Abstract
We first discuss the relationship between the SL(2; R)/U (1) supercoset and N = 2 Liouville theory and make a precise correspondence between their representations. We shall show that the discrete unitary representations of SL(2; R)/U (1) theory correspond exactly to those massless representations of N = 2 Liouville theory which are closed under modular transformations and studied in our previous work [18] .
Abstract
It is known that toroidal partition functions of SL(2; R)/U (1) theory (2D Black Hole) contain two parts, continuous and discrete representations. The contribution of continuous representations is proportional to the space-time volume and is divergent in the infinitevolume limit while the part of discrete representations is volume-independent.
Abstract
In order to see clearly the contribution of discrete representations we consider elliptic genus which projects out the contributions of continuous representations: making use of the SL(2; R)/U (1), we compute elliptic genera for various non-compact space-times such as the conifold, ALE spaces, Calabi-Yau 3-folds with A n singularities etc. We find that these elliptic genera in general have a complex modular property and are not Jacobi forms as opposed to the cases of compact Calabi-Yau manifolds.
Introduction
Study of superstring vacua in irrational superconformal theories has been a challenging problem. These theories describe superstrings propagating in non-compact curved space-time often developing isolated singularities. Applications of the study of this subject include the string theory on Calabi-Yau singularities and its holographic description, or the little string theory on NS5-branes in the T-dual picture [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] . The analysis is also important for backgrounds with non-trivial time dependence studied in the context of time-like Liouville theory [14, 15, 16] . Much of the efforts of understanding irrational (super)conformal theories have centered around the study of Liouville field theory. See, for instance, [17] for a recent review including a detailed list of literature.
Introduction
One of the outstanding difficulties in these non-compact models is the coexistence of both discrete and continuous spectra of primary fields. Characters of these representations mix in a non-trivial manner under modular transformations in contrast with rational theories.
Introduction
As an example, let us recall the superstring vacua of the type Minkowski space-time ⊗ N = 2 minimal ⊗ N = 2 Liouville .
Introduction
Then, we had the following puzzle 1 ;
Introduction
1. These backgrounds correspond to isolated singularities in Calabi-Yau spaces [1, 2, 3, 4] , and one expects the existence of massless excitations describing the deformation of the singularities in the string spectra.
Introduction
2. However, the modular invariant partition functions for such superconformal systems (in the infinite-volume limit) contain only contributions from continuous representations which possess the mass gap and no discrete representations corresponding to massless excitations occur, as was studied for instance in [7] .
Introduction
In our previous work [18] we have partially resolved this puzzle by showing that 3. Discrete massless modes representing non-trivial cycles appear in the open string channels of cylinder amplitudes associated with supersymmetric boundary states. We have obtained candidate Cardy boundary states which correspond to ZZ and FZZT branes in the bosonic Liouville theory [19, 20, 21] . See also [22] .
Introduction
In this paper in order to further study these problems, we analyze the SL(2; R) k /U(1) Kazama-Suzuki model [23] , which is known to be T-dual (mirror) to the N = 2 Liouville theory [24, 5, 25, 26] . After identifying the SL(2; R) k /U(1) conformal blocks (branching functions) with the N = 2 characters, we perform the character expansion of the toroidal partition function of the Kazama-Suzuki model in order to study its closed string spectrum. We will follow the analysis given in [27] (see also [28, 29] ) of the partition function of the bosonic SL(2; R)/U(1)model [30, 31] . It is well-known that the gauged WZW model for SL(2; R)/U(1) describes the geometry of a 2-dimensional black hole [32] .
Introduction
We find an infra-red divergence in the partition function of the theory corresponding to the infinite volume V of the geometry of 2D black hole. In the limit V → ∞ the partition function becomes simply the diagonal sum of continuous representations. When we suitably regulate the IR divergence, however, we find a non-trivial weight function for the continuous series and also a set of discrete representations with spin 1/2 ≤ j ≤ (k + 1)/2. We find that these discrete series are exactly the massless representations in N = 2 Liouville theory which are closed under modular transformations [18] .
Introduction
Discrete states are localized around the tip of the cigar in 2D black hole [33] . Thus they are suppressed in the infinite-volume limit as compared with the continuous representations.
Introduction
In order to see clearly identify the contributions of the discrete states not buried under the continuous representations, we propose to study the elliptic genus of the theory [34] to which the continuous representations do not contribute. We consider coupled superconformal systems: SL(2; R) k /U(1) ⊗ M, where M are various N = 2 RCFT's. By taking suitable choices for M we have computed elliptic genera for the conifold, ALE spaces, Calabi-Yau 3-folds with A n singularities etc. It turns out that in general elliptic genera have a complicated modular property and are not Jacobi forms as in the case of compact Calabi-Yau manifolds. They are instead expressed in terms of the Appell function which features in the study of higher rank vector bundles over elliptic curves [35, 36] .
Introduction
While preparing this manuscript, we became aware of an interesting paper [37] on the eprint Archive, where the authors studied the modular properties of the extended characters in the bosonic SL(2; R)/U(1)-model defined in the similar manner as in [18] , and there are some overlaps with the present work.
