4.1. First order approximate metric [044G]

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4.1. First order approximate metric

We plan to construct an approximate Calabi-Yau metric using the generalised Gibbons-Hawking ansatz, on a singular $S^{1}$ -bundle $M^{-}$ over an open neighbourhood of the origin inside the real 5-dimensional base $\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{R}_{\mu}$ , whose discriminant locus is

S=\{z_{1}+z_{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}.

Let

\eta_{1}=\frac{1}{2\pi\sqrt{-1}}\log z_{1},\quad\eta_{2}=\frac{1}{2\pi\sqrt{-1}}\log z_{2}

be complex variables with period 1, and denote $\eta_{p}=x_{p}+\sqrt{-1}y_{p}$ for $p=1,2$ . The topological situation is described in Section 1.1.5 and the expected complex structure is discussed in Section 1.1.6. A more historical view can be found in Section 1.1.4.

The basic heuristic idea is again to perturb the constant solution (cf. Example 1.6) while incorporating the topology. The information of the constant solution is contained in the base metric

(4.1)

g_{a}=\text{Re}(a_{p\bar{q}}d\eta_{p}\otimes d\bar{\eta}_{q})+A|d\mu|^{2},

where $(a_{p\bar{q}})$ is a Hermitian $2\times 2$ matrix referred to as coupling constants, with determinant $A=\det a$ , and $(a^{p\bar{q}})$ is the transposed inverse matrix such that $a_{j\bar{q}}a^{p\bar{q}}=\delta_{j}^{p}$ . The associated volume measure is

d\text{Vol}_{a}=A^{3/2}dx_{1}\wedge dy_{1}\wedge dx_{2}\wedge dy_{2}\wedge d\mu.

In order for the perturbative way of thinking to be effective, we impose

(4.2)

C^{-1}A^{1/2}\delta_{p\bar{q}}\leq a_{p\bar{q}}\leq CA^{1/2}\delta_{p\bar{q}},\quad A\gg 1.

In this Chapter all constants in estimates depend on $a_{p\bar{q}}$ only through the above scale-invariant uniform ellipticity constant.

Notation.

The $g_{a}$ -distance to the origin is $|(\eta_{1},\eta_{2},\mu)|_{a}=\sqrt{a_{p\bar{q}}\eta_{p}\bar{\eta}_{q}+A\mu^{2}}$ . A variant

\varrho=|(y_{1},y_{2},\mu)|_{a}^{\prime}=(\frac{A}{\mathbb{A}}a_{p\bar{q}}y_{p}y_{q}+A\mu^{2})^{1/2},\quad\mathbb{A}=A+|\text{Im}(a_{1\bar{2}})|^{2}.

stands for the distance function for the Euclidean metric $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}_{y}$

(4.3)

\begin{split}g_{a}^{\prime}=\frac{A}{\mathbb{A}}(a_{1\bar{1}}dy_{1}^{2}+2\text{Re}(a_{1\bar{2}})dy_{1}dy_{2}+a_{2\bar{2}}dy_{2}^{2})+A|d\mu|^{2}.\end{split}

Let $S$ is $R=\text{dist}_{g_{a}}(\cdot,S)$ . The parameter $R+A^{-1/2}$ is relevant for regularity scales.

Now in terms of the local potential $\Phi$ the Calabi-Yau condition (1.9) reads

\det(-4\frac{\partial^{2}\Phi}{\partial\eta_{p}\partial\bar{\eta}_{q}})=\frac{\partial^{2}\Phi}{\partial\mu\partial\mu},

whose linearised equation at the constant solution is the Laplace equation

\Delta_{a}\phi=A^{-1}\frac{\partial^{2}\phi}{\partial\mu\partial\mu}+4a^{p\bar{q}}\frac{\partial^{2}\phi}{\partial\eta_{p}\partial\bar{\eta}_{q}}=0.

Here $\Delta_{a}$ is unsurprisingly the Laplacian of $g_{a}$ . This suggests that at least away from the discriminant locus, the first order correction to $V$ and $W^{p\bar{q}}$ from the constant solution

(4.4)

v=\frac{\partial^{2}\phi}{\partial\mu\partial\mu},\quad w^{p\bar{q}}=-4\frac{\partial^{2}\phi}{\partial\eta_{p}\partial\bar{\eta}_{q}}

ought to be given by $\Delta_{a}$ -harmonic functions,

(4.5)

\Delta_{a}w^{p\bar{q}}=0,\quad\Delta_{a}v=0,\quad v=Aa^{p\bar{q}}w^{p\bar{q}}.

