Chapter 3 The Positive Vertex

3.1. First order approximate metric

We plan to construct an approximate Calabi-Yau metric using the generalised Gibbons-Hawking ansatz, on a singular $T^{2}$ -bundle $M^{+}$ over an open neighbourhood of the origin inside the real 4-dimensional base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ , whose discriminant locus is

\begin{split}\mathfrak{D}&=\mathfrak{D}_{1}\cup\mathfrak{D}_{2}\cup\mathfrak{D}_{3}\cup\{0\}=\{\mu_{1}=0,\mu_{2}>0\}\cup\{\mu_{2}=0,\mu_{1}>0\}\cup\{\mu_{1}=\mu_{2}<0\}\cup\{0\}\\ &\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\{0\}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}.\end{split}

Here $\eta=x+\sqrt{-1}y$ is a complex variable with period 1. The topological situation is described in Section 1.1.3, Example 1.9 and the expected complex structure can be found in Section 1.1.6.

This situation has very strong similarity with the Taub-NUT type metric on $\mathbb{C}^{3}$ in Chapter 2, the only difference being the periodicity condition on $\eta$ . The basic heuristic idea is to perturb the constant solution (cf. Example 1.6) after incorporating the topology. The information in the constant solution is encoded by the base metric

(3.1)

g_{a}=a_{ij}d\mu_{i}\otimes d\mu_{j}+A|d\eta|^{2}

with $a_{ij}$ being a real symmetric positive definite matrix and $A=\det a$ , analogous to Section 2.1. We call $a_{ij}$ the coupling constants and emphasize that $a_{ij}$ are parameters we would like to vary. We impose the scale invariant ellipticity bound

(3.2)

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

The $A\gg 1$ assumption is essential for the perturbative way of thinking to be effective; this assumption was absent in the $\mathbb{C}^{3}$ case because there was no intrinsic scale provided by periodicity. The appearance of the gluing parameter $A$ means we need to carefully track down $A$ -dependence in our estimates; in this Chapter all constants in estimates depend on $a_{ij}$ only through the above scale invariant ellipticity constant unless stated otherwise.

Notation.

We denote $\vec{\mu}=(\mu_{1},\mu_{2},\eta)$ and $|\vec{\mu}|_{a}=\sqrt{a_{ij}\mu_{i}\mu_{j}+A|\eta|^{2}}$ is the $g_{a}$ -distance to the origin. A variant $\varrho=|(\mu_{1},\mu_{2},y)|_{a}^{\prime}=\sqrt{a_{ij}\mu_{i}\mu_{j}+Ay^{2}}$ stands for the distance in the $g_{a}^{\prime}$ -metric on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}_{y}$

(3.3)

g_{a}^{\prime}=a_{ij}d\mu_{i}d\mu_{j}+Ady^{2}=a_{ij}d\mu_{i}d\mu_{j}+A|d\text{Im}(\eta)|^{2}.

Another useful length parameter is $\ell=\text{dist}_{g_{a}}(\cdot,\mathfrak{D})+A^{-1/4}$ which is relevant for regularity scales.

Exactly the same discussions as in Section 2.1 lead us to consider the linearised equations (2.2)(2.3)(2.4), which describe the first order corrections we need to make to the constant solution. The key difference is the periodicity requirement. The principle of superposition allows us to immediately produce the solution from Proposition 2.2. We recall from there the functions $\alpha_{1},\alpha_{2},\alpha_{3}$ .

Proposition 3.1.

We define the functions $\tilde{\alpha}_{i}(\mu_{1},\mu_{2},\eta)$ by

(3.4)

\begin{cases}\tilde{\alpha}_{1}=\alpha_{1}(\mu_{1},\mu_{2},\eta)+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{22}}}\}\\ \tilde{\alpha}_{2}=\alpha_{2}(\mu_{1},\mu_{2},\eta)+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\alpha_{2}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{11}}}\}\\ \tilde{\alpha}_{3}=\alpha_{3}(\mu_{1},\mu_{2},\eta)+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\alpha_{3}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{11}+2a_{12}+a_{22}}}\}\end{cases}

Then $\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\alpha}_{3}$ are convergent away from $\mathfrak{D}$ , 1-periodic in $\eta$ , and $\Delta_{a}$ -harmonic away from $\mathfrak{D}$ . Morever the functions

\tilde{v}^{11}=\tilde{\alpha}_{1}+\tilde{\alpha}_{3},\quad\tilde{v}^{22}=\tilde{\alpha}_{2}+\tilde{\alpha}_{3},\quad\tilde{v}^{12}=\tilde{v}^{21}=-\tilde{\alpha}_{3},\quad\tilde{w}=Aa^{ij}\tilde{v}^{ij}

provide a solution in the periodic setting to (2.2)(2.3) away from $\mathfrak{D}$ , which also solves the distributional equation (2.4) globally.

