`Analytic Continuation' of N=2 Minimal Model

In this paper we discuss what theory should be identified as the `analytic continuation' with $N \rightarrow -N$ of the ${\cal N}=2$ minimal model with the central charge $\hat{c} = 1 - \frac{2}{N}$. We clarify how the elliptic genus of the expected model is written in terms of {\em holomorphic\/} linear combinations of the `modular completions' introduced in [arXiv:1012.5721 [hep-th]] in the $SL(2)_{N+2}/U(1)$-supercoset theory. We further discuss how this model could be interpreted as a kind of {\em `compactified'} model of the $SL(2)_{N+2}/U(1)$-supercoset in the $(\tilde{R},\tilde{R})$-sector, in which only the discrete spectrum appears in the torus partition function and the potential IR-divergence due to the non-compactness of target space is removed. We also briefly argue on possible definitions of the sectors with other spin structures.


Introduction
The N = 2 minimal model is one of the most familiar rational superconformal field theories in two dimensions [1][2][3][4][5][6]. It is defined by the supercoset theory of SU (2) k /U (1) with level k = N − 2 [7] and also described as the IR fixed point of the N = 2 Landau-Ginzburg (LG) model with the superpotential W (X ) = X N + [lower powers] [8][9][10]. One of the good features of N = 2 minimal models is a very simple formula for elliptic genus [11], which behaves nicely under modular transformations as well as spectral flows. Namely, the function Z (N −2) min (τ, N z) is a weak Jacobi form [12] of weight 0 and index N (N −2) 2 . An old question is what theory should be identified as the "analytic continuation" of the N = 2 minimal model under N → −N ? A naive guess of the answer would be the SL(2) N +2 /U (1) supercoset theory that has the expected central chargeĉ ≡ c 3 = 1 + 2 N . However, this is not completely correct, for the following reasons: • The SL(2)/U (1) supercoset theory contains both discrete and continuous spectra of primary fields, while the N = 2 minimal model only has discrete spectra. It is not likely to be the case that two such theories are directly connected by an analytic continuation of the parameter of theory. • It has been shown that the elliptic genera of the SL(2)/U (1) supercoset theories show nonholomorphicity with respect to the modulus τ of the world-sheet torus [13,14], whereas the elliptic genera of the minimal model (1.1) are manifestly holomorphic.

Holomorphic linear combinations of modular completions
In this section we address some mathematical results. The main claims will be expressed in (2.24) and (2.31).

Preliminary
We first introduce the relevant notations. To begin with, we introduce the symbol of "IR part" just for convenience: where f (τ, z) is assumed to be holomorphic around the cusp τ = i∞.

A holomorphic Jacobi form.
We consider a holomorphic function defined by (N ) (τ, z) := θ 1 τ, N +1 N z θ 1 τ, 1 N z . (2.2) This is obtained by the formal replacement N → −N in the elliptic genus of the N = 2 minimal model (1.1). The function (2.2) possesses the next modular and spectral flow properties witĥ c ≡ 1 + 2 N : We next introduce the "spectral flow operator" s (a,b) (a, b ∈ Z) defined by s (a,b) · f (τ, z) := (−1) a+b qˆc 2 a 2 yĉ a e 2π i ab N f (τ, z + aτ + b), (2.7) and set Since having a periodicity , we may assume a, b ∈ Z N . Its modular property is written as The IR part of and also (for a = 0), which proves (2.10).

Modular completions.
Let us introduce the "modular completion" of the extended discrete characters (of the R sector) [15][16][17] in the SL(2)/U (1) supercoset according to [14]. For the casê c = 1 + 2 N , (∀ N ∈ Z >0 ), this function is defined as where χ (N ,1) dis denotes the extended discrete character introduced in [15][16][17] (written in the convention of [14]): The first line of (2.11) naturally appears through the analysis of the partition function of the SL(2)/U (1) supercoset [14], and the second line just comes from the contour deformation. The modular completion of the Appell function (or the "Appell-Lerch" sum) K (2N ) (τ, z) [18,19] is given as [20]: 1 where we set which is generically non-holomorphic due to explicit τ 2 (≡ Im τ ) dependence. The next "Fourier expansion relation" [14] will be useful for our analysis: which is the "hatted" version of the similar relation between K (2k) and χ dis (v, a) given in [17]. The modular transformation formulas for the modular completions (2.11) and (2.13) are written as [14,20] The spectral flow property is summarized as follows: More detailed formulas in general cases of χ 1 The relation to the notation given in Chapter 3 of [20] is as follows:

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(a,b) (τ, z); (2.22) in other words, The main formula that we would like to prove in this section is In order to achieve this formula we first consider the elliptic genera of the SL(2) N +2 /U (1) supercoset withĉ = 1 + 2 N [13,14]. We set and Here, Z(τ, z) is identified with the elliptic genus of the axial supercoset of SL(2) N +2 /U (1) ("cigar" [21][22][23][24][25]), while Z(τ, z) is associated with the vector supercoset: as shown in [26]. The following identities play crucial roles: 1) dis (N − v, a + v; τ, z); (2.28) these are proven in [14,26]. 5 We also note that the IR parts of Z (a,b) and Z (a,b) are given as With these preparations we shall prove the next identity, from which the identity (2.24) is readily derived by using (2.27) and (2.28) as well as the definition Proof of (2.31).
(i) Modular and spectral flow properties: We first note that (N ) (τ, z) possesses the expected modular and spectral flow properties: These are shown from the same properties of Z(τ, z) as well as the fact that s (a,b) acts modular covariantly. (ii) Holomorphicity: Recall the fact that Z(τ, z) is written as and the second term is non-holomorphic. Let us consider the "Z N -orbifold action," that is to this non-holomorphic correction term. Because of the simple identity Therefore, potential non-holomorphic terms in (N ) (τ, z) are strictly canceled out, and we can rewrite it as This is manifestly holomorphic. (iii) IR part: Recall and thus In this way, we can conclude that both of (N ) (τ, N z) and (N ) (τ, N z) are holomorphic Jacobi forms of weight 0 and index N 2ĉ which share the IR part, namely, Consequently, if setting we can conclude F(τ, z) ≡ 1 and the proof will be completed.
in the fundamental region of double periodicity of F. Hence the function F(τ, z) at most possesses simple poles at z = z a,b in the fundamental region, and the following lemma is enough for completing the proof: [Lemma] All the residues of F(τ, z) at z = z a,b vanish.
where C a,b (τ ) is a small contour encircling z a,b . Because of the modular invariance of F(τ, z), we find the modular properties of R a,b as In fact, the first formula is trivial, and the second formula is proven as follows: with the help of (2.38) and (2.39). The proof of (2.44) is straightforward, and we shall present it in Appendix C. Now, we define the "modular orbit" Then, should be a modular form of weight −2k. Therefore, R (r,k) (τ ) has to vanish everywhere over H ∪ {i∞} for arbitrary r ∈ D(N + 1) and k ∈ Z >0 . This is sufficient to conclude that R a,b (τ ) ≡ 0 for every pole z a,b .
In this way, the identity (2.31) has been proven, leading to the main formula (2.24).

Physical interpretation
In this section we try to make a physical interpretation of the main result given above, that is, the identities ( In other words, As a consistency check, one may also derive these formulas directly from the "Fourier expansion relation" (2.23). 2. Manifestly holomorphic expression: In place of (2.24), X (N ) (v, a) can be rewritten in a manifestly holomorphic expression: In fact, the non-holomorphic correction terms in the right-hand side of (2.24) are found to be canceled out precisely, as in (2.36). One can achieve this identity by making the Fourier expansion of (2.36) and recalling the identity (N ) (τ, z) = (N ) (τ, z) proven in the previous section. 3. Interpretation as "analytic continuation" of the character of N = 2 minimal model: It would be worthwhile to recall the basic facts of the N = 2 minimal model, namely, the SU (2) N −2 /U (1) supercoset which hasĉ = 1 − 2 N . As we mentioned at the beginning of this paper, the elliptic genus of the minimal model is given as [11] where ch ,m (τ, z) denotes the R-character of the N = 2 minimal model which has the Witten index Moreover, the spectrally flowed elliptic genus is Fourier expanded in terms of the R-characters as follows (a, b ∈ Z N ): z). (3.8) In this way, by comparing (2.23) with (3.8) one would find a similarity between the function X (N ) (v, a; τ, z) and the minimal character ch ,m (τ, z) with the correspondence Furthermore, let us recall the modular transformation formula of ch ,m (τ, z): ,m (τ, z). (3.11) The first line of (3.11) is familiar formula and we have made use of the property of minimal character, ,m (τ, z), (3.12) to derive the second line. 2 The modular transformation formulas (3.2) and (3.1) nicely correspond to (3.10) and (3.11) under the formal replacement (3.9). In this way, one would regard X (N ) (v, a; τ, z) as a formal "analytic continuation" of the minimal character ch ,m (τ, z).

