1. Creation of D9-brane — anti-D9-brane Pairs from Hagedorn Transition of Closed Strings Hokkaido Univ. Kenji Hotta

2. 1. Introduction Hagedorn Temperature Hagedorn Temperature maximum temperature for perturbative strings A single energetic string captures most of the energy.

3. Open String Gas on D9-brane Open String Gas on D9-brane We need an infinite energy to reach. is a `limiting temperature'. is a `limiting temperature'. Closed String Gas Closed String Gas We can reach by supplying a finite energy. for (Matsubara Method) for (Matsubara Method) Winding modes in Euclidean time direction Winding modes in Euclidean time direction become tachyonic. become tachyonic. Hagedorn Transition (Sathiapalan, Kogan, Atick-Witten) Hagedorn Transition (Sathiapalan, Kogan, Atick-Witten) A phase transition takes place due to the condensation of tachyon fields. (stable minimum?) A phase transition takes place due to the condensation of tachyon fields. (stable minimum?)

4. D p -D p pairs and Non-BPS D p -brane type II D p -D p pairs and Non-BPS D p -brane type II unstable at zero temperature unstable at zero temperature open string tachyon tachyon potential finite temperature system of unstable branes finite temperature effective potential (BSFT) finite temperature effective potential (BSFT) D9-D9 pairs and Non-BPS D9-branes become stable near the Hagedorn temperature. Hotta Hotta  D9-D9 pairs A phase transition occurs and Dp-Dp pairs become stable.  D p -D p pairs with No phase transition occurs.

5. Thermodynamic Balance on D ９ -D ９ Pair Thermodynamic Balance on D ９ -D ９ Pair (open string closed string)  open strings We need an infinite energy to reach.  closed strings We can reach by supplying a finite energy. Energy flows from closed strings to open strings. Open strings dominate the total energy Open strings dominate the total energy

6. ‘Winding Tachyon’ Condensation Atick-Witten ‘Winding Tachyon’ Condensation Atick-Witten For sphere world sheet does not contribute to free energy. For sphere world sheet contributes to free energy. insertion of ‘winding tachyon’ vertex creation of a tiny hole in the world sheet which wraps around Euclidean time creation of a tiny hole in the world sheet which wraps around Euclidean time boundary of a hole boundary of open string on D ９ - D ９ pair on D ９ - D ９ pair Relation between two phase transitions? Relation between two phase transitions? From above arguments we conjecture that D ９ -D ９ Pairs are created by the Hagedorn transition of closed strings.

7. Contents 1. Introduction 1. Introduction 2. Hagedorn Transition 3. BSFT Approach 4. BCFT Approach 5. Conclusion and Discussion

8. 2. Hagedorn Transition Matsubara Method Matsubara Method Euclidean time with period ideal closed string gas 1-loop (torus world sheet) Proper Time Form of Free Energy GSO projection GSO projection

9. Dual-representation Dual-representation ‘ winding tachyon' in the Euclidean time direction ‘ winding tachyon' in the Euclidean time direction

10. 3. BSFT Approach BSFT (Boundary String Field Theory) BSFT (Boundary String Field Theory) solution of classical master eq. (superstring) : effective action : 2-dim. partition function : effective action : 2-dim. partition function D p- D p Pair D p- D p Pair open string tachyon tachyon potential Sen ’ s conjecture potential height=brane tension

11. tree amplitude (up to two derivative terms) tree amplitude (up to two derivative terms) 1-loop amplitude 1-loop amplitude Conformal invariance is broken by the boundary terms. ambiguity in choosing the Weyl factors ambiguity in choosing the Weyl factors

12. Cylinder Boundary Action Andreev-Oft Cylinder Boundary Action Andreev-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 function low energy part of = 1-loop partition function derived from = 1-loop partition function derived from

13. Free Energy of Open Strings on D9-D9 Pair Free Energy of Open Strings on D9-D9 Pair

14. Free Energy near the Closed String Vacuum Free Energy near the Closed String Vacuum propagator of ‘ winding tachyon ’ if we ignore part momentum 0 momentum conservation for Neumann direction

15. 3. BCFT Approach BCFT (Boundary Conformal Field Theory) BCFT (Boundary Conformal Field Theory) boundary action (non-BPS D9-brane) Euclidean version of rolling tachyon non-BPS D9-brane non-BPS D9-brane closed string vacuum closed string vacuum Matsubara Formalism

16. 1-loop Free Energy Sugawara 1-loop Free Energy Sugawara SU(2) current algebra

17. Closed String Vacuum Case Closed String Vacuum Case propagator of 'winding tachyon ‘ ( ‘ winding tachyon' become massless at the Hagedorn temperature.)

18. 5. Conclusion and Discussion BSFT Approach BSFT Approach 1-loop free energy of open strings on D9-D9 pair similar to propagator of ‘ winding tachyon ‘ similar to propagator of ‘ winding tachyon ‘ of closed strings of closed strings BCFT Approach BCFT Approach 1-loop free energy of open strings on non-BPS D9-brane propagator of `winding tachyon' (massless) propagator of `winding tachyon' (massless) of closed strings of closed strings

19. Hagedorn Transition Hagedorn Transition Creation of D9-D9 Pair? Creation of D9-D9 Pair? higher 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? cf) BCFT with imaginary D-branes Gaiotto-Itzhaki-Rastelli disk scattering amplitude with m closed string insertion disk scattering amplitude with m closed string insertion sphere amplitude with m+1 closed string insertion sphere amplitude with m+1 closed string insertion Dilaton? Dilaton?Atick-Witten first order phase transition first order phase transition Non-perturbative Calculation? Non-perturbative Calculation? Matrix model (IKKT, BFSS), K-matrix Model