Topological recursion and variations of spectral curves for twisted Higgs bundles

Abstract

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 $L$-twisted Higgs bundle for some fixed holomorphic line bundle $L$ on the surface. We decorate the Higgs bundle with the choice of a section s of $K^{⁎}\otimes L$, 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 $\text{Tot}\left(L\right)$ in which the curve is interpreted to reside. While our focus is almost exclusively geometric, we include some preliminary thoughts on connections to questions in theoretical condensed matter physics.