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Gepner-like Description of a String Theory on a Noncompact Singular Calabi-Yau Manifold

We investigate a Gepner-like superstring model described by a combination of multiple minimal models and an N=2 Liouville theory. This model is thought to be equivalent to the superstring theory on a singular noncompact Calabi-Yau manifold. We construct the modular invariant partition function of this model, and confirm the validity of an appropriate GSO projection. We also calculate the elliptic genus and Witten index of the model. We find that the elliptic genus factorises into a rather trivial factor and a non-trivial one, and the non-trivial one has the information on the positively curved base manifold of the cone.


Introduction
The correspondence between a string theory on a Kähler manifold and an N = 2 Landau-Ginzburg theory is interesting and is very largely investigated [1,2,3,4]. But, the most results are limited to the cases of compact Calabi-Yau manifolds. Recently, it is conjectured that in the case of a noncompact Calabi-Yau manifold, the associated CFT consists of an N = 2 Liouville theory and a Landau-Ginzburg theory [5]. They claimed that when the Calabi-Yau n-fold X is written as a hypersurface F (z 1 , . . . , z n+1 ) = 0 in C n+1 by a quasi-homogeneous polynomial F , then the string theory on X is equivalent to the CFT on Here R φ is a real line parametrized by φ with linear dilaton background and LG(W = F ) is the IR theory of the Landau-Ginzburg model with the superpotential F . In this case, the background charge Q of R φ is determined by a condition of the total central charge. From the condition that Q is a non-zero real number, we find that the base manifold X/C × should be curved positively.
The boson with linear dilaton background is strongly coupled in the region φ → −∞, so we should introduce the Liouville potential or consider an SL(2)/U(1) Kazama-Suzuki model to avoid the strong coupling singularity [6,7,8]. But we do not care about this point in this paper.
In [9,5,7,8], they also claim that the string theory on this singular noncompact Calabi-Yau manifold X is holographic dual to the "little string theory".
In the case of a compact Calabi-Yau manifold, the string theory is "solved" in the description of Gepner model in a special point of the moduli space. We want to describe also the string theory on the noncompact Calabi-Yau manifold X by the Gepner-like solvable model. If we can do it successfully, it will be possible to analyze more deeply a noncompact Calabi-Yau manifold and the little string theory.
In [10], they treat the string theories with ADE simple singularities. They construct the modular invariant partition functions, and show the consistency of these string theories.
In this paper, we consider more general cases, in which the Landau-Ginzburg part is described by a direct product of a number of minimal models. A typical example of ours is the Calabi-Yau n-fold X described in the form We construct the modular invariant partition functions and show the string theory actually exists consistently in these cases. We also calculate the elliptic genus, and find that it is factorised into two parts -a rather trivial one and a rather non-trivial one. We analyze the non-trivial one in detail, and find that it has the information on the cohomology of the positively curved base manifold X/C × except the elements generated by cup products of a Kähler form. The organization of this paper is as follows. In the next section, we explain the setup and review the correspondence between a noncompact Calabi-Yau manifold and an N = 2 Liouville theory × Landau-Ginzburg theory. In section 3, we construct the modular invariant partition function. In section 4, we calculate the elliptic genus and compare it with the geometric property of the associated Calabi-Yau manifold X. In the last section, we summarize the results and discuss the problems and prospects. In Appendix A. we collect some useful equations of theta functions and characters that we use in this paper.

