Rolling D-brane in Lorentzian 2D-black hole @YITP

1. Rolling D-brane in Lorentzian 2D-black hole@YITP Yu Nakayama, S. J. Rey and Y. Sugawara hep-th/0507040,JHEP09(2005)020 hep-th/0605013, JHEP08(2006)014

2. Outline • Introduction • What’s 2D black hole? • Boundary states for falling D-brane • Bulk theory, Ishibashi states • Wick rotation • Contour choice • Radiation from falling D-brane • Tachyon-Radion correspondence • String/Black hole transition • Summary

3. Purpose of the talk • Large charge(+BPS) vs. Small Charge (+non-BPS) • Black hole / String phase transition • Hawking temperature vs. Hagedorn temperature • Is 2D (pure) Black hole really black? • Analyticity vs. Non-analyticity • Universality of Tachyon-Radion correspondence • Wick rotation in curved space • Unitarity vs. Open/Closed duality • Optical theorem • Lorentzian world-sheet vs. Euclidean world-sheet

4. What’s 2D Black Hole ? • 2D blackhole is the simplest black hole geometry as an exact string background: SL(2,R)/U(1) (Witten) • Global metric looks Schwarzshild-like

5. Euclidean geometry In Euclidean geometry, 2D black hole is cigar geometry: SL(2,R)/U(1) coset • String theory in the Euclidean 2D black hole is much better understood (Euclidean SL(2,R)/U(1) coset) • Even exact matrix model construction is proposed (KKK)

6. Applications • Near horizon limit of nonextremal NS5 brane • Taking the limit with keeping • Level k corresponds to number of NS5 branes. • k  ∞ is the semiclassical (supergravity) limit. 2D black hole is important for holographic dual of NS5 branes (Little String Theory)

7. Tachyon-Radion correspondence • D-brane near NS5-brane shows resemblance to rolling tachyon (Kutasov): rolling D-brane • Rolling tachyon has similar form. Is tachyon-radion correspondence universal? Artifact at the level of effective action?

8. 2. Boundary states for falling D-brane

9. Bulk theory, Ishibashi states (Euclidian) Euclidean system is much better understood. Wick rotation is needed to obtain Lorentzian theory. • Spectrum is classified by SL(2,R)/U(1) coset primary states. • Continuous representation Wick rotation is possible, but not one to one. • Discrete representation (winding)  ? Lorentzian interpretation? • Idenetity representation Non in closed string sector • Reflection is unitary: |R| = 1.

10. SL(2,R)/U(1) coset, Hawking temperature vs Hagedorn temperature Exact CFT background is described by SL(2,R)/U(1) coset • Central charge: • is correction • Euclidean radius gives Hawking temperature • Central charge determines Hagedorn temperature: • k = 1 seems special (c.f. Aharony et al.)

11. Bulk theory, Ishibashi states (Lorentzian) Hilbert space is twice as large as in Euclidean case. • Essentially, left modes and right modes are independent. • Reflection is nonunitary. • Physical boundary condition is required. • Ex. No radiation from white hole (V=0).

12. Branes in 2D Black Hole geometries Classical D-branes are classified by solutions of DBI action. Class 1 (D0): identity rep D0 at horizon ? Infalling brane Class 2’ (D1): continuous rep Class 3 (D2): discrete rep Space-time filling D1?

13. Euclidean boundary states (Ribault-Shomerus) • Class 2’ boundary states in Euclidean BH • Effect of 1/k correction • Delta function localized trajectory  smeared wavefunction Poisson distribution: • The steeper the hairpin, the wider the trajectory (NRPT).

14. Wick rotation: rolling D-brane boundary states Naïve momentum space Wick rotation does not work. • Performing Wick rotation in coordinate space, or choosing the contour integral properly, • Finite k correction: • Trajectory is smeared (NPRT) • Rolling D-brane gathers moss  analytic continuation of winding tachyon? Infalling brane

15. Other solutions Contour choice and boundary condition gives many solutions • Falling (absorbed) solution (V=0) • Emitted solution (U=0) : time reversal of Falling solution • Time reversal symmetric solution (essentially Falling + Emitted) • All boundary states are consistent with reflection relation. • Analogy to different S-brane solutions in rolling tachyon.

16. 3. Radiation from falling D-brane

17. Radiation from falling brane From boundary states, we can compute closed string emission from falling D-branes. • From the optical theorem, imaginary part of one-loop amplitude gives total emission rate.

18. Saddle point evaluation 1 Radiation consists of infalling part and outgoing part • With fixed transverse mass M, the radiation shows a structure. • “Gray-body” factor is different for in and out.

19. Saddle point evaluation 2 Integration over p can be done by saddle point approximation • Let us assume k>1. • Hagedorn temperature (with correction) appeared in infalling mode! • Outgoing mode is still at Hawking temperature (Hawking radiation?).

20. Tachyon-radion correspondence. • We can sum over all the final states • Density of states exactly cancels with the radiation density shows the same behavior in rolling tachyon (LLM) • Remarkable cancellation of stringy corrections  universal property of rolling (falling) D-brane? Tachyon-radion correspondence is true at the stringy level.

21. String-Black hole transition at k = 1 Evaluation changes drastically at k=1 • There is no nontrivial saddle point for k<1 • Emission rate is UV convergent (exponentially). • “Black hole” interpretation for 2D BH has been doubted. • SL(2,R)/U(1) description is worse. N=2 Liouville is better. • Width of trajectory diverges at k=1. • Dual LST also shows phase transition. It is really a challenging problem whether the genuine 2D BH is really black. Matrix model description helps?

22. Unitarity and Open/Closed duality (NRS) • Is unitarity consistent with open/closed duality? • Open string channel? • Euclidean V.S. Lorentzian worldsheet • Gives the same answer in rolling tachyon (KLMS), but… (Okuyama-Rozalli, NRS)

23. Open string computation • Modular transform is (only) well-defined in Lorentzian signature world sheet. • Imaginary part consists of two parts • Naïve part corresponds to contribution easily guessed in the Euclidean approach (but not enough) closed open

24. Unitarity meets open/closed duality • Pole part comes from poles in Euclidean (Wick) rotation • Both contributions are imperative to understand • Unitarity • Tachyon-Radion correspondence • Summary • In Euclidean approach, no apparent reason to include/exclude pole contributions. • Unitarity demands its existence, and Lorentzian theory automatically knows it. • Fortunately no pole contribution in rolling tachyon

25. 4. Discussion/Summary

26. Summary • Boundary states for falling D-brane in 2D BH geometries has been constructed. • Subtlety concerning Wick rotation is taken into consideration properly. • Different contour and different boundary condition gives different solutions. • Emission rate is very similar to rolling tachyon. • Established tachyon-radion correspondence at the stringy level. (universality of decaying brane) • String-Black hole transition at k=1 is observed.

27. Outlook • Construction of other D-brane solutions in 2DBH. • Application to cosmology. Dimensional selection?