Closed Superstrings at Finite Temperaturein the Framework of Thermo Field Dynamics

arXiv:1509.xxxxx

Hokkaido Univ.
Kenji Hotta1. Introductionn Hagedorn Temperature

Type II superstring in 10 dim.
maximum temperature for perturbative strings
A single energetic string captures most of the energy.

n 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?)n 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n 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?

n 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-equilibriumsystem.
(real time formalism)

TFD has been applied to string theory
string 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2. Closed Superstringn Light-Cone Momentum

light-cone momentum

partition function

n Mass Spectrum
space time boson

space time fermion

We show only the NS-NS mode case.
GSO projectionlevel-matching condition3. Second Quantized Closed Superstringn 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

4. Thermal Vacuum State and Free Energyn Thermal Vacuum State for NS-NS Strings

generator of Bogoliubov tr.

thermal vacuum state for multiple strings

n Free Energy for Multiple NS-NS String

Hamiltonian

entropyRelation betweenand.

domain of integration Sn 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 F-representation
or Dual-representationby using modular transformation.

5. Conclusion and Discussionn 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.
n 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n D9-D9 Pair Case

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

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

n Half-formed D9-D9 Pairs

D９-D９ pairs are half-formed before the Hagedorn transition.
Closed strings in the bulk and open strings
on D９-D９ pairs are in thermal equillibrium.

Do D９-D９ pairs and open strings play a roleof ‘hole’ for closed strings?