Abstract
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 path-integral evaluation. This new expression is written in a form like a Poincare 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 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.
Highlights
Completed mock modular form fu(k)(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 Θm,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
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]
We shall propose a method of deriving compact expressions for the completed mock modular forms that makes their modular properties manifest
Summary
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. |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. The partition function Zreg(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. Substituting (2.18) back into the ‘spectral flow formula’ (2.14), we obtain the following simple expression of elliptic genus of cigar model;.
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