Discrete D Branes in AdS 3 and in the 2d Black Hole

Transcription

1 ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Discrete D Branes in AdS 3 and in the d Black Hole Sylvain Ribault Vienna, Preprint ESI 1781 (006) February 0, 006 Supported by the Austrian Federal Ministry of Education, Science and Culture Available via anonymous ftp from FTP.ESI.AC.AT or via WWW, URL:

2 Preprint typeset in JHEP style - HYPER VERSION hep-th/05138 DESY Discrete D-branes in AdS 3 and in the d black hole Sylvain Ribault Deutsches Elektronen-Synchrotron Theory Division Notkestrasse 85, Lab a Hamburg 603 Germany sylvain.ribault@desy.de ABSTRACT: I show how the AdS D-branes in the Euclidean AdS 3 string theory are related to the continuous D-branes in Liouville theory. I then propose new discrete D-branes in the Euclidean AdS 3 which correspond to the discrete D-branes in Liouville theory. These new D-branes satisfy the appropriate shift equations. They give rise to two families of discrete D-branes in the d black hole, which preserve different symmetries.

3 Contents 1. Introduction and overview 1. AdS D-branes from Liouville theory 3.1 Comparison of one-point functions 3. Comparison of conformal blocks 4.3 The bulk regime 6 3. More branes in the Euclidean AdS An ansatz for new discrete D-branes 7 3. Verification of the shift equation Checks and interpretations à la Cardy 9 4. More branes in the d black hole Known branes and new branes in the d black hole Checks and interpretations A matrix model suggestion Introduction and overview Liouville theory and string theories with an affine ŝl symmetry have played an important rôle in recent studies of time-dependent string theory, two-dimensional quantum gravity, and the AdS/CFT correspondence. The features of these theories which are well-understood suggest that they share many important properties. This is not surprising considering that the Virasoro algebra of Liouville theory can be obtained from the ŝl affine Lie algebra by a quantum Hamiltonian reduction [1]. This suggests that Liouville theory can be found as a subsector of theories with an ŝl symmetry. For example, AdS 3 string theory can be reduced to Liouville theory via a topological twist [, 3]. Conversely, it would be interesting to reconstruct the full AdS 3 string theory in terms of the better-understood Liouville theory. A hint that this can be done comes from Zamolodchikov and Fateev s relation [4] between the Knizhnik Zamolodchikov(KZ) and Belavin Polyakov Zamolodchikov (BPZ) systems of differential equations, which reflect the ŝl and Virasoro symmetries respectively. More recently, all correlation functions of the H 3 + model (the Euclidean version of AdS 3 string theory) on a sphere have been written in terms of Liouville correlation functions [5]. Proving this relation relied on the prior knowledge of these H 3 + correlation functions in terms of well-characterized objects, namely the three-point structure constants and the conformal blocks. However, the H Liouville relation would be most useful if it allowed the construction of previously unknown objects in the H 3 + model from known objects in Liouville theory. One purpose of this article is to demonstrate that it indeed does. 1

4 The new objects in the H 3 + model which I plan to construct are discrete D-branes (in both meanings of having a discrete open string spectrum and coming in a discrete family) which correspond to the Zamolodchikov Zamolodchikov (ZZ) D-branes in Liouville theory [6]. I will first determine a relation between the known continuous AdS branes in the H 3 + model [7] and the continuous Fateev Zamolodchikov Zamolodchikov Teschner (FZZT) D-branes in Liouville theory [8, 9]. The main feature of this relation is the correspondence (.8) between the parameters of these families of D-branes, which associates two different FZZT branes to one AdS brane. Moreover, it is possible to relate the correlators of bulk fields in the presence of FZZT and AdS branes eq. (.10), but only in a particular regime which I will call the bulk regime. This is due to singularities in the H 3 + conformal blocks, which have a clear interpretation but so far no resolution in terms of Liouville theory. The relation between FZZT and AdS branes will then suggest a natural ansatz for a family of discrete branes in H 3 + parametrized by two integers (m, n), related to the ZZ branes of Liouville theory. The most useful characterization of these branes, which I will call AdS d branes, is the relation to the AdS branes eq. (3.4). (The name AdS d refers to that relation and not to the geometry of the new discrete branes.) These AdS d branes will be shown to be solutions of the same shift equation that was checked for the AdS branes [7]. How to modify this equation for the case of discrete branes will be suggested by Liouville theory. I will then propose a tentative relation between ŝl representations and D-branes in H 3 +, inspired by the Cardy relation which holds in rational conformal field theories, and which may help understand which H 3 + D-branes can be related to Liouville branes and which ones cannot. From the new discrete D-branes in the H 3 + model, two families of compact D-branes in the d cigar black hole SL(, R)/U(1) obeying two different gluing conditions can be constructed along the lines of [10]. Some of these D-branes have a geometrical interpretation as D0-branes at the tip of the cigar, the others do not have any geometrical interpretation. These new D-branes in the d black hole can then easily be translated into D-branes in the N = Liouville theory in Hosomichi s formalism [11], which provides a second independent shift equation. Moreover, the new D-branes might play an important rôle in the identification of non-perturbative corrections predicted by the matrix model for the d black hole. Most of the D-branes which appear in this article are represented in the following picture: Liouville H + 3 SL(, R)/U(1) N = Liouville FZZT (s) s= r πb ± i 4b AdS (r) s= i (mb 1 ±nb) r=iπ(m 1 ±nb ) r=iσ D1 (r) D (σ) B-branes chiral A-branes ZZ (m, n) (m,n) (m 1,n) AdS d (m, n) D1 d (m, n) D d (m, n) new B-type branes new A-type branes S (n) D0 (n) chiral degenerate A-branes