Introduction
2 Toroidal Partition Function for the SL(2; R)/U (1) Kazama-
Suzuki Model

Preliminaries
The Kazama-Suzuki supercoset model [23] for SL(2; R) k /U(1) is defined as the coset CFT
Preliminaries
which is an N = 2 SCFT with the central charge and level 2
Preliminaries
More explicitly, the total world-sheet action is written as S(g, A, ψ ± ,ψ ± ) = κS gWZW (g, A) + S ψ (ψ ± ,ψ ± , A) , (2.3)
Preliminaries
κS gWZW (g, A) = κS SL(2;R)
4)
S SL(2;R)
5)
S ψ (ψ ± ,ψ ± , A) = 1 2π
5)
where the complex fermions ψ ± (andψ ± ) have charge ±1 with respect to the U(1)-gauge group. The bosonic part κS gWZW (g, A) is the gauged WZW model for the coset SL(2; R) κ /U(1) A [38, 30] , where U(1) A indicates the gauging the axial U(1)-symmetry; g → ΩgΩ, Ω(z,z) = e iu(z,z)σ 2 (u(z,z) ∈ R, σ 2 is the Pauli matrix.) It is well-known that this model describes string theory on 2D Euclidean black-hole with the cigar geometry [32] . The WZW action κS
5)
WZW (g), and has a negative signature in iσ 2 -direction. Since we have H 3 (SL(2; R)) = 0, the action κS SL(2;R) WZW (g) can be rewritten as a purely two dimensional form and the level κ need not be an integer.
5)
The chiral currents
5)
satisfy the affine SL(2; R) κ current algebra (we write the left-mover only);
5)
and the pair of free fermions ψ + , ψ − satisfy the OPE's ψ
5)
The explicit realization of N = 2 SCA is given by
5)
where we set J 3 ≡ j 3 + ψ + ψ − , which is the unique U(1)-current commuting with all the generators of N = 2 SCA and hence defines the denominator of the SL(2; R) k /U(1)-supercoset.
5)
To close this preliminary subsection, we summarize irreducible representations of SL(2; R) κ current algebra. We concentrate on the representations with conformal weights bounded from below 3 and corresponding to unitary representations of the zero-mode subalgebra {j 3 0 , j ± 0 }. The parameter j is related to the conformal weight of vacuum states by the well-known formula
5)
in all cases.
continuous series :Ĉ
They are non-degenerate representations and all the states lie above the mass gap h ≥ 1 4(κ − 2)
continuous series :Ĉ
. The vacua have the j 3 0 -spectrum; j 3 0 = α + n, n ∈ Z. The character formula is simply given by (q ≡ e 2πiτ , y ≡ e 2πiu );
continuous series :Ĉ
In the following arguments we often use formal identities such as
continuous series :Ĉ
For a more rigorous treatment, one may consider the "regularized characters" χ ξ (τ, u; ǫ) defined by replacing e 2πi(n+α)u with e 2πi(n+α)u e −|n|ǫ (ǫ > 0, n ∈ Z, 0 ≤ α < 1). 3 We have more general representations constructed by the spectral flows;
continuous series :Ĉ
However, they generically have unbounded spectra of conformal weights.
continuous series :Ĉ
2. discrete series :D ± j (j ∈ R >0 ) The superscript + indicates that these are the lowest weight representations and − does the highest weight ones. The spin parameter j is allowed to be continuous despite the name "discrete series" (while, in the SL(2; R)/U(1)-coset theory j takes discrete values).
continuous series :Ĉ
The vacua have the j 3 0 -spectrum; j 3 0 = ±(j + n), n ∈ Z ≥0 forD ± j respectively. The character formula is written as
continuous series :Ĉ
.
continuous series :Ĉ
(2.13)
identity representation :
This is the representation generated by the vacuum h = j 3 0 = 0 corresponding to the identity operator. Since this vacuum is both highest and lowest weight, the character formula becomes
identity representation :
These are non-degenerate representations below the mass gap. The vacua again have the j 3 0 -spectrum; j 3 0 = α + n, (n ∈ Z). The character formula has the same form as the continuous series. The range of j comes from the unitarity of zero-mode subalgebra.
identity representation :
At the "boundary" of this range j = 0, α = 0, the representation with h = 0 becomes reducible and is decomposed aŝ
Branching Functions
We start our analysis by identifying the conformal blocks for the toroidal partition function in the SL(2; R) k /U(1) Kazama-Suzuki model [23] . Although this task has been already implicitly carried out in [39] 4 , it is helpful for our later analysis to provide the explicit formulas of conformal blocks. It turns out that these blocks are identified with the irreducible characters of N = 2 SCA. This fact will be the simplest evidence for T-duality with the N = 2 Liouville theory.
