Creation of D9-brane—anti-D9-brane Pairs Creation of D9-brane—anti-D9-brane Pairs from Hagedorn Transition of Closed Stringfrom Hagedorn Transition of Closed Stringss

Hokkaido Univ.Hokkaido Univ. Kenji HottaKenji Hotta

1. Introduction1. Introduction

Hagedorn TemperatureHagedorn Temperaturemaximum temperature for perturbative maximum temperature for perturbative

stringsstringsA single energetic string captures most of A single energetic string captures most of

the energy.the energy.

Open String Gas on D9-braneOpen String Gas on D9-braneWe need an infinite energy to reachWe need an infinite energy to reach 　　　　　　 ..　　　　　　　　　　　　　　 is a `limiting temperature'.is a `limiting temperature'.

Closed String GasClosed String GasWe can reach We can reach 　　　　　　 by supplying a finite energy.by supplying a finite energy.　　　　　　　　　　　　　　　　　　　　 for for (Matsubara M(Matsubara M

ethod)ethod)

Winding modes in Euclidean time directionWinding modes in Euclidean time direction become tachyonic.become tachyonic.

Hagedorn Transition Hagedorn Transition (Sathiapalan, Kogan, Atick-Witten)(Sathiapalan, Kogan, Atick-Witten)

A phase transition takes place due to the condensatA phase transition takes place due to the condensation of tachyon fields. ion of tachyon fields. (stable minimum?)(stable minimum?)

DDpp-D-Dpp pairs pairs andand Non-BPS DNon-BPS Dpp-brane-brane type IItype II unstable at zero temperatureunstable at zero temperatureopen string tachyon tachyon potentialopen string tachyon tachyon potential

finite temperature system of unstable branesfinite temperature system of unstable branes finite temperature effective potential (BSFT)finite temperature effective potential (BSFT)

D9-D9 pairs and Non-BPS D9-branes become stablD9-D9 pairs and Non-BPS D9-branes become stable near the Hagedorn temperature.e near the Hagedorn temperature.

HottaHotta

D9-D9 pairsD9-D9 pairs A phase transition occurs and DA phase transition occurs and Dpp-D-Dpp pairs beco pairs beco

me stable.me stable.

DDpp-D-Dpp pairs pairs with with No phase transition occurs.No phase transition occurs.

Thermodynamic Balance on DThermodynamic Balance on D ９９ -D-D ９ ９ PairPair

(open string closed string)(open string closed string)

open stringsopen stringsWe need an infinite energy to reach .We need an infinite energy to reach .

closed stringsclosed stringsWe can reach by supplying a finite energy.We can reach by supplying a finite energy.

Energy flows from closed strings to open Energy flows from closed strings to open strings.strings.

Open strings dominate the total energyOpen strings dominate the total energy

‘‘Winding Tachyon’ CondensationWinding Tachyon’ Condensation 　　 Atick-WittenAtick-Witten

For For 　　　　 　　　　　　　 　　　 sphere world sheet does not contribute to free enersphere world sheet does not contribute to free energy.gy.

For For 　　　　　　　　　　　　　　　　 sphere world sheet contributes to free energy.sphere world sheet contributes to free energy.

insertion of ‘winding tachyon’ vertexinsertion of ‘winding tachyon’ vertex　　　　　　　　 creation of a tiny hole in the world sheet which wraps creation of a tiny hole in the world sheet which wraps

around Euclidean timearound Euclidean time

boundary of a holeboundary of a hole 　　　　　　　　 boundary of open stringboundary of open string　　　　　　　　　　　　　　　　　　　　　　 　　　　　　　　　　　　　　　　　　　　　　 on Don D ９９ -D-D

９ ９ pairpair

Relation between two phase transitions?Relation between two phase transitions?From above arguments we conjecture thatFrom above arguments we conjecture that

DD ９９ -D-D ９ ９ Pairs are created by the Hagedorn transition of cloPairs are created by the Hagedorn transition of closed strings.sed strings.