5 . AdS D-branes from Liouville theory The aim of this section is to generalize the relation between H 3 + and Liouville bulk correlators on the Riemann sphere [5] to correlators of bulk fields in the presence of worldsheet boundaries described by continuous D-branes: the AdS branes on the H 3 + side, the FZZT branes on the Liouville side. The relation between bulk correlators on the sphere can be decomposed into relations between bulk conformal blocks on the one hand, and bulk three-point structure constants on the other hand [1]. The introduction of a worldsheet boundary implies a modification of the conformal blocks, and the introduction of extra structure constants, namely the one-point functions (which must vanish when no boundary is there to break the worldsheet translation invariance). Let me briefly recall that Liouville theory is a two-dimensional conformal field theory on a worldsheet parametrized by a complex number z. The theory may be defined in terms of a field φ(z) by the action: ( S Liouville = d z z φ + µ L e bφ). (.1) The H 3 + φ, γ, γ: model describes strings in a three-dimensional space and therefore requires three fields S H+ 3 = k d z ( z φ + e φ γ γ ). (.) A more complete review with relevant references can be found in [5]..1 Comparison of one-point functions Consider one-point functions of the closed string worldsheet fields V α (z) in Liouville theory and Φ j (x z) in the H 3 + model. From the bulk H+ 3 -Liouville relation, the Liouville momentum α and the H 3 + spin j are related by: α = b(j + 1) + 1 b, (.3) where the Liouville parameter b is related to the H 3 + model level k by b = 1 k. In terms of j, the one-point function for the Liouville FZZT brane parametrized by the real number s is [8, 9] V α=b(j+1)+ 1 (z) b s = ΨFZZT s z z, α Ψ FZZT s = (πµ L γ(b )) j 1 1 π 1 4b Γ(j + 1)Γ(1 + b (j + 1)) coshπbs(j + 1), (.4) where z is the complex worldsheet coordinate and µ L the Liouville interaction strength. The onepoint function for an AdS brane in H 3 + with real parameter r is: [7, 13] Ψ AdS r (x) = ν j+1 b (8b ) 1 4 x + x j Γ(1 + b (j + 1))e r(j+1)sgn(x+ x), (.5) where ν b = π Γ(1 b ) Γ(1+b ) so that Φj=0 is the identity field (in slight contrast to [7]), and x is a complex isospin variable which labels states within a continuous SL(, C) representation of spin j. 3

6 This one-point function of the x-basis fields is written here for later use, but is not clearly related to the one-point function in Liouville theory. Instead, the bulk H 3 + -Liouville relation suggests to consider the µ-basis fields Φ j (µ z) = µ j+ d x e µx µ x Φ j (x z), (.6) whose one-point functions are obtained from eq. (.5) after a straightforward calculation: Φ j (µ z) r = ΨAdS r z z j, Ψ AdS C r = µ δ(rµ)ν j+1 π(8b ) 1 4 Γ(j + 1)Γ(1 + b (j + 1)) cosh(j + 1)(r i π sgniµ). b It is now obvious that the AdS D-brane one-point function is essentially the same as that of an FZZT brane (.4), but depending on sgniµ two different boundary parameters may appear: (.7) s ± = r πb ± i 4b. (.8) Such a relation could have been expected on several grounds. First, the FZZT branes are invariant under s s whereas the AdS branes are not invariant under r r, so there cannot be a one-to-one relation between the parameters r and s. Second, the SL(, R) symmetry of the AdS brane, which acts on the x parameter, does not completely determine the x dependence of the onepoint function, but allows an arbitrary dependence on sgn(x + x) [7]. Therefore the one-point function for an AdS brane involves two structure constants (instead of one in Liouville theory), which in the µ basis are encoded in the sgniµ dependence. Third, the difference s + s = i b is the jump in Liouville boundary condition induced by a boundary degenerate field B 1. This b is not surprising in view of the appearance of such degenerate fields in the H 3 + -Liouville relation beyond the one-point function discussed below.. Comparison of conformal blocks The H 3 + bulk conformal blocks are controlled by the Knizhnik Zamolodchikov equations [14], which are enough to determine their relation with Liouville conformal blocks [1]. Let me determine the KZ equations satisfied by the conformal blocks involved in the correlator of n bulk fields in the presence of an AdS brane Φ j 1(µ 1 z 1 ) Φ jn (µ n z n ). In Wess Zumino Witten r models with symmetry-preserving boundary conditions, such KZ equations are identical to the KZ equations satisfied by a correlator of n bulk fields on the sphere (at points z 1, z n, z 1 z n ), modulo a twist of the currents acting on the reflected fields if the gluing conditions are non-trivial. In the case of AdS branes, the gluing conditions are trivial as I will now show. Let me call J a (z), J a ( z) the left- and right-moving currents of the H 3 + model [15]. Their modes generate an ŝl (C) ŝl (C) affine Lie algebra. Their zero modes act on the fields Φ j (x z) or Φ j (µ z) as differential operators with respect to the isospin variables x or µ: J0 = x = µ, J0 0 = x x = µ µ J 0 + = x x jx = µ j(j+1) µ µ, (.9) 4

