1. Introduction Motivation to study N = 2 Liouville or SL(2 R )=U(1) : Irrational (super)conformal theories are still challenging problems. innite pri

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1 N = 2 Liouville, SL(2 R )=U (1) and Related Topics Yuji Sugawara (Univ. of Tokyo) collaborated with T. Eguchi, Y. Nakayama, H. Takayanagi, S. Rey Based on hep-th/ , (JHEP 0401, 025 (2004)) hep-th/ , (JHEP 0405, 014 (2004)) hep-th/ , (JHEP 0501, 027 (2005)) hep-th/ , (JHEP 0407, 020 (2004)) + February 2005, at Rikkyo Univ.

2 1. Introduction Motivation to study N = 2 Liouville or SL(2 R )=U(1) : Irrational (super)conformal theories are still challenging problems. innite primary elds belonging to both continuous and discrete spectra N = 2 theories are more desirable than N = 0 or N = 1 models if applying to superstring theory. no \c = 1 barrier", (non-compact) Gepner type construction of superstring vacua Possibility to leading interesting time-dependent, curved backgrounds in string theory by Wick rotation N = 2 Liouville : f, Y, +, ; g () 2D (fermionic) Blackhole SL(2 R )=U (1) (FZZ, Giveon-Kutasov, Hori-Kapustin)

3 Contents Introduction Conformal Blocks in SL(2 R )=U(1) Kazama- Suzuki Models : Modular Data Boundary States for D-branes in SL(2 R )=U(1) or N = 2 Liouville : Modular Bootstrap Application to Non-compact Calabi-Yau Models Application to D0-brane Falling Down to Near-extremal Blackhole Some Comments

4 2. Conformal Blocks of SL(2 R )=U (1) Kazama-Suzuki model : Modular Data Kazama-Suzuki supercoset for SL(2 R )=U(1) SL(2 R ) SO(2) 1 U(1) ;(;2) ^c = k ( = k + 2) We focus on the rational level models k = N K N K : relatively prime (^c = 1 + 2K N ) Coset characters (branching functions) dened by the branching relation : ; 2 k z + w 3 k+2 k z + w ( ) = X ( ( u) : character of SL(2 R ) k+2 ) m (NS) m m 2 k ( z)q; e 2imw ( ) : branching functions = N = 2 irreducible characters

5 continuous series ^C p () massive character : c (NS) p m ( z) q p 2 +m 2 k e 2i2mz 3( z) k ( ) 3 discrete series ^D j (j 6= 0) () massless matter character : d (NS) j j+n ( z) q j+n 2 +2nj ; 1 k 4ke 2i2(j+n) k ( ) 3 (n 2 Z : the spectral ow paramter) 1 + e 2iz q n+1=2 3 ( z) z identity rep. ^D 0 () graviton character : (NS) 0 n ( z) (1 ; q)q n 2 k +n; 1 2 ; 1 4ke 2i ( 2n k +1 )z ; 1 + e 2iz q n+1=2; 1 + e 2iz q n;1=2 3( z) ( ) 3 We can also consider the spectrally owed reps.

6 `Extended Characters' dened by spectral ow sums over n 2 N Z c (NS) (p m z) X n2nz q Kp 2 N c (NS) p m=2k+n ( z) 2z 3 ( z) N ( ) 3 m NK d (NS) (s m z) ( P n2nz d (NS) s 2K m 2K +n( z) m ; s 2 2KZ 0 m ; s 62 2KZ (NS) 0 (m z) ( P n2nz (NS) 0 m 2K +n( z) m 2 2KZ 0 m 62 2KZ =) Good modular properties! compatible with space-time SUSY (;1= ) = Z dp 0 X N +K s 0 =K X m 0 2Z 2NK S (c) (p 0 m 0 j)c(p 0 m 0 ) X some modular coe- S (c) (p 0 m 0 j), S (d) (s 0 m 0 j) : cients m 0 2Z 2NK S (d) (s 0 m 0 j) d (s0 m 0 ) :

