(Positive vertex and Taub-NUT type $\mathbb{C}^{3}$) Recall from Section 1.1.3 that $e_{1},e_{2}$ are the homology classes of the two circle factors in the $T^{2}$-fibre, or equivalently an integral basis in $\mathfrak{t}$. The $\mathfrak{t}$-valued curvature 2-form $F=F_{1}e_{1}+F_{2}e_{2}$ satisfies (cf. (1.12))

$$\frac{1}{2\pi}dF_{j}\otimes e_{j}=\frac{\sqrt{-1}}{4\pi}\left(\frac{\partial^{2}W}{\partial\mu_{i}\partial\mu_{j}}+4\frac{\partial^{2}V^{ij}}{\partial\eta\partial\bar{\eta}}\right)d\mu_{i}\wedge d\eta\wedge d\bar{\eta}\otimes e_{j},$$

which is a $\mathfrak{t}$-valued 3-current supported on the codimension 3 discriminant locus $\mathfrak{D}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$. Now take small 3-balls transverse to $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ respectively. The integrals of $\frac{1}{2\pi}dF_{j}\otimes e_{j}$ over the balls are equal to the integrals of the Chern class representative $\frac{1}{2\pi}F$ over the $S^{2}$ linking $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$, which by Section 1.1.3 are $e_{1},-e_{2},-e_{1}+e_{2}$ up to orientation issues. Thus

(1.16)

$$\frac{\sqrt{-1}}{4\pi}\left(\frac{\partial^{2}W}{\partial\mu_{i}\partial\mu_{j}}+4\frac{\partial^{2}V^{ij}}{\partial\eta\partial\bar{\eta}}\right)d\mu_{i}\wedge d\eta\wedge d\bar{\eta}\otimes e_{j}=\mathfrak{D}_{1}\otimes e_{1}-\mathfrak{D}_{2}\otimes e_{2}+\mathfrak{D}_{3}\otimes(e_{2}-e_{1}),$$

where the RHS is a $\mathfrak{t}$-valued codimension 3 cycle. The orientation here is decided by comparing with the Taub-NUT example. If $d\mu_{1}\wedge d\mu_{2}\wedge d\text{Re}\eta\wedge d\text{Im}\eta$ is an orientation form on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$, then the orientation forms on $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ are $d\mu_{2},d\mu_{1},-d\mu_{1}$, compatible with the directions pointing to infinity.

In the variant situation where $\eta\in\mathbb{C}$ instead of being periodic, to which previous discussions still apply, the generalised Gibbons-Hawking ansatz has a scaling symmetry compatible with the distributional equation (1.16): a new solution $\Phi_{\Lambda}$ may be constructed from an old solution $\Phi$ by

(1.17)

$$\begin{cases}\Phi_{\Lambda}(\mu_{1},\mu_{2},\eta)=\Lambda^{-1}\Phi(\Lambda\mu_{1},\Lambda\mu_{2},\Lambda^{1.5}\eta),\\ V^{ij}_{\Lambda}(\mu_{1},\mu_{2},\eta)=\Lambda V^{ij}(\Lambda\mu_{1},\Lambda\mu_{2},\Lambda^{1.5}\eta),\\ W_{\Lambda}(\mu_{1},\mu_{2},\eta)=\Lambda^{2}W(\Lambda\mu_{1},\Lambda\mu_{2},\Lambda^{1.5}\eta)\end{cases}$$

These solutions are isometric up to a scaling factor, analogous to Taub-NUT metrics with different asymptotic circle lengths. The presence of the periodicty condition (or more abstractly an integral lattice structure) breaks down scaling symmetry by singling out a special scale.

(Negative vertex) By a similar argument, in the negative vertex setting (cf. Section 1.1.5) the curvature 2-form $F$ satisfies

(1.18)

$$-\frac{1}{2\pi}dF=-\frac{\sqrt{-1}}{4\pi}\left(\frac{\partial^{2}W^{p\bar{q}}}{\partial\mu\partial\mu}+4\frac{\partial^{2}V}{\partial\eta_{p}\partial\bar{\eta}_{q}}\right)d\mu\wedge d\eta_{p}\wedge d\bar{\eta}_{q}=S,$$

where $S=\{z_{1}+z_{2}=1\}=\{e^{2\pi i\eta_{1}}+e^{2\pi i\eta_{2}}=1\}\subset\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\{0\}\subset\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{R}_{\mu}$ defines a codimension 3 cycle. Here $S$ is endowed with the complex orientation, and the orientation on $\mathbb{C}^{*}_{z_{2}}\times\mathbb{R}_{\mu}$ is defined by the form $d\mu\wedge d\text{Re}\eta_{1}\wedge d\text{Im}\eta_{1}\wedge d\text{Re}\eta_{2}\wedge d\text{Im}\eta_{2}$.