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

Yu Nakayama, S. J. Rey and Y. Sugawara hep-th/0507040, JHEP09(2005)020hep-th/0605013, JHEP08(2006)014

Outline1. 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

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

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

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

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)

Tachyon-Radion correspondence D-brane near NS5-brane shows resemblance to rolling ta

chyon (Kutasov): rolling D-brane

Rolling tachyon has similar form.

Is tachyon-radion correspondence universal? Artifact at the level of effective action?

2. Boundary states for falling D-brane

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.

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

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.

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?

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).

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

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

3. Radiation from falling D-brane

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.

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

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

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.

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?

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)

Open string computation Modular transform is (only) well-defined in Lorentzian sig

nature world sheet.

Imaginary part consists of two parts

Naïve part corresponds to contribution easily guessed in the Euclidean approach (but not enough)

closed

open

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 automa

tically knows it. Fortunately no pole contribution in rolling tachyon

4. Discussion/Summary

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.

Outlook

Construction of other D-brane solutions in 2DBH.

Application to cosmology. Dimensional selection?