Compact formulas for the completed mock modular forms

In this paper we present a new compact expression of the elliptic genus of SL(2)/U(1)-supercoset theory by making use of the ‘spectral flow method’ of the pathintegral evaluation. This new expression is written in a form like a Poincaré series with a non-holomorphic Gaussian damping factor, and manifestly shows the modular and spectral flow properties of a real analytic Jacobi form. As a related problem, we present similar compact formulas for the modular completions of various mock modular forms which appear in the representation theory of N=2,4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2,\;4 $$\end{document} superconformal algebras. We further discuss the generalization to the cases of arbitrary spin-structures, that is, the world-sheet fermions with twisted boundary conditions parameterized by a continuous parameter. This parameter is naturally identified with the ‘u-variable’ in the Appell-Lerch sum.


Introduction and summary
It is known that superconformal field theories with non-compact target space possess interesting theoretical issues; for instance, in order to regulate its IR divergences we may sometimes have to sacrifice its holomorphy property. A good laboratory to study them is the SL(2)/U(1)-supercoset theory, which is interpreted as the supersymmetric extension of the model of 2-dimensional black hole [1][2][3][4]. One of the important progress in the study of this model has been the appearance of non-holomorphicity in the elliptic genus [5][6][7]. This originates from the existence of gapless continuous spectrum of non-BPS states causing the IR-divergence in the theory. Quite interestingly, these studies have revealed the physical origin of a mathematical construction of 'modular completions' of mock modular forms [8].

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and has an anomalous modular transformation law (this function with u = 0 is closely related to the character of massless representations of N = 4 superconformal algebra [9,10]). Anomalous transformation law is compensated by the second term in the r.h.s. of (1.1) with which has a non-holomorphic dependence on τ . Completed mock modular formf (k) u (τ, z) has a well-defined transformation law as a Jacobi form of weight 1 and index k. (See appendix A for the convention of error function Erf(x), theta function Θ m,k (τ, z) and Jacobi forms.) This means that the elliptic genus of SL(2)/U(1)-supercoset is described by a non-holomorphic generalization of a Jacobi form.
Moreover, the torus partition function has also been analyzed in [6], and is shown to be expanded in terms of the modular completions of extended discrete characters (see appendix D) as well as continuous (non-BPS) characters. More recently, in literatures [11][12][13], the elliptic genera of the SL(2)/U(1)-theory as well as some generalized models have been analyzed using the formulation of the gauged linear sigma model as proposed in [14].
In this paper, we shall propose a method of deriving compact expressions for the completed mock modular forms that makes their modular properties manifest. To this end, we adopt the technique of spectral flow method, which was partly studied in [15], rather than the method of character decomposition in [6]. Using the new method, we can simplify relevant calculations.
(ĉ is defined in (2.1)). Therefore, this expression manifestly exhibits modular invariance as in the case of Poincaré or Eisenstein series. The two integers m, n are identified with spectral parameters along the spatial and temporal cycles of a torus. Let us summarize the main results of this paper; (i) The main formula we will derive is given in (2.21). This is written in the form of nonholomorphic Poincaré type series mentioned above, and will be a most tranparent expression of elliptic genus of the cigar model. We will reach this formula in two ways: one is a direct evaluation of path-integration with the help of spectral flow expansion. The other is based on the properties of the modular completions of extended or

