Modular invariance and uniqueness of conformal characters

Bits and Pieces in Logarithmic Conformal Field Theory

“…The characters of our theories in question are solutions of certain degenerate modular differential equations, obtained in a unique way by analytic continuation. Therefore, we conjecture that the result of [23] should also hold for logarithmic rational CFTs. Thus, we should only be concerned with n = 1 in our case.…”

“…One can show [23] that regular rational theories with c eff ≤ 1 can only have one power η(τ )η(τ ) in the denominator of the partition function. Regular means that the characters are modular forms.…”

Structure of the module of vector-valued modular forms

“…Vector-valued modular forms have been a part of number theory for some time, but a systematic development of their properties has begun only relatively recently [1,5,6,7,11,12]. One motivation for this comes from rational and logarithmic field theories, where vector-valued modular forms arise naturally [3,4,13,14]. Modular forms on noncongruence subgroups are a special case of vector-valued modular forms, and one of the goals of both this case and the general theory is to find arithmetic conditions that characterize classical modular forms (that is, on a congruence subgroup) among all vector-valued modular forms (cf.…”

Partition Functions for Supersymmetric Black Holes

“…[143]. Dimension formulas for vector-valued modular forms are also mentioned in [139], [144] and [145]. Ref.…”

“…This terminology appears naturally in the derivation of the Selberg trace formula, see for example [146]. The three con-tributions are given by [143,144] A s = 1 − w 12 χM(1), (4.68)…”