Proof.

The only issue worth checking is convergence, which follows from the fact that $|\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{22}}}|\leq\frac{C(a_{ij},\mu_{1},\mu_{2},\eta)}{n^{2}}$ , and likewise for $\alpha_{2},\alpha_{3}$ . ∎

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

\tilde{V}^{ij}_{(1)}=a_{ij}+\tilde{v}^{ij},\quad\tilde{W}_{(1)}=A+\tilde{w}.

A subtlety here is that the connection $\vartheta=(\vartheta_{1},\vartheta_{2})$ can be twisted by a flat connection. This choice is parametrised by $H^{1}(\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times S^{1}\times\mathbb{R}\setminus\mathfrak{D},T^{2})=H^{1}(\mathbb{R}^{2}\times S^{1}\times\mathbb{R},T^{2})=T^{2}$ , since the codimension 3 subset $\mathfrak{D}$ 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 $T^{2}$ -invariant tensors, which are rarely sensitive to the flat connection.

The Kähler structure is well defined over the region where the matrix $\tilde{V}_{(1)}^{ij}$ is positive definite and $\tilde{W}_{(1)}$ is positive, except at the singular point $(\mu_{1},\mu_{2},\eta)=0$ . A sufficient condition for positive definiteness will be given in (3.9). The Kähler structure extends smoothly across $\mathfrak{D}_{i}$ , where the local structure is modelled on the Taub-NUT fibration described by $g_{\text{Taub}}$ for $|\vec{\mu}|_{a}\gtrsim A^{-1/4}$ (cf. Section 2.3).

Remark 3.1.

The series definition of $\tilde{\alpha}_{i}$ involves ‘subtracting a logarithmic infinity from a logarithmic infinity’, as in the usual Ooguri-Vafa metric.

Remark 3.2.

Compared to the Taub-NUT type $\mathbb{C}^{3}$ case in Chapter 2, the $T^{2}$ -symmetry and the discrete symmetry persist, while the scaling symmetry and the additional $U(1)$ -symmetry are now broken.

Remark 3.3.

There is some freedom to add some additive constants to the definition of $\tilde{\alpha}_{1},\tilde{\alpha}_{2},\tilde{\alpha}_{3}$ , which does not affect the validity of the linearised equations. Our choice ensures that $\tilde{\alpha}_{i}-\alpha_{i}$ vanishes at the origin, which is need later for gluing in the Taub-NUT type metric on $\mathbb{C}^{3}$ . A more quantitative statement is:

Lemma 3.2.

Let $|x|=|\text{Re}(\eta)|\leq\frac{1}{2}$ . The difference $\tilde{\alpha}_{i}-\alpha_{i}$ satisfies the estimate

|\tilde{\alpha}_{i}-\alpha_{i}|\leq CA^{-1/4}\log(1+A^{-1/2}|\vec{\mu}|_{a}).

Proof.

In the series (3.4) defining $\tilde{\alpha}_{1}$ , we can separate the sum into two ranges $|n|\gtrsim A^{-1/2}|\vec{\mu}|_{a}+1$ and $1\leq|n|\lesssim A^{-1/2}|\vec{\mu}|_{a}$ . In the first range, using elementary Taylor expansion of arctan,

|\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{22}}}|\leq\frac{C|\vec{\mu}|_{a}}{A^{3/4}|n|^{2}},

which implies after summation

\begin{split}\sum_{|n|\gtrsim A^{-1/2}|\vec{\mu}|_{a}+1}|\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{22}}}|\leq CA^{-1/4}\min\{1,A^{-3/4}|\vec{\mu}|_{a}\}.\end{split}

The second range only appears if $1\lesssim A^{-1/2}|\vec{\mu}|_{a}$ . This sum is crudely estimated by

\begin{split}&\sum_{1\leq|n|\lesssim A^{-1/2}|\vec{\mu}|_{a}}|\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\frac{1}{4|n|\sqrt{a_{22}}}|\leq CA^{-1/4}\sum_{1\leq|n|\lesssim A^{-1/2}|\vec{\mu}|_{a}}\frac{1}{n}\\ &\leq CA^{-1/4}\log(A^{-3/4}|\vec{\mu}|_{a}).\end{split}