7 and the currents J 0 a are defined by repacing x, µ with x, µ. Note that this definition of the J 0 a currents is incompatible with the change of basis (.6) and is therefore basis-dependent. As a result, the gluing conditions will also be basis-dependent. The µ-basis one-point function of the AdS brane satisfies (J0 a + J 0 a)ψads r (µ) = 0, which corresponds to the trivial gluing condition J = J (see for instance [16]). Thus, it satisfies the same KZ equations as the bulk two-point function Φ j (µ z)φ j ( µ z). Indeed, the µ-dependences are similar: µ δ () (µ + µ) for the bulk two-point function, µδ(µ + µ) for the one-point function. In contrast, the x-basis one-point function has an x + x j factor which contrasts with the bulk two-point function x x 4j. This reflects the fact that the gluing conditions are non-trivial in the x-basis. Since the correlator Φ j 1 (µ 1 z 1 ) Φ jn (µ n z n ) satisfies the same KZ equations as a bulk r correlator with n fields, these equations are equivalent to BPZ equations via Sklyanin s separation of variables, as explained in [5, 17]. This leads to the following relation between H 3 + and Liouville correlators in the presence of worldsheet boundaries, where the equality so far means satisfies the same differential equations as : n Φ j l (µ l z l ) l=1 r b = π ( 1)n P n i=1r(µ i z i ) δ (R( P n i=1µ i )) Θ n k n l=1 n 1 V αl (z l ) V 1 (y a ) b a=1 s= r πb i 4b sgn P n i=1 Iµ i, (.10) In this equation the following conventions are used: the momenta and spins are related as in eq. (.3), I assume µ L = b, the function Θ π n is defined by l<l n z ll l l n (z l z l ) a<a n 1 y aa Θ n = n l=1 n 1 a=1 (z l y a )(z l ȳ a ) a a n 1 (y a ȳ a ), (.11) and most importantly the y a are the roots with positive imaginary parts of the real polynomial P(t) defined by: n l=1 ( µl t z l + µ ) [ n ] l = (µ l z l + µ l z l ) t z l l=1 In the case n = 3, the equation (.10) can be represented as: P(t) n l=1 (t z l)(t z l ). (.1) +z 1 +z +z 3 r +z 1 +z +z 3 y1 ys= r πb ± i 4b (.13) + z 1 + z H 3 + model + z 3 + z 1 ȳ1 + z ȳ Liouville theory + z 3 The reflected fields at ( z 1 z n ) in the lower half-plane are not physical, but they are indicated in this picture because they appear in the KZ or BPZ equations satisfied by the physical correlators of eq. (.10). 5

8 In this subsection I only argued that both sides of equation (.10) satisfy identical systems of differential equations. This amounts to a relation between the conformal blocks from which the correlators are built. In the next subsection I will complete the argument for equation (.10) and show that it holds in a certain regime..3 The bulk regime From the explicit expressions for the one-point functions (.4), (.7) it is easy to check that the equation (.10) holds in the case n = 1, which does not involve any insertion of degenerate Liouville fields V 1. One could then think that it is possible to prove equation (.10) by a recursion on b n, using the bulk operator product expansion to reduce the case of the n-point function in the limit z 1 z to the case of the n 1-point function. (The bulk OPEs in the H 3 + model and Liouville theory are indeed related in a way which would suit such an argument [5].) Then one would rely on the KZ equation to extend the relation (.10) to all values of z i, away from the limit z 1 z. However, this argument does not work because the conformal blocks which solve the KZ equations have singularities. These singularities are most easily seen in the corresponding Liouville theory conformal blocks: they occur whenever one of the y a becomes real. Indeed the y a are defined as the roots of the real polynomial P(t) (.1). Such a polynomial can have real roots and pairs of complex conjugate roots. Let me call the bulk regime the range of values of µ l, z l such that all the roots of P(t) are complex. The repeated use of the bulk OPE z 1 z z n (as required by the recursion above) is possible only in the bulk regime, because the definition of P(t) (.1) implies that for z 1 z some root y 1 of P(t) will also move close to z 1, and therefore in the bulk. Thus, the equation (.10) holds only in the bulk regime. Unfortunately, this prevents the easy determination of an H 3 + -Liouville relation in other bases like the x basis, which would involve an integration over all values of µ l. Let me illustrate the singularities of the conformal blocks in the case of a two-point function Φ j 1 (µ 1 z 1 )Φ j (µ z ). In this case the polynomial P(t) has degree two and its roots are r complex provided z z 1 z z 1 z > µ 1 + µ. (.14) µ 1 + µ The cross-ratio z varies from 1 when the two H 3 + bulk fields are far apart or close to the boundary, to + when they are close together or far from the boundary. The corresponding Liouville configurations are: +z 1 + z 1 y 1 y +z + z +z 1 + z 1 +z + z + z 1 + z Boundary regime Singularity Bulk regime 1 + z = z 1 z z 1 z µ 1 + µ µ 1 +µ + z 1 y ȳ + z (.15) In the boundary regime, the relation between the KZ and BPZ equations still holds. However it is not clear that a relation between H 3 + and Liouville correlators can be found. Such a relation would 6