The string theory on a noncompact singular Calabi-Yau manifold
We consider the string compactification to a noncompact, singular Calabi-Yau n-fold X.
The total target space is expressed by a direct product of a d dimensional flat spacetime and the manifold X Here, n is related to d by the constraint on the total dimension 2n + d = 10. For simplicity, we concentrate the case that the noncompact singular Calabi-Yau manifold X is realized as the hypersurface in C n+1 determined by the algebraic equation with a quasi homogeneous polynomial F F (z 1 , . . . , z n+1 ) = 0.
It is conjectured in [5] that the string theory on the space (2.1) is equivalent to the theory including flat spacetime R d−1,1 , a line with the linear dilation background R φ , S 1 , and the Landau-Ginzburg theory with a superpotential W = F ; The part (R φ × S 1 ) has a world sheet N = 2 superconformal symmetry. Let φ be the parameter of R φ , Y be the parameter of S 1 , and ψ + , ψ − be the fermionic part of (R φ × S 1 ). The N = 2 superconformal currents are written in terms of the above fields The associated central charge of this algebra isĉ(= c/3) = 1 + Q 2 .
In this paper, we consider the case in which the Landau-Ginzburg theory with superpotential W = F can be described by a direct product of N = 2 minimal models. Let M G,N be the minimal model corresponding to simply laced Lie algebra G = A, D, E with dual Coxeter number N. We consider the theory in the following; where R is the number of the minimal models. The cases with R = 1 are treated in [10] and R = 0 in [11,12] A typical example is the case that all G j are A type. In this example, the quasihomogeneous polynomial is written as The background charge of R φ is determined by the criticality condition. To cancel the conformal anomaly, the total central charge is to be 0. The central charge of the ghost sector is −15, so the total central charge of the matter sector is to be 15. The central charge of the flat spacetime is 3/2 for each pair of a boson and a fermion, and that of the R φ × S 1 is 3 + 3Q 2 as mentioned above, and that of a minimal model M G,N is 3(N −2) N . Therefore, the criticality condition leads to the equation From this criticality condition, we obtain the value of Q 2 as By the condition Q 2 > 0 for a real number Q, the right-hand side should be positive; It is equivalent to a condition that the singularity is in finite distance in the moduli space of deformation of singular Calabi-Yau manifold X [5,13]. In view of the base manifold X/C × , the finite distance condition is equivalent to that X/C × is positively curved.

Modular invariant partition function
Now, let us construct the modular invariant partition function. We take the light-cone gauge, then the associated CFT to consider is The toroidal partition function can be separated into 2 parts: the one Z GSO concerning to the GSO projection and the other Z 0 not concerning to it. We construct the total partition function Z as where τ = τ 1 + iτ 2 is the moduli parameter of the torus, and d 2 τ /τ 2 2 is the modular invariant measure.
First, we study the rather easy part Z 0 , then we investigate the rather complicated part Z GSO .

GSO independent part of the partition function
In this subsection, we discuss the Z 0 : the GSO independent part. It is completely the same as that in [10].
The Z 0 includes the contribution from the flat spacetime bosonic coordinates X I , (I = 2, . . . , d − 1) and the linear dilation φ.
The partition function of each flat spacetime boson is represented by the Dedekind eta function η(τ ) as The partition function of φ is defined as Here the trace is taken over delta function normalizable primary fields and their excitations by oscillators. Then we obtain Z L as where the region of the integral of p is as (3.1). As a result, the partition function of φ is the same as that of an ordinary boson. So we obtain Z 0 as the partition function of effectively (d − 1) free bosons; The primary fields of (3.1) correspond to "Principal continuous series" in terms of the representation of SL (2). To include the other sectors is an interesting problem and is postponed as a future work.

