SYZ mirror conjecture for Hitchin-type integrable systems

Establish that mirror partners for two-dimensional N=(4,4) sigma models on hyperkähler targets—especially Hitchin moduli spaces for Langlands-dual groups G and ^LG—arise by dualizing the typically torus fibres of the corresponding algebraically completely integrable systems (the Hitchin fibrations), in accordance with the Strominger–Yau–Zaslow (SYZ) mirror conjecture.

Background

Hitchin moduli spaces of solutions to Hitchin’s equations on a Riemann surface are hyperkähler and admit the Hitchin fibration, an algebraically completely integrable system whose generic fibres are abelian varieties. For Langlands-dual groups G and LG, the Hitchin systems have dual abelian fibres over a common base.

The SYZ mirror conjecture predicts that mirror symmetry can be realized by fiberwise dualization (T-duality) along special Lagrangian torus fibres. In the Hitchin setting, this suggests that sigma models with targets the Hitchin moduli spaces for G and LG are mirror via dualization of the Hitchin fibres, a conjectural statement highlighted in the paper’s discussion of 2D mirror symmetry.

References

Mirror symmetry for such theories is naturally formulated as a hyperk\"ahler version of SYZ mirror conjecture: mirror partners are expected to arise by dualising the (typically torus) fibres of an integrable system.

— From Yang-Mills to Yang-Baxter: In Memory of Rodney Baxter and Chen--Ning Yang  (2512.24494 - Wang, 30 Dec 2025) in Appearance in 2D mirror symmetry, Section 3.2 (Dimensional reductions of ASD Yang–Mills)