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 $\mathcal{L}$-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.

Figures (5)

• Figure 1: Generic fibres of $\mathcal{M}_{X}^{L}$ are tori. There is a locus of degenerate singular tori.
• Figure 2: Local picture of a spectral curve $S$ on $\mathbb{P}^{1}$.
• Figure 3: $\mathcal{B}_{eff}$ mapped into $\mathcal{B}$ by $s=\det\phi$.
• Figure 4: Spectral curve $S$ inside of $K(Z)$.
• Figure 5: Vector bundle over $H^{0}(X,K^{*}\otimes L)$ whose fibres are $\mathcal{B}$ (in blue). Each fibre has a distinguished subspace $\widetilde{\mathcal{B}}_{s}$ (in red). For a chosen Higgs bundle, there is a constant section given by the characteristic coefficients (in green).

Theorems & Definitions (71)

• Theorem 1.1: Baraglia-Huang,BaragliaHuang17
• Definition 1.2
• Definition 1.3
• Theorem 1.4: Variational Formula for twisted-E-O invariants
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5