September, 2011
Comments on Nonholomorphic Modular Forms
and
Noncompact Superconformal Field Theories
Yuji Sugawara^{*}^{*}*
Department of Physical Science, College of Science and Engineering,
Ritsumeikan University, Shiga 5258577, Japan
We extend our previous work [1] on the noncompact defined as the supersymmetric gauged WZW model. Starting from pathintegral calculations of torus partition functions of both the axialtype (‘cigar’) and the vectortype (‘trumpet’) models, we study general models of the orbifolds and fold covers with an arbitrary integer . We then extract contributions of the degenerate representations (‘discrete characters’) in such a way that good modular properties are preserved. The ‘modular completion’ of the extended discrete characters introduced in [1] are found to play a central role as suitable building blocks in every model of orbifolds or covering spaces. We further examine a large limit (the ‘continuum limit’), which ‘deconstructs’ the spectral flow orbits while keeping a suitable modular behavior. The discrete part of partition function as well as the elliptic genus is then expanded by the modular completions of irreducible discrete characters, which are parameterized by both continuous and discrete quantum numbers modular transformed in a mixed way. This limit is naturally identified with the universal cover of trumpet model. We finally discuss a classification of general modular invariants based on the modular completions of irreducible characters constructed above.
1 Introduction
In this paper we try to extend our previous work [1] on the supersymmetric (SUSY) gauged WZW model, that is, the SUSY nonlinear model on 2dimensional blackhole [2]. In spite of its simplicity there are several intriguing features which originate from the noncompactness of target space. Among other things it would be surprising enough that this model could lead to a nonholomorphic elliptic genus [3, 1, 4], in other words, wouldbe lack of holomophic factorization in the torus partition function [1]. In more detail the torus partition function of this model is found to be expressed in the form as [1]^{1}^{1}1Previous studies closely related to this subject have been given e.g. in [5, 6, 7, 8, 9].
(1.1) 
The ‘continuous part’ is mainly contributed from free strings propagating in the asymptotic region of 2D blackhole background, and is written in a holomorphically factorized form composed of characters of nondegenerate representations. On the other hand, the ‘discrete part’ , which includes strings localized near the tip of 2D blackhole, would not be written in a holomorphically factorized form. It is modular invariant and formally expressible in an analogous way to rational conformal field theories (RCFTs);
However, the building blocks are no longer holomorphic. They are written such as
(1.2) 
where denotes the ‘extended discrete character’ defined by spectral flow orbits of irreducible characters generated by BPS states [10, 7]. Although shows the same IR behavior (around ) as , it is never expressible in terms only of characters of superconformal algebra due to the dependence in the subleading term. We emphasize that, while the discrete characters themselves are not, the functions show simple modular behaviors mimicking RCFTs and are closed under modular transformations. When performing the Stransformation of , a continuous term of ‘Mordell integral’ [11, 12] emerges in the similar manner to the characters [13]. However, the subleading term exactly cancels it out, simplifying considerably the modular transformation formulas of . Therefore, we shall call them the ‘modular completions’ of discrete characters.
Related mathematical studies of nonholomorphic modular and Jacobi forms [14] seem to have been still in new area. (See e.g. [15, 16].) Very roughly speaking, one finds correspondences such as
Another possible application of the theory of mock modular forms to superconformal field theories has been presented in [17].
What we would like to clarify in this paper is addressed as follows;
 (1) General orbifolds with arbitrary :

For the parafermion theory [18, 19], which is a ‘compact analogue’ of theory, general modular invariants have been classified and interpreted as some orbifolds [19]. Inspired by this fact, we will examine general orbifolds of SUSY coset with arbitrary . This would be a natural extension of the analysis given in our previous work [1]. We especially would like to clarify the roles played by modular completions introduced in [1] in general models of orbifolds.
 (2) Modular completion of irreducible discrete character :

