Modular and number-theoretic properties of the modified character

Ascertain whether the modified character I_R(q)=I_M(qe^{πi},e^{πi}) enjoys modular transformation properties analogous to those of ordinary VOA characters and characterize such properties if they exist.

Background

Ordinary VOA characters often exhibit modular properties that play a key role in understanding module structures and classifications. The paper’s modified character I_R(q) is again a q‑series (possibly in q^{1/2}), suggesting potential modular behavior.

The authors explicitly raise the question of number‑theoretic and modular aspects of I_R(q), but no results are established, leaving a concrete direction for future work.