
## 1. Open problems

1. #### Problem 1.02.

[Boalch/Neitzke] What is the precise relation between the complete hyperkahler metrics on the moduli spaces $\mathcal{M^{\ast }}$ and those on the full (wild) Hitchin moduli space $\mathcal{M}$? Can one view the simpler metric on $\mathcal{M^{\ast }}$ as an approximation?
1. Remark. For genus zero, $\mathcal{M^{\ast }}=\left \{ \left ( V,\triangledown \right )| V \text{ holom. trivial } \right \}$ for fixed $\Sigma =\left ( \Sigma ,\alpha,Q \right )$, and is an open part of the degree zero component of the full moduli space, at least for generic parameters.
• Remark. The open part is often isomorphic a Nakajima quiver variety (arXiv:0806.1050) and more generally a finite dimensional symplectic quotient of a flat space and some copies of T^*G (cf. section 2 of Boalch’s 2001 Adv. Math. paper), which Kronheimer showed to be hyperkahler.
• #### Problem 1.04.

Prove: [Hitchin’s projectively flat connection over $\mathcal{M_{g}}$] $\cong$ [Faltings’ projectively flat connection over $\mathcal{M_{g}}$]
1. Remark. $\tt SU_{n}$+? extensions
• Remark. Example, motivation: Extend Hitchin to other $G$
• #### Problem 1.06.

Extend Laszlo’s proof of the relation between Hitchin’s connection and TUY connection, to the case with punctures. Laszlo looks at the case with no marked points and group $SU_n$, genus at least 3.
1. Remark.
• #### Problem 1.08.

Let $X$ be a surface and fix topological data. Then: $X\rightarrow Bun_{G}\left(X\right)\rightarrow H^{0}\left( Bun_{G},\alpha\right)$ $\underline{X} \rightarrow Bun_{G}\left(X\right)\rightarrow U \,\,\,\,\,\,\,\, (\star)$ Is $(\star)$ naturally projectively flat?
1. Remark. Reference: Peter Scheinost.
• #### Problem 1.1.

[Hoskins] Let $X$ be a complex symplectic algebraic manifold and $G$ a complex reductive group with $G\curvearrowright X$ in a Hamiltonian fashion, $\mu:X\rightarrow \mathfrak{g}^{\ast}$ a moment map and $Y=X//_{\lambda}G=\mu^{-1}(\lambda)/G$. Find examples where:

i) $Y$ is an algebraic completely integrable Hamiltonian system.

ii) $Y\ncong \mathcal{M^{\ast}}(\Sigma)$ (some moduli of tame/wild Higgs on decorated surface)
• #### Problem 1.12.

Extend Hitchin section $B\rightarrow\mathcal{M}_{H}$ to a map $T^{\ast}B\rightarrow\mathcal{M}_{H}$.

Does there exist a formal version?
• #### Problem 1.14.

i) What is the Krull dimension of $\mathcal{O}(T^{\ast}\overline{\mathcal{M}}_{g,n})$?

ii) Is the affinization map an algebraic completely integrable Hamiltonian system?
• #### Problem 1.16.

What happens on singular curves? e.g.

i)$\mathcal{O}(T^{\ast}Bun_{G}(X))$

ii) relate to meromorphic Higgs on $\tilde{X}$

iii) spectral data

iv) extending moduli space $/\mathcal{M}_{g,n}$ to $/\mathcal{\overline{M}}_{g,n}$
•     Let $\Sigma$ a surface and $H$ a handlebody. Then: $\mathcal{L}\subset \mathcal{M}_{B} (holomorphic) \Rightarrow\mathcal{L}\subset \mathcal{M}_{Dol} (real \, Lagrangian)$ for $X^{3},\partial(X)=\Sigma$.

#### Problem 1.18.

Questions:

i) $H(\mathcal{L}_X)$ is a subvariety of $B$ of half dimension. What is the intersection of $\mathcal{L}$ and fibres of the Hitchin map, i.e. the connected components of $H^{-1}(b)\cap \mathcal{L}_X$? [do genus 1 at least]

ii) $RFM(\mathcal{L})=?$
1. Remark. True for handlebodies.
• #### Problem 1.2.

