Convergence of non-abelianization under the Novikov specialization T = e^{-1/ħ}

Determine whether the local system produced by the non-abelianization map NA(σ) over the Novikov field Λ converges under the substitution T = e^{-1/ħ} for sufficiently small positive ħ when the coefficient field is the complex numbers, thereby establishing a convergent complex local system obtained from the Novikov-valued one.

Background

The authors construct a non-abelianization map NA(σ) that sends rank-1 local systems on the spectral curve L to rank-K local systems on the base C\D over the Novikov field Λ, using their sheaf-quantization framework and an almost-embedding into Λ_0-module sheaves.

The remark formulates a concrete convergence question: whether the Novikov-valued local systems can be specialized via T = e^{-1/ħ} to produce bona fide complex local systems for small ħ. This convergence would bridge formal-Novikov constructions with analytic structures and is motivated by exact WKB considerations.