−
wτ
|
d2 τ
(4π 2 α! τ2 )−13 |η(τ )|−48
exp −
= V25 β
!τ
4τ
4πα
2
2
F0
m,w=−∞
#
$
! 2
∞
2 2
"
β r
d τ
2 !
−13
−48
exp −
= V25 β
(4π α τ2 ) |η(τ )|
4τ
4πα! τ2
2
R
r=−∞
!
Extract Hagedorn Behaviors
• Using free string gas (obeying Maxwell-Boltzmann statistics) approximation,
the free energy of string gas is
1 Z1 (β)
F (β) = −
β V25
i.e. Zstring gas = eZ1 (β)
• Using the saddle-point approximation
F (β) ! −
!
∞
"
#
2
dτ2
β τ2
2 "
−13
(4π α τ2 )
exp(4πτ2 ) exp −
4τ2
4πα"
• There exists a Hagedorn critical temperature ( τ2
βH
√
= 4π α!
∼")
Near Hagedorn T
2
e4π(βH −β
2
2
)τ2 /βH
∼ e8πβH (βH −β)τ2 /βH ∼ e(βH −β)R
Thermal Winding Scalar
• The Hagedorn phase transition can be understood as the appearance of the
thermal tachyonic winding mode as T>TH
• Or by the observation
!
−d/2
log Z(β) ∼ V
∼
!
0
!
∞
dd k −!k2
e
(2π)d
d" e−(β−βs )"
2
2
∼
−T
r
log[−∇
+
M
(β)]|Rd
d/2
"
"
Hagedorn Thermodynamics I
• From free energy we can evaluate the other thermodynamic quantities.
However, the canonical ensemble may break down if it is not equivalent to
micro-canonical ensemble. For d>2,
Ω(E) =
!
L+i∞
L−i∞
eβH E+γ0 V
dβ βE
e Z(β) = CV (d/2+1) (1 + O(1/E))
2πi
E
• The specific heat from micro-canonical ensemble density of states is negative
and divergent at Hagedorn temperature, so the Hagedorn thermodynamics is
not well-defined.
Hagedorn Thermodynamics II
• If all the space are compact, the micro-canonical and canonical ensembles are
equivalent but the specific heat from micro-canonical ensemble is divergent for
all temperature, therefore, the Hagedorn temperature is limiting.
S = log Ω = βH E ,
∂β −1
CV = −β (
)V = ∞
∂E
2
Hagedorn Thermodynamics III
• We can regularize the Hagedorn thermodynamics by compactifying the large
spatial dimensions, this introduce the sub-leading singularities in partition
function,
Hagedorn Cosmology
• In Brandenberger et al, they proposed a model using Hagedorn thermal
fluctuation to produce CMB spectrum
However, the power spectrum is not scale-invariant!
String/Black Hole Correspondence
Bekenstein-Hawking v.s. Hagedorn
• The Bekenstein-Hawking entropy for AdSd+1-charged Schwarzschild black hole
(d>2)
SBH
2
r+
d
d−2 l2AdS
d−21+
=
d−1 1+
2
r+
l2AdS
+
− µ2
µ2
βBH M,
βBH
2
4πlAdS
r+
=
2
2 ,
(d − 2)(1 − µ2 )lAdS
+ dr+
µ=
• BTZ
SBH
1 − (lAdS µ)2
=2
βBH M,
1 + (lAdS µ)2
βBH
2
2πlAdS
1
=
,
r+ 1 − (lAdS µ)2
• Hagedorn entropy
Sstring = βs (k)M,
2
,
µ = 4GN J/r+
wd+1 Q
d−2
2r+
Horowitz-Polchinski
• It suggests that there is a correspondence between Hagedorn strings and
black hole when
βs = βBH
• This leads Horowitz and Polchinski (first by Susskind) to conjecture that a black
hole will turn into a single string when the size of the horizon is of order of
string length, e.g. d=3 (true for other d’s)
2
MBH
SBH
r02
α!
N
2
∼ 2 ∼ 2 ∼ Ms ∼ !
G
G
α
c.p.
√
r02
α!
∼
∼
∼ N ∼ Ss
G
G
Hagedorn long string
• At the correspondence point, the black hole temperature is of order of string
scale, so the strings are in the Hagedorn regime. However, as in the earlier
discussions we see that the long string configurations dominate at Hagedorn
regime. This is also consistent with the random walk picture, i.e. diffusion of a
thermal scalar.
• At zero string coupling, the typical size of a Hagedorn string (diffusion length)
!∼N
1/4
ls
Self interaction
• If we start with a long string, we may expect the string to condense into a
black hole. However, this is not the case if the string coupling is weak as
required by the Hoop conjecture.
• It is interesting to see if there is a critical string coupling so that the long string
will collapse by self interaction and turn into a black hole as the size of the
string ball is of order of the Schwarzschild radius.
• The correspondence principle implies
√
α!
∼ N , G ∼ g 2 α!
G
gc ∼ N −1/4
• However, some preliminary study for a thermal scalar coupled to gravity yields
g0 ∼ N
(d−6)/8
!∼
ls
g 2 N 1/2
(d=3)
Hagedorn strings on AdS
Hagedorn Strings and Correspondence Principle in AdS(3).
