Closed Superstrings at Finite Temperaturein the Framework of Thermo Field Dynamics

arXiv:1503.xxxxx

Hokkaido Univ.
Kenji Hotta1. Introduction
 Hagedorn Temperature(type II)

maximum temperature for perturbative strings
A single energetic string captures most of the energy.

The finite temperature systems of strings havebeen mainly investigated in Matsubara formalism.

 Hagedorn Transition of Closed Stringsfor(Matsubara Method)winding tachyon in the Euclidean time direction
Hagedorn Transition (Sathiapalan, Kogan, Atick-Witten)
A phase transition takes place due to thecondensation of tachyon fields. (stable minimum?) Brane-antibrane Pair Creation Transition

Dp-Dp Pairs are unstable at zero temperature

finite temperature system of Dp-Dp Pairs
D9-D9 pairs become stable
near the Hagedorn temperature.
Hotta Relation between two phase transitions?

we have conjectured that

D９-D９ Pairs are createdby the Hagedorn transition of closed strings.
These works are based on Matsubara Method.
One of the method to investigate finite temperature systemis thermo field dynamics (TFD).
finite temperature system of Dp-Dp based on TFD
finite temperature system of closed superstring
based on TFD?

 Thermo Field Dynamics (TFD) Takahashi-Umezawa

statistical average

We can represent it as
by introducing a fictitious copy of the system.
thermal vacuum state
The fictitious state is interpreted as `hole’ state.
We cannot represent it asfor ordinary number
cannot be satisfied.

, sinceHawking-Unruh effect can be described by TFD.
It is expected that TFD is available to non-equilibrium system.
(real time formalism)

TFD has been applied to string theorystring field theory
Leblanc, Fujisaki, etc.
D-braneVancea, Cantcheff, etc.
closed bosonic string
Abdalla-Gadelha-Nedel
AdS background
Grada-Vancea, etc.
pp-wave background
Nedel-Abdalla-Gadelha, etc.
Unruh Effect in bosonic open string theory
Hata-Oda-Yahikozawa
closed string which can propagate bulk spacetime
finite temperature system of closed superstring
based on TFD?

Contents
1. Introduction 
2. Thermo Field Dynamics
3. First Quantized Closed Superstring
4. Second Quantized Closed Superstring
5. Conclusion and Discussion

2. Thermo Field Dynamics(Takahashi-Umezawa)

 Ensemble of Free Fermions (example)

Hamiltoniananti-commutation relation

We introduce fictitious system.
Hamiltoniananti-commutation relation

generator of Bogoliubov tr.

thermal vacuum state

Bogoliubov tr. of annihilation ops.

Thermal vacuum state satisfies

Fermi distribution

fictitious system as `holes’

free energy

entropy

relation betweenand .

3. First Quantized Closed Superstring Light-Cone Momentum

We consider a single first quantized string.
light-cone momentum

partition function Mass Spectrum
space time boson

space time fermion

We show only the NS-NS mode case.
GSO projectionlevel-matching condition Thermal Vacuum Stategenerator of Bogoliubov tr.

thermal vacuum state for a single string Free Energy for a Single String

Hamiltonianlevel-matching condition

GSO projection

entropyRelation betweenand.

 Partition Function for a Single String

domain of integration S

 Free Energy for Multiple Strings

Free energy for multiple strings can be obtained from the following eq.

This equals to the free energy in the S-representation
based on Matsubara formalism.
We can transform this to the F-representationor the Dual-representationby using modular transformation.

4. Second Quantized Closed Superstring Light-Cone Gauge SFT cf) Kaku-Kikkawa, Hua

We consider free string case.
actioneq. of motion

We show only the NS-NS mode case.

mode expansion

solutionsecond quantization

commutation relation

Hamiltonianlevel-matching condition

GSO projection Thermal Vacuum State for NS-NS Strings

generator of Bogoliubov tr.

thermal vacuum state for multiple strings Free Energy for Multiple NS-NS String

Hamiltonian

entropyRelation betweenand.

domain of integration S Free Energy for Multiple Strings

Summing over the free energy for all sectors, we obtain

This equals to the free energy in the S-representation
based on Matsubara formalism.
We can transform this to the F-representation
or Dual-representation by using modular transformation.

5. Conclusion and Discussion
 First Quantized Closed Superstring Case

We computed thermal vacuum state and partition function
for a single closed superstring based on TFD.
The free energy for multiple strings agrees with thatbased on the Matsubara formalism.
We need some gimmicks for calculation.
 Second Quantized Closed Superstring Case

We computed thermal vacuum state and free energy
for multiple closed superstrings based on TFD.
The free energy for multiple strings agrees with thatbased on the Matsubara formalism.

 D9-D9 Pair Case

We need to use second quantized string field theory in orderto obtain the thermal vacuum state for multiple open strings.
 D-brane boundary state of closed string cf) Cantcheff

The thermal vacuum state is reminiscent of
the D-brane boundary state of a closed string.

 Hawking-Unruh Effectclosed strings in curved spacetime
Unruh Effect in bosonic open string theory
Hata-Oda-Yahikozawa
black hole firewall Almheiri-Marolf-Polchinski-Sully
Planck solid model Hotta

A. Brane-antibrane Pair in TFD
 Light-Cone Momentum

We consider a single first quantized string.
partition function for a single string BSFT (Boundary String Field Theory) (BV formalism)

solution of classical master eq. (superstring): effective action: 2-dim. partition function Mass Spectrum

We consider an open string
on a Brane-antibrane pair.
mass spectrum
space time bosonspace time fermion

number ops.
oscillation mode of world sheet boson
oscillation mode of world sheet fermion (NS b. c)oscillation mode of world sheet fermion (R b. c)

We will show only the NS mode case.

 Bogoliubov Transformation

generator of Bogoliubov tr.

 Thermal Vacuum State

thermal vacuum state for a single string Free Energy for a Single String

Hamiltonian
entropy

This is not useful for analysis of thermodynamical system of strings.
free energy for a single string
partition function for a single string
free energy for multiple strings(string gas) Partition Function for a Single String

 Free Energy for Multiple Strings

Free energy for multiple strings can be obtained from the following eq.

This equals to the free energy based on Matsubara formalism.
This implies that our choice of Weyl factors
in the case of Matsubara formalism is quite natural.