Irreducible vector-valued modular forms of dimension less than six

Abstract

An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T -unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vectorvalued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert-Poincaré series is given, using the fact that the space of vector-valued modular forms is a free module over the ring of classical modular forms for the full modular group. A mild restriction is then placed on the class of representation considered in these dimensions, and this again yields an explicit determination of the associated Hilbert-Poincaré series.

Structure of the module of vector-valued modular forms

“…This condition necessarily holds in a number of cases when V is an irreducible CΓ-module of small dimension, including all irreducible V with dim V 3. (See [10,12] for further details.) We establish the following theorem.…”

Towards a classification of two-character rational conformal field theories

“…These procedures were implemented in considerable detail in [12,18] and some recent work in this direction can be found in [19]. There is also now a considerable mathematical literature on using MLDE to find vector-valued modular forms of CFT type, of which some relevant works are [20][21][22][23][24][25][26][27][28] and additional ones will be described in what follows.…”

Curiosities above c = 24

“…(iii) The number of lattices for dimension ≤ 24 is small, namely (1, 2, 24) for dimension (8,16,24) respectively. This property actually follows from the two above.…”

“…The third property listed above for lattices in dimension ≤ 24 -that their number is small -is also related to, though does not immediately imply, a comparably small number of meromorphic CFT's with c ≤ 24. The actual number turns out to be (1, 2, 71) for c = (8,16,24).…”