To incorporate the topology we recall the distributional equation (1.18). Since $v$ and $w^{p\bar{q}}$ are linearisations, it makes sense to require the equation on currents

(4.6)

-\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.

The task is to find a compatible solution to (4.4)(4.5)(4.6). As in the last two Chapters, the functions $v$ and $w^{p\bar{q}}$ are global quantities while $\phi$ is only locally defined. The existence of the local potential $\phi$ in (4.4) should be read as imposing some integrability on $v$ and $w^{p\bar{q}}$ (cf. (1.10)(1.11)).

Remark 4.1.

(Motivational Discussion on singularities) We denote

f_{S}=1-z_{1}-z_{2}=1-e^{2\pi i\eta_{1}}-e^{2\pi i\eta_{2}}

and write the 3-current $S$ as

S=\delta(f_{S})\frac{\sqrt{-1}}{4\pi^{2}}df_{S}\wedge d\bar{f}_{S}\wedge d\mu,

which defines a generalised function $\delta(f_{S})$ satisfying the measures identities:

\int\frac{\sqrt{-1}}{4\pi^{2}}d\eta_{p}\wedge d\bar{\eta}_{q}\wedge\delta(f_{S})\wedge df_{S}\wedge d\bar{f}_{S}\wedge d\mu=\int_{S}d\eta_{p}\wedge d\bar{\eta}_{q},

where the notation $\int_{S}d\eta_{p}\wedge d\bar{\eta}_{q}$ is the shorthand for the complex measure $f\mapsto\int_{S}fd\eta_{p}\wedge d\bar{\eta}_{q}$ , and similarly for the LHS. Now $df_{S}=-2\pi\sqrt{-1}(z_{1}d\eta_{1}+z_{2}d\eta_{2})$ , so

\begin{cases}-\int_{S}|z_{2}|^{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{1}\wedge d\bar{\eta}_{1},\\ -\int_{S}|z_{1}|^{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{2}\wedge d\bar{\eta}_{2},\\ \int_{S}\bar{z}_{1}z_{2}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{1}\wedge d\bar{\eta}_{2},\\ \int_{S}\bar{z}_{2}z_{1}\delta(f_{S})d\eta_{1}\wedge d\bar{\eta}_{1}\wedge d\eta_{2}\wedge d\bar{\eta}_{2}\wedge d\mu=\int_{S}\sqrt{-1}d\eta_{2}\wedge d\bar{\eta}_{1}.\\ \end{cases}

The distributional equation (4.6) is written in components as

-\frac{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)=\delta(f_{S})z_{p}\bar{z}_{q}.

Multiplying these equations by $a^{p\bar{q}}$ and summing up, we obtain

(4.7)

-\frac{1}{4\pi}\Delta_{a}v=\delta(f_{S})a^{p\bar{q}}z_{p}\bar{z}_{q},

or equivalently the measure equality

(\Delta_{a}v)d\text{Vol}_{a}=-\int_{S}\pi\sqrt{-1}A^{1/2}a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}=-\int_{S}2\pi A^{1/2}d\mathcal{A},

where $d\mathcal{A}=\frac{\sqrt{-1}}{2}a_{p\bar{q}}d\eta_{p}\wedge d\bar{\eta}_{q}$ is the natural area form on $S$ . A natural guess for $w^{p\bar{q}}$ is then

(4.8)

-\frac{1}{4\pi}\Delta_{a}w^{p\bar{q}}=A^{-1}\delta(f_{S})z_{p}\bar{z}_{q},

or equivalently

(\Delta_{a}w^{p\bar{q}})d\text{Vol}_{a}=-\int_{S}2\pi\frac{A^{-1/2}z_{p}\bar{z}_{q}}{a^{i\bar{j}}z_{i}\bar{z}_{j}}d\mathcal{A},

where summation convention is used. The singularity around $S$ to leading order looks like (cf. Section 4.4 below)

v\sim\frac{A^{1/2}}{2R},\quad w^{p\bar{q}}\sim\frac{A^{-1/2}z_{p}\bar{z}_{q}}{2Ra^{i\bar{j}}z_{i}\bar{z}_{j}},\quad R\sim(\frac{|f_{S}|^{2}}{4\pi^{2}a^{i\bar{j}}z_{i}\bar{z}_{j}}+A\mu^{2})^{1/2},

which is compatible with the singularity in the distributional equation (4.6).