Let $\Sigma, H$ as above.

i) $\mathcal{O}(\mathcal{M}_{B})\xrightarrow{def. quant.}\mathcal{A}_{q}$

ii) $\mathcal{L}\subset\mathcal{M}_{B}\Rightarrow \mathcal{I_{L}}\subset\mathcal{O} (\mathcal{M}_{B})\xrightarrow{def. quant.}\mathcal{I_{L}^{q}}<\mathcal{A}_{q}$ (left ideal)

iii) $\mathcal{I_{L}^{q}}$ acts on $H^{0}(\mathbb{L}^{k},Bun_{G}(\Sigma))$ and this algebra annihilates $v_{H}\in H^{0}(\mathbb{L}^{k},Bun_{G}(\Sigma))$
• #### Problem 1.22.

[L. Anderson] Study confluence of simple poles, e.g. hyperkahler metric on limit as limit of hyperkahler metrics.
1. Remark. This means the type of confluence where one obtains an irregular pole, rather than a tame connection on a stable curve.
• #### Problem 1.24.

Describe the Hitchin-Witten connection on the space of holomorphic sections on the moduli space of Higgs bundles.
• #### Problem 1.26.

Find the vector $v_{H}\in Z(\Sigma)$ associated to a handlebody $H$ bounding $\Sigma$.
• #### Problem 1.28.

[Boalch] Describe the relation between the WKB, Eynard-Orantin, Dumitrescu-Mulase topological recursion and the (wild) non-abelian Hodge correspondence, at least in one example.
• #### Problem 1.3.

Determine the deformation families for branes in the Hitchin system (complex Lagrangian or hyper-holomorphic?).
• ### Langlands duality for meromorphic Higgs bundles

#### Problem 1.32.

[Boalch] The existence of the hyperkahler metrics on the wild Hitchin spaces means the moduli spaces of meromorphic connection on curves (and the corresponding wild character varieties) have natural special Lagrangian torus fibrations, by hyperkahler rotation of the Hitchin fibration (shown to be proper in this generality by Nitsure).

Study the SYZ mirror symmetry proposal in this context. For example suppose we take the dual holomorphic Lagrangian fibration of the smooth fibres of the meromorphic Higgs bundle moduli space. Is the result another Hitchin fibration? What is the involution on the set of data (group, irregular curve, residue data) that corresponds to this T-duality?

For example for holomorphic Higgs bundles it is known one should just replace the group by the Langland dual group, so we are asking for an involution generalizing this.
1. Remark. The approach of Donagi-Pantev works for the Poisson moduli spaces of meromorphic Higgs bundles, so the question is about matching symplectic leaves.
• Remark. Gukov-Witten studied the tame case and described interesting phenomenon related to rigid coadjoint orbits, but it does not seem that a precise involution was established
• #### Problem 1.34.

Ways to construct hyper-holomorphic sheaves on branes, their mirrors and spectral data.
• #### Problem 1.36.

Construct and quantize Lagrangian in the moduli space of tame/wild parabolic Higgs bundles/quiver varieties.
• #### Problem 1.38.

Understand the intersection of Lagrangians coming from 3-manifolds bounding the surface with fibres of the Hitchin fibration.
• ### Hilbert schemes of points on $K2$-surfaces.

There are $12$ known deformation classes of complete hyperkahler manifolds of real dimension four that arise as moduli spaces of meromorphic Higgs bundles. Lets call them $K2$ surfaces. For each of these $K2$ surfaces there is a sequence of Higgs bundle moduli space of dimension $4n$ for each $n$.

#### Problem 1.4.

[Boalch] Is each of these $4n$ dimensional hyperkahler manifolds diffeomorphic to (or a deformation of) the Hilbert scheme of $n$-points on the underlying $K2$ surface?
1. Remark. M. Groechenig (arXiv:1206.5516) has proved this for the tame cases (D4,E6,E7,E8,T^*E) but his methods do not extend to the 7 wild cases
• Remark. At the level of the open parts M^* this is known to work in most cases via the identification of the open parts with ALE spaces (i.e. affine ADE quiver varieties) and a result of Nakajima.
• Remark. This conjecture appears in remark 11.3 of arXiv:1107.0874. The $12$ classes are listed in section 3.2 of arXiv:1203.6607 and the underlying complex manifolds first appeared in the theory of Painleve equations.
• #### Problem 1.42.

