Brane-Antibrane at Finite Temperature in the Framework of Thermo Field Dynamics Hokkaido Univ. Kenji Hotta

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

Brane-antibrane Pair Creation Transition Brane-antibrane Pair Creation Transition Dp-Dp Pairs are unstable at zero temperature Dp-Dp Pairs are unstable at zero temperature finite temperature system of Dp-Dp based on Matsubara formalism based on Matsubara formalism D9-D9 pairs become stable D9-D9 pairs become stable near the Hagedorn temperature. Hotta near the Hagedorn temperature. Hotta Thermo Field Dynamics (TFD) Takahashi-Umezawa Thermo Field Dynamics (TFD) Takahashi-Umezawa expectation value We can represent it as by introducing a fictitious copy of the system. thermal vacuum state thermal vacuum state finite temperature system of Dp-Dp based on TFD? finite temperature system of Dp-Dp based on TFD?

Dp-Dp Pair Dp-Dp Pair unstable at zero temperature unstable at zero temperature open string tachyon tachyon potential Sen ’ s conjecture potential height=brane tension Tachyon Potential of Dp-Dp (BSFT) Tachyon Potential of Dp-Dp (BSFT) tree level tachyon potential (disk worldsheet) : complex scalar field : tension of Dp-brane : complex scalar field : tension of Dp-brane : coupling of strings : p-dim. volume : coupling of strings : p-dim. volume 2. Brane-anti-brane Creation Transition

Free Energy of Open Strings on Dp-Dp Pair Free Energy of Open Strings on Dp-Dp Pair Euclidean time with period ideal open string gas 1-loop (cylinder worldsheet) Finite Temperature Effective Potential Finite Temperature Effective Potential cf) We cannot trust the canonical ensemble method microcanonical ensemble method microcanonical ensemble method

Brane-anti-brane Pair Creation Transition Brane-anti-brane Pair Creation Transition  D9-D9 Pairs term of finite temperature effective potential term of finite temperature effective potential Critical temperature Above becomes the potential minimum. A phase transition occurs and D9-D9 pairs become stable. A phase transition occurs and D9-D9 pairs become stable. is a decreasing function of. is a decreasing function of. Multiple D9-D9 pairs are created simultaneously. Multiple D9-D9 pairs are created simultaneously.  D -D Pairs with No phase transition occurs.

3. Brane Antibrane Pair in TFD Light-Cone Momentum Light-Cone Momentum We consider a single first quantized string. light-cone momentum light-cone Hamiltonian partition function

Mass Spectrum Mass Spectrum We consider an open string on a Brane-antibrane pair with. on a Brane-antibrane pair with. mass spectrum space time boson space time boson space time fermion space time fermion number ops. oscillation mode of world sheet boson oscillation mode of world sheet boson oscillation mode of world sheet fermion (NS b. c) oscillation mode of world sheet fermion (NS b. c) oscillation mode of world sheet fermion (R b. c) oscillation mode of world sheet fermion (R b. c)

Thermal Vacuum State Thermal Vacuum State generator of Bogoliubov tr. thermal vacuum state for a single string cf) D-brane in bosonic string theory Vancea et al., Cantcheff

Partition Function Partition Function partition function for a single string Free energy of multiple strings can be obtained from the following eq. This equals to the free energy with based on Matsubara formalism. based on Matsubara formalism.

4. Conclusion and Discussion Brane-antibrane in TFD Brane-antibrane in TFD We computed thermal vacuum state and partition function of a single string on a Brane-antibrane pair based on TFD. of a single string on a Brane-antibrane pair based on TFD. D-brane boundary state of closed string D-brane boundary state of closed string This thermal vacuum state is reminiscent of the D-brane boundary state of a closed string. the D-brane boundary state of a closed string. String Field Theory String Field Theory We need to use second quantized string field theory in order to obtain the thermal vacuum state for multiple strings. to obtain the thermal vacuum state for multiple strings. Open String Tachyon Open String Tachyon We need to introduce open string tachyon in order to see the phase structure of the system. in order to see the phase structure of the system.

-functions -functions modular transformation