| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

9. Other problems

This is the section for problems which (a) were not multi-part and (b) I had fewer notes on.
    1. Simplify the fibres of the Hodge bundle.

      Problem 9.1.

      Seek notion of genericity for complex structures simplifying the stratification of the singular locus of the Hitchin fibration (for $SL_2(\mathbb{C})$).
        • Relation to Hilbert schemes of points.

          Problem 9.2.

          Better understand the relationship between the moduli of Higgs bundles on $C$ and the Hilbert scheme of points $\text{Hilb}^{\text{pt}}(T^*C)$.
            • Compactifications of the moduli of Higgs bundles.

              Problem 9.3.

              Compare compactifications of the moduli space of Higgs bundles for $GL_n(\mathbb{C})$ of Hausel, Schmitt, and MSWW-F.
                • Deformations from Hochschild homology.

                  Problem 9.4.

                  Give a geometric realisation of objects given by deforming a Hitchin pair by a non-zero class in $HH_0(C)$ (c.f. Keller).
                    • Branes in the moduli of Higgs bundles.

                      Problem 9.5.

                      Review techniques for constructing Lagrangians in hyperkahler spaces, see which can be applied to the moduli space of Higgs bundles, and explicitly translate/apply those techniques.
                        • $tt^*$-equations and Higgs bundles.

                          Problem 9.6.

                          Make explicit the relationship between the $tt^*$-equations and Higgs bundles in a specific example.

                              Cite this as: AimPL: Singular geometry and Higgs bundles in string theory, available at http://aimpl.org/singularhiggs.