Topological recursion and variations of spectral curves for twisted Higgs bundles

Prior works relating meromorphic Higgs bundles to topological recursion, in particular those of Dumitrescu-Mulase, have considered non-singular models that allow the recursion to be carried out on a smooth Riemann surface. We start from an $\mathcal{L}$-twisted Higgs bundle for some fixed holomorphic line bundle $\mathcal{L}$ on the surface. We decorate the Higgs bundle with the choice of a section $s$ of $K^*\otimes\mathcal{L}$, where $K$ is the canonical line bundle, and then encode this data as a $b$-structure on the base Riemann surface which lifts to the associated Hitchin spectral curve. We then propose a so-called twisted topological recursion on the spectral curve, after which the corresponding Eynard-Orantin differentials live in a twisted cotangent bundle. This formulation retains, and interacts explicitly with, the singular structure of the original meromorphic setting -- equivalently, the zero divisor of $s$ -- while performing the recursion. Finally, we show that the $g=0$ twisted Eynard-Orantin differentials compute the Taylor expansion of the period matrix of the spectral curve, mirroring a result of Baraglia-Huang for ordinary Higgs bundles and topological recursion. Starting from the spectral curve as a polynomial form in an affine coordinate rather than a Higgs bundle, our result implies that, under certain conditions on $s$, the expansion is independent of the ambient space $\mbox{Tot}(\mathcal{L})$ in which the curve is interpreted to reside.