9 have to specify which boundary parameters appear in Liouville theory. The boundary degenerate fields induce jumps of the boundary parameter s by the quantity b i [8]. The fact that the two boundary parameters s ± (.8) differ by this quantity is very suggestive, but more work needs to be done. This issue is however not relevant to the present article, whose purpose is to find new discrete D-branes in the H 3 + model. 3. More branes in the Euclidean AdS 3 In this section I will show that the relation between Liouville FZZT branes and AdS branes in the H 3 + model suggests a natural ansatz for new discrete D-branes in the H+ 3 model, which will be related to the discrete ZZ-branes in Liouville theory. This ansatz will then be subjected to a number of tests. Let me first briefly review the ZZ branes and their relation to the continuous FZZT branes. The ZZ branes are parametrized by two strictly positive integers (m, n) and are described by the one-point functions [6] V α=b(j+1)+ 1 (z) b (m,n) = ΨZZ (m,n) z z, α Ψ ZZ (m,n) = (πµ Lγ(b 3 )) j 1 4 πb Γ(j + 1)Γ(1 + b (j + 1)) sinπm(j + 1) sinπnb (j + 1). A well-known property of these ZZ branes which will be most useful in the following is: 3.1 An ansatz for new discrete D-branes Ψ ZZ (m,n) = ΨFZZT i (mb 1 +nb) ΨFZZT i (3.1) (mb 1 nb). (3.) The previous section demonstrated that an AdS brane with boundary parameter r is related to FZZT branes with boundary parameters s = πb r 4b i sgniµ. It is natural to look for discrete branes in the H 3 + model which would preserve the same symmetries as the AdS branes (in other words, they would obey the same gluing conditions) and which would be related in a similar manner to ZZ branes with parameters depending on sgniµ. In addition, I have explained that the difference of the two possible boundary parameters has an interpretation as the jump induced by a boundary degenerate field, which is quite natural considering the appearance of such fields in the boundary regime. Through the relation between ZZ and FZZT branes eq. (3.), this jump corresponds to a jump m m 1 of the parameter m of the ZZ branes. This suggests the following relation: New discrete brane in H + 3 (m, n) strictly positive integers { Liouville ZZ branes (m 1, n) if sgniµ > 0 (m, n) if sgniµ < 0 I will call discrete AdS branes or AdS d branes these new branes. Their above definition in terms of ZZ branes can be translated into a relation with AdS branes via the ZZ-FZZT relation eq. (3.) and the AdS -FZZT relation of the previous section: (3.3) Ψ AdSd (m,n) = ΨAdS iπ(m 1 +nb ) ΨAdS iπ(m 1 nb ). (3.4) 7

10 The essential feature of this relation is the shift 1, which directly corresponds to the shift of the boundary parameters in the AdS -FZZT relation eq. (.8). Let me write explicitly the one-point function of the AdS d branes in the x basis: Ψ AdSd (m,n) (x) = νj+1 b (8b ) 1 4 x + x j Γ(1 + b (j + 1)) isgn(x + x) e iπ(m 1 )(j+1)sgn(x+ x) sin πb n(j + 1). (3.5) Naturally, the relation (3.4) provides a simple way to derive the one-point functions of the AdS d branes in any basis. I have used the x basis because the shift equation of the next subsection is formulated in this basis. 3. Verification of the shift equation The one-point function for an AdS brane was found in [7] by solving a shift equation indicating how it should behave under shifts j j ± 1. A modified version of the shift equation is expected to hold for discrete branes preserving the same symmetries. This expectation is based on the study of shift equations for ZZ and FZZT branes in Liouville theory [8, 6] which I now review. The shift equations for ZZ and FZZT branes are of the type: R a sψ a s(α) = F Ψ a s(α b ) + F +Ψ a s(α + b ), (3.6) where the index a means ZZ or FZZT, with brane parameters generically called s. The coefficients F ± on the right-hand side do not depend on the type or parameter of the D-brane because they are fusing matrix elements. However, the quantity R a s depends on the type of brane: 1 R FZZT s = R FZZT ( b, Q s) = π µl sinπb Γ( 1 b ) Γ( b ) cosh πbs, (3.7) R ZZ (m,n) = ΨZZ Ψ ZZ (m,n) (α = b ) (m,n) (α = 0), (3.8) where the bulk-boundary structure constant value R FZZT ( b, Q s) was derived in [8] by a free field computation and can also be deduced from the general formula for the bulk-boundary structure constant [18] by carefully taking the relevant limit as sketched in [19]. One may wonder how the shift equations (3.6) can be compatible with the relation (3.) between ZZ and FZZT branes. The compatibility actually requires the non-trivial relation R ZZ R FZZT s=i mb 1 +nb = R FZZT s=i mb 1 nb Ψ ZZ (m,n) ( b) Ψ ZZ (m,n) (0) = R FZZT i mb 1 +nb. A direct computation shows that this relation is indeed obeyed: = R FZZT i mb 1 nb (m,n) = µl Γ( 1 b ) = π sinπb Γ( b ) ( 1) m cos πnb.(3.9) This analysis of the Liouville branes shift equations shows that once the shift equation for AdS branes is known, the shift equation for the associated discrete AdS branes boils down to: Ψ AdSd (m,n) (j = 1 ) Ψ AdSd (j = 0) (m,n)! = R AdS r=iπ(m 1 ±nb ). (3.10) 1 In the article [6] the denominator Ψ ZZ (α = 0) is absent from R ZZ (m,n) because the one-point function is normalized so that Ψ ZZ (α = 0) = 1. 8