In [1] we only introduced the modular completions of the extended discrete characters. It may be a natural question what is the modular completions of the irreducible discrete characters. We also would like to clarify the model including these new completions as natural building blocks.
This paper is organized as follows;
In section 2, we demonstrate the pathintegral evaluations of torus partition functions of both axialtype (‘cigar’) and vectortype (‘trumpet’) SUSY models, with an IRregularization preserving good modular behaviors.
In section 3, we study general models of the orbifolds of cigar and the fold covers of trumpet with an arbitrary integer , related with each other by the Tduality as expected [20]. We then extract contributions of the degenerate representations (‘discrete characters’), which are captured by the modular completions of extended discrete characters. Especially, the discrete parts of partition functions of general orbifolds are defined so that the modular invariance is preserved. The ‘twisted’ discrete partition functions are also introduced and they show the modular covariance. We further discuss a ‘continuum limit’ by suitably taking , which leads us to the modular completion of irreducible discrete characters.
In section 4, we study the elliptic genera of relevant models. It will turn out that they are rewritten as linear combinations of the modular completions in all the cases.
In section 5, we discuss general forms of modular invariants when assuming the modular completions to be fundamental building blocks. Fourier transforms of the irreducible modular completions play a crucial role, and we will see that all the discrete partition functions and elliptic genera presented above are reexpressed in a unified way based on them.
We will summarize the main results and give some discussions in section 6.
2 Variants of SUSY Coset Conformal Field Theories
2.1 Preliminaries : SUSY Gauged WZW Actions
We shall first introduce the model which we study in this paper, summarizing relevant notations. We consider the SUSY gauged WZW model with level ^{2}^{2}2 is the level of the total current including fermionic degrees of freedom, whose bosonic part has the level . , which is quite familiar [21] to have superconformal symmetry with central charge;
(2.1) 
We restrict ourselves to cases with rational level () for the time being, and will later discuss the models with general levels allowed to be irrational. Note here that we do not necessarily assume that and are coprime.
The worldsheet action of relevant SUSY gauged WZW model in the present convention is written as
(2.2)  
(2.3)  
(2.4)  
(2.5)  
In (2.3) and (2.5), the sign/ sign is chosen for the axiallike/vectorlike gauged WZW model, which we shall denote as / (/ ) from now on. The chiral gauge transformation is defined by
(2.6) 
where we set for the axial, vector model, respectively. The gauged WZW action / is invariant under the axial/vector type gauge transformations that correspond to in (2.6). Both of the classical fermion actions , (2.5) are invariant under general chiral gauge transformations , , and we assume the absence of chiral anomalies when holds.
It is wellknown that this model describes the string theory on 2D Euclidean blackhole [2]. The axialtype corresponds to the cigar geometry, while the vectortype does to the ‘trumpet’, which is Tdual to the cigar [20]. We will later elaborate their precise relation from the viewpoints of torus partition functions.
It will be convenient to introduce alternative notations of gauged WZW actions;
(2.7)  
(2.8) 
They are indeed equivalent with (2.3) under the identification of gauge field;
(2.9) 
where we set for the axial (vector) model as before, as one can confirm by using the PolyakovWiegmann identity;
(2.10) 
2.2 Axial Coset : Euclidean Cigar
We shall first focus on the axial model. We are interested in the torus partition function. We define the worldsheet torus by the identifications (, , and use the convention , ). We call the cycles defined by these two identifications as the and cycles as usual.
Detailed calculations of the torus partition function have been carried out in [7, 1] based on the Wick rotated model (i.e. supercoset, with ). Especially, the partition function of sector (Rsector with insertion) with the moduli , (i.e. the insertion of , where , are currents) has been presented in our previous work [1]. We shall just sketch it here.
In the Wick rotated model , the gauge field should be regarded as a hermitian 1form. Following the familiar treatment of gauged WZW models (see e.g. [22, 23, 24]), we decompose the gauge field as follows;
(2.11) 
where , are real scalar fields parameterizing the chiral gauge transformations (in the Wick rotated model);
(2.12) 
and , is the modulus of gauge field. To emphasize the modulus dependence of gauge field we took the notation ‘’. Note that the modulus parameter is normalized so that it correctly couples with the zeromodes of currents , which should be gauged;
(2.13) 
where we set
(2.14) 
satisfying the twisted boundary conditions;
(2.15) 
We also introduce the notation;
(2.16) 
Then, the modulus part of gauge field is expressed as
(2.17) 
Including the ‘angle parameter’ which couples with the symmetry in superconfomral symmetry, the torus partition function is written as
(2.18) 
where is the modular invariant measure of modulus parameter , and we work in the sector for worldsheet fermions. We can explicitly evaluate this pathintegration by separating the degrees of freedom of chiral gauge transformations (real scalar fields and ) according to the standard quantization of gauged WZW models [22, 23, 24], which renders this model ‘almost’ a free conformal system. Namely, interactions among each sector are caused only through the integration of modulus . One can easily confirm that the complex parameter precisely corresponds to the insertion of an operator , where and are the currents in the KazamaSuzuki model [21]. (See [1] for more detail.)
To proceed further we have to pathintegrate the compact boson , while the noncompact boson is decoupled as a gauge volume. By using the definitions of (2.7), (2.8) and a suitable change of integration variables, we obtain
(2.19)  
where the ghost variables have been introduced to rewrite the Jacobian factor. It is most nontrivial to evaluate the pathintegration of the compact boson . Its worldsheet action is evaluated as
(2.20)  
where we set . Note that the linear couplings between the currents , and the moduli , are precisely canceled out^{3}^{3}3Of course, this cancellation is expected by construction of the superconformal algebras in the KazamaSuzuki supercoset.. Since satisfies the following boundary condition;
(2.21) 
the zeromode integral yields the summation over winding sectors weighted by the factor determined by the ‘instanton action’. After all, we achieve the next formula of partition function;
(2.22)  
where is a normalization constant. This is identified as the Euclidean cigar model whose asymptotic circle has the radius .
To be more precise, one should make a suitable regularization of (2.22), since it shows an IR divergence that originates from the noncompactness of target space. In other words, the integral of modulus logarithmically diverges due to the quadratic behavior of integrand near the point . According to [1], we take the regularization such that the integration region of modulus is replaced with
(2.23) 
where we set , , and denotes the regularization parameter. Then, the regularized partition function is defined as
One of the main results of [1] is the ‘character decomposition’ of (LABEL:part_fn_A_reg). Namely, it has been shown that the partition function can be uniquely decomposed in such a form as
(2.25) 
where denotes the ‘modular completion’ of extended discrete character [1], while does the extended continuous character [10, 7] only attached with a real ‘Liouville momentum’ (above the ‘mass gap’, in other words). Their precise definitions and relevant formulas are summarized in Appendix C.
Important points are addressed as follows;