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irreducible discrete characters given in the previous papers [6,15]. Combining these two approaches, we can rewrite the function f (k) u=0 (τ, z) (1.1) as given in (2.29). We can also express the modular completions of discrete characters in the forms of double series mentioned above. These are given in (2.31) and (2.32).
(ii) We also discuss the generalization to the cases of arbitrary spin-structures, that is, the world-sheet fermions with twisted boundary conditions parameterized by a continuous parameter. This continuous parameter is naturally identified with the 'u-variable' in the Appell-Lerch sum. The complete formula of rewriting f (k) u (τ, z) in terms of the non-holomorphic Poincaré series is given in (3.15).
2 Revisiting elliptic genus of SL(2)/U(1)-supercoset model as a non-holomorphic Poincaré series 2.1 A refined calculation of elliptic genus based on the spectral flow method We shall start with the torus partition function with the general twist angles z,z ∈ C (in the R-sector) of the cigar SL(2)/U(1)-supercoset model, which has been evaluated in [6]. 1 Throughout this paper we set the level of SL(2) super-WZW model to be a real positive number k (the level of bosonic part is k + 2), which means that the central charge of this superconformal system isĉ Because of the IR-divergence of cigar geometry, we have to introduce a suitable regularization. Here we shall adopt a 'refined' scheme of regularization. Namely, we start with the following regularized partition function 2 (µ ≡ µ 1 + iµ 2 , z ≡ z 1 + iz 2 , µ 1 , µ 2 , z 1 , z 2 ∈ R) 2) where we introduced a regularization factor σ(τ, µ, z,z; ǫ) (ǫ > 0) defined as σ(τ, µ, z,z; ǫ) := Note that all the singularities of the integrand located at µ + 2 k z ∈ Zτ + Z that originate from the θ 1 -factor are removed by inserting (2.3) and the µ-integral converges as long as ǫ > 0. We will later regard σ(τ, µ, z,z; ǫ) as a holomorphic function with respect to complex variables s 1 , s 2 .
Let us very briefly sketch how the torus partition function (2.2) is derived. See [6] for the details.

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• According to the standard analysis of gauged WZW models [19][20][21], the relevant calculation reduces to that of free fields, except for the integral of modulus µ that is coupled to the anomaly-free U(1)-current we should gauge; where j 3 denotes the contribution from the bosonic SL(2)-WZW with level k + 2 and ψ + , ψ − are the world-sheet fermions. The compact boson Y corresponds to the degrees of freedom of gauge field which cannot be gauged away. This obeys the twisted boundary condition parameterized by the µ-variable, and the Gaussian factor is the contribution of its zero-modes. Note that the range of µ-integral is extended from Jac(Σ) ∼ = Σ ∼ = C/Λ, (Λ ≡ Zτ + Z) to the whole complex plane due to the winding modes of Y .
• The z-parameter ('chemical potential') couples to the left-moving N = 2 U(1)-current given by [22] Notice that the Gaussian factor e − πk τ 2 |µ| 2 includes no dependence on z in the torus partition function (2.2). This should be the case, since J possesses no contribution from the boson Y . Now, the µ-integral in (2.2) is finite as long as ǫ > 0, and because of the modular invariance of σ(τ, µ, z,z; ǫ), Z reg (τ, z,z; ǫ) is defined so as to preserve the modular property manifestly; Z reg (τ + 1, z,z; ǫ) = Z reg (τ, z,z; ǫ), The partition function Z reg (τ, z,z; ǫ) logarithmically diverges in the limit ǫ → +0, and the divergent part is identified with the contributions from the strings propagating in the asymptotic cylindrical region of the cigar. This implies that the characteristic behavior around ǫ → +0 is given by where C is some positive constant independent of ǫ. The leading part proportional to |ln ǫ| (the 'asymptotic part') is expressible in terms of the extended continuous characters [18,23] as in the free theory when the level k is rational. We here denote the term of order O(ǫ 0 ) as

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Z finite (τ, z,z), which is still finite after taking the ǫ → +0 limit and is modular invariant. This part can be directly extracted from Z reg (τ, z,z; ǫ) as in [24]; We emphasize that Z finite (τ, z,z) is uniquely determined irrespective of the adopted regularization scheme, even though the asymptotic part (the overall constant C) as well as the correction terms of O(ǫ, ǫ ln ǫ) will depend on the method of regularization. The elliptic genus is obtained by formally settingz = 0 while fixing z at a general complex value in Z finite (τ, z,z). We also have to divide the partition function by a factor in order to include the anomaly factor correctly. In fact, we can uniquely determine this factor by requiring the following conditions due to the analysis given in [6]; • the elliptic genus should have the correct modular property;
In this way, the elliptic genus is written as The equality of second line is due to the simple fact that the asymptotic term in (2.7) drops off when settingz = 0. We thus obtain the path-integral expression of elliptic genus as (2.13) The µ-integral is not easy to perform since the Gaussian factor e − πk τ 2 |µ| 2 breaks the periodicity of the integrand. If this factor was absent, the relevant integral would reduce to a simple period integral over a torus Σ ≡ C/Λ, (Λ ≡ Zτ + Z).

