In 2002 Hausel - Thaddeus interpreted SYZ conjecture in the context of Hitchin system and Langlands duality. Let briefly explain it

Let $\pi : E \to Σ$ a complex vector bundle of rank $r$ and degree $d$ equipped with a hermitian metric on Riemann surface $\Sigma$ . Take th moduli space $$M(r, d) = \{(A, Φ) \text{ solving }(\star)\}/\mathcal G $$

(which is a finite-dimensional non-compact space carrying a natural hyper-Kähler metric)


$$F^0_A + [Φ ∧ Φ^∗] = 0 ,\; \; \bar ∂AΦ = 0\; \; (\star)$$

Here $A$ is a unitary connection on $E$ and $Φ ∈ Ω^{1,0}(End E)$ is a Higgs field. $F^0$ denotes the trace-free part of the curvature and $\mathcal G$ is the unitary gauge group.

$M(r, d)$ is the total space of an integrable system(which can be interpreted by the non-abelian Hodge theory due to Corlette), the Hitchin fibration, together with Langlands duality between Lie groups provides a model for mirror symmetry in the Strominger-Yau and Zaslow conjecture.

As beginner, what is the recent progress on SYZ conjecture and is there any result on Hausel-Thaddeus interpretation in context of framed Riemann surface $(\Sigma, D)$

As more motivation: Yau-Vafa introduced semi Ricci flat metric which play important role in the study of Strominger-Yau and Zaslow conjecture

In fact rank 2 Hitchin fibration $$\det: M(2, d) → H^0(Σ, K^2_Σ),\; [(A, Φ)]→ \det Φ$$. gives rise to semi Ricci flat metric $\omega_{SRF}$ on $ M(2, d)_{reg}$ which still is open to be semi-positive as current. Study of analysis of such semi-Ricci flat metric led to solution of a lot of conjectures in mirror symmetry


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