Branching Functions
We focus on the NS-sector and the formulas for other spin structures are obtained by using the 1/2-spectral flow. According to the standard treatment of coset CFT, the conformal blocks are defined as the following branching functions;
Branching Functions
where ξ indicates each of the irreducible representations of SL(2; R) κ classified above. The angle variables u, v, z, w are associated to the U(1)-currents j 3 , ψ + ψ − , J, J 3 respectively, and we can easily read off the relations among them as
Branching Functions
The summation of m runs over the possible J 3 0 -spectrum of representation ξ (tensored with the free fermion system ψ ± ) The branching functions for continuous series (and the complementary representations) are easily obtained due to the absence of null states;
Branching Functions
Here ch (NS) denotes the N = 2 irreducible character for the massive (non-degenerate) representation (B.1). We sometimes allow the pure imaginary values of p corresponding to the branching functions for the complementary representationsÊ j,α (j = 1 2 + ip). The unitarity condition [41] is derived from those for (the zero-mode parts of)Ĉ p,α ,Ê j= 1 2 +ip,α ;
Branching Functions
Derivation of the branching functions for the discrete series is more non-trivial. We will focus onD + j becauseD − j is obtained by the spectral flow
Branching Functions
The desired branching function is expressed as
Branching Functions
We first consider the following shifts of angular variables in (2.21) (or (2.16));
Branching Functions
which leaves u invariant and causes v −→ v + nτ . Using the property θ 3 (τ, v + nτ ) = q − n 2 2 e −2πinv θ 3 (τ, v), we find the relation
Branching Functions
Therefore, it is enough to calculate χ dis (NS) j,j (τ, z). The easiest way to evaluate it is to use of the character relation
Branching Functions
R.H.S is the character of complementary representationÊ j,α=j .
Branching Functions
We act by q j 2 k 1 0 dw θ 3 (τ, v)e −2πijw on both sides of (2.24). We also need the branching
Branching Functions
We then obtain 
Branching Functions
One may check directly the validity of our branching relation
Branching Functions
by comparing the residues at poles e 2πiu q m = 1 (m ∈ Z) of both sides. It is useful to note:
Branching Functions
• n ≥ 0: The vacuum state of χ dis (NS) j,j+n (τ, z) is (j + 0 ) n |j, j ⊗ |0 ψ , which possesses the quantum numbers
Branching Functions
As is proved in [39] , the unitarity bound for the Casimir parameter j is given as
Branching Functions
which reproduces all the (spectrally flowed) massless matter representations of N = 2 SCA lying on the "unitarity segments" given in [41] .
Branching Functions
Branching functions for the identity representation χ dis
Branching Functions
j=0,m=0 (τ, z) may be derived in a similar manner with the help of (2.15). As one may expect, it is given by the graviton representation h = Q = 0 (B.3) of N = 2 SCA. We find that
Branching Functions
Spectral-flowed version is given by
Branching Functions
The corresponding vacua are slightly non-trivial;
Branching Functions
• n = 0 : The vacuum is |0, 0 ⊗ |0 ψ with h = Q = 0.
Branching Functions
• n ≥ 1 : The vacuum is (j + −1 ) n−1 |0, 0 ⊗ ψ + −1/2 |0 ψ , which has the quantum numbers
Branching Functions
• n ≤ −1 : The vacuum is (j − −1 ) |n|−1 |0, 0 ⊗ ψ − −1/2 |0 ψ , which has the quantum numbers
Branching Functions
We finally introduce the branching functions of other spin structures to fix the convention in this paper. Let χ (NS) * ,m (τ, z) be the abbreviated notations of the branching functions considered above (m = J 3 0 ). We define
Branching Functions
(2.37)
Toroidal Partition Function
Let us analyze the toroidal partition function of SL(2; R) k /U(1) Kazama-Suzuki model. It can be evaluated by the path-integral approach as described in [30, 31] for the bosonic 2D black-hole model [32] . Here we present only the result and leave the detailed calculations to Appendix C. For the NS sector of the theory, we obtain
Toroidal Partition Function
where C is a normalization constant to be fixed later. The partition functions for other spin structures are obtained by simply replacing
Toroidal Partition Function
Assuming the diagonal modular invariant for spin structures, we obtain the partition function
Toroidal Partition Function
is the modulus of gauge field A. One can view the sum over m, n as summing over the momentum and winding modes of a compact boson Y which parameterizes the 2-dimensional gauge field A. Y has a radius √ 2k which is the size of the asymptotic circle far from the tip of the cigar. With the canonical normalization Y (z)Y (0) ∼ − ln z for the field Y , total anomaly free current defining the BRST charge (see [38] ) is given as
Toroidal Partition Function
Of course, these currents have no singular OPE's with the N = 2 SCA generators (2.10), assuring their BRST-invariance.
Toroidal Partition Function
The partition function (2.39) is modular invariant in a formal sense since the modulus integral ds 1 ds 2 is logarithmically divergent due to the double pole of 1/|θ 1 (τ, s 1 τ − s 2 )| 2 . The appearance of divergence is not surprising since the target space has an infinite volume. The evaluation of modulus integral with a suitable IR cut-off in (2.38) is quite useful in determining the closed string spectrum, as shown in [27] for the bosonic SL(2; R)/U(1) model. We turn to this analysis from now on.
Expansion into Branching Functions
We expand the toroidal partition function into branching functions of SL(2; R) k /U(1) following the procedure of [27] . (See also [28, 29] .) Although our result will be quite similar to the bosonic case [27] , we will present our analysis for the supersymmetric case for the sake of completeness.