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where s (κ) λ denotes the spectral flow (Eichler-Zagier [25]) operator defined by Recall that the elliptic genus of a complex D-dimensional manifold is a Jacobi form with index D 2 [25]. Hereĉ (2.1) is the effective dimension of a target manifold described by a superconformal field theory with a central charge c. Thus the suffixĉ 2 of the flow operator s (ĉ 2 ) λ denotes the index of the elliptic genus Z(τ, z) describing the cigar geometry. Since (2.14) is a crucial step in our analysis, we will present its proof in appendix B. What is important here is the fact that the action of the operator s (κ) λ (as well as the sign factor (−1) n 1 +n 2 +n 1 n 2 ) preserves the modular covariance, whose precise meaning is given in appendix A.
On the other hand, Z (∞) (τ, z) is defined schematically as the elliptic genus of 'Z ∞orbifold' of the cigar model as given in (B.7), or equivalently the universal cover of trumpet in the T -dual picture. Namely, In the second line, due to the periodicity of the integrand under µ → µ + ν, ν ∈ Λ except for the factor e − πk τ 2 |µ+ω| 2 , we made use of an obvious relation Σ , and carried out the Gaussian integral over ω. Computation is now easy. Since the integrand of (2.16) is holomorphic and periodic with respect to the integration variables s 1 , s 2 with µ ≡ s 1 τ + s 2 , one may regard it as a double period integral.
Thus, by deforming the integration contour as we can directly evaluate (2.16) with the helps of the identity (A.9) as In the second line, we have used as well as the fact that the µ-integral converges even in the absence of σ-factor. We also introduced the symbol (A.5); Substituting (2.18) back into the 'spectral flow formula' (2.14), we finally obtain the following simple expression of elliptic genus of cigar model; In the last line, we introduced the notation More explicitly, (2.19) is rewritten like a Poincaré series Here the double power series of λ ∈ Λ absolutely converges due to the Gaussian factor, and thus (2.19) exhibits good modular behavior. One can easily confirm that the elliptic genus (2.19) possesses the modular property as a weak Jacobi form with weight 0 and index c/2 given in (2.10), and also for the case of k = N/K, N, K ∈ Z >0 ,

Relations between the new and old expressions of modular completions
Now, let us discuss on the relations between present results and our previous analyses given in [6,15], which were based on the 'character decomposition' of the torus partition function.
First of all, we would like to point out that Z (∞) (τ, z) is expanded in terms of modular completions of the irreducible discrete characters,

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as was shown in [15]. Here, the modular completion ch dis (ν, n; τ, z) (0 ≤ ν ≤ k, n ∈ Z) has been defined by (D.1). It would be helpful to confirm the consistency of the formulas (2.18) and (2.23). In fact, by using (D.1), the r.h.s. of (2.23) is explicitly written as The first term corresponds to the holomorphic part, which just yields On the other hand, the double integral in the second term can be evaluated by using the polar coordinate re iθ ≡ p + iν and elementary properties of Bessel function as follows; We next recall the formula of the elliptic genus of cigar model in the case of k = N/K, (N, K ∈ Z >0 ) given in [6]; is the function given in (1.1). Equating this formula with our present result (2.19) in the special case of N = 1, K = 1/k ∈ Z >0 , one can rewrite f (K) in a compact form as We shall present a direct proof of this formula in appendix C, with the 'u-parameter' included.

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We also recall the relations [6,15] denotes the modular completion of the discrete extended character (D.4) (for the u = 0 case). We then likewise obtain We would like to make a comment. The above manipulations leading to (2.19) is easily generalized to the 'twisted' case; where ω ∈ C/Λ(≡ Σ) is the twist parameter. (We have an obvious periodicity under As is mentioned in appendix B, the formula of 'spectral flow decomposition' (2.14) is generalized to the present case, which is naturally interpreted as the Fourier expansion written as Here, we introduced which is also expressed in terms of the modular completion ch dis (λ, n) (D.1) in the same way as (2.23); The identity (2.34) looks nice. This schematically means that the spectral flow with ν ∈ Λ is the Fourier transform of the twisting by ω ∈ C/Λ.
Each manipulation given here keeps the modularity intact. We can readily confirm

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One can also evaluate the elliptic genus of the Z M -orbifold of cigar with ∀ M ∈ Z >0 as This result is consistent with the identity (B.5), which is directly checked based on the path-integral expression (2.33) for the special cases k = N/K, M = N , as is mentioned in appendix B. It would be worthwhile to note that (2.40) is always correct for ∀ M ∈ Z and ∀ k ∈ R >0 (not necessarily rational).