GSO dependent part of the partition function : d = 2, 6 cases
Now, let us proceed to the GSO dependent part Z GSO . In this subsection, we treat d = 2, 6 cases.
Let us turn to Verma modules of N = 2 minimal models. The Verma module of an N = 2 minimal model is specified with three indices (ℓ, m, s), which satisfy the following conditions [2] ℓ = 0, . . . , N − 2, We denote χ ℓ,s m (τ, z) as the character of the Verma module labeled by the set (ℓ, m, s). Some properties of this character is collected in Appendix A.
The Verma module of the whole GSO dependent parts is specified by the index s 0 of SO(d) 1 representation, the indices (ℓ j , m j , s j ), (j = 1, . . . , R) of the minimal models, and the S 1 momentum p. We combine these indices except p into two vectors λ, µ.
We shall introduce the inner product between µ and µ ′ as in [2] Also it is convenient to introduce special vectors β 0 , β j (j = 1, . . . , R) β 0 := (1; 1, . . . , 1; 1, . . . , 1), Here the β 0 is the vector with all components 1, and β j is the vector with s 0 and s j components 2 and the others zero. With these notations, the criticality condition (2.4) can be written in a rather simple form as When we define an integer K := lcm(2, N j ), KQ 2 is shown to be an even integer because of the equations (2.4) and (3.3). Therefore, it is convenient to define an integer J by the equation In terms of J, the finite distance condition (2.5) can be expressed as J > 0. Now, let us consider the character of the Verma module (λ, µ, p), where χ λ µ (τ ) is the product of characters of the minimal models and the SO(d) 1 character χ s 0 (τ ) of the s 0 representation In this character, χ λ µ (τ ) has good modular properties, but q 1 2 p 2 /η(τ ) has bad ones. So, we will sum up the characters with respect to certain values of p and make the modular properties good [11] [10].
Let us consider the GSO projection. By the GSO projection, we pick up the states with odd integral U(1) charges of the N = 2 superconformal symmetry. The U(1) charge of the states in the above Verma module is expressed as From the condition that this U(1) charge must be an odd integer (2u + 1) with u ∈ Z, the S 1 momentum p is written as If we sum up the characters for all u ∈ Z, we obtain the theta function with a fractional level [10], which does not have good modular properties. So we perform the following trick. Let us write u = Jv + w with integers v, w and sum up the characters for v ∈ Z. Then the sum leads to the following theta function Note that −2Kβ 0 • µ + K(2w + 1) is an integer, and the above theta function has good modular properties. Now, including oscillator modes and other sectors, we can define the building blocks We should use only the building blocks f λ µ with the conditions The condition (3.6) comes from the formula of (3.5), and the condition (3.7) implies that the boundary condition of the fermionic currents are the same in all the sub-theories, i.e. they must be all in the NS sector, or all in the R sector. The modular invariant partition function can be systematically obtained by "the beta method" [2].
The inner product between of two vectors µ , µ ′ is defined as We also extend the vectors β 0 , β j to β 0 , β j as and evaluate the inner products of these β 0 , β j vectors Note that β 0 • β 0 is an odd integer, β j • β j are even integers (Recall that we consider the cases d = 2, 6), and β j • β 0 are integers. Using these special vectors, the conditions (3.6) and (3.7) are written in a simple form We call this condition "the beta condition". Using these notations, and the modular transformation laws of theta functions and N = 2 characters written in the Appendix A, we can calculate the modular transformation laws of f λ µ as where the sums of the λ ′ , µ ′ are taken only for the range (3.2) and for M = 0, . . . , 2KJ −1.
Especially we must impose the condition ℓ j + m j + s j ≡ 0 mod 2 for each minimal model.
and we use here and the rest of this paper the notation e[x] = exp(2πix). Let us note that if a vector µ satisfies the beta condition (3.9), the vector µ + b 0 β 0 + j b j β j for b 0 , b j ∈ Z, (j = 1, . . . , R) also satisfies the beta condition by virtue of (3.8). Using this fact, we define the function F λ µ for (λ, µ ) which satisfies the beta conditions (3.9) as a sum of f λ µ +b 0 β 0 + j b j β j 's as where the sum is taken for b 0 ∈ Z 2K and b j ∈ Z 2 . The sign (−1) s 0 +b 0 is (−1) for the Ramond sector. These functions have very good modular properties. Especially by S transformation, the functions are mixed among those which satisfy the beta condition; where the sums of λ ′ , µ ′ is taken for restricted subclass that satisfies the conditions (3.2) and the beta condition (3.9). With this function F λ µ , we obtain the modular invariant Z GSO as [14,15,16]. We can check the modular invariance of the above partition function. We expect from spacetime supersymmetry that the partition function vanishes, or equivalently F λ µ (τ ) = 0. It is a future work to check that it actually vanishes. Here, we find a solution, but it may not be the only solution, and there can be some variety of modular invariant partition function. Actually, for R = 1 case, there are many other solutions associated with the other modular invariants of the theta system [10].