Expansion into Branching Functions
We start with the partition function of NS sector (2.38). Using the Poisson resummation formula, we can rewrite it as;
Expansion into Branching Functions
(2.41)
Expansion into Branching Functions
The s 2 -integral is easily performed since s 2 appears only linearly in the exponent. The qexpansion of the theta function terms is expressed as the trace over the Hilbert space of various oscillators. We introduce the oscillator levels N,Ñ and also the operators l,l defined as
Expansion into Branching Functions
The relevant Hilbert space is
Expansion into Branching Functions
where F * denotes the Fock space of each sector. Especially, in the SL(2; R)-sector, F + SL(2) and F − SL(2) means respectively the ones associated to the lowest and highest weight representations of zero-modes: namely we have
Expansion into Branching Functions
We thus obtain
Expansion into Branching Functions
where the trace is constrained by the condition l −l = n imposed by the s 2 -integral. The s 1 -integral is non-trivial since s 1 appears quadratically in the exponent. Following [28, 27] , we linearize it by means of the Fourier transformation;
Expansion into Branching Functions
(2.46)
Expansion into Branching Functions
The s 1 -integral is then easy to compute and gives
Expansion into Branching Functions
In the last line we set c = 2τ 2 p − iτ 2 k in the first term, and set c = 2τ 2 p in the second term.
Expansion into Branching Functions
The integration contours are defined as C 1 : Im p = 0, C 2 : Im p = k/2.
Expansion into Branching Functions
To proceed further it is useful to make use of the spectral flow associated to the total currents (2.40), defined symbolically as 5
Expansion into Branching Functions
It is easy to see that U 1 acts as
Expansion into Branching Functions
and maps the Fock space H + to H − . We thus find that
Expansion into Branching Functions
Substituting (2.47) and (2.50) to (2.45), we can show (ĉ ≡ (k + 2)/k)
Expansion into Branching Functions
As in [28, 27] , let us perform the contour deformation; C 2 → C 1 , which picks up extra contributions from simple poles within the range 0 ≤ Im p ≤ k/2. The partition function is now divided into two parts;
Expansion into Branching Functions
where the first term includes the p-integration on the real axis (C 1 ) and the second corresponds to the sum of pole residues. The first term Z con (NS) (τ ) is rewritten as
Expansion into Branching Functions
Since the factor l +l appears only in the denominators, the traces are logarithmically divergent when one sums over the states of the form
Expansion into Branching Functions
. This divergence comes from the pole s 1 = s 2 = 0 in (2.38), that is, the infinite volume effect. Since the exponent
Expansion into Branching Functions
is the correct weights for the continuous representations, it is natural to expect that Z con (NS) (τ ) can be expressed in a form
Expansion into Branching Functions
with a suitable spectral density ρ(p, w, n). Here χ con (NS) p,m (τ, 0) is the branching function of continuous series (2.18) and is an irreducible massive character of N = 2 SCA. Although there appear some ambiguities in regulating the IR divergence, a candidate expression for ρ(p, w, n) has been proposed in [28, 27] .
Expansion into Branching Functions
where ǫ > 0, ǫ ≈ 0 is the IR cut-off. The first term in (2.56) is interpreted as the volume factor. The second term has a non-trivial momentum dependence and is related to the reflection amplitudes of Liouville theory as is discussed in [28] .
Expansion into Branching Functions
On the other hand, the pole contributions yield the sum over the branching functions of discrete series (2.27) (and (2.28)). We take the identification j = −ip+ 1 2 for the spin parameter j so that we have e −2πτ 2 p 2 k = e 2πτ 2 1 k (j− 1 2 ) 2 (with this identification relevant poles are located in the region j ≥ 0). The pole occurs in the 2nd term of (2.51) at
Expansion into Branching Functions
Since only the poles located on the interval between C 1 and C 2 can contribute, we must impose
Expansion into Branching Functions
Note that this range (2.58) coincides with the one derived in the bosonic model [27] (with respect to κ) and is strictly smaller than the unitarity bound (2.32) [39] for generic values of k. This bound also coincides with the one required by the analysis of reflection coefficients (two point functions on sphere) [5] , and also with that obtained from the no-ghost theorem for the parent SL(2; R) theory [42] (see also [43, 44, 45] ).
Expansion into Branching Functions
We also would like to emphasize that the restricted range (2.58) agrees exactly with the range of massless matter representations Ch
Expansion into Branching Functions
Liouville theory discussed in [18] . In fact under the correspondence of notations
Expansion into Branching Functions
which is exactly the range of massless representations closed under modular transformations.
Expansion into Branching Functions
Recalling the branching relation (2.25), the desired character expansion is obtained as
Expansion into Branching Functions
where m andm are defined as above (2.55). The factor a(j) is necessary to give a correct weight to the poles on the boundary, j = 1/2 and (k + 1)/2. 6 We choose the normalization constant C = k. 6 One may object to a fractional factor 1/2 in the weight of discrete representations in the partition function.
Expansion into Branching Functions
However, there exists a character identity
Expansion into Branching Functions
which enables us to write (2.61) in a form (set C = k)
Expansion into Branching Functions
We finally make a comment with respect to the modular invariance. The regularized partition function is written as
Expansion into Branching Functions
where we have indicated the dependence on IR cut-off ǫ explicitly. Strictly speaking this expression is not modular invariant because of the non-trivial p-dependence of the spectral density ρ(p, w, n; ǫ) (2.56), even though the original formula coming from the path-integral (2.38) appears modular invariant. In fact, the IR regularization would spoil the modular invariance, as we often face in general non-compact curved backgrounds. In order to recover invariance, the best one can do is to consider the partition function per unit volume;
Expansion into Branching Functions
The modular invariance of (2.65) follows from that of a free compact boson with a radius R = √ 2k. To summarize, the partition function is decomposed into two parts: (1) the continuous part Z con (τ ), and (2) the discrete part Z dis (τ ). The continuous part Z con (τ ) includes dominant contributions proportional to the volume factor, and correspond to the modes freely propagating in the bulk. Its precise definition depends on the regularization scheme.