Combining the discussions gives the result. ∎

3.2. Asymptotes for the first order ansatz

This Section is concerned with obtaining refined asymptotes of $\tilde{\alpha}_{1}$ , $\tilde{\alpha}_{2}$ , $\tilde{\alpha}_{3}$ ; a summary can be found at the end of the Section. We define the average functions

(3.5)

\begin{cases}\bar{\alpha}_{1}(\mu_{1},\mu_{2},y)=\int_{0}^{1}\tilde{\alpha}_{1}(\mu_{1},\mu_{2},x+\sqrt{-1}y)dx,\\ \bar{\alpha}_{2}(\mu_{1},\mu_{2},y)=\int_{0}^{1}\tilde{\alpha}_{2}(\mu_{1},\mu_{2},x+\sqrt{-1}y)dx,\\ \bar{\alpha}_{3}(\mu_{1},\mu_{2},y)=\int_{0}^{1}\tilde{\alpha}_{3}(\mu_{1},\mu_{2},x+\sqrt{-1}y)dx.\end{cases}

The main goal in this Section is to prove exponential decay estimate for $\tilde{\alpha}_{i}-\bar{\alpha}_{i}$ outside a tubular neighbourhood of $\mathfrak{D}_{i}$ .

Proposition 3.3.

(Leading order asymptote) The formulae for $\bar{\alpha}_{i}$ are explicitly given as

(3.6)

\begin{cases}\bar{\alpha}_{1}=&\frac{1}{2\sqrt{a_{22}}}\{\log 2-\gamma_{E}-\log\left(\frac{1}{\sqrt{A}}|(\mu_{1},\mu_{2},y)|_{a}^{\prime}-\frac{\sqrt{a_{22}}}{\sqrt{A}}(\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1})\right)\}\\ \bar{\alpha}_{2}=&\frac{1}{2\sqrt{a_{11}}}\{\log 2-\gamma_{E}-\log\left(\frac{1}{\sqrt{A}}|(\mu_{1},\mu_{2},y)|_{a}^{\prime}-\frac{\sqrt{a_{11}}}{\sqrt{A}}(\mu_{1}+\frac{a_{12}}{a_{11}}\mu_{2})\right)\}\\ \bar{\alpha}_{3}=&\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}}\{\log 2-\gamma_{E}\\ &-\log\left(\frac{1}{\sqrt{A}}|(\mu_{1},\mu_{2},y)|_{a}^{\prime}+\frac{a_{11}\mu_{1}+a_{12}\mu_{2}+a_{21}\mu_{1}+a_{22}\mu_{2}}{\sqrt{A}\sqrt{a_{11}+2a_{12}+a_{22}}}\right)\}\end{cases}

where $\gamma_{E}=\lim_{n\to\infty}\sum_{k=1}^{n}\frac{1}{k}-\log n$ is the Euler constant.

Proof.

We will focus on $\bar{\alpha}_{1}$ . The periodic version of equation (2.7) on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ is the measure equation

(\Delta_{a}\tilde{\alpha}_{1})A^{3/2}d\mu_{1}\wedge d\mu_{2}\wedge dx\wedge dy=-2\pi\sqrt{A}\int_{\mathfrak{D}_{1}}d\mu_{2}.

Integrating in the periodic $x$ -variable from 0 to 1,

(3.7)

(\Delta_{a}^{\prime}\bar{\alpha}_{1})d\text{Vol}_{a}^{\prime}=-2\pi\int_{\mathfrak{D}_{1}}d\mu_{2},

where $\Delta_{a}^{\prime}$ is the Laplacian of the metric $g_{a}^{\prime}=a_{ij}d\mu_{i}d\mu_{j}+Ady^{2}$ on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}$ , whose volume form is $d\text{Vol}_{a}^{\prime}=Ad\mu_{1}\wedge d\mu_{2}\wedge dy$ .

Now the basic strategy is to build a function satisfying the same measure equation and then compare. For a large positive cutoff $\Lambda$ , we calculate the Green representation

\begin{split}&-\frac{1}{4\pi}\int_{0}^{\Lambda}\frac{-2\pi}{|(\mu_{1},\mu_{2}-s,y)|_{a}^{\prime}}ds=\frac{1}{2\sqrt{a_{22}}}\sinh^{-1}\left(\frac{s}{\sqrt{\frac{A}{a_{22}^{2}}\mu_{1}^{2}+\frac{A}{a_{22}}y^{2}}}\right)|_{s=-\mu_{2}-\frac{a_{12}}{a_{22}}\mu_{1}}^{s=-\mu_{2}-\frac{a_{12}}{a_{22}}\mu_{1}+\Lambda}.\end{split}

If we subtract $\frac{1}{2\sqrt{a_{22}}}\log(2\Lambda)$ and take the limit $\Lambda\to\infty$ , we obtain the function

-\frac{1}{2\sqrt{a_{22}}}\log\left(\frac{1}{\sqrt{a_{22}}}|(\mu_{1},\mu_{2},y)|_{a}^{\prime}-\mu_{2}-\frac{a_{12}}{a_{22}}\mu_{1}\right)

which by construction satisfies the same measure equation as (3.7).