The modular completion is nonholomorphic with respect to modulus , but possesses simple modular properties: the Stransformation is closed by themselves, whereas the extended discrete character is not.

The second term in (2.25) shows a logarithmic divergence under the limit, which corresponds to the contribution from strings freely propagating in asymptotic region. On the other hand, the first term remains finite under , and we denote it as (the ‘discrete part’ of partition function). is modular invariant by itself, as we will later elaborate on it.

It is worth pointing out that the decomposition (2.25) itself uniquely determines the functional form of modular completion . In fact, the subleading terms in (1.2) are unambiguously determined by making ‘completion of the square’ for terms including the extended discrete characters (C.15) in the decomposition of .
2.3 Vector Coset : Euclidean Trumpet
The partition function of the vectortype model is defined in the same way as (2.18), with the vectortype gauged WZW action . In the Wickrotated model, we should again regard as , while the gauge field is now parameterized as
(2.26) 
Again, the noncompact direction is anomaly free (vectorlike), and the compactdirection is anomalous (axiallike). We should note that the gauge field is neither a hermitian nor an antihermitian 1form. This fact originates from the sign difference of modulus compared with (2.11), which has been chosen so that it leads to the same coupling to current zeromodes as given in (2.13).
Now, the wanted partition function is written as
(2.27) 
We can again evaluate it by using the formulas (2.7) and (2.8) as follows^{4}^{4}4A caution: after separating chiral gauge transformations, the fermion action should get , rather than . It is due to our parameterization of gauge field . (Recall how (2.26) includes the modulus .) This fact leads us to the correct fermion factor in the partition function (2.31). ;
(2.28)  
In deriving (2.28), we assumed that the pathintegral measure of fermions is anomalous along the axial direction () as opposed to the axial model (2.19).
The worldsheet action of compact boson is now evaluated as
(2.29)  
Note that the dependence is completely canceled out contrary to the axial case (2.20). Moreover, the absence of quadratic term of modulus is characteristic for the vectortype model. The second term in (2.29) is nondynamical and contributes to the pathintegral just through ‘winding numbers’;
In this way we obtain
(2.30) 
However, we face a subtlety since is fractional in general. We recall , and assume that and are coprime from now on. The periodicity of moduli parameters , would be violated unless . In other words, one should impose this restriction of winding numbers to assure the consistency of functional integration.
Combining all the pieces and by taking the regularization: (2.23), we finally achieve the following expression for the vectortype coset;
(2.31)  
One can also make the character decomposition for (2.31). We will work on this subject in the next section. Before that, let us first discuss aspects of the orbifolds of cosets with an arbitrary integer systematically.
3 General Orbifold of
We next consider the general orbifold of model. We again assume a model of rational level; () and let be an arbitrary divisor of , setting , ^{5}^{5}5Here we do not assume that and are coprime integers. Therefore, in case is not a divisor of , one may just replace , with , , and all the following arguments are applicable. .
3.1 Orbifold and fold Cover
We start with the axial model. Since the twisted boson introduced in (2.20) is associated with the (asymptotic) angle coordinate of cigar geometry, one may consistently define the orbifold by introducing fractional winding sectors , leading to the torus partition function;
(3.1)  
(3.2) 
Here we again took the regularization (LABEL:part_fn_A_reg) and chose the normalization constant as so that
We also included an modular invariant factor by hand to avoid unessential complexity of equations below^{6}^{6}6In convention adopted in this paper, the axial coset includes the factor , while the vector coset does the different factor
The twisted partition function (3.2) behaves ‘almost’ modular covariantly;
(3.3) 
as is directly checked by the definition (3.2). Note that the violation of covariance in (3.3) is at most at the order of .
It is convenient to introduce the ‘Fourier transform’ of by the next relations;
(3.4) 
Using the identity
(3.5) 
which is proven by the Poisson resummation formula, we obtain the explicit form of (3.4) as