Expansion into Branching Functions
On the other hand, the discrete part Z dis (τ ) only contains representations of (anti-)chiral primaries and their spectral flows within the range (2.58). They describe excitations localized around the tip of 2D black-hole that could be identified as the bound states [33] (see also [46] ). The absence of IR divergence in Z dis (τ ) is in accord with this expectation. The part of discrete representations is universal: insensitive to the choice of regularization scheme and stable under marginal deformations preserving N = 2 SUSY. We will make use of this piece to compute elliptic genus in the next section which captures the geometrical information of the singular space-time.
Expansion into Branching Functions
It seems that a strictly modular invariant partition function is obtained only after dividing by the infinite volume factor, which inevitably contains only the continuous representations. All the states appearing in this partition function (2.64) lie above the mass gap h ≥ 1/(4k), which corresponds to the decoupling of gravity in such a space-time.
Coupling to RCFT's
In this section we investigate the type II string vacua of the forms;
Coupling to RCFT's
where M is an arbitrary rational N = 2 SCFT withĉ =ĉ M . The criticality condition is
Coupling to RCFT's
and we assume d is even. We expect that the superconformal system M ⊗ SL(2; R)/U (1) describes a non-compact CY n with n = 5 − d/2. We assume a modular invariant of M-sector with conformal blocks F I as
Coupling to RCFT's
where z is the angle associated to the U(1)-charge of N = 2 SCA and σ denotes the spin structures as before. In this paper we use the conventions; 
Coupling to RCFT's
with some positive integer K 7 . Throughout this section we shall assume (3.6) with fixed positive integers N, K in the SL(2; R)/U(1)-sector. 7 Note that both of the pairs N 0 , M and N , K are not necessarily relatively prime. For example, in the case
Coupling to RCFT's
we find N 0 = N = 2n + 2, M = 2n, K = n + 2. Therefore, N 0 , M are not relatively prime for any n, and N , K are also not for even n.
Coupling to RCFT's
A typical example is the Gepner model [47] ;
Coupling to RCFT's
where M n is the level n N = 2 minimal model. The relevant conformal blocks are the products of minimal characters (A.6) 
Coupling to RCFT's
(3.10)
Modular Invariant Partition Functions per Unit Volume
We first consider the modular invariant partition function for the coupled system M ⊗ SL(2; R)/U(1) withĉ = n. We assume N and K are relatively prime for the time being.
Modular Invariant Partition Functions per Unit Volume
Let us recall the partition function of the SL(2; R) k /U(1)-sector defined by dividing by the volume factor (2.65). Now in the case of a rational level k = N/K, the partition function (2.65) can be rewritten in terms of the extended characters
Modular Invariant Partition Functions per Unit Volume
This function is identified with the extended massive character (B.6) introduced in [18]; 14) and the modular invariant has the same form as the level NK theta system given in [48] .
Modular Invariant Partition Functions per Unit Volume
Z N -periodicity under integral spectral flows is easy to see;
Modular Invariant Partition Functions per Unit Volume
(3.15) χ con (σ) (p, m; τ, z) for other spin structures are defined by the 1/2-spectral flows in the same way as (2.37). Now, the task we have to carry out is the chiral projection onto integral U(1)-charges as in the Gepner models [47] while taking account of the twisted sectors generated by integral spectral flows (see [49] ). Because of the periodicities (3.5), (3.15) , this is reduced to a Z N -orbifoldization. The desired conformal blocks are thus defined as the flow invariant orbits [49] 
Modular Invariant Partition Functions per Unit Volume
and recall the assumption (3.9). Then the factor 1/N in (3.21) is necessary to remove the N-fold overcounting of states.
Modular Invariant Partition Functions per Unit Volume
Incorporating the R d−1,1 -sector ( d 2 + n = 5), the supersymmetric conformal blocks are constructed as
Modular Invariant Partition Functions per Unit Volume
where θ [σ] again denotes θ 3 , θ 4 , θ 2 , iθ 1 for σ = NS, NS, R, R respectively, and we set ǫ(NS) = ǫ( R) = +1, ǫ( NS) = ǫ(R) = −1. The conformal blocks (3.24) actually vanish for arbitrary τ , z [50] , as is consistent with the space-time SUSY. It is not difficult to confirm that the conformal blocks (3.24) reproduce the results obtained in [7] for the special cases M = M n−2 . (Precisely speaking, in the d = 4 case we need some further orbifoldization in the SL(2; R)/U(1)-sector to reproduce the formula of [7] .)
Elliptic Genera
Let us next study the discrete spectrum of the theory which carries geometrical information of the target space geometry. As we have seen in the previous section, contributions of continuous representations dominate the partition functions and it is difficult to isolate the contributions of discrete states of the theory by inspecting the partition functions. We thus propose to study the elliptic genera from which continuous series decouple and one can clearly see the contents of discrete states in the theory.