ContentsContents

1.1. Introduction Introduction 2.2. Hagedorn TransitionHagedorn Transition3.3. BSFT ApproachBSFT Approach4.4. BCFT ApproachBCFT Approach5.5. Conclusion and DiscussionConclusion and Discussion

2. Hagedorn Transition2. Hagedorn Transition

Matsubara MethodMatsubara MethodEuclidean time with periodEuclidean time with period ideal closed string gasideal closed string gas 1-loop1-loop (torus world sheet) (torus world sheet)

Proper Time Form of Free EnergyProper Time Form of Free Energy

GSO projectionGSO projection

Dual-representationDual-representation

‘ ‘winding tachyon' in the Euclidean time winding tachyon' in the Euclidean time directiondirection

3. BSFT Approach3. BSFT Approach

BSFT BSFT (Boundary String Field Theory)(Boundary String Field Theory)solution of classical master eq. (superstring)solution of classical master eq. (superstring)

: effective action : 2-dim. partition function: effective action : 2-dim. partition function

DDp-p-DDp p PairPairopen string tachyon tachyon potentialopen string tachyon tachyon potentialSen’s conjectureSen’s conjecture potential height=brane tension potential height=brane tension

tree amplitude tree amplitude (up to two derivative terms)(up to two derivative terms)

1-loop amplitude1-loop amplitudeConformal invariance is broken by the boundary terms.Conformal invariance is broken by the boundary terms.

ambiguity in choosing the Weyl factorsambiguity in choosing the Weyl factors

Cylinder Boundary Action Cylinder Boundary Action Andreev-OftAndreev-Oft

Both sides of the cylinder world sheet are treated on an equal footing.Both sides of the cylinder world sheet are treated on an equal footing.

2-dim. 1-loop partition function2-dim. 1-loop partition function

low energy part of low energy part of = 1-loop partition function derived from= 1-loop partition function derived from

Free Energy of Open Strings on D9-D9 Free Energy of Open Strings on D9-D9 PairPair

Free Energy near the Closed String Free Energy near the Closed String VacuumVacuum

propagator of ‘winding tachyon’ if we ignore partpropagator of ‘winding tachyon’ if we ignore partmomentum 0 momentum 0 momentum conservation for Neumann momentum conservation for Neumann

directiondirection

3. BCFT Approach3. BCFT Approach

BCFT BCFT (Boundary Conformal Field Theory)(Boundary Conformal Field Theory)

boundary actionboundary action (non-BPS D9-brane)(non-BPS D9-brane)

Euclidean version of rolling tachyonEuclidean version of rolling tachyon

non-BPS D9-branenon-BPS D9-brane

closed string vacuumclosed string vacuum

Matsubara Formalism Matsubara Formalism

1-loop Free Energy 1-loop Free Energy SugawaraSugawaraSU(2) current algebraSU(2) current algebra

Closed String Vacuum CaseClosed String Vacuum Case

propagator of 'winding tachyon‘propagator of 'winding tachyon‘(‘winding tachyon' become massless at the Hagedorn temperature.)(‘winding tachyon' become massless at the Hagedorn temperature.)

5. Conclusion and 5. Conclusion and DiscussionDiscussion

BSFT ApproachBSFT Approach1-loop free energy of open strings on D9-D9 pair1-loop free energy of open strings on D9-D9 pair

similar to propagator of ‘winding tachyon‘similar to propagator of ‘winding tachyon‘ of closed stringsof closed strings

BCFT ApproachBCFT Approach1-loop free energy of open strings on non-BPS D9-brane1-loop free energy of open strings on non-BPS D9-brane propagator of `winding tachyon' (massless)propagator of `winding tachyon' (massless) of closed stringsof closed strings

Hagedorn TransitionHagedorn Transition Creation of Creation of D9-D9D9-D9 Pair? Pair?

higher loop free energy of open stringshigher loop free energy of open strings 'tree level' free energy of closed strings with ‘winding tachyon' vertexes?'tree level' free energy of closed strings with ‘winding tachyon' vertexes?

ambiguity in choosing the Weyl factors?ambiguity in choosing the Weyl factors?

cf) BCFT with imaginary D-branes cf) BCFT with imaginary D-branes Gaiotto-Itzhaki-Rastelli Gaiotto-Itzhaki-Rastelli disk scattering amplitude with disk scattering amplitude with mm closed string insertion closed string insertion sphere amplitude with sphere amplitude with mm+1 closed string insertion+1 closed string insertion

Dilaton?Dilaton?Atick-WittenAtick-Witten first order phase transitionfirst order phase transition

Non-perturbative Calculation?Non-perturbative Calculation?Matrix model (IKKT, BFSS), K-matrix ModelMatrix model (IKKT, BFSS), K-matrix Model