Now we move on to a more formal construction. The main idea is to write down the solution via a periodic version of Green’s representation. The series

(4.9)

\gamma(\eta_{1},\eta_{2},\mu)=-\frac{1}{8\pi^{2}}\sum_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\frac{1}{|(\eta_{1}+n_{1},\eta_{2}+n_{2},\mu)|_{a}^{3}},

converges absolutely away from $\mathbb{Z}^{2}\times\{0\}\subset\mathbb{C}_{\eta_{1}}\times\mathbb{C}_{\eta_{2}}\times\mathbb{R}_{\mu}$ and is $\mathbb{Z}^{2}$ -periodic, so descends to a function on $\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{R}_{\mu}$ which is the periodic Newtonian potential. We shall extract the asymptote for $\gamma(\eta_{1},\eta_{2},\mu)$ :

Lemma 4.1.

For $\varrho\gtrsim A^{1/4}$ , we have

|\gamma(\eta_{1},\eta_{2},\mu)+\frac{1}{4\pi\varrho\sqrt{\mathbb{A}}}|\leq C\varrho^{-3}.

Proof.

We consider the closely related integral

\bar{\gamma}(y_{1},y_{2},\mu)=-\frac{1}{8\pi^{2}}\int\frac{1}{(a_{p\bar{q}}\eta_{p}\bar{\eta}_{q}+A\mu^{2})^{3/2}}dx_{1}dx_{2}.

After substituting the variables

\begin{cases}x_{1}^{\prime}=x_{1}-\frac{\text{Im}(a_{2\bar{1}})}{a_{1\bar{1}}}y_{2}+\frac{\text{Re}(a_{2\bar{1}})}{a_{1\bar{1}}}x_{2},\\ x_{2}^{\prime}=x_{2}+\frac{\text{Im}(a_{2\bar{1}})\text{Re}(a_{2\bar{1}})}{\mathbb{A}}y_{2}+\frac{a_{1\bar{1}}\text{Im}(a_{2\bar{1}})}{\mathbb{A}}y_{1},\end{cases}

we complete the square

a_{p\bar{q}}\eta_{p}\bar{\eta}_{q}+A\mu^{2}=a_{1\bar{1}}x_{1}^{\prime 2}+\frac{\mathbb{A}}{a_{1\bar{1}}}x_{2}^{\prime 2}+|(y_{1},y_{2},\mu)|_{a}^{\prime 2}=a_{1\bar{1}}x_{1}^{\prime 2}+\frac{\mathbb{A}}{a_{1\bar{1}}}x_{2}^{\prime 2}+\varrho^{2}.

This allows us to evaluate using polar coordinates

\bar{\gamma}=-\frac{1}{4\pi\varrho\sqrt{\mathbb{A}}}.

For fixed $y_{1},y_{2},\mu$ , we can compare the integral $\bar{\gamma}$ with the series $\gamma$ , by estimating the difference using the mean value inequality

\begin{split}&\frac{1}{|(\eta_{1},\eta_{2},\mu)|_{a}^{3}}-\int_{[x_{1}-\frac{1}{2},x_{1}+\frac{1}{2}]\times[x_{2}-\frac{1}{2},x_{2}+\frac{1}{2}]}\frac{1}{|(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{3}}ds_{1}ds_{2}\\ \leq&\frac{CA^{1/2}}{|(\eta_{1},\eta_{2},\mu)|_{a}^{5}}.\end{split}

Summing over all square regions, and applying Cauchy integral test,

\begin{split}|\gamma-\bar{\gamma}|&\leq CA^{1/2}\sum_{n,m}\frac{1}{|(\eta_{1}+n,\eta_{2}+m,\mu)|_{a}^{5}}\\ &\leq CA^{1/2}\int\frac{1}{|(s_{1}+\sqrt{-1}y_{1},s_{2}+\sqrt{-1}y_{2},\mu)|_{a}^{5}}ds_{1}ds_{2}\\ &\leq C\varrho^{-3},\end{split}

as required. ∎

Lemma 4.2.

For $\varrho\lesssim A^{1/4}$ and $|x_{1}|,|x_{2}|\leq\frac{1}{2}$ we have

|\gamma(\eta_{1},\eta_{2},\mu)+\frac{1}{8\pi^{2}|(\eta_{1},\eta_{2},\mu)|_{a}^{3}}|\leq CA^{-3/4}.