"Compactified SL(2)/U (1) supercoset" in the ( R, R) sector
Now, let us discuss what the superconformal model is whose modular invariant is built up from the functions X (N ) (v, a)  A corresponds to the cigar-type 2 We have an obvious identity for X (N ) (v, a; τ, z): as already mentioned in Sect. 2. This means that the torus partition functions in the ( R, R) sector of these theories are schematically written as (ĉ ≡ 1 + 2 N , τ ≡ τ 1 + iτ 2 , z ≡ z 1 + iz 2 , and F L , F R denotes the fermion number operators mod 2): (3.14) The discrete part of (3.13) obviously looks like the diagonal modular invariant, whereas that of (3.14) can be regarded as the anti-diagonal one with respect to the "minimal-model-like" quantum number m ≡ v + 2a adopted in [26]. 3 The "anomaly factors" e −2πĉ z 2 2 τ 2 , e 2πĉ z 2 1 τ 2 , which ensure the modular invariance, originate precisely from the path integrations, and they differ due to the gauged WZW actions of vector and axial types [14,26]. Note that these factors just get the common form e 2πĉ z 2 1 τ 2 , if we replace z with −z in the axial model. Thus it would be useful to rewrite (3.14) as where we made use of the identity in the second line. 3 The quantum number m labels the U (1) R charges appearing in the spectral flow orbit defining χ (N ,1) dis (v, a). As mentioned in [26], the axial type (3.14) is anti-diagonal only in the case of integer levels (i.e. K = 1), while the vector type (3.13) is always diagonal. Moreover, it is found that the continuous terms appearing in (3.13) and (3.15) are written in precisely the same functional form with inverse sign. We will prove this fact in the next subsection, and here address our main result in this section:

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(3.17) Note that, even though both Z A include contributions of continuous characters with nontrivial coefficients showing IR divergence, the combined partition function (3.17) is written in terms only of a finite number of holomorphic building blocks X (N ) (v, a; τ, z) given in (2.24) or (3.5). This fact strongly suggests that the modular invariant partition function (3.17) would define an N = 2 superconformal model withĉ = 1 + 2 N associated with some compact background. Therefore, we tentatively call this model the "compactified SL(2)/U (1) supercoset model" here. It is obvious that the elliptic genus of this model is given by the holomorphic function (N ) (τ, z) (2.2):

Cancellation of continuous terms
In this subsection, we prove the precise cancellation of the continuous terms potentially appearing in the first line of (3.17). We start with the explicit form of the partition function Z V , which is evaluated in [26] by means of the path integration (u ≡ s 1 τ + s 2 ): Here we introduced the IR regularization adopted in [14,26], which removes the singularity of the integrand originating from the non-compactness of target space. Namely, we set where we set z ≡ ζ 1 τ + ζ 2 , ζ 1 , ζ 2 ∈ R, and (> 0) denotes the regularization parameter.
In the same way, the axial partition function Z