We claim that this function differs from $\bar{\alpha}_{1}$ by a constant. By the Liouville theorem, it suffices to show that the function $\bar{\alpha}_{1}$ on $\mathbb{R}^{2}\times\mathbb{R}$ has the logarithmic growth estimate

\bar{\alpha}_{1}\leq CA^{-1/4}\{\log(1+A^{-3/4}\varrho)+|\log(\frac{1}{a_{22}^{-1}\mu_{1}^{2}+y^{2}})|+1\}

which is easy to deduce from Lemma 3.2.

Now to pin down the constant, we can evaluate $\bar{\alpha}_{1}$ for $\mu_{1}=\mu_{2}=0,y\neq 0$ . Then the $\arctan$ term drops out, and

\begin{split}\bar{\alpha}_{1}(0,0,y)&=\frac{1}{2\sqrt{a_{22}}}\lim_{\Lambda\to+\infty}\{\int_{0}^{\Lambda}\frac{1}{\sqrt{x^{2}+y^{2}}}dx-\sum_{n=1}^{\Lambda}\frac{1}{n}\}\\ &=\frac{1}{2\sqrt{a_{22}}}\lim_{\Lambda\to\infty}\{\sinh^{-1}(\frac{\Lambda}{|y|})-\log\Lambda-\gamma_{E}\}\\ &=\frac{1}{2\sqrt{a_{22}}}(\log(\frac{2}{|y|})-\gamma_{E})\end{split}

Comparing the expressions give the formula for $\bar{\alpha}_{1}$ . ∎

Lemma 3.4.

The difference $\tilde{\alpha}_{1}-\bar{\alpha}_{1}$ satisfies the following estimate: if either $y^{2}+a_{22}^{-1}\mu_{1}^{2}\gtrsim 1$ or $\mu_{2}\lesssim-A^{1/4}$ , namely if $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\gtrsim A^{1/2}$ , then $|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|\leq CA^{-1/4}$ . Similar bounds hold for $\tilde{\alpha}_{i}-\bar{\alpha}_{i}$ for $i=1,2,3$ .

Proof.

We notice in advance that $\tilde{\alpha}_{1}$ and $\bar{\alpha}_{1}$ are periodic in $\eta$ , so it suffices to assume $|x|\leq 1/2$ . The main idea of a variant of Cauchy’s integral test for convergence.

Using the fact that $|\frac{\partial^{2}\alpha_{1}}{\partial x^{2}}|\leq C\frac{a_{22}}{(\mu_{1}^{2}+a_{22}|\eta|^{2})^{3/2}}$ , and the mean value type inequality

f(0)-\int_{-1/2}^{1/2}f(s)ds\leq C\sup_{|s|\leq 1/2}|f^{\prime\prime}(s)|,

we deduce that for $|\eta|^{2}+\frac{\mu_{1}^{2}}{a_{22}}\gtrsim 1$ ,

|\alpha_{1}(\mu_{1},\mu_{2},\eta)-\int_{x-1/2}^{x+1/2}\alpha_{1}(\mu_{1},\mu_{2},s+\sqrt{-1}y)ds|\leq CA^{-1/4}(|\eta|^{2}+\frac{\mu_{1}^{2}}{a_{22}})^{-3/2}

Thus for $|x|\leq 1/2$ ,

\begin{split}&|\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\int_{x-1/2}^{x+1/2}\alpha_{1}(\mu_{1},\mu_{2},s+n+\sqrt{-1}y)ds\}|\\ &\leq CA^{-1/4}\sum_{|n|\neq 0}(|\eta+n|^{2}+\frac{\mu_{1}^{2}}{a_{22}})^{-3/2}\leq CA^{-1/4},\end{split}

and the sum converges to zero as $\mu_{1}^{2}+|\eta|^{2}\to\infty$ .