Proof.

Modify the above proof to control the series for $(n_{1},n_{2})\in\mathbb{Z}^{2}\setminus\{0\}$ . ∎

Before proceeding further we recall that topologically $S$ is a thrice punctured 2-sphere. The 3 punctures correspond to 3 ends of $S$ :

\begin{cases}y_{2}>1,\quad\eta_{1}=\frac{1}{2\pi i}\log(1-e^{2\pi i\eta_{2}}),\\ y_{1}>1,\quad\eta_{2}=\frac{1}{2\pi i}\log(1-e^{2\pi i\eta_{1}}),\\ y_{1}<-1,\quad y_{2}<-1,\quad\eta_{2}-\eta_{1}=\frac{1}{2\pi i}\log(-1+e^{-2\pi i\eta_{1}}).\end{cases}

At infinity these are respectively asymptotic to $\mathfrak{D}_{1}\times S^{1}$ , $\mathfrak{D}_{2}\times S^{1}$ , $\mathfrak{D}_{3}\times S^{1}$ where

\mathfrak{D}_{1}=\{y_{1}=0,y_{2}>0,\mu=0\},\quad\mathfrak{D}_{2}=\{y_{2}=0,y_{1}>0,\mu=0\},\quad\mathfrak{D}_{3}=\{y_{1}=y_{2}<0,\mu=0\}.

The image of $S$ under the log map $\mathbb{C}^{2}_{z_{1},z_{2}}\to\mathbb{R}^{2}_{y_{1},y_{2}}$ (called the ‘amoeba’) is

(4.10)

\text{Image}(S)=\{e^{-2\pi y_{1}}+e^{-2\pi y_{2}}\geq 1,e^{-2\pi y_{1}}+1\geq e^{-2\pi y_{2}},e^{-2\pi y_{2}}+1\geq e^{-2\pi y_{1}}\},

which is a thickening of the trivalent graph $\mathfrak{D}=\mathfrak{D}_{1}\cup\mathfrak{D}_{2}\cup\mathfrak{D}_{3}\cup\{0\}.$ This is the simplest case of a general picture for amoebas of algebraic varieties [31].

We can now make the following definitions, involving a cutoff and limiting procedure for logarithmically divergent integrals.

(4.11)

\begin{cases}\gamma_{1}(\eta_{1},\eta_{2},\mu)=&-\text{Re}\lim_{\Lambda\to\infty}\{\pi A^{1/2}\int_{S\cap\{y_{2}^{\prime}<\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime})+\frac{1}{2\sqrt{a_{2\bar{2}}}}\log 2\Lambda\}\\ \gamma_{2}(\eta_{1},\eta_{2},\mu)=&-\text{Re}\lim_{\Lambda\to\infty}\{\pi A^{1/2}\int_{S\cap\{y_{1}^{\prime}<\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\\ &\sqrt{-1}d\eta_{1}^{\prime}\wedge(d\bar{\eta}_{1}^{\prime}-d\bar{\eta}_{2}^{\prime})+\frac{1}{2\sqrt{a_{1\bar{1}}}}\log 2\Lambda\}\\ \gamma_{3}(\eta_{1},\eta_{2},\mu)=&-\text{Re}\lim_{\Lambda\to\infty}\{\pi A^{1/2}\int_{S\cap\{y_{1}^{\prime}>-\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\\ &\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}+\frac{1}{2\sqrt{a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}}}\log 2\Lambda\}\\ \gamma_{4}(\eta_{1},\eta_{2},\mu)=&-\text{Im}\{\pi A^{1/2}\int_{S}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\sqrt{-1}d\eta_{2}^{\prime}\wedge d\bar{\eta}_{1}^{\prime}\}.\end{cases}

The desired first order corrections $v$ and $w^{p\bar{q}}$ are constructed as linear combinations:

(4.12)

\begin{cases}v=Aa^{p\bar{q}}w^{p\bar{q}},\\ w^{1\bar{1}}=\gamma_{1}+\gamma_{3},\\ w^{1\bar{2}}=-(\gamma_{3}+\sqrt{-1}\gamma_{4}),\\ w^{2\bar{1}}=-(\gamma_{3}-\sqrt{-1}\gamma_{4}),\\ w^{2\bar{2}}=\gamma_{2}+\gamma_{3}.\end{cases}

The advantage of $\gamma_{i}$ is that they only involve divergence issues at one end. This is because the measures $\text{Re}(\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime}))$ etc decay exponentially along all but one end, with respect to the Lebesgue measure on the three asymptotic cylinders.