Elliptic Genera
We first recall that the elliptic genera are defined by the partition functions in theR sector of the theory [34, 51] ,
Elliptic Genera
where F ≡ F L − F R denotes the world-sheet fermion number. When one sets z = 0 above, elliptic genus is reduced to the Witten index. It is well-known that, in any rational N = 2 SCFT the elliptic genus is a good supersymmetric index stable under arbitrary chiral marginal deformations. Furthermore, it possesses simple modular and spectral flow properties. It is identified as a (weak) Jacobi form [52] in mathematical terminology (see, e.g. [34, 53] ).
Elliptic Genera
As we shall see in the following, in the case of singular non-compact manifolds elliptic genera are no longer Jacobi forms and have some complicated modular properties: they are in general described by Appell functions which feature in the study of vector bundles of higher rank over elliptic curves [35, 36] .
Elliptic Genera
The evaluation of elliptic genus is almost parallel to the previous analysis of the modular invariant partition functions: we just replace the continuous extended characters χ con (σ) (p, m; τ, z) j,m (τ, z) are the branching functions for discrete series (2.28) and identified with the N = 2 massless matter characters. Ch 
Elliptic Genera
In the following let us assume that K and N are relatively prime for the sake of simplicity. Performing the Z N -orbifoldization, we can construct the conformal blocks in the same way as which describes the SL(2; R)/U(1)-part of elliptic genera in the prescription of Z N -orbifoldization. With the help of (B.10) we can further rewrite it as
Elliptic Genera
where K ℓ (τ, ν, µ) is the "level ℓ Appell function" [35, 36] defined by
Elliptic Genera
We present examples of concrete calculations: We only have the boundary terms of the range (2.58). The following identity is presented in [54] and quite useful;
Elliptic Genera
where Ch
Elliptic Genera
is the characters ofĉ = 3 extended chiral algebra [54, 49] (we use the notation given in Appendix C of [18] ). Applying the 1/2-spectral flow z → z + τ 2 + 1 2 to both sides of (3.37), we obtain the identity (see also [36] )
Elliptic Genera
To derive the last line we used the Watson's quintuple product identity (A.7) (see e.g. [55] .) In this way we obtain the simple formula
Elliptic Genera
It may be worthwhile to point out the following fact: The elliptic genus for the level n − 2 N = 2 minimal model was calculated in [51] by using the free field method in N = 2 Landau-Ginzburg theory with superpotential W (X) = X n . The result is expressed as
Elliptic Genera
, (3.40) where ch
Elliptic Genera
is the character of minimal model M n−2 (A.6). It is curious to see that the "analytic continuation" of this formula to the inverse power potential W (X) = X −1 (n = −1), which is often used to describe the conformal system of conifold [1] , correctly reproduces our result (3.39) (up to normalization).
Elliptic Genera
This is the conformal system first analyzed in [2] and is considered to describe the ALE space obtained by deforming the A n−1 -type singularity (in the case of the diagonal modular 43) where ch N =4 ( R) 0 (ℓ; τ, z) denotes the level 1 N = 4 massless character of spin ℓ(= 0, 1/2) (related with ch N =4 (NS) 0
Elliptic Genera
(1/2 − ℓ; τ, z) by the spectral flow) [56] . Using the formulas (3.34) and (3.40) , we can also rewrite (3.41) in the form that makes the orbifold structure manifest as in [53] ;
Elliptic Genera
Here we used the relation θ 1 (τ, z + aτ + b) = (−1) a+b q − a 2 2 e −i2πaz θ 1 (τ, z). Note that the theta function term in (3.34) is eliminated by the Z N -orbifoldization.
Elliptic Genera
One can also perform similar calculations in the cases of CY 4 -singularity of A n−1 -type (N = n, K = n + 1). We obtain
Elliptic Genera
Example 3. General cases of non-compact CY 3 : Lastly we consider the general models of n = 3, where M is an arbitrary N = 2 RCFT withĉ M < 2. We make a natural assumption of the "charge conjugation symmetry" in the M-sector. Namely, we postulate Rather surprisingly we can show that the elliptic genera for these models have a simple and universal form;
Elliptic Genera
where χ ≡ lim z → 0 Z(τ, z) is the Witten index that counts the Ramond ground states in the total system. We sketch how one can derive this formula. Thanks to the above assumption (3.46) , the elliptic genus is found to have only contributions of the symmetrized forms G 
Elliptic Genera
(3.49) (We again use the notations given in Appendix C of [18] .) Note that the graviton representation (h = Q = 0) cannot occur owing to the constraint K ≤ s ≤ N + K (or (2.58)). Consequently, recalling the identity (3.37), (3.38), one can find
Elliptic Genera
The massive part is generically an infinite sum of the terms of the forms q * Θ 0 (1),1 (τ, 2z)iθ 1 (τ, z)/η(τ ) 3
Elliptic Genera
and hence odd functions of z. It does not contribute because of the above remark. In this way we arrive at the general formula (3.48).
Elliptic Genera
It has been shown in [53] that the elliptic genera for arbitrary Gepner models (or the LGorbifolds) of compact CY 3 can be written in the form
51)
where h 1,2 , h 1,1 are the numbers of (c, c), (c, a)-type chiral primaries of h =h = 1/2 respectively, which should be identified with the Hodge numbers of CY 3 . Our result (3.48) is the generalization of this formula to the non-compact models. Note that χ = 2(h 1,2 − h 1,1 ) is an even number for any compact CY 3 , while χ in (3.48) is allowed to be odd. Recall the conifold case, Example
1.