7 Note : The existene of discrete terms of massless reps. in R.H.S is a characteristic feature of N = 2 case. (They are absent for N = 0 (Virasoro) and N = 1 (supervirasoro) cases.) However, S (d) (s 0 m 0 j) = 0 for 8 = massive rep.. Not all the massless representations appear in the discrete terms. Especially, the graviton rep. does not appear. Nevertheless, our modular transformation formulas are consistent. (One can prove e.g. S 2 = C, by carefully using the `contour deformation technique'.) The range K s N + K for the discrete rep. is generally smaller than the unitarity range: 1 s N + 2K ; 1 (if K 6= 1). This range matches with the spectrum from the toroidal partition function.

8 toroidal partition function (evaluated by path-integration) Z Z (NS) ( ) = D[g A ~ ] e ;S gw ZW (g A);S ( ~ A) = C Z 1 ds 1 Z 1 0 X 0 w m2z j 3 ( s 1 ; s 2 )j 2 ds 2 j 1 ( s 1 ; s 2 )j 2 ; k j(w + s 1 ) ; (m + s 2 )j 2 2 exp : Include IR divergence = innite volume eect of the 2D BH background (`cigar geometry') Regularized partition function can be decomposed into two parts Eguchi-Y.S hep-th/ , Israel-Kounnas-Pakman-Troost, (see also Hanany- Prezas-Troost ) Z (NS) ( ) = Zc (NS) ( ) + Zd (NS) ( ) : IR cut-o, j log j volume Each part Zc (NS) ( ), Zd (NS) ( ) are expanded by the extended characters. Modular invariance is achieved only after dividing by the volume factor log Z( ) lim! 0 Z( ) log lim! 0 Zc( ) log

9 which only includes continuous representations. (propagating modes in the bulk) The discrete part Zd( ) is universal (insensitive to the regularization scheme), and detemines the allowed discrete representaions in the closed string spectrum. The range K s N + K is successfully reproduced. (\bound states" localized around the tip of cigar)

10 3. Boundary States for D-branes in SL(2 R )=U (1) or N = 2 Liouville : Modular Bootstrap (Eguchi-Y.S hep-th/ ) Generality of modular bootstrap Consistent D-brane = Cardy state jb i characterized by Cardy condition (open/closed duality) : Z hb 1 je ;TH(c) jb 2 i = + X I N (Ij 1 2 ) I (it) : dp (pj 1 2 ) p (it) (H (c) L 0 + L ~ 0 ; c=12 : closed string Hamiltonian, T : closed string modulus, t 1=T : open string modulus, (pj 1 2 ) : spectral density, N (Ij 1 2 ) : non-negative integer) jb i = Z dp (p)jpii + X I C (I)jIii where jiii, jpii are the Ishibashi states associated to the characters I (it ) (discrete rep.), p (it ) (continuous rep.)

11 The coecients (p), C (I) (\boundary wave functions") should be determined from the Cardy condition. Generally, Cardy condition is hard to solve. =) We start with an Ansatz. Pick up a special Cardy state jb Oi s.t. hb Oje ;TH(c) jb Oi = 0 (it) (identity rep., h = 0) \modular bootstrap equa- and solve the tions" hb Oje ;TH(c) jb i = (it) ( 8 ) : ( runs over some range of rep. labelling Cardy states) =) determine the candidates of Cardy states jb i from the modular data. Based on the modular data, we can solve the modular bootstrap equations for N = 2 Liouville (or SL(2 R )=U(1))) : Eguchi-Y.S, hep-th/ , Ahn-Stanishkov-Yamamoto, hep-th/

