
## 3. 2k integrable Hitchin equations

1.     Consider the space of “$2k$ integrable Hitchin equations”, with the Higgs fields possibly satisfying some extra conditions (e.g. maybe they commute, maybe they are in the same Borel, etc.).

For instance, these can come from Higgs bundles on a higher dimensional space where the Higgs field satisfies the Simpson condition $\phi\wedge\phi=0$. If the higher dimensional space is of general type and we restrict to a curve in this space, we obtain two Higgs fields $(E,\phi_1,\phi_2)$ where $\phi_1$ and $\phi_2$ are required to commute.

#### Problem 3.1.

Formulate a non-abelian Hodge theorem and mirror symmetry for the moduli spaces of $2k$ integrable Hitchin systems.
There is apparently some study of these in the sense of “generalised Hitchin systems”, coming from higher dimensional versions of Yang-Mills. Ward has constructed solution spaces (reference?).
•     These $2k$ integrable Hitchin systems are supposed to correspond to 4d $\mathcal{N}=1$ SUSY quantum field theories.

#### Problem 3.2.

Classify exact solutions to these $2k$ integrable Hitchin equations, and interpret them in terms of 4d $\mathcal{N}=1$ SUSY theories using marked points on the Riemann surface $C$.

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