11 The left-hand side can be directly computed from the ansatz (3.5): Ψ AdSd (m,n) (j = 1 ) Ψ AdSd (m,n) (j = 0) = i x + x sgn(x + x) Γ(1 + b ) νb Γ(1 + b ) ( 1)m cosπnb. (3.11) (Note that the factor sgn(x + x) comes from the shift 1 in the m 1 dependence.) The right-hand side of eq. (3.10) appears in the shift equation for AdS branes studied in [7]. In the notations of that article, R AdS r = (x + x)b( 1 )A(1, 0 r) (see in particular the equation (3.8) therein). The explicit computation of this quantity for r = iπ(m 1 ±nb ) gives the same result as eq. (3.11). Therefore, the equation (3.10) holds. 3.3 Checks and interpretations à la Cardy Let me discuss how the proposed discrete AdS branes help to complete the list of D-branes in the Euclidean AdS 3. Cardy has shown that in rational two-dimensional conformal field theories, symmetry-preserving D-branes are naturally associated to representations of the relevant symmetry algebra [0]. This idea has a natural extension in Liouville theory, where the continuous FZZT branes would then be associated to the continuous Virasoro representations appearing in the physical spectrum whereas the ZZ branes would be associated to the degenerate representations appearing in the Kac table, with momenta α mn Q = mb 1 + nb, (3.1) where (m, n) are still strictly positive integers, and I ignore the reflected degenerate representations α mn Q = (mb 1 + nb) because the reflection symmetry of Liouville theory makes them redundant. Now, the relation (.3) between the Liouville momentum and the H 3 + spin relates the Virasoro degenerate representations to ŝl degenerate representations with spins j mn + 1 = mb + n. (3.13) The discrete AdS branes should be considered as associated with such representations, whereas the ordinary AdS branes would be associated with the physical continuous representations. This interpretation of the AdS branes was already considered in [7] (section 4.), where the computation of some fusing matrix elements suggested the relation j(r) = 1 1 4b + i r πb. (3.14) However, as observed in [7], this relation does not give physical values j 1 + ir for r real due to the term 1. But this term precisely corresponds to the shift in the AdS 4b -FZZT relation (.8), and now seems rather natural. The reflection symmetry of the spectrum j j 1 then corresponds to the invariance of the FZZT branes under s s. Now replacing r in eq. (3.14) Note that ν b = π Γ(1 b ) Γ(1+b ) now has an extra factor π wrt [7] so that Φj=0 is the identity field, see eq. (.5) of the present article and footnote 7 of [7]. Also note that the requirement B(j = 0) = 1 leads to a different sign for B(j) as compared to [7]. 9

12 with the values appropriate for discrete AdS branes (3.4) gives the spins of the ŝl degenerate representations with null vector at nonzero level: j(r = iπ[m 1 + nb ]) + 1 = (mb + n). There is another series of degenerate representations of ŝl with m = 0, which do not correspond to Virasoro degenerate representations because they have a null vector at level zero [1]. D-branes corresponding to these representations are therefore not expected to be simply related to Liouville theory objects. There exist natural candidates for such D-branes: the S branes with imaginary radius of [7]. In contrast to the AdS branes which preserve an SL(, R) symmetry out of the SL(, C) of the H 3 + model, the S branes preserve an SU() symmetry. This enables the degenerate representations with level zero null vectors to be unitary and to appear in the physical spectrum of these D-branes. The representations mentioned so far are summarized in the following table, which should be compared to the picture of the moduli spaces of D-branes in H 3 + and Liouville theory in the Introduction: Virasoro ŝl α Q + ir α mn Q = mb 1 + nb j 1 + ir j mn + 1 = mb + n j n + 1 = n The shift equation studied in the previous subsection only provides a rather formal check of the consistency of the D-branes. A physically more significant condition is the consistency of the spectrum of open strings on the D-branes [0]. In non rational conformal field theories, the spectrum should consist of continous states with a positive density and/or discrete states with positive integer multiplicities. The consistency of the AdS branes has been checked in this way in [7, ]. Let me now consider the open strings on the discrete AdS branes. The open-string spectrum can be encoded in the annulus amplitudez AdSd (m 1,n 1 )(m,n ) = Tr ql 0 c 4 where the powers of q are the energies of the open-string states 3. Like the annulus amplitude for AdS branes, the annulus amplitude for open strings stretched between two AdS d branes is most easily computed in the µ basis. (A naive x-basis computation would give a wrong result due to an improper treatment of the divergences [7].) Z AdSd (m 1,n 1 )(m,n ) = d µ ( dj 1 +ir C µ ΨAdSd (m 1,n 1 ) (µ) = δ(0) 1 0 n n 1 n m m 1 m + ) b q (m,n ) ( µ) 4 (j+1) l=1 (1 ql ) 3(3.15) χ mn ( q).(3.16) Ψ AdSd m (m 1 1) (m 1) In this formula, m m 1 m means m 1 m < m < m 1 + m while m 1 + m m is an odd integer (like in sinm 1xsin m x sinx = m m 1 m sin mx), and χ mn (q) is an ŝl degenerate character [1]. The infinite prefactors (which come from the integral C d µ) result from the SL(, R) symmetry of the AdS d branes and are similar to infinite prefactors appearing in the annulus amplitude 3 With standard conventions: q = exp πi τ and q = exp πiτ where τ is the modular parameter of the annulus 10