A few remarks are in order:
1.
1. The elliptic genus includes contributions of Ramond ground states that are naively supposed to describe massless excitations in string theory. However, to identify them with the massless spectrum we must take account of the GSO condition for spin structures. The simplest way to do so is to look for the corresponding NS (anti-)chiral states with h = 1 2 |Q| = 1 2 . For example, in the ALE(A n−1 ) case (Example 2), we have n − 1 Ramond ground states and all of them correspond to massless string states. They are identified as each of normalizable deformations of A n−1 -singularity ("moduli"). However, even though the A n−1 -model of the Calabi-Yau 4-fold n = 4 (N = n, K = n + 1) still has the Witten index Z(τ, 0) = n − 1, they cannot define NS massless courtneykent@example.com. This feature has its origin in the restriction K ≤ s ≤ N + K. We thus find the inequality
1.
for the chiral primary states in the SL(2; R)/U(1)-sector, implying no massless states appear in the superstring spectrum. These missing massless states would be associated to the nonnormalizable deformations of A n−1 -singularity ("coupling constants"). In the Calabi-Yau 3-fold n = 3 case the aspect of massless states is more complex: the half of them appears as massless states in the closed string spectrum and the remaining ones are missing. These aspects of marginal fields in singular CY n have been discussed in [57, 58, 4, 6] .
2.
As we already mentioned, the elliptic genera of N = 2 RCFT's for compact CY n are known to be the (weak) Jacobi form with weight 0 and index n/2. This means that the elliptic genera have the following properties;
2.
It is easy to confirm that our elliptic genera for non-compact models (3.31) satisfy (3.53) and (3.54) . However, the third condition for the S-transformation is not necessarily obeyed. For the n = 3 cases, (3.55) is also satisfied because of the general formula (3.48) . However, in the n = 2, 4 cases, the elliptic genera are proportional to the Appell function which transforms as [36] 
2.
The above transformation law corresponds to the mixing of discrete and continuous representations in N = 2 Liouville theory [56, 54, 59, 18] (see Appendix B). Appell functions (3.35) are interpreted as sections of higher rank vector bundles over elliptic curves as opposed to thetafunctions which are sections of line bundles [35] . It is interesting to see if one can achieve a precise geometrical interpretation of the elliptic genera in the non-compact space-time.
Summary
In this paper we have confirmed the correspondence between SL(2; R) k /U(1) supercoset and N = 2 Liouville theory and computed the elliptic genera for various singular space-times. We summarize the main ingredients of this paper.
Summary
1. Partition functions of SL(2; R) k /U(1) theories are decomposed into two pieces: (1) the part consisting of continuous (massive) representations and (2) the part consisting of discrete (massless matter) representations.
Summary
2. The continuous part is proportional to the volume factor, since it describes the propagating modes in the bulk, and gives the leading contribution to the partition function. It seems that strictly modular invariant partition functions are obtained after the division by the infinite volume factor. Then they contain only the continuous representations (massive modes), and reproduce the results obtained previously in [7] .
Summary
3. The discrete part describes excitations localized around the tip of cigar [33] and thus appears without the volume factor. Embedded in superstring vacua it could be interpreted as contributions from massless matter fields corresponding to the deformations of Calabi-Yau singularities.
Summary
4. Continuous representations do not contribute to the elliptic genera and thus elliptic genus clearly exhibits the contributions of the discrete states. In generic cases (ofĉ tot = 2, 4) the elliptic genera possess complex modular behaviors, and they are not Jacobi forms (section of line bundles) but sections of higher rank vector bundles. On the other hand, in models withĉ tot = 3 the elliptic genera behave in the same way as rational conformal theories.
5.
When embedded in superstring vacua by means of the Gepner-like method, the extended characters defined in [18] emerges quite naturally in continuous and discrete series (see (3.12) , (3.26) ). We have also confirmed that discrete representations in N = 2 Liouville theory closed under modular transformations are mapped to unitary discrete representations in the range 1/2 ≤ j ≤ (k + 1)/2 which appear in the regularized partition function of SL(2, R) k /U(1) theory. This justifies our Ansatz for the basis of Ishibashi states of N = 2 Liouville theory we have proposed in [18] .
5.
6. It appears quite likely that N = 2 Liouville and SL(2, R) k /U(1) theories are in fact exactly mapped into each other (T-duality) and will essentially be one and the same theory with identical physical contents. This is gratifying since in the Liouville approach it has been extremely difficult to incorporate the effects of the Liouville potential terms nonperturbatively into the theory. We note that in the SL(2, R) k /U(1) supercoset theory, on the other hand, the space-time is curved into a 2D black hole but the (cosmological constant) parameter µ does not appear explicitly. Thus it seems that SL(2, R) k /U(1) theory has deformed the space-time by absorbing the Liouville potential terms. Agreement of our
5.
Liouville results [18] with those of SL(2, R) k /U(1) theory is encouraging and indicates that we have incorporated properly the effects of Liouville potential terms in the analysis.