[J. Sawon] For $G$ other than $GL(n)$, do we have a smoothing of $\overline{G-Hitchin}$?
• #### Problem 1.44.

[D. Baraglia] What are the algebraic/analytic automorphisms, $\mathcal{M}_{n,d}^{dR}, \mathcal{M}_{n,d}^{Betti}=?$
• #### Problem 1.46.

[S. Gukov] Compute $R=K_{\mathbb{C}^{\ast}}^{quant}(V^{\otimes k}\oplus Adj//_{U(N)})$ and describe what this ring is.
• #### Problem 1.48.

[M. Thaddeus] Does $\mathcal{M}_{Higgs}^{ss}(SL(n,\mathbb{C}))$ have a symplectic resolution?
• #### Problem 1.5.

Is there a topological recursion formula for the $SL(2, \mathbb{C})$-Verlinde’s formula?
• #### Problem 1.52.

Given a measured foliation can we construct the Hitchin system?
• #### Problem 1.54.

Torelli theorems on $\mathcal{M}_{dR}$ with parabolic structure.
• #### Problem 1.56.

What is the mirror of the action $\mathbb{C}^{\ast}\curvearrowright \mathcal{M}_{G_{\mathbb{C}}}$ in the Langlands dual side $\mathcal{M}_{^{L}G_{\mathbb{C}}}$?
• ### The Lax project

#### Problem 1.58.

[Boalch] For each classical integrable system there is often a "Lax pair", which is basically a meromorphic Higgs bundle on the trivial bundle on the Riemann sphere. Thus we can consider the corresponding wild Hitchin moduli space which is a complete hyperkahler manifold. Thus we have a map from integrable systems to complete hyperkahler manifolds.

The project is to go though the integrable systems literature and compile a list Lax pairs and the corresponding irregular curve and the resulting hyperkahler manifold. Further in the cases where integrable systems have alternative Lax pairs, show that the corresponding hyperkahler manifolds are isomorphic.
• ### Modularity conjecture

#### Problem 1.6.

[Boalch] Amongst the real 4d complete hyperkahler manifolds (gravitational instantons) all the known examples of type ALG and ALH are modular, for example the ALG examples are all moduli spaces of meromorphic Higgs bundles (although this has not been proved yet—the examples match precisely). Prove that this holds in higher dimensions, i.e. define classes of complete hyperkahler manifolds (for example by their asymptotics), higherdiemsnional analogues of ALG and ALH, and show that all examples are modular (spaces of solutions of Hitchin’s equations, or the Bogomolnyi equations or other reduction of the ASDYM equations)
• ### Nonlinear representation theory

#### Problem 1.62.

[Boalch] Suppose we define an abstract notion of "wild nonabelian Hodge structure", consisting of a hyperkahler manifold with its various algebraic structures, such that it is a wild Hitchin moduli space for some choice of a group, an irregular curve and residue data. The choice of such a realisation of the abstract space can be viewed as a "representation" or "realisation" of the space. Study this representation theory.

Here we are viewing these moduli spaces as global/nonlinear analogues of algebraic groups (as the Riemann–Hilbert problem suggests).

For example, take one simple example (such as the $D_4$ Hitchin space, the Painleve VI space, $GL_2$ with four simple poles on $P^1$), of complex dimension $2$ and find all the possible representations of this space.
1. Remark. Via the Fourier-Laplace transform (or Katz’s middle convolution) all the moduli spaces on the Riemann sphere have an infinite number of faithful representations, with arbitrarily high rank. (One can think of this phenomenon as analogous to linear representation theory, e.g. the abstract Lie group $SL_2(C)$, has an infinite number of finite dimensional faithful representations, of arbitrarily high rank.).
• Remark. For certain parameters the $D_4$ space also has a representation with structure group $G_2$ (see arXiv:1305.6594).

Cite this as: AimPL: Spectral data for Higgs bundles, available at http://aimpl.org/spectralhiggs.