GSO dependent part of the partition function : d = 4 case
In this subsection, we comment on the d = 4 case. To construct the modular invariant partition function in the d = 4 case, we combine the four fermions to construct the affine currents SO(2) 1 × SO(2) 1 and label the the Verma module by indices s −1 and s 0 . Then, the modular invariant partition function can be constructed in almost the same way as the d = 2, 6 cases.
First we define the vectors µ 's and the inner product between them as µ := (s −1 , s 0 ; s 1 , . . . , s R ; m 1 , . . . , m R ; M), It is convenient to introduce special vectors β 0 , β j and β −1 Using these vectors, we can construct the building blocks f λ µ (τ ) as where χ s −1 (τ ) and χ s 0 (τ ) are SO(2) 1 characters. Then the GSO conditions and the condition of fermionic sectors are and we can construct the modular invariant partition function by the beta method in this case. Next we introduce the function F λ µ (τ ) as then we obtain the GSO dependent part of the modular invariant partition function Z GSO We can check the modular invariance of the above partition function.

Elliptic genus
In this section, we calculate the elliptic genus of the theory [17]. The definition of the elliptic genus is where the trace is taken for the RR states, and y = exp(2πiz). This elliptic genus has the following modular properties; Z(τ + 1,τ + 1, z) = Z(τ,τ , −z) = Z(τ,τ , z), Here, we omit the contribution from the flat space time and consider only the internal part describing the Calabi-Yau n-fold X. We calculate its elliptic genus and the Witten index, and discuss its geometrical interpretation.
Let us consider again "the criticality condition", in other words "the Calabi-Yau condition" of X. Here the totalĉ should be n because we want the theory that describes a Calabi-Yau n-fold. Therefore, the totalĉ of the N = 2 Liouville and the minimal models should satisfy the relations We introduce the following vectors with R components {m j }, and the inner product between them as We also introduce the special vector γ 0 with all components 2 γ 0 := (2, . . . , 2).
With these notations, the condition (4.2) becomes Next we let N := lcm(N j ), and define J as In this paper, we concentrate only the case that (n − 1 − R) is even, then in this case, J is an integer. In terms of J, the finite distance condition Q 2 > 0 can be written as J > 0. Because we want a Calabi-Yau CFT, we have to pick up only the states with integral U(1) charges. This condition is realized as the condition (4.4) From this, p can be written with an arbitrary integer u Following the same manner as in the previous section, we let u = 2Jv + w and sum up for v ∈ Z. It leads to the theta function Note that N(w − γ 0 • ν) is an integer and Θ N (w−γ 0 •ν),N J (τ, 2z/N) has good modular properties.
Collecting these, we define the building blocks g λ ν as where ν := (ν, M). In the sign (−1) 2 ) represents contributions of ordinary U(1) charges from the indices s 0 , s j , and the rest γ 0 • ν + M N = w = u − 2Jv reflects contributions from the indices m j and S 1 momentum. Let us define the inner product between ν and ν ′ as and the special vector We also introduce the functions I ℓ m and I λ . With these notations, the building blocks g λ ν can be written as where we omit the overall irrelevant phase. The condition (4.4) can be rewritten as and again we call this condition "the beta condition". Now, we construct elliptic genus using the above building blocks g λ ν which satisfy the condition (4.5).
Note that if ν satisfies the beta condition, then ν + b 0 γ 0 for b 0 ∈ Z also satisfies the beta condition, because is an integer. 1 Here we used the definition of J (4.3). So let us define the new functions G ℓ ν as follows; where ν satisfies the beta condition (4.5). Then, from the modular properties of g λ ν g λ ν (τ + 1, z) = e j ℓ j (ℓ j + 1) G ℓ ν have very good modular properties; Using these functions, we obtain the elliptic genus in the following form; Here L λλ is the product of SU(2) modular invariants, and the factor 1/ √ τ 2 |η(τ )| 2 is contribution of φ. We can check that the above elliptic genus has the right modular properties (4.1) withĉ = n. Actually, this elliptic genus is 0 because it has an overall factorθ 1 (τ , 0) = 0.