Elliptic genus with general spin structures
In this section, we shall extend the analyses in the previous section to the cases of general spin structure with the continuous parameter of twisting u ≡ ατ + β, ( ∀ α, β ∈ R). Namely, we assume the world-sheet fermions of super-gauged WZW model ψ ± (w),ψ ± (w) satisfy the twisted boundary condition (with respect to the cylinder coordinate w,w); ψ ± (w + 1) = e ∓2πiα ψ ± (w), ψ ± (w + 2πτ ) = e ∓2πiβ ψ(w), In the end, this twist parameter u will turn out to be identified with the 'u-variable' of the function f ( * ) u (τ, z) introduced in [8]. Closely related studies including such a twisting based on a different approach have been given in [7,26].
We shall again start our analysis from the torus partition function. The partition function for the fermions with boundary condition (3.1) is written concisely by using the spectral flow operator s In the second line we set X := z − u for convenience of calculation given below. Note that we use the notations The elliptic genus is again defined by settingz to a particular value while keeping z at a generic complex value. It is easy to find that we have to setz =ū (⇐⇒X = 0) to realize the supersymmetric cancellation. We still have to divide by a suitable anomaly factor corresponding to (2.9) in the u = 0 case. To determine this factor, we require the following conditions which extend (2.10) and (2.11) to the u = 0 cases: • the elliptic genus should have the modular property;

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In the second line we used the identity s (1/2) n 1 τ +n 2 · ϑ u (τ, z) = (−1) n 1 +n 2 +n 1 n 2 e 2πi 1 τ 2 Im(λū) ϑ u (τ, z), which is equivalent with (A.14), and we also made use of the product relation of the spectral flow operators (A.11) in the third line. The main result (3.12) is an extension of the formula (2.19). This obviously shows that Z(τ, z, u) possesses the expected modularity (3.4), and also the spectral flow property for the case of k = N/K, N, K ∈ Z >0 expressed as s (ĉ 2 ) n 1 τ +n 2 · Z(τ, z, u) = (−1) n 1 +n 2 +n 1 n 2 Z(τ, z, u), ∀ n 1 , n 2 ∈ N Z . One can also write down the extension of the formulas (2.28) and (2.30); where the modular completion of extended character χ is the function given in (1.1). Comparing (3.12) with (3.14), we obtain a formula, which is the extension of (2.29); We present a direct proof of this identity in appendix C. Based on (3.15), one can readily confirm the following identities for the function f We also note Zτ + R . 3 It would be useful to rewrite (3.15) as in order to check the spectral flow relations (3.17) and (3.18).

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Several remarks are in order; 1. Let us assume the rational level k = N/K, N, K ∈ Z >0 . In the similar manner to the analysis given in [6] for the u = 0 case, one can make the 'character decomposition' of regularized partition function Z reg (τ, z,z, u; ǫ), and extract the modular completions χ dis (D.4). Although necessary computations are slightly cumbersome, one can achieve the expected result; Thus, (3.19) correctly reproduces the elliptic genus (3.14) by settingz =ū and dividing it by the anomaly factor (3.6).
2. The 'twisted elliptic genus' Z ω (τ, z, u) can be also defined by the replacement in the path-integral expression (3.7). We can analyze Z ω (τ, z, u) in a parallel manner as the u = 0 case and find that Based on this result, we can also derive the modular transformation formulas 3. Recall that the parameter u expresses the general spin structure of world-sheet fermions. However, one should keep it in mind that for generic u, despite the relation (3.11). This feature comes from the noncommutativity of the spectral flow operator s ( * ) λ . (See (A.11) and also appendix D). Using the twisted elliptic genus Z ω (τ, z, u) given above, we rather obtain  (3.12). We thus obtain the useful formulas which extend (2.31), (2.32).

