Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?

Question

Is the mirror of a hyperkaehler manifold always a hyperkaehler manifold?

What I know so far is as follows:

In this paper (https://arxiv.org/pdf/hep-th/9512195.pdf) by Verbitsky, it is claimed that every hyperkaehler manifold is mirror to itself (which would meant the answer to my question is 'yes').

On the other hand, from the answer to this question - Mirror symmetry for hyperkahler manifold , it seems that this is not always the case, i.e., a K3 surface can have a nontrivial mirror.

Answer 1

I think Verbisky proves a refined form of the Mirror Conjecture for hyperkaehler manifolds, not the conjecture in the strict form. This is explained at page 3 of the paper that you link.

In fact, the assumptions for the conjecture are known to hold only for a hyperkaehler manifold which is generic in its deformation class. More precisely, Verbitsky proves the following result, see page 26.

Theorem 5.4. Let $M$ be a compact holomorphically symplectic manifold, which is generic in its deformation class. Then the Mirror Conjecture holds for $M$, which is mirror dual to itself.

For instance, $\mathrm{Pic}(M)=0$ is enough, see the footnote in the same page.

Answer 2

The following example from Hausel–Thaddeus say that mirror hyperKahler is HyperKahler. Lets explain it

Let $\mathcal M$ be the moduli space of stable $GL_n$-Higgs bundles, non-singular and hyperkahler and $\tilde {\mathcal M}$ moduli space of stable $SL_n$-Higgs bundles, non-singular and hyperkahler and ${\hat{\mathcal{M}}}:= \tilde{\mathcal M} /Γ $ is $PGL_n$-Higgs moduli space which is an orbifold($\Gamma\cong \mathbb Z_n^{2g}$)

Hausel–Thaddeus , proved the following theorem about Strominger-Yau-Zaslow conjecture for the hyperKahler mirror pair $(\hat{\mathcal M}, \tilde {\mathcal M})$

\begin{array} ^\tilde{\mathcal M} & \stackrel{}{\longrightarrow} & \hat{\mathcal M}\\ \downarrow{\tilde\chi} & & \downarrow{\hat\chi} \\ \mathcal A & \stackrel{\cong}{\longrightarrow} & \mathcal A \end{array}

The generic fibers $\tilde\chi^{-1}(a)$ and $\hat \chi^{-1}(a)$ are dual Abelian varieties. The pair of hyperkahler manifolds $(\tilde{\mathcal M} , J)$ and $(\hat{\mathcal M} , J)$ satisfy SYZ conjecture and mirror to each other

See

Mirror symmetry, Langlands duality, and the Hitchin system, Inventiones mathematicae, July 2003, Volume 153, Issue 1, pp 197–229

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 $$\mathcal M(r, d) = \{(A, Φ) \text{ solving }(\star)\}/\mathcal G $$

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

where

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

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.

$\mathcal M(r, d)$ 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.