Limiting metric conjecture for mathfrak{sl}(2)-type symplectic Higgs bundles

Let (E,Φ,ω)\M\Sp(2n,\C)(X,K)(E,\Phi,\omega) \in \M_{\Sp(2n,\C)}(X,K) be a symplectic Higgs bundle with irreducible spectral curve of \sl(2)\sl(2)-type. Let hth_t solve the rescaled Hitchin equation

Fht+t2[ΦΦht]=0,tR+.F_{h_t}+t^2[\Phi\wedge\Phi^{*{h_t}}]=0, \quad t \in \R_+.

Limiting metric conjecture. The solution hdc(E,Φ,ω)h_{dc}(E,\Phi,\omega) to the decoupled Hitchin equation is a limiting metric: hth_t converges to hh_\infty in CC^\infty on every compact subset of XZ(\disc\sp(E,Φ,ω))X\setminus Z(\disc_\sp(E,\Phi,\omega)) as tt\rightarrow\infty.

The conjecture asserts that the explicitly constructed decoupled solution describes the large-tt limit away from the discriminant locus. The local models at the relevant singularities are known to be approximated by results for \SL(2,\C)\SL(2,\C) and regular Hitchin fibres, but the global limiting statement is not established in the source.

Sources & referencesView supporting material

Primary source

Johannes Horn, “sl(2)-type singular fibres of the symplectic and odd orthogonal Hitchin system”, arXiv:2011.02192 (2021).

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