Feng-Li Lin, Toshihiro Matsuo, Dan Tomino (Taiwan, Natl. Normal U.) . Jun 2007. 28pp.
e-Print: arXiv:0705.4514 [hep-th]
Thermal AdS(3), BTZ and competing winding modes condensation.
Micha Berkooz, Zohar Komargodski, Dori Reichmann (Weizmann Inst.) . WIS-06-07-JUN-DPP, Jun 2007. 40pp.
e-Print: arXiv:0706.0610 [hep-th]
Winding tachyons in BTZ.
Mukund Rangamani, Simon F. Ross (Durham U. & Durham U., Dept. of Math.) . DCPT-07-21, Jun 2007. 37pp.
e-Print: arXiv:0706.0663 [hep-th]
Hawking-Page v.s. Hagedorn
• AdS-Schwarzschild metric
ds2d+1 = f (r)d2 τ + f −1 (r)d2 r + r2 dΩ2d−1
2
f (r) = 1 + r2 /lAdS
− wd+1 M/rd−2 ,
wd+1 = 16πGd+1 /(d − 1)Ωd−1
• There is Hawking-Page transition from thermal AdS to AdS-black-hole, which
could be different Hagedorn transition.
BTZ black hole
• We need to superheat to reach Hagedorn regime without collapsing into a
black hole, i.e.
THP ∼ 1/lAdS ,
Ts ∼ 1/ls
• The nature of string/black hole transition is not clear in AdS space.
• For d=2 case, it is called BTZ black hole. In this case, there is no ``small black
hole” phase.
• The asymptotic torus of thermal AdS(3) and BTZ are related by the SL(2,R)
transformation. The BTZ black hole is topological in nature.
AdS(3) Strings
• String theory on AdS(3) is exactly solvable. This is not the case for higher
dimensional AdS space.
• The spectrum is the unitary representations of the SL(2,R) (isometry of AdS(3))
algebra
short strings (discrete)
long strings (continuous)
Maldacena-Ooguri-Son
• The AdS(3) thermal string partition function is
Z(β) =
=
!
d2 τ
Zgh Zint ZAdS
4τ
2
F0
!
d2 τ β(1 − 2/k)1/2 |η(τ )|4−2cint
Vint
(4π 2 α" τ2 )(cint +1)/2
F0 4τ2
∞
"
n,m=−∞
e−β
2
|m−nτ |2 /4πα! τ2 +2π(ImUn,m )2 /τ2
|ϑ1 (τ, Un,m )|2
• We have assumed the internal space to be trivial, and the central charges ar
cSL(2,R) = 3 +
6
,
k−2
cint = 26 − cSL(2,R)
• For internal CFT to be unitary, we need
k ≥ k0 = 2 +
6
23
,
pole structures & saddle-point
• The Theta function has complicated pole structure, which makes the saddle
point approximation more subtle than in the flat space.
• These poles are IR divergence, we need to introduce the IR cutoff before doing
saddle-point approximation, i.e.,
!
0
∞
dτ2 · · · = lim
!→0
∞ !
"
w=0
β
1
√
2π k (w+#)
β
1
√
2π k (w+1−#)
dτ2 · · · .
Extract Hagedorn behavior I
• We perform the saddle-point approximation for each sub-strip separately, and
we find that the sub-strip near the τ2 -axis dominates.
• We perform the saddle-point approximation of τ1-integral with fixed τ2, and
find out the dominant saddle-point at τ1 = 0. The resulting partition function is
Extract Hagedorn behavior II
• We can extract the IR divergence of the AdS volume, and the partition function
becomes
• Near the Hagedorn temperature, we can extract the long string spectrum by
introducing the momentum integration
Extract Hagedorn behavior III
• Taking the continuum limit for the w-sum, the partition function can be put in a
suggestive form
• The form for flat space Hagedorn string is
• We see that they have the same form except that d is replaced by 1+cint where
``1” corresponds to the radial direction. Therefore, the Hagedorn
thermodynamics is similar to the flat space one.
Correspondence principle on AdS(3)
• Our Hagedorn temperature in AdS(3) is monotonically decreasing as k grows,
and note that k>k0=2.26>9/4 so that the maximal Hagedorn temperature is
finite about 0.388 ls-1
• At the correspondence point
βAdS = βBH
2
2πlAdS
:=
r+
it yields
Large black hole condenses at flat space limit.
Unitarity cutoff the black size to string scale.
Strings on BTZ
• By performing the SL(2,Z) duality on the thermal AdS(3) partition function and
swap winding modes associated with the thermal and angular circles, we can
obtain the string partition function on BTZ.
• This implies that the low temperature BTZ black hole (stringy size) is unstable
due to the appearance of the tachyonic angular winding modes since it
diverges as
• However, zBTZ is pathological since it has wrong Boltzmann weight.
Conclusion
• In this talk, we have discussed the various aspects of the Hagedorn strings,
especially on the implication to the strings/black hole correspondence.
• In the AdS(3) case, we have analyzed the string partition function, and the
results illuminate the effect of the stringy effect on the Hagedorn
thermodynamics.
• Some subtleties cannot be captured by the thermal partition function such as
(1) self-interaction (2) strings on BTZ (3) k=3 transition for fermionic AdS(3)
string at which TcBTZ=TAdS.
• Hagedorn strings may have implication at earlier Universe.