Theta functions
We here summarize our notations of theta functions. We set q ≡ e 2πiτ and y ≡ e 2πiz ,
Character Formulas for the N = 2 Minimal Models
The easiest way to represent the character formulas of the level k N = 2 minimal model (ĉ = k/(k + 2)) is to use its realization as the coset SU(2) k × SO(2) 1 U(1) k+2 . We then have the following branching relation;
Character Formulas for the N = 2 Minimal Models
The branching function χ ℓ,s m (τ, z) is explicitly calculated as follows;
Character Formulas for the N = 2 Minimal Models
ℓ,m−s+4r (τ )Θ 2m+(k+2)(−s+4r),2k(k+2) (τ, z/(k + 2)) .
Character Formulas for the N = 2 Minimal Models
Then, the desired character formulas are written as
Character Formulas for the N = 2 Minimal Models
By definition, we restrict to ℓ + m ∈ 2Z in NS and NS sectors, and to ℓ + m ∈ 2Z + 1 in R and R sectors.
Useful identity
The following identity is often useful in checking the modular invariance
Useful identity
We also note that the combination |u| 2 /τ 2 is modular invariant.
Useful identity
Appendix B N = 2 (Extended) Character Formulas for c > 1
Useful identity
We denote the conformal weight and U(1)-charge of the highest weight state as h, Q and again set q ≡ e 2πiτ , y ≡ e 2πiz . The irreducible characters of N = 2 SCA withĉ > 1 are summarized as follows [60] ;
Useful identity
(1) massive representations :
Useful identity
(2) massless matter representations :
Useful identity
They correspond to the (anti-)chiral primary state with h = |Q|/2, (0 < |Q| <ĉ).
Useful identity
(3) graviton representation :
Useful identity
They correspond to the vacuum h = Q = 0, which is the unique state being both chiral and anti-chiral primary.
Useful identity
More general unitary representations are generated by the integral spectral flows and classified in [41] . The spectral flow generator U η with a real parameter η is defined by
Useful identity
Half-integral spectral flows η ∈ where ch (NS) * ( * ; τ, z) is the abbreviated notation of (B.1)-(B.3). For the theory ofĉ = 1 + 2K/N (N, K ∈ Z >0 ), we introduce the "extended characters" [18] which should be the characters of unitary representations of the extended chiral algebra defined by adding the spectral flow generators U ±N [56, 54, 49, 50] .
Useful identity
where the ranges of parameters r, α, s are given as
Useful identity
In the calculations of elliptic genera, we use the following formula (r ∈ For convenience of readers we also present the modular transformation formulas given in [18] in terms of the new notations (3.12), (3.26) . They are written as
Useful identity
(B.13)
Appendix C Explicit Calculation of Partition Function
In this Appendix we present the explicit derivation of (2.38) by the path-integration. This is almost parallel to the analysis of the bosonic models given in [30, 31] . We define the worldsheet torus by the identifications (w,w) ∼ (w + 2π,w + 2π) ∼ (w + 2πτ,w + 2πτ ) (τ ≡ τ 1 + iτ 2 , τ 2 > 0, and use the convention z = e iw ,z = e −iw ). We call the cycles defined by these two identifications as the α and β-cycles as usual.
Appendix C Explicit Calculation of Partition Function
The desired partition function is written as
Appendix C Explicit Calculation of Partition Function
Although the Euclidean 2D BH is manifestly positive definite (since the time-like U(1) is gauged away), the calculation of partition function could be subtle due to the Lorentzian signature in the parent SL(2; R) theory. Therefore, it is better to start with the Wick rotated model H + 3 /R, where H + 3 ∼ = SL(2; C)/SU(2) is the Euclidean AdS 3 . The Wick rotation is defined by the replacement; g ∈ SL(2; R) → g ∈ SL(2; C)/SU(2) and the gauge field A ≡ (Azdz + A z dz) σ 2 2 To calculate the partition function (C.1) it is convenient to reexpress the gauged WZW action (2.4) in the form
Appendix C Explicit Calculation of Partition Function
where we used the abbreviated notation S WZW (g) ≡ S SL(2;R) WZW (g). After the Wick rotation we must suppose h L = h † R (≡ h) ∈ exp (Cσ 2 ). We also introduce the vector-like gauged WZW action Here b, c (b,c) are the spin (1,0) ghost system to rewrite the Jacobian of path integral measure. The level shift κ → κ − 2(≡ k) for the action S (A) (e iY σ 2 , h u , h u † −1 ) in (C.9) is owing to the chiral anomaly of the fermion determinant, regularized so that it is anomaly free along the axial direction.
Appendix C Explicit Calculation of Partition Function
We next evaluate each sector separately:
Appendix C Explicit Calculation of Partition Function
• H + 3 -sector : This non-trivial sector has been already evaluated in [30, 31] (see the comment below); Again this is manifestly modular invariant.
Appendix C Explicit Calculation of Partition Function
• fermion and ghost sectors :
Appendix C Explicit Calculation of Partition Function
The remaining fermionic sectors are easy to evaluate. They are the standard fermion determinants with anti-periodic and periodic boundary conditions respectively (for the included in (C.14) is the correct anomaly factor to assure the modular invariance.
Appendix C Explicit Calculation of Partition Function
Gathering all the contributions (C.10), (C.13), (C.14) and (C.15), we finally obtain the desired partition function (2.38).
Appendix C Explicit Calculation of Partition Function
We further make a comment on the several path-integral formulas for the H + 3 (gauged) WZW models presented in [30, 31] ; 

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