The SEENET-MTP Workshop JW2011 Scientific and Human Legacy of Julius Wess 27-28 August 2011, Donji Milanovac, Serbia

1. Thermal duality and non-
singular superstring cosmology
Hervé Partouche
Ecole Polytechnique, Paris
Based on works in collaboration with :
C. Angelantonj, I. Florakis, C. Kounnas and N. Toumbas
arXiv: 0808.1357, 1008.5129, 1106.0946
Balkan Summer Institute, « Particle Physics from TeV to Planck Scale », August 29, 2011.

2. Introduction
In General Relativity : Cosmological evolution leads to an
initial Big-Bang singularity.
There is a belief : Consistency of String Theory is expected to
resolve it.
In this talk, we would like to discuss this question.

3. Consistency of a string vacuum means what ?
There is no tachyon in the spectrum i.e. particles with
negative masses squared.
A tachyon would mean we sit at a maximum of a potential.
An Higgs mechanism should occur to bring us to a true
vacuum.
It is an IR statement but string theory has the property to
relate IR and UV properties into each other.
From a UV point of view, consistency means the 1-loop
vacuum-to-vacuum amplitude Z is convergent in the
« dangerous region » l → 0 :
+∞ +∞
dl d
d k −(k2 +M 2 )l/2
Z= Vd d
dM ρB (M ) − ρF (M ) e
0 2l (2π) 0

4. +∞ +∞
dl d
d k −(k2 +M 2 )l/2
Z= Vd dM ρB (M ) − ρF (M ) e
0 2l (2π)d 0
In fact, there must be a cancellation between the densities of
bosons and fermions.
Highly non-trivial statement, because they are βH M .
∼ e
Cancellation such that the effective density of states is that of
a 2-dimensional Quantum Field Theory [Kutasov, Seiberg].
Effectively, there is a finite number of particles in
the UV.

5. For cosmological pusposes, we should reconsider this
question of consistency for string models at finite T.
The 1-loop vacuum amplitude Z is computed in a space where
1
time is Euclidean and compactified on S (R0), (β = 2πR0 ).
−βH
Z = ln Tr e =
d−1 +∞ √
d k 1 −β k2 +M 2
Vd−1 dM ρB (M ) ln √ + ρF (M ) ln 1 + e
(2π)d−1 0 1− e−β k2 +M 2
βH M −βM
∼e ∼e
When β βH i.e. T TH = 1/βH , Z diverges.

6. This break down of the canonical ensemble formalism at high
temperature is not believed to be an inconsistency of string
theory in the UV.
Instead, it is interpreted as the signal of the occurence of a
phase transition := Hagedorn phase transition.
This UV behavior can be rephrased in an IR point of view by
observing the appearance of tachyons when T = TH.
Usually, one gives a vacuum expectation value to these
tachyons in order to find another string background which is
stable.
However, the dynamical picture of this process, in a
cosmological set up, is unknown.

7. So, the logics in string theory at finite temperature is
the following :
When we go backward in time and the temperature is growing
up, we incounter the Hagedorn transition before we reach an
eventual initial Big Bang singularity.
The hope is that if we understand how to deal with it, we
may evade the initial curvature and infinite temperature
divergences of the Big Bang present in General Relativity.

8. Before annoucing what we are going to do in this talk,
let us define some notations :
To describe a canonical ensemble, the fields are imposed (−)F
boundary conditions along S 1(R0).
In string theory, a state corresponds to the choice of two waves
propagating along the string : A Left-moving one (from right
to left) and a Right-moving one (from left to right) :
0 or 1 mod 2
F =a+a
¯
In this talk : We are going to see that string models yield
deformed canonical partition functions
−βH a
¯
Z(R0 ) = ln Tr e (−)
Total Right-moving fermion number of the multiparticle states

9. It is not a canonical partition function, but has good properties :
Converges for any R0, due to the alternative signs
(no tachyon).
T-duality symmetry
1 1
R0 → with fixed point Rc = √
2R0 2
1
T = 1/β has a maximal value Tc = .
2πRc
−βH a
¯ −βH
T Tc =⇒ Tr e (−) Tr e
This allows to interpret T as a temperature.

10. In fact, the alternative signs imply the effective number of
string modes which are thermalized is finite.
Thanks to the consistency of these thermal backgrounds, we
obtain cosmological evolutions with :
No Hagedorn divergence.
No singular Big Bang.

11. Type II superstring models
Compactification on S 1 (R0 ) × T d−1 (V ) × T 9−d × S 1 (R9 )
Supersymmetries generated by Right-moving waves are broken
by imposing (−)a boundary conditions along S 1(R9).
¯
Supersymmetries generated by Left-moving waves are broken by
imposing (−)a+¯ or (−)a boundary conditions along S 1(R0).
a
Both models are non-supersymmetric 1-loop vacuum
amplitude Z is non-vanishing and may diverge :
Lightest scalar masses satisfy :
√
M 2 = R0 − 2 =⇒ RH =
2
2
2
2 1
M = − R0 ≥0
2R0

12. Moreover, the tachyon free model has two kinds of fermions :
Spinors and Conjugate Spinors : S8, C8.
Admits a T-duality symmetry :
1
R0 → with S8 ↔ C 8
2R0
At the self-dual point R0=Rc: Enhanced SU(2)L + adj matter.