13 of AdS branes [7]. In the case of AdS branes, there was an extra divergence of the integral dj at j = 1. This zero radial momentum divergence reflected the infinite extension of the AdS branes in the radial direction and is absent in the case of the AdS d branes. Therefore, the spectrum of open strings on the AdS d branes is consistent. However, the spectrum of open strings between AdS and AdS d branes is suspect (although it is discrete with integer multiplicities) because it contains states with imaginary conformal dimensions, as is clear from the formula: Z AdS AdS d r,(m,n) dj [ sin π(m 1)(j + 1) sin π(j + 1) q b 4 (j+1) sinπnb (j + 1) l=1 (1 ql ) 3 sinπb (j + 1) cosh(r i π sin πm(j + 1) )(j + 1) + sin π(j + 1) cosh(r + iπ )(j + 1) ]. (3.17) The Gaussian integral on j will indeed yield powers of q which are not real. In such cases I will say that Z AdS AdS d has an imaginary spectrum pathology. r 0,(m,n) The AdS d branes should however not be discarded because they might still play an important formal rôle in the theory, like the ZZ branes with (m, n) (1, 1) do in Liouville theory in spite of a similar pathology. In addition, the pathology can be absent in the case of some branes constructed from the AdS d branes as I will argue in the context of the d black hole SL(, R)/U(1). 4. More branes in the d black hole D-branes in the d cigar Euclidean black hole SL(, R)/U(1) can be obtained from D-branes in the Euclidean AdS 3 by a descent procedure [10]. On the one hand this will yield more consistency checks for the new D-branes constructed in the present article, and on the other hand this will suggest a comparison with matrix model results. 4.1 Known branes and new branes in the d black hole Let me now recall the one-point functions of the SL(, R)/U(1) bulk fields Φ j n,w in the presence of boundary conditions defined by the D-branes descending from S and AdS branes in H 3 +. A D0-brane in the cigar descends from an S branes in H 3 + labelled by a strictly positive integer n: Ψ D0 n = δ n 0ν j+1 b k 1 4b 1 Γ( kw j)γ( kw j) π( 1) nw+1 Γ( j) Γ(1 + b (j + 1)) sinπnb (j + 1) (4.1). A D1-brane in the cigar descends from an AdS brane in H 3 + with a real parameter r and an angle θ 0 : Ψ D1 r = δ w,0 e in θ 0 ν j+1 k 1 4 b 1 Γ(j + 1)Γ(1 + b (j + 1)) b Γ(1 + j + n n )Γ(1 + j ) ( e r(j+1) + ( 1) n e r(j+1)) (4.). A D-brane in the cigar also descends from an AdS brane in H 3 +, whose parameter r now has to be taken pure imaginary r = iσ. The real parameter σ of the D-branes is quantized in units of 11

14 πb and bounded σ < π (1 + b ). Ψ D σ = δ n,0ν j+1 k 4b 1 1 b π Γ(j + 1)Γ(1 + b (j + 1)) ( Γ( j + kw ) Γ(j kw + Γ( j ) kw ) )eiσ(j+1) Γ(j + 1 kw ) e iσ(j+1). (4.3) New discrete branes can be obtained in SL(, R)/U(1) from the discrete AdS branes in H + 3. Like the original AdS branes, the discrete AdS branes give rise to two families of D-branes in the coset. Their one-point functions can be obtained from D1- and D-branes one-point functions thanks to the formula (3.4). Let me first consider the D1 d -branes obtained from the D1-branes: Ψ D1d (m,n) = δ w,0e in (θ 0 + π ) ν j+1 Γ(j + 1)Γ(1 + b b k 1 4b (j + 1)) 1 Γ(j n n )Γ(j + 1 ) sinπ [(j + 1)(m 1 ) + n ] sin πnb (j + 1). (4.4) The spectrum encoded in the annulus amplitude Z(m D1d 1,n 1 )(m,n ) contains a finite number of discrete representations with positive integer multiplicities and is therefore consistent. (However, I did not find the marginal field which might have been expected from the existence of a modulus θ 0.) Note also that the amplitude Z D1 D1d suffers from the same imaginary spectrum pathology as the r 0,(m,n) amplitude Z AdS AdS d r 0,(m,n) in H + 3. The D d -branes obtained from the D-branes are characterized by the one-point function: (m,n) = δ n,0ν j+1 b i k 1 4b 1 Γ(j + 1)Γ(1 + b (j + 1)) sinπnb (j + 1) ( π Γ( j + kw ) 1 Γ(j kw )(j+1) Γ( j kw ) )eiπ(m Γ(j + 1 kw ) e iπ(m Ψ Dd ) 1 )(j+1). (4.5) The spectrum encoded in the annulus amplitude Z(m Dd 1,n 1 )(m,n ) contains a finite number of discrete representations with positive integer multiplicities and is therefore consistent. 4 The spectrum Z D Dd σ,(m,n) is also consistent and in particular free from the imaginary spectrum pathology, because the D-brane parameter σ comes from pure imaginary values of the AdS brane parameter r. However, it might be more relevant to examine the amplitude Z D1 Dd r,(m,n), which is more difficult to compute because of the difference in gluing conditions between D1- and D d -branes. This difficulty is no obstacle to finding that Z D1 Dd 5 has the imaginary spectrum pathology except if (m, n) = (1, 1), like the amplitude Z 0,(m,n) FZZT ZZ s,(m,n) in Liouville theory. Notice that Z D1 D0 r 0,n 4 The detailed computation of this spectrum for m 1 m would require a non-trivial generalization of the calculations in [10]. Note also that the multiplicities are positive in contrast to the D-brane case [3], due to the sign difference between the second lines of eqs (4.3) and (4.5). 5 The imaginary spectrum pathology is absent from such a discrete annulus amplitude if and only if Ψ D1 r Ψ Dd (m,n) is a linear combination of a finite number of terms of the type cos λ(j + 1) with λ either real or pure imaginary. The pathology results from such terms with a generic complex λ. 1