Lemma 4.3.

The limits defining $\gamma_{i}$ converge as $\Lambda\to+\infty$ .

Proof.

We focus on $\gamma_{1}$ . The 2-form $\sqrt{-1}d\eta_{2}\wedge(d\bar{\eta}_{2}-d\bar{\eta}_{1})$ on $S$ is exponentially small along the $\mathfrak{D}_{2},\mathfrak{D}_{3}$ ends, so the only divergence problem happens at infinity along the $\mathfrak{D}_{1}$ end.

Applying Lemma 4.1 allows us to replace $\gamma$ by the much simpler function

-\frac{1}{4\pi\sqrt{\mathbb{A}}}|(y_{1}-y_{1}^{\prime},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}.

The integral

\int_{S\cap\{y_{2}^{\prime}<\Lambda\}}\gamma(\eta_{1}-\eta_{1}^{\prime},\eta_{2}-\eta_{2}^{\prime},\mu)\sqrt{-1}d\eta_{2}^{\prime}\wedge(d\bar{\eta}_{2}^{\prime}-d\bar{\eta}_{1}^{\prime})

has the same divergence behaviour as

\int^{\Lambda}-\frac{1}{2\pi\sqrt{\mathbb{A}}}|(y_{1},y_{2}-y_{2}^{\prime},\mu)|_{a}^{\prime-1}dy_{2}^{\prime}\sim-\frac{1}{2\pi\sqrt{Aa_{2\bar{2}}}}\log\Lambda,

which is cancelled by the log term we put in the limit. ∎

By the construction of the Green representations,

Proposition 4.4.

The functions $v$ and $w^{p\bar{q}}$ satisfy the decoupled Laplace equations with distributional terms (4.7) and (4.8).

However the original linearised equations we set off to solve is an overdetermined coupled system, not just the decoupled Laplace equations. We still need to check the integrability equation (4.4) and the distributional equation (4.6).

Lemma 4.5.

The following integrability condition is satisfied globally

\frac{\partial w^{p\bar{q}}}{\partial\eta_{r}}=\frac{\partial w^{rq}}{\partial\eta_{p}},\quad\frac{\partial w^{p\bar{q}}}{\partial\bar{\eta}_{r}}=\frac{\partial w^{pr}}{\partial\bar{\eta}_{q}},\quad p,q,r=1,2.

Proof.

We consider the Laplacian

\begin{split}\Delta_{a}(\frac{\partial w^{p\bar{q}}}{\partial\eta_{r}}-\frac{\partial w^{rq}}{\partial\eta_{p}})=&\frac{\partial}{\partial\eta_{r}}\Delta_{a}w^{p\bar{q}}-\frac{\partial}{\partial\eta_{p}}\Delta_{a}w^{rq}\\ =&-4\pi A^{-1}\{\frac{\partial}{\partial\eta_{r}}(\delta(f_{S})z_{p}\bar{z}_{q})-\frac{\partial}{\partial\eta_{p}}(\delta(f_{S})z_{r}\bar{z}_{q})\}\\ =&-4\pi A^{-1}\delta(f_{S})\{\frac{\partial}{\partial\eta_{r}}(z_{p}\bar{z}_{q})-\frac{\partial}{\partial\eta_{p}}(z_{r}\bar{z}_{q})\}=0,\end{split}

where we have crucially used that $S$ is an algebraic cycle to deduce $\partial\delta(f_{S})=0$ . Thus $\frac{\partial w^{p\bar{q}}}{\partial\eta_{r}}-\frac{\partial w^{rq}}{\partial\eta_{p}}=0$ would follow from a Liouville theorem argument, by checking some a priori growth condition

|\frac{\partial w^{p\bar{q}}}{\partial\eta_{r}}|\lesssim\begin{cases}R^{-1},\quad&\varrho\gtrsim A^{1/4},\\ A^{1/4}R^{-2},\quad&\varrho\lesssim A^{1/4},\end{cases}

which is easy to derive using the techniques in the previous lemmas in this Section. The $\bar{\eta}$ derivatives can be treated similarly. ∎

Corollary 4.6.