12 The solutions are classied corresponding to the representations \class 1 states"! (extended) graviton characters localized brane, `ZZ-brane like' (`Dirichlet' along the Liouville direction) \class 2 states"! massive character extending brane, `FZZT-brane' like (`Neumann' along the Liouville direction) \class 3 states"! massless matter character Note : extending brane, (subtlety in unitarity for class 3 brane) We also have another type of FZZT-type brane, which does not seem to be derived by modular bootstrap. (\class 2 0 ") (descent from AdS 2 - brane in AdS 3 ) Ribault-Schomerus, Israel-Pakman-Troost, Fotopulos-Niarchos-Prezas We here only consider the `(1,1)-type' ZZ-brane (only one corresponding to unitary rep.). More general ZZ-type branes have been constructed in Ahn-Stanishkov-Yamamoto hep-th/ , Hosomichi hep-th/

13 4. Application to Non-compact Calabi- Yau Models (Eguchi-Y.S. hep-th/ , ) Construct superstring vacua of the type (`Non-compact Gepner model') : Mk1 M knm L N1 K 1 L NNL K NL U (1)-projection M ki : level k i N = 2 minimal, L Nj K j : SL(2 R )=U(1) Kazama-Suzuki of level N j =K j ( = N = 2 Liouville) XN M i=1 k i k i XN L j= K j N j = ^c ^c = : We set N L:C:Mfk i + 2 N j g. expected to describe non-compact Calabi- Yau manifolds. In particular, ALE spaces, `singular' CY 3, CY 4, and also, M N;2 L N K1 L N KNL type vacua identied as ALE-brations over (weighted) projective spaces (or wrapped NS5-branes in the T-dual picture) (c.f. Hori- Kapustin hep-th/ )

14 Conformal blocks can be constructed in the same way as compact Gepner models. The extended characters play the similar roles to the minimal characters in compact models. It is straightfoward to construct: Modular invariant partition function (only include continuous rep. of L Nj K j sectors) Elliptic genus (discrete rep. can only contribute.) Note : Elliptic genera in non-compact Gepner models are closely related with the \higher level Appell functions". (Semikhatov- Taormina-Tipunin, math.qa/ ) In the cases of ^c = 3, elliptic genera are Jacobi forms (section of line bundles) as in compact Gepner models, but they are not for ^c = 2 4 (could be identied with sections of higher rank vector bundles).

15 Massless Closed String Spectrum Serch for (anti) chiral prmaries with (h L h R ) = (1=2 1=2) in the spectral ow invariant orbit For ^c = 3 4, there exists at most one chiral primary of the (a c), (c a)-type (Kahler deformation), and no (c c), (a a)-type in each orbit. (complex structure deformation). For ^c = 2, we at most have a quartet of the (c c), (a a), (c a), (a c)-type chiral primaries in each orbit. =) in a sharp contrast with compact Gepner models. (In the ^c = 3 4 cases) the moduli spaces are one-sided. =) characteristic for the conifold type models We expect : resolved geometry! SL(2 R )=U (1)-side deformed geometry! N = 2 Liouville-side

16 Compact BPS D-branes in non-compact CY models compact BPS brane = supersymmetric cycle L N K -sector =) described by the class 1 state. The total Cardy states for the BPS compact branes should be written as 2 jb fl i g Mi = N P closed 4 Y jl i M i i Y i j M suitable (weighted) sum of M i, R i jr j i N : normalization const. P closed : projection to the allowed closed string spectrum, Note : For the compact branes only the B-branes are possible. (characteristic for the conifold type models) 3 5

17 The number of compact BPS branes (`supersymmetric cycles') shows apparent mismatch with the number of massless states (`cohomology'). (asymmetry between the closed and open string spectra) For non-compact BPS branes, both of the A and B-type branes are possible (in contrast to the compact branes.) We can also construct non-bps branes in the similar manner to Maldacena-Moore-Seiberg hep-th/ (`unstable B-branes') Also regarded as the `Z N -extension' of the Sen's non-bps D-branes in at background. We obtain consistent results with the DBI analysis by Kutasov, hep-th/ (Precise agreement of the tachyon mass evaluation)

