A Construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n}$
Dashen Yan
TL;DR
This paper constructs explicit non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n}$ for $n\ge 3$ with branching along a codimension-2 ellipsoid, via a double cover endowed with a modified set of ellipsoidal coordinates. The approach uses separation of variables in the modified coordinates to produce $\mathbb{Z}_{2}$-harmonic functions, then combines basic solutions to realize a non-degenerate quadric asymptotics $f_{\boldsymbol{h}}=a_{0}-\sum_{j=1}^{n} a_{j} x_{j}^{2}+O(|\boldsymbol{x}|^{2-n})$, with explicit integral formulas for $a_j$ and surjectivity in parameter space. The work further shows that, at $n=3$, the construction matches Donaldson’s twistor-based $\mathbb{Z}_{2}$-harmonic function via a 1-1 correspondence with harmonic polynomials, and connects $d f_{\boldsymbol{h}}$ to a 2-valued graph that limits to Lawlor’s necks, tying the local model to exact special Lagrangian geometry. Overall, the results provide concrete local models for non-degenerate $\mathbb{Z}_{2}$-harmonic 1-forms with shrinking branching sets and have potential gluing applications in gauge theory and calibrated geometry.
Abstract
We discover an explicit construction of non-degenerate $\mathbb{Z}_{2}$-harmonic functions on $\mathbb{R}^{n},n\geq 3$, using a variant of ellipsoidal coordinates on $\mathbb{R}^{n}$. The branching set of these examples is a codimension-$2$ ellipsoid, providing the first known family of non-degenerate $\mathbb{Z}_{2}$-harmonic $1$-forms on $\mathbb{R}^{n}$ with compact branching sets. Moreover, the graph of the related $\mathbb{Z}_{2}$-harmonic one form in $T^{*}\mathbb{R}^{n}$ can be obtained as a certain limit of a specific sequence of Lawlor's necks in $\mathbb{C}^{n}=T^{*}\mathbb{R}^{n}$.