( R)
A , which describes the cigar background, is written as [14]: As demonstrated in [14], one can make "character decompositions" of the partition functions (3.19) and (3.21). They are schematically expressed as where χ (N ,1) con ( p, m) denotes the extended continuous character (B.1) explicitly written as In the continuous part, the "density functions" ρ 1 (ρ 1 ) and ρ 2 (ρ 2 ) have rather complicated forms. As expected, ρ 1 (ρ 1 ) includes the logarithmically divergent term as the leading contribution, with some constant C, which corresponds to the strings freely propagating in the asymptotic region, and C |ln | is roughly identified as an infinite volume factor. However, both ρ 1 (ρ 1 ) and ρ 2 (ρ 2 ) include subleading, non-diagonal terms with p L = p R and considerably non-trivial dependence on m, as mentioned in [14]. Now, we would like to prove the equalities If this is the case, by using the "charge conjugation relation" 4 as well as (3.16), we obtain Proof of (3.24). We start by recalling the "orbifold relation" between Z ( R) V , which is shown in [14,26]: (3.27) where we introduced the "twisted partition function" 5 in the right-hand side of (3.27): We can rewrite (3.28) by means of the Poisson resummation: where we set ε := 2π τ 2 N . A closely related analysis is given in [14]. We obviously find where s L (α 1 ,α 2 ) (α i ∈ Z) denotes the spectral flow operator (2.7) acting only on the left-mover: Finally, by using the identities which are easily proven by direct calculation, 6 we can achieve the required identities (3.24). 5 The relation of the notations here and in [26] is as follows: where we have explicitly indicated the spin structure. The explicit definitions of modular completions with general spin structures are presented in [31]. We remark that the functions X (v, a) appearing in (3.35) are generically non-holomorphic except for the R-sector.
This model shares the asymptotic cylindrical region with the radius 1 N √ N α ≡ 1 N α with the standard SL(2)/U (1) (of the vector type) and the aspect of the propagating strings is almost the same. However, we have non-trivial modifications in the discrete spectrum, which leads us to the holomorphic elliptic genus (3.18) and would be non-geometric since they are never realized only within the cigar theory (or the trumpet theory). (ii) ε(σ ) = −1 for σ = NS, NS, R, and ε( R) = 1 : "Compactified SL(2)/U (1) supercoset": This second possibility is more curious. In this case, the continuous sectors are canceled out for all 7 Since we are now assuming the non-chiral GSO projection, we do not need to choose the parameter aŝ c = 1 + 2K N , K ∈ 2Z >0 (N and K are not necessarily co-prime) as in [31]. This assumption is necessary when considering the chiral GSO projection. This result is exhibited in terms of the formulas (2.24) or (2.31), equivalently. This is similar to the fact that the elliptic genus of N = 2 minimal model Z
(2) The superconformal system corresponding to (1.2) is identified with a "compactified" model of the SL(2)/U (1) supercoset, as is given by (3.17). (3) Two possibilities of extension to general spin structures have been presented. One is a noncompact model regarded as a "non-geometric deformation" of the SL(2)/U (1) supercoset, and the other is the natural extension of the compactified model (3.17). The latter is quite similar to the N = 2 minimal model, although it is not a unitary theory.
The partition function (3.17) (and (3.37)) looks very like those of RCFTs. We only possess finite conformal blocks that are holomorphically factorized in the usual sense. However, there is a crucial difference from generic RCFTs defined axiomatically. The partition function (3.17) or the elliptic genus (3.18) does not include the contributions from the Ramond vacua saturating the unitarity bound Q = ±ĉ 2 . This implies that the Hilbert space of normalizable states does not contain the NS vacuum (h = Q = 0) which should correspond to the identity operator. Of course, this feature is common with the spectrum of the original SL(2)/U (1) supercoset read off from the torus partition function evaluated in [16] (see also [27][28][29][30]). It may be an interesting question whether or not the finiteness of conformal blocks without the identity representation, which is observed in our "compactified SL(2)/U (1) model", unavoidably leads to a non-unitarity of the spectrum in general conformal field theories.
A natural extension of this work would be the study of the cases of "fractional levels"ĉ = 1 + 2K N (K ≥ 2, GCD{N , K } = 1). In other words, one may seek a theory of which the elliptic genus would be which has the Witten index Z(τ, z = 0) = N + K . However, the function (N /K ) (τ, z) is only meromorphic with respect to the angle variable z, and such a function is not likely to be realized as the elliptic genus of any superconformal field theory. We also point out that the cancellation of continuous parts such as (3.17) does not seem to happen in that case. This fact suggests that the "compactification" of the SL(2)/U (1) supercoset works only for integer levels, that is,ĉ = 1 + 2 N .
PTEP 2014, 043B02 Y. Sugawara We also set The spectral flow properties of theta functions are summarized as follows (m, n, a ∈ Z, k ∈ Z >0 ): Extended discrete characters [15][16][17] s (a,b) . Therefore, it is enough to compare the behaviors of On the other hand, by using the decomposition (C.2), we can evaluate (N ) (τ, N z) around z = z N ,b as follows: (N ) (τ, N z) = (N ) (τ, N (ξ N ,b +z N ,b )) In this way, we have shown that the residue function R a,b (τ ) is finite at the cusp τ = i∞ for every simple pole z = z a,b .