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 1. Introduction What’s 2D black hole? 2. Boundary states for falling D-brane Bulk theory, Ishibashi states Wick rotation Contour choice 3. Radiation from falling D-brane Tachyon-Radion correspondence String/Black hole transition 4. 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 black hole 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 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) In Euclidean geometry, 2D black hole is cigar geometry: SL(2,R)/U(1) coset

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) 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. Euclidean system is much better understood. Wick rotation is needed to obtain Lorentzian theory.

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

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

12. Branes in 2D Black Hole geometries Classical D-branes are classified by solutions of DBI action. Class 1 (D0): identity rep Class 3 (D2): discrete rep Class 2’ (D1): continuous rep D0 at horizon ? Infalling brane 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 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? Naïve momentum space Wick rotation does not work. Infalling brane

15. Other 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. Contour choice and boundary condition gives many solutions

16. 3. Radiation from falling D-brane

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

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

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

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 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. Evaluation changes drastically at k=1 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?