13. Physical interpretation
In QFT at finite T, the particles have momentum m0/R0
along S 1(R0).
In String Theory, the strings have momentum m0/R0 and wrap
n0 times along S 1(R0).
However, it is possible to use symetries on the worldsheet
(modular invariance) to reformulate the theory in terms of an
infinite number of point-like particles i.e. with n0 = 0.
In the thermal model, for R0 RH, the string 1-loop
amplitude can be rewritten as a canonical partition function :
−βH
Z= = ln Tr e where β = 2πR0

14. In the tachyon-free model, for R0 Rc :
Since (−)a = (−)a+¯ (−)a the same procedure yields a
a ¯
deformed canonical partition function :
Z −βH a
¯
e = Tr e (−)
where β = 2πR0 and the fermions are Spinors S8,
|m0 |
1
the excitations along S (R0) momenta .
R0
for R0 Rc : T-duality leads the same expression but
1
β = 2π and the fermions are Conjugate Spinors C8,
2R0
the excitations along S 1(R0) have mass contributions |n0 | 2R0 .

15. This contribution is proportional to the lenght of
S 1(R0) : They are winding modes.
(solitons in QFT language)
Thus, we have two distinct phases :
a momentum phase R0 Rc and a winding phase R0 Rc .
In both : β 2πRc
√
2
Therefore : T = 1/β Tc =
2π

16. Z
−βH a
¯
Moreover, e = Tr e (−)
may differ from a canonical partition function for multiparticle
states containing modes with a = 1 .
¯
However, in these models, these modes have
1 1
M≥ = R 9 = √ Tc (R9 is stabilized there)
2R9 2
Since T Tc , the multiparticle states which contain them are
suppressed by Boltzmann’s factor :
Z −βH a
¯ −βH
e = Tr e (−) Tr e
The tachyon free model is essentially a thermal
model.

17. Then, why is there no Hagedorn divergence ?
In fact, the convergence of eZ = Tr e−βH (−) when T ≈ Tc is
a
¯
due to the alternative signs.
For a gas of a single Bosonic (or Fermionic) degree of freedom,
with Right-moving fermion number a , ¯
√
2
ln Tr e−βH (−)a = ∓
¯
ln 1∓(−)a e−β
¯ k+M
k
(Boson, a) and (Fermion, 1 − a) have opposite contributions
¯ ¯
when they are degenerate.
This reduces the effective number of degrees of freedom which
are thermalized.

18. The pairing of degenerate (Boson, a) and (Fermion, 1 − a)
¯ ¯
can even be an exact symmetry of models.
This symmetry can exist only in d = 2 and concerns massive
modes only [Kounnas].
In this case, exact cancellation in Z of all contributions from
the massive modes : They remain at zero temperature.
We are left with thermal radiation for a finite number
of massless particles (in 2 dimensions) :
F Z nσ2
≡− =− 2
V βV β
In general, in any model and dimension, the cancellation is only
approximate :
F nσd
− d
V β

19. What happens at R0 = Rc ?
There are additional massless states in the Euclidean model :
gauge bosons U(1) → SU(2) + matter in the adjoint.
They have non-trivial momentum and winding numbers
m0 = n0 = ±1 along S 1(R0).
This is exactly what is needed to transform the states of the
« winding phase » R0 Rc into the states of the
« momentum phase » R0 Rc .
(0, −N)
(N, 0)
(N, N)
They trigger the Phase Transition between the two
worlds.

20. Cosmology
In each phase, the Euclidean low energy effective action in
1
S (R0) ×T d−1
is, up to 1-loop and 2-derivatives,
0 d−1 d−1 −2φ R 2 Z
dx d x βa e + 2(∂φ) + d−1
2 βa
1
2πR0 or 2π Dilaton := String coupling
2R0
At a fixed such that R0 = Rc, there are additional massless
0
xc
scalars ϕ with m0 = n0 = ±1. They don’t even exist at other
α
x0, since in each phase there are only pure momentum or pure
winding modes. Their action depends on space only :
d−1 0 d−1 −2φ 1 α α α 0 α
d x a(xc ) e − ∂ ϕ ∂i ϕ
0 )2 i
¯ =⇒ ∂i ϕ = a(xc )γi
a(xc

21. In total, the action is
0 d−1 d−1 −2φ R Z d−1 −2φ
dx d x βa e + 2(∂φ)2 + d−1 − 0 d−1
dx d xa e 0
δ(x − x0 )
c
α
|γi |
2 βa i,α
The analytic continuation x0 → i x0 is formally identical :
The gradients yield a negative contribution to the
pressure and vanishing energy density,
PB = − |γi |2 δ(x0 − x0 ) ,
α
c ρB = 0
i,α
This is an unusual state equation. It arises from the phase
transition and not from exotic matter.
Z nσd
If we approximate d to solve explictly the
βad−1 β
equations of motion, we find in conformal gauge :

22. a Tc 1 ω|τ | ω|τ |
ln = ln = η+ ln 1 + − η− ln 1 +
ac T d−2 η+ η−
√
d−1 ω|τ | ω|τ |
φ = φc + ln 1 + − ln 1 +
2 η+ η−
d−2√ √
where ω= √ nσd ac Tc eφc ,
d/2
η± = d−1±1
2
d−2 T φ
a e
τ τ τ
Bouncing cosmology, radiation dominated at late/early times.
Perturbative if eφc is chosen small.
∂τ ≤ O(eφc ) =⇒ Ricci curvature is small and higher
derivative terms are negligeable.
NB: A similar solution with spatial curvature k=−1 also exists. It is
curvature dominated.

23. Summary
We have seen string models describe systems at finite T, where
the effective number of states which are thermalized is finite.
These models have two thermal phases, where the thermal
excitations are either momentum or winding modes along S 1(R0).
Additional massless states with both windings and momenta
trigger the phase transition at the maximal temperature Tc.
They induce a negative contribution to the pressure,
which implies a bounce in the string coupling and Ricci curvature
during the cosmological evolution.