15 is also free from the imaginary spectrum pathology only for n = 1. The D0- and D d -branes with parameters n and (1, n) respectively behave identically in this respect because their overlaps with D1-branes only involve closed strings with winding zero, which make no difference between them: Ψ DO n (w = 0) = Ψ Dd (1,n) (w = 0). 4. Checks and interpretations Let me first discuss whether the new D1 d - and D d -branes have a geometric interpretation. A geometric description of the d black hole is possible in the limit k which corresponds to small string length l s = α (while kα is a fixed length). First recall the geometric interpretation of the known D0-, D1- and D-branes [10] as zero-, one- and two-dimensional geometric objects in the d black hole. This can be seen in the large k behaviour of the one-point functions, Ψ D0 n k 1, Ψ D1 (r,θ 0 ) 1, ΨD σ k 1. (4.6) How this behaviour depends on the dimensionality of the D-branes is indeed consistent with the dependence of the D-branes tensions T (α ) p with respect to the D-branes dimensions p. Now the observation (from the previous subsection) that closed strings with zero winding couple identically to D0-branes and to D d -branes implies that the D d -branes should be interpreted as pointlike branes at the tip of the cigar like the D0-branes. The behaviour of D1 d -branes is different: Ψ D1d (m,n) k 1, (4.7) thus their one-point functions decrease too fast at large k to allow a geometric interpretation. It is possible to call the D1 d -branes anisotropic localized branes at the tip of the cigar only in a heuristic sense. Let me nevertheless compare this heuristic geometric picture to the situation in Liouville theory. The localization of the D1 d -branes at the tip of the cigar, and the existence of continuous D1-branes extending from infinity up to some finite distance from the tip (a distance determined by their parameter r), are similar to the localization of the ZZ branes at strong Liouville coupling, together with the existence of FZZT branes extending to infinity. The situation of the D d - and D- branes is quite different since the D-branes extend to the tip where the D d -branes are located. However, a species of branes with the same gluing conditions as the D-branes and a behaviour similar to that of the FZZT branes has been predicted to exist [4, 5]: the D-branes descending from ds branes in AdS 3 [6], which I will also call ds branes: D1 d D1 D d ds ZZ FZZT 13

16 The geometry of the ds branes in AdS 3 suggests that the ds branes in the cigar are parametrized by a real number r related to the D-brane s parameter σ by σ = π + ir. The identifications σ = ir and r = πbs i π (from eq. (.8)) then imply the relation r = πbs between the ds brane parameter r and the FZZT brane parameter s. In addition, the ds brane is expected to be invariant under r r like the FZZT brane under s s. This supports the idea of a close relationship between ds branes in the cigar and FZZT branes in Liouville theory. Let me now compare the D-branes in the d black hole with D-branes in the N = supersymmetric Liouville theory. The N = Liouville theory is indeed equivalent to the N = supersymmetric d black hole theory [7], which is itself very similar to the bosonic d black hole theory which has been considered in this section. (On the other hand, the N = Liouville theory is considerably more complicated than bosonic Liouville theory.) The comparison of D-branes is relevant to this article because it will provide an independent shift equation for the one-point functions of the new D1 d -branes. This is based on the article on N = Liouville theory by Hosomichi [11], which among many interesting results formulates a shift equation with j-shift by k in addition to the shift equation with j-shift by 1 considered in subsection 3.. These two possible shifts are independent if k is not rational, and in contrast to the two elementary α-shifts in Liouville theory (by b 1 and b ) they are not related by a duality of the theory. The D1-branes in the d black hole (4.) correspond to Hosomichi s B-branes [11] (4.55). According to the principles of subsection 3., the D1 d -branes should therefore satisfy Ψ D1d (m,n) (j = k, n ) Ψ D1d (m,n) (j = 0, n = 0) )! = c t ( n, n r=iπ(m 1 ±nb ), (4.8) where c t is explicitly known [11] (4.55), and the N = Liouville degenerate spin k becomes k in SL(, R)/U(1) after k k and reflection. If proper care is taken of the other differences of conventions, this equation is found to hold. The D-branes in the d black hole (4.3) correspond to Hosomichi s chiral or anti-chiral A- branes [11] (3.6). The k -shift equation for these branes [11] (4.33) has a vanishing left-hand side, leading to the condition: Ψ Dd (m,n) (j = k, w)! Ψ Dd (m,n) (j = 0, w = 0) = 0. (4.9) Surprisingly, this equation holds due to the denominator being infinite. It therefore provides a rather trivial check of the discrete D-branes s one-point function Ψ Dd (m,n). To summarize, translating the new AdS d branes to the D1d -branes in the d black hole and then to N = Liouville theory has yielded a strong independent check of their consistency. 4.3 A matrix model suggestion The ZZ branes in Liouville theory are likely to account for some non-perturbative contributions to the partition function of two-dimensional string theories. Such contributions can in certain cases be independently computed using a matrix model approach, and can reasonably be attributed to ZZ branes. (See for instance [8, 9] and [30] for a review.) The matrix model can also predict 14