18 Cylinder (annulus) amplitudes : hep-th/ ) (Eguchi-Y.S. Especially, `open string Witten index' is important: We have obtained, consistent results with geometric interpretations rigourous derivation of the formula conjectured by Lerche, hep-th/ (natural generalization of the formula of Witten indices of B-branes in compact Gepner models)

19 5. Application to D0-brane Falling Down to Near-extremal Blackhole Nakayama, Rey, Y. S, in preparation, closely related with Nakayama-Y.S-Takayanagi hepth/ Near extremal black NS5 brane : ds 2 = ; 1 ; r2 0 e 2 = g 2 s dt 2 + r N r N r 2 dr 2 1 ; r2 0 r 2 + r 2 d 2 3! + dy 2 5 Non-extremal =) break SUSY, nite Hawking temperature We want to study the D0-brane falling down to the black NS5 (5-dim. blackhole) Near horizon limit : (e.g. Maldacena-Strominger hep-th/ ) r 0 g s! 0 with keeping r2 0 g 2 s 0 nite. =) (energy density above extremality) ds 2 = ; tanh 2 dt 2 + N 0 d 2 + N 0 d dy2 5 N e 2 = cosh 2

20 where we set r = r 0 cosh, =) SL(2 R ) N +2 U(1) SU(2) N;2 (R 5 ) (Lorentzian 2D BH SU(2) WZW ) =) need to study the D0-brane solution falling into 2D BH expected trajectory : cosh t sinh = sinh r (r : parameter) obtained by the Wick-rotation of the D1-brane in Euclidean 2D BH ( \hairpin shape" ) : cos sinh = sinh r (descent from the AdS 2 brane in AdS 3 ) How to Wick rotate the boundary state for the Euclidean D1-brane? Non-trivial points : We start from the coordinate space boundary wave function, and choose suitably the integration contour.

21 =) improved UV behavior (Nakayama-Y.S-Takayanagi hep-th/ ) Choice of boundary conditions of the wave functions in the Lorentzian theory. =) We shall choose : ( t)! 0 at the past horizon (seems physically plausible condition) =) boundary state for the \falling D0-brane" is constructed Evaluation of closed string radiation : We nd N(p M) (p) exp ; Hawking!(p M) 2 q M 0!(p M) = p 2 + M 2 (p : radial momentum, M : `transverse mass' (including the mass gap)) =) closed string excitations are in thermal equilibrium at Hawiking temperature (quite

22 expected) (p) is some `grey body factor' After integrating out p (saddle point approx.), we nd N out exp ; Hawking 2 M M 0 N in exp ; Hagedorn M M 0 2 Hawking = 2 p q 2N, Hagedorn = 4 1 ; 1, 2N ( Hawking Hagedorn if N suciently large) outgoing radiation : UV nite, (`Hawking radiation') incoming radiation : Hagedorn like divergence, =) `eective Hagedorn behavior' analogous to the rolling tachyon (c.f. Lambert-Liu-Maldacena, Maloney-Strominger- Yin) \tachyon-radion correspondence" in NS5 background (Kutasov hep-th/ )

23 Comments : Our boundary state can only describe latter half of classical trajectory : due to the boundary condition we chose. =) apparently breaks the time-reversal symmetry. (The `detailed balance' of radiation rates failes.) Dierent boundary conditions are also possible to dene boundary states, (although they seem physically unacceptable) : `D0-brane emitted from whitehole', `time-reversal symmetric solution' (recover the full trajectory), etc.

24 6. Some Comments In the non-compact CY models, extended characters play the similar roles as the minimal characters in the compact Gepner models. =) quite satisfactory However, topologically twisted models are still non-trivial for non-compact models. (They do not seem to be simply related with the Ramond vacua of the original models, contrary to the compact cases. ) Study of time-dependent D-branes in curved backgrounds is quite stimulating, and our analysis of the `falling D-branes' could yield a good starting point. The boundary state approach is important, since we often detect radiations of closed string massive modes. Is the `eective Hagedorn behavior' a universal feature of these systems? (c.f. Yin) Lambert-Liu-Maldacena, Maloney-Strominger-

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