Spectral interpretation of the deformation parameter

Let $(A,\phi,\psi)\in\mathcal{M}_\alpha^G$ be a solution and let $b=\mathrm{Hit}_\alpha^G(A,\phi,\psi)$. Denote by $\Sigma_b$ the associated spectral curve and by $L$ the corresponding line bundle in the classical sense. Let $\mathfrak{L}$ be the relevant spectral line bundle, let $\pi^{-1}(\partial\Sigma)\subset\Sigma_b$ be the boundary divisor, and let $\tau(\psi)$, $\iota_*$, $\partial$, $\omega_\Sigma$, $g(\lambda,\alpha)$, and $\mathrm{GeoLang}_0$ have the meanings determined by the coupled Hitchin–He construction. Spectral interpretation of the deformation. The deformation parameter $\alpha$ and the field $\psi$ are encoded on the spectral curve as follows: the boundary condition $\tau(\psi)$ together with the deformation term $\alpha\cdot\iota_*(\partial(\tau(\psi)))\cdot\omega_\Sigma$ induce, on the restriction of $\mathfrak{L}$ to $\pi^{-1}(\partial\Sigma)$, a $\bar{\partial}$-connection with a singularity supported on that divisor, whose residue or weight is $\alpha$; the gauge transformation $g(\lambda,\alpha)$ that absorbs the boundary term corresponds to a specific holomorphic automorphism of $\mathfrak{L}$ near $\pi^{-1}(\partial\Sigma)$ removing the singular $\bar{\partial}$-potential and converting the deformed line bundle into an ordinary holomorphic line bundle; and under the classical correspondence $\mathrm{GeoLang}_0$, the singular $\bar{\partial}$-potential on the $G$-side is transformed into prescribed monodromy around the boundary points for the associated local system on the $^LG$-side, so that $\alpha$ appears there as a deformation of the monodromy data of the dual flat connection. This conjecture proposes a spectral-geometric interpretation of the deformation parameter: it changes the presentation and boundary data of the spectral object rather than its intrinsic moduli-space geometry, while the precise analytic and geometric realization of the induced singular connection, its removal by gauge transformation, and the resulting dual monodromy remain to be established.