Brane-Antibrane at Finite TemperatureBrane-Antibrane at Finite Temperature in the Framework of Thermo Field in the Framework of Thermo Field DynamicsDynamics

Hokkaido Univ.Hokkaido Univ.

Kenji HottaKenji Hotta

1. Introduction1. Introduction

Hagedorn TemperatureHagedorn TemperatureType II superstring in 10 dim.Type II superstring in 10 dim.

maximum temperature for perturbative maximum temperature for perturbative stringsstrings

A single energetic string captures most of A single energetic string captures most of the energy.the energy.

Brane-antibrane Pair Creation TransitionBrane-antibrane Pair Creation Transition DDpp-D-Dpp Pairs are unstable at zero temperature Pairs are unstable at zero temperature

finite temperature system of Dfinite temperature system of Dpp-D-Dpp

based on Matsubara based on Matsubara formalismformalism

D9-D9 pairs become stableD9-D9 pairs become stable near the Hagedorn temperature.near the Hagedorn temperature.

HottaHotta

Thermo Field DynamicsThermo Field Dynamics (TFD) Takahashi-Umezawa (TFD) Takahashi-Umezawa

expectation valueexpectation value

We can represent it asWe can represent it as

by introducing a fictitious copy of the system.by introducing a fictitious copy of the system.

thermal vacuum statethermal vacuum state

finite temperature system of Dfinite temperature system of Dpp-D-Dpp based on TFD? based on TFD?

DDp-p-DDp p PairPair unstable at zero temperatureunstable at zero temperatureopen string tachyon tachyon potentialopen string tachyon tachyon potentialSenSen’’s conjectures conjecture potential height=brane tensionpotential height=brane tension

Tachyon Potential of DTachyon Potential of Dpp-D-Dp p (BSFT)(BSFT)

tree level tree level tachyon potential tachyon potential (disk worldsheet)(disk worldsheet)

: complex scalar field : tension of D: complex scalar field : tension of Dpp--branebrane

: coupling of strings : : coupling of strings : pp-dim. volume-dim. volume

2. Brane-anti-brane Creation 2. Brane-anti-brane Creation TransitionTransition

Free Energy of Open Strings on DFree Energy of Open Strings on Dpp-D-Dpp PairPairEuclidean time with period Euclidean time with period

ideal open string gasideal open string gas 1-loop1-loop (cylinder worldsheet) (cylinder worldsheet)

Finite Temperature Effective PotentialFinite Temperature Effective Potential

cf) We cannot trust the canonical ensemble methodcf) We cannot trust the canonical ensemble method

microcanonical ensemble microcanonical ensemble methodmethod

Brane-anti-brane Pair Creation TransitionBrane-anti-brane Pair Creation Transition

D9-D9 PairsD9-D9 Pairs term of finite temperature effective term of finite temperature effective

potentialpotential

Critical temperatureCritical temperature

Above becomes the potential minimum.Above becomes the potential minimum.

A phase transition occurs and D9-D9 pairs become A phase transition occurs and D9-D9 pairs become stable.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 D -D Pairs with

No phase transition occurs.No phase transition occurs.

3. Brane Antibrane Pair in TFD3. Brane Antibrane Pair in TFD

Light-Cone MomentumLight-Cone MomentumWe consider We consider a single first quantized string.a single first quantized string.

light-cone momentum light-cone light-cone momentum light-cone HamiltonianHamiltonian

partition functionpartition function

Mass SpectrumMass SpectrumWe consider an open stringWe consider an open string

on on a Brane-antibrane pair with a Brane-antibrane pair with . .

mass spectrummass spectrum

space time bosonspace time boson

space time fermionspace time fermion

number ops.number ops.

oscillation mode of world sheet bosonoscillation mode of world sheet boson

oscillation mode of world sheet oscillation mode of world sheet fermion (NS b. c)fermion (NS b. c)

oscillation mode of world sheet oscillation mode of world sheet fermion (R b. c)fermion (R b. c)

Thermal Vacuum StateThermal Vacuum Stategenerator of Bogoliubov tr.generator of Bogoliubov tr.

thermal vacuum state for a single stringthermal vacuum state for a single string

cf) D-brane in bosonic string theory Vancea et al., Cantcheffcf) D-brane in bosonic string theory Vancea et al., Cantcheff

Partition FunctionPartition Functionpartition function for a single stringpartition function for a single string

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

This equals to the free energy withThis equals to the free energy with

based on Matsubara based on Matsubara formalism.formalism.

4. Conclusion and 4. Conclusion and DiscussionDiscussion Brane-antibrane in TFDBrane-antibrane in TFD

We computed thermal vacuum state and partition functionWe 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 stringD-brane boundary state of closed stringThis thermal vacuum state is reminiscent ofThis 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 TheoryString Field TheoryWe need to use second quantized string field theory in orderWe 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 TachyonOpen String TachyonWe need to introduce open string tachyonWe 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 transformationmodular transformation