Independence of the twisted period-matrix expansion from the choice of section s in H^0(X,K^*⊗L)

Determine whether the equality in Theorem ThmBigThm, which identifies the partial derivatives of the period matrix of the spectral curve with integrals of the g=0 twisted Eynard–Orantin differentials computed via coordinates on the image \widetilde{\mathcal{B}} of the effective Hitchin base, is independent of the choice of section s ∈ H^0(X, K^* ⊗ L) used to define the twisted recursion, for all choices of s whose zero divisor is compatible with the chosen symplectic basis and disjoint from ramification points. In particular, ascertain that the Taylor expansion of the period matrix produced by the twisted Eynard–Orantin differentials is invariant under changing s within this admissible class.

Background

The paper defines twisted Eynard–Orantin differentials on Hitchin spectral curves embedded in K(Z) via a choice of section s ∈ H^0(X, K^* ⊗ L), imposing a b-geometry on both the base curve X and the spectral curve S. Using a variational formula adapted to this twisted setting, Theorem ThmBigThm shows that the g=0 twisted differentials W_{0,m} compute the Taylor expansion of the period matrix of S with respect to local coordinates induced by s on the image \widetilde{\mathcal{B}} of the effective Hitchin base.

The authors discuss that, while the period matrix itself depends only on S and the chosen symplectic basis, the coordinates used in Theorem ThmBigThm depend on s. They note that deformations are intrinsically controlled by the effective Hitchin base \mathcal{B}_{eff}, which is independent of s, and conjecture that the full interpretation of Theorem ThmBigThm should be independent of s. Establishing this requires clarifying how the constructions used in the twisted recursion (including the relevant kernels and coordinates) depend on s.