Hodge number and Witten index
To get some nontrivial information from the above elliptic genus, we factor out the trivial parts and defineẐ by the equations , Now, we take the limit τ → i∞ and consider the ground states. In this limit, Θ M,N J becomes Θ M,N J (i∞, z) = δ mod NJ M , so, theĜ's can be evaluated aŝ Then,Ẑ is expressed in the formula When we replace the ν + M 2J γ 0 by ν, then we can perform the sum of M ∈ Z 2N J . Moreover, from the fact it can be seen that the even condition ℓ j + m j ≡ 1 mod 2 is included in this factor. We obtain a formula of theẐ in this limit So far, we treat rather general cases, but from now, we take an example and restrict ourselves to calculations in the example. We consider the example which satisfies all the following conditions.
In other words, this example is the case where the associated Calabi-Yau manifold X is the hypersurface of the form We can write the finite distance condition as N < n + 1, which is equivalent to the condition that first Chern number of X/C × is positive. In this case, nontrivial factor of the elliptic genus can be calculated as  Table 1: The values of the coefficients h p for n = 3, 4, 5, N = 3, . . . , n + 1. We include the N = n + 1 case in the table, despite it is suppressed by the finite distance condition.
We obtain theẐ in this limit We show several examples of h p for lower n, N in Table 1. These coefficients h p seem to coincide with the middle dimensional Hodge numbers of X/C × except for the cohomology elements generated by cup products of a Kähler form of X/C × , on which we mention below. In our model, the Witten index can also be calculated as On the other hand, the Euler number of the (n − 1)-dimensional manifold X/C × is expressed in the formula The Witten index of the CFT almost coincide with the Euler number χ X/C × of X/C × . One of the differences of the two is the sign (−1) n+1 , but this is not relevant. Except this difference of the sign, the Witten index is smaller by n than the Euler number of X/C × in our case. This difference may correspond to the cohomology elements generated by cup products of the Kähler form of X/C × . On X, these cohomology elements are absent because they appear when we take the quotient of X by C × . This seems the reason why the Witten index is smaller by n than the Euler number of X/C × .

Conclusion and discussion
We construct the toroidal partition function of the string theory described by the combination of an N = 2 Liouville theory and multiple N = 2 minimal models. This partition function is actually modular invariant, and we can conclude that the theory exists consistently. This string theory is thought to describe the string on a noncompact singular Calabi-Yau manifold. To check this proposition, we also calculate the elliptic genus of this theory and the Witten index.
The Euler number defined from non-trivial factor of the Witten index in the CFT seems to be that of the non-vanishing elements of the cohomology. In the case of a singular manifold, there are vanishing elements of the cohomology, which are supported on the singular point and reflect the structure of singularities.
The fact that the vanishing elements of the cohomology cannot be seen, is probably related to our method of construction in which we include only the states in "principal continuous series" of the SL(2) theory. If we can include some "discrete series" (but it is difficult [18]) , the structure of the singularities might be seen in the CFT.
Another reason is that we treat the N = 2 Liouville theory as free field theory in this paper. It is mentioned in [19] that if we treat appropriately the effect of Liouville potential, the Witten index does not vanish and gives the Euler number including the vanishing elements of the cohomology. In this paper, since we treat the case of µ = 0 and not deformed singularity, the vanishing elements of the cohomology actually vanish and it is consistent with the vanishing Witten index.
We may not be able to use our result to analyze the structure of the singularities, but we can use it to analyze the string on the positively curved manifold X/C × . Especially it is interesting to analyze the D-branes wrapped on infinite cycle in this noncompact Calabi-Yau manifold through the recipes of boundary states in the CFT [20,21,22,19] as the case of the ordinary Gepner models [23,24,25,26].
This work is supported in part by JSPS Research Fellowships for Young Scientists.
Appendix A.

Theta functions and characters
We use the following notations in this paper. (1 − q n ).