Higher level appell functions, modular transformations and non-unitary characters

Abstract

In this thesis, we firstly extend elements and periodicity properties of the theta function theory to functions that represent a wider domain of symmetries and properties, graded with different amounts of p ≥ 1, p ϵ N. Unlike theta functions, these generalised, "higher-level Appell functions" K(_p) satisfy open quasiperiodicity relations, with additive theta function terms emerging as violating terms of open quasiperiodic K(_p)’s. We evaluate the S and T modular transformations of these functions and show that the S-transform of K(_p) does not just give back K(_p), but also includes p additional 0-functions which are precisely those violating the quasiperiodicity of Appell functions. This sets a new pattern of modular group representations on functions that are not double quasiperiodic. While calculating the S-transform of K(_p), a newly arising function, namely ɸ(T, μ) will be also thoroughly analysed. As two interesting applications, we firstly study the modular group action on unitary and on an admissible class of non-unitary N = 2 characters which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. Secondly we continue this study for the admissible representation of the affine Lie superalgebra sl (2|l). We see in the final result for both cases that the functions A(T, V) are the "violating" terms of unitary calculations. We lastly confirm all our results by some sets of consistency checks including an essential residue calculation. We believe this new way of using Appell functions, could be used for any other algebraic structure whose characters can be rewritten in terms of higher-level Appell functions.