17 some non-perturbative contributions in the d black hole string theory at level k = 9 4. It is natural to expect that some of these contributions reflect the presence of the new discrete branes in the d black hole, because they correspond to the ZZ branes in Liouville theory. In particular, some non-perturbative contributions should come from objects which do not couple to the sine-liouville interaction [31]. Natural candidates for these objects are the D1 d -branes, because they do not couple to closed strings with nonzero winding. However, the new discrete D-branes are far too numerous compared to the non-perturbative corrections predicted by the matrix model. Like in Liouville theory, the problem is now to find which ones should be considered physical. Acknowledgments I am supported by a fellowship from the Alexander von Humboldt Stiftung. I am grateful to Sergei Alexandrov, Thomas Quella, Andreas Recknagel, Volker Schomerus and Jörg Teschner for interesting conversations. In addition, I wish to thank Sakura Schäfer-Nameki and Volker Schomerus for helpful comments on the draft of this article. I benefitted from the hospitality of the Erwin- Schrödinger Institut in Vienna. References [1] P. Forgacs, A. Wipf, J. Balog, L. Feher and L. O Raifeartaigh, Liouville and toda theories as conformally reduced wznw theories, Phys. Lett. B7 (1989) 14. [] S. Mukhi and C. Vafa, Two-dimensional black hole as a topological coset model of c = 1 string theory, Nucl. Phys. B407 (1993) [hep-th/ ]. [3] L. Rastelli and M. Wijnholt, Minimal ads(3), hep-th/ [4] A. B. Zamolodchikov and V. A. Fateev, Operator algebra and correlation functions in the twodimensional wess-zumino su() x su() chiral model, Sov. J. Nucl. Phys. 43 (1986) [5] S. Ribault and J. Teschner, H(3)+ correlators from liouville theory, JHEP 06 (005) 014 [hep-th/050048]. [6] A. B. Zamolodchikov and A. B. Zamolodchikov, Liouville field theory on a pseudosphere, hep-th/ [7] B. Ponsot, V. Schomerus and J. Teschner, Branes in the Euclidean AdS(3), JHEP 0 (00) 016 [hep-th/011198]. [8] V. Fateev, A. B. Zamolodchikov and A. B. Zamolodchikov, Boundary liouville field theory. i: Boundary state and boundary two-point function, hep-th/ [9] J. Teschner, Remarks on liouville theory with boundary, hep-th/ [10] S. Ribault and V. Schomerus, Branes in the -d black hole, JHEP 0 (004) 019 [hep-th/031004]. [11] K. Hosomichi, N = liouville theory with boundary, hep-th/ [1] S. Ribault, Knizhnik-zamolodchikov equations and spectral flow in ads(3) string theory, JHEP 09 (005) 045 [hep-th/ ]. 15

18 [13] P. Lee, H. Ooguri and J.-w. Park, Boundary states for AdS() branes in AdS(3), Nucl. Phys. B63 (00) [hep-th/011188]. [14] V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and wess-zumino model in two dimensions, Nucl. Phys. B47 (1984) [15] K. Gawedzki, Noncompact wzw conformal field theories, hep-th/ [16] V. Schomerus, Lectures on branes in curved backgrounds, Class. Quant. Grav. 19 (00) [hep-th/00941]. [17] A. V. Stoyanovsky, A relation between the knizhnik zamolodchikov and belavin- -polyakov zamolodchikov systems of partial differential equations, math-ph/ [18] K. Hosomichi, Bulk-boundary propagator in liouville theory on a disc, JHEP 11 (001) 044 [hep-th/ ]. [19] V. Schomerus, Non-compact string backgrounds and non-rational cft, hep-th/ [0] J. L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B34 (1989) 581. [1] M. Kato and Y. Yamada, Missing link between virasoro and sl() kac-moody algebras, Prog. Theor. Phys. Suppl. 110 (199) [] S. Ribault, Two AdS() branes in the Euclidean AdS(3), JHEP 05 (003) 003 [hep-th/01048]. [3] D. Israel, A. Pakman and J. Troost, D-branes in n = liouville theory and its mirror, Nucl. Phys. B710 (005) [hep-th/040559]. [4] A. Fotopoulos, Semiclassical description of D-branes in SL()/U(1) gauged WZW model, Class. Quant. Grav. 0 (003) S465 S47 [hep-th/ ]. [5] S. Ribault, Strings and d-branes in curved space-times. (in french), hep-th/ [6] C. Bachas and M. Petropoulos, Anti-de-Sitter D-branes, JHEP 0 (001) 05 [hep-th/00134]. [7] K. Hori and A. Kapustin, Duality of the fermionic d black hole and N = Liouville theory as mirror symmetry, JHEP 08 (001) 045 [hep-th/01040]. [8] E. J. Martinec, The annular report on non-critical string theory, hep-th/ [9] S. Y. Alexandrov, V. A. Kazakov and D. Kutasov, Non-perturbative effects in matrix models and d-branes, JHEP 09 (003) 057 [hep-th/ ]. [30] S. Alexandrov, Matrix quantum mechanics and two-dimensional string theory in non-trivial backgrounds, hep-th/ [31] S. Alexandrov. unpublished note. 16