The distributional equation (4.6) is satisfied. In component form,

-\frac{1}{4\pi}(\frac{\partial^{2}w^{p\bar{q}}}{\partial\mu\partial\mu}+4\frac{\partial^{2}v}{\partial\eta_{p}\partial\bar{\eta}_{q}})=\delta(f_{S})z_{p}\bar{z}_{q}.

Proof.

Let’s focus on $p=q=1$ . By Proposition 4.4,

\frac{1}{4\pi}\frac{\partial^{2}w^{1\bar{1}}}{\partial\mu\partial\mu}+\frac{1}{\pi}Aa^{i\bar{j}}\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}}=-\delta(f_{S})|z_{1}|^{2}.

But by Lemma 4.6 we have $\frac{\partial^{2}w^{i\bar{j}}}{\partial\eta_{1}\partial\bar{\eta}_{1}}=\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}},$ so $Aa^{i\bar{j}}\frac{\partial^{2}w^{1\bar{1}}}{\partial\eta_{i}\partial\bar{\eta}_{j}}=\frac{\partial^{2}v}{\partial\eta_{1}\partial\bar{\eta}_{1}},$ hence the claim. ∎

Lemma 4.5 and Corollary 4.6 combine to imply the local existence of the potential away from $S$ as is required in (4.4). Taking stock of our progress,

Proposition 4.7.

(First order linearised solution) The functions $v$ and $w^{p\bar{q}}$ solve the integrability condition (4.4) and the harmonicity condition (4.5) away from $S$ , and the distributional equation (4.6) globally.

Remark 4.2.

It will turn out in the next few Sections that $v$ and $w^{p\bar{q}}$ have logarithmic growth at infinity bounded away from $S$ . If we restrict to solutions to (4.4)(4.5)(4.6) with the same growth properties, then $v$ and $w^{p\bar{q}}$ are unique up to additive constants. The choices of these constants are not completely canonical, related to the philosophy that the Ooguri-Vafa type metrics are only effective descriptions admitting a certain amount of small fluctuation.

We obtain by the generalised Gibbons-Hawking construction a Kähler ansatz $(g^{(1)},\omega^{(1)},J^{(1)},\Omega^{(1)})$ associated to

(4.13)

V_{(1)}=A+v,\quad W^{p\bar{q}}_{(1)}=a_{p\bar{q}}+w^{p\bar{q}}.

A subtlety here is that the $S^{1}$ -connection $\vartheta$ can be twisted by a flat connection. This choice is parametrised by $H^{1}((\mathbb{C}^{*})^{2}\times\mathbb{R}\setminus S,\mathbb{R}/\mathbb{Z})=H^{1}((\mathbb{C}^{*})^{2}\times\mathbb{R},\mathbb{R}/\mathbb{Z})=T^{2}$ , since the codimension 3 subset $S$ inside the base does not affect the fundamental group. We sometimes suppress mentioning this choice since it does not have a strong impact on the geometry, especially because we will exclusively work with $S^{1}$ -invariant tensors, which are rarely sensitive to the flat connection. The Kähler structure is well defined away from $S$ , over a bounded region where $V_{(1)}>0$ and $W^{p\bar{q}}_{(1)}$ is positive definite; the metric is incomplete. We will specify more precisely the ambient space $M^{-}$ of the Kähler ansatz once we obtain sufficiently accurate asymptotes on $V_{(1)}$ and $W^{p\bar{q}}_{(1)}$ to check positive definiteness (cf. Corollary 4.17).

The family of ansatzs admit $S_{3}$ -discrete symmetries, generated by

\eta_{1}\leftrightarrow\eta_{2},\quad\eta_{2}\leftrightarrow\eta_{2}-\eta_{1}+\frac{1}{2},\quad\eta_{1}\leftrightarrow\eta_{1}-\eta_{2}+\frac{1}{2},\quad\mu\to\mu.

These actions on $(\mathbb{C}^{*})^{2}\times\mathbb{R}_{\mu}$ preserve $S$ , and respectively interchange $\mathfrak{D}_{1}$ with $\mathfrak{D}_{2}$ , $\mathfrak{D}_{1}$ with $\mathfrak{D}_{3}$ , and $\mathfrak{D}_{2}$ with $\mathfrak{D}_{3}$ . The induced action on coupling constants permute $a_{1\bar{1}},a_{2\bar{2}},a_{1\bar{1}}+a_{1\bar{2}}+a_{2\bar{1}}+a_{2\bar{2}}$ , and act on $\text{Im}(a_{1\bar{2}})$ by $\pm 1$ depending on the sign of the permutation.