Modular completions of superconformal characters
As an interesting application of the inclusion of spin structure parameter u, let us discuss the modular completion of characters in N = 2 and 4 superconformal algebras.
We first look at the modular completions of the N = 4 massless characters of general level k(∈ Z >0 ).
The character formula of the massless representation of isospin ℓ/2, level k (ℓ = 0, 1, . . . , k) is given [9,10,27] (in the R-sector) as Therefore, it is natural to define its modular completion by replacing f However, because of the identity (3.18), we just obtain (3.29) Hence we find This means that after the modular completion all of the N = 4 massless characters for different spins (ℓ) collapse to a single function up to an overall constant which is just the
where we allow λ to be a general real number.

A Notations and useful formulas
In appendix A we summarize the notations adopted in this paper and related useful formulas. We assume τ ≡ τ 1 + iτ 2 , τ 2 > 0 and set q := e 2πiτ , y := e 2πiz .
Namely, ϑ(τ, z) is a weak Jacobi form of weight −1 and index 1 2 . The next identity is quite useful for our calculations; See e.g. [6] for the proof.
Spectral flow operator. (See also [25].) An important property of the spectral flow operator s (κ) λ is the modular covariance, which precisely means the following.

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Assume that f (τ, z) is an arbitrary function with the modular property; The next 'product formula' is also useful; in other words, We should note that the spectral flow operators do not commute with each other in general. The next notation has been repeatedly used in the main text; which satisfies ϑ u (τ, z + mτ + n) = (−1) m+n e 2πi(mβ−nα) q − m 2 2 y −m ϑ u (τ, z). (A.14) Error function.
The next identity is elementary but useful; Weak Jacobi forms. The weak Jacobi form [25] for the full modular group Γ(1) ≡ SL(2, Z) with weight k (∈ Z ≥0 ) and index r ∈ 1 2 Z ≥0 is defined by the conditions (i) modularity: (ii) double quasi-periodicity:

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In this paper, we shall use this terminology in a broader sense. We allow a half integral index r, and more crucially, allow non-holomorphic dependence on τ , while we keep the holomorphicity with respect to z. 4 B Proof of (3.8) In this appendix we prove the formula (3.8). Even though this identity (for the u = 0 case) has been already shown in [15] by using the modular completion of irreducible characters, we shall here directly derive it from the path-integral expression of elliptic genus with an arbitrary parameter of continuous spin structure u. Of course, we can readily obtain the identity (2.14) by setting u = 0. We shall assume k = N/K with some positive integers N and K (not necessarily coprime) for the time being. We will later take the large N -limit with keeping k a fixed value. We denote Λ ≡ Zτ + Z as in the main text.
Let's start with the orbifolding associated with the Z N -symmetry acting on the angle coordinate θ of cigar as θ → θ + 2π a N , ∀ a ∈ Z N . The corresponding elliptic genus is given as where each twisted sector is explicitly written as We further introduce their 'Fourier transformation' by the relation as Moreover, by using the above definitions as well as that of spectral flow operator (A.10), one can directly show the next identity for ∀ λ ≡ n 1 τ + n 2 ∈ Λ/N Λ. 4 According to the original terminology of [25], the 'weak Jacobi form' of weight k and index r (k, r ∈ Z ≥0 ) means that Φ(τ, z) should be Fourier expanded as

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The identity (B.4) holds for an arbitrarily large N ∈ Z >0 as long as k is fixed, since Z(τ, z, u) depends only on k. Therefore, taking the large N -limit, and combining the identities (B.4) with (B.5), we eventually obtain in the second line.

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Using the identity n∈Z e 2πinν = r∈Z δ(ν − r), we can further rewrite (C.1) as n∈Z R+i dp − R dp yw −1 q m e −πτ 2 p 2 +(ν+2kα) 2 k p − i(ν + 2kα) × e 2πinν yw −1 q m ν 1 − yw −1 q m y 2km q km 2 . (C.2) Here, it is not obvious whether the infinite n-summation commutes with the ν-integral as well as the m-summation. We thus define In appendix D, we summarize the definitions of modular completions of the irreducible and extended characters of N = 2 superconformal algebra with general spin structures parameterized by a continuous parameter u ∈ C. They are natural extensions of those given in [6,15] for the u = 0 case. We assume τ ∈ H and set q ≡ e 2πiτ , y ≡ e 2πiz , w ≡ e 2πiu ≡ e 2πi(ατ +β) .