Establish convergence of the similarity-transformation series e^{ad_D}

Determine whether the series e^{ad_D}({\cal Q}) = {\cal Q} + [D,{\cal Q}] + (1/2)[D,[D,{\cal Q}]] + ⋯ converges in the vertex-operator-algebra setting considered, for operators D that commute with L_0, and specify conditions under which this conjugation produces well-defined deformations of the charges without altering traces.

Background

To explain non-uniqueness of the operator implementing the modular transform, the authors introduce a similarity transformation e^{D} with [D,L_0]=0, which preserves traces and generates deformed charges via the formal series e^{\text{ad}_D}({\cal Q}).

While this construction captures a natural ambiguity, its validity depends on the convergence of the Baker–Campbell–Hausdorff-type series. The authors explicitly note that convergence is unclear, leaving open whether the similarity transformation is well defined beyond a purely formal level.