In particular if $y^{2}+a_{22}^{-1}\mu_{1}^{2}\gtrsim 1$ , then adding the above two inequalities already implies the bound

\begin{split}|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|&=|\sum_{n\in\mathbb{Z}}\{\alpha_{1}(\mu_{1},\mu_{2},\eta+n)-\int_{x-1/2}^{x+1/2}\alpha_{1}(\mu_{1},\mu_{2},s+n+\sqrt{-1}y)ds\}|\leq CA^{-1/4},\end{split}

and that $|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|$ converges to zero as $\mu_{1}^{2}+|\eta|^{2}\to\infty$ .

If however $y^{2}+a_{22}^{-1}\mu_{1}^{2}\ll 1$ but $\mu_{2}\lesssim-A^{1/4}$ , then we can make

\frac{a_{22}\mu_{2}+a_{12}\mu_{1}}{A^{1/2}\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\lesssim-1,

and the Taylor expansion of $\arctan$ will ensure $|\alpha_{1}(\mu_{1},\mu_{2},\eta)|\leq\frac{C}{-\mu_{2}}$ , so

|\alpha_{1}(\mu_{1},\mu_{2},\eta)-\int_{x-1/2}^{x+1/2}\alpha_{1}(\mu_{1},\mu_{2},s+\sqrt{-1}y)ds|\leq C\mu_{2}^{-1}\leq CA^{-1/4},

from which we again deduce $|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|\leq CA^{-1/4}$ . ∎

Proposition 3.5.

(Exponential decay for higher Fourier modes in the first order ansatz) If $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\gtrsim A^{1/2}$ , then

(3.8)

|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|\leq CA^{-3/4}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\exp(-2\pi A^{-1/2}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})).

Similar bounds hold for $\tilde{\alpha}_{i}-\bar{\alpha}_{i}$ for $i=1,2,3$ .

Proof.

We focus on the region $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\gtrsim A^{1/2}\}$ . The key idea is that $\tilde{\alpha}_{1}-\bar{\alpha}_{1}$ is $\Delta_{a}$ -harmonic , bounded and has no zero Fourier mode in the $S^{1}$ direction defined by the $x$ -variable, so the exponential decay follows from Fourier analysis. We remark that similar ideas have appeared in the recent paper [13].

We perform Fourier decomposition in the $S^{1}$ direction

\tilde{\alpha}_{1}-\bar{\alpha}_{1}=\sum_{n\neq 0}h_{n}(\mu_{1},\mu_{2},y)e^{2\pi inx},

Parseval identity combined with Lemma 3.4 shows

\sum_{n}|h_{n}|^{2}=\int_{0}^{1}|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|^{2}dx\leq CA^{-1/2}.

Now $\Delta_{a}$ -harmonicity translates into the 3-dimensional Helmholtz equations:

\Delta_{a}^{\prime}h_{n}-4\pi^{2}n^{2}A^{-1}h_{n}=0.

The remaining task is conceptually speaking to estimate the Dirichlet Green’s function for the Helmholtz equation on the noncompact 3-dimensional domain $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\gtrsim A^{1/2}\}$ . In practice, building an upper barrier for the Green’s function suffices for our purpose.

Recall $\varrho=|(\mu_{1},\mu_{2},y)|_{a}^{\prime}$ is the distance function for the Euclidean metric $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}_{y}$ . By simple direct computation, for any $\kappa>0$ ,

(\Delta_{a}^{\prime}-4\pi^{2}n^{2}A^{-1})e^{-\kappa\varrho}\leq(\kappa^{2}-4\pi^{2}n^{2}A^{-1})e^{-\kappa\varrho},

so for $k_{n}=2\pi|n|A^{-1/2}$ , the function $e^{-k_{n}\varrho}$ is a supersolution of the Helmholtz equation. Now we build a barrier function

h_{n}^{\prime}(\mu_{1},\mu_{2},y)=A^{-1/2}\int_{0}^{\infty}e^{-k_{n}|(\mu_{1},\mu_{2}-s,y)|_{a}^{\prime}}ds,

whose singularity lies on $\mathfrak{D}_{1}$ . Since $h_{n}^{\prime}$ is a positive superposition of supersolutions, it must be itself a supersolution. Other basic properties are:

•

On $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\geq A^{1/2}\}$ , using the saddle point method for Laplace type integrals

0\leq h_{n}^{\prime}\leq CA^{-3/4}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\exp(-k_{n}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})).

•

On the boundary of $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\geq A^{1/2}\}$ , we have $h_{n}^{\prime}\geq\frac{A^{-1/4}}{C|n|}$ .

Since $|h_{n}|\leq CA^{-1/4}$ by the Parseval identity, the comparison principle implies

\begin{split}h_{n}\leq C|n|h_{n}^{\prime}\leq CA^{-3/4}|n|\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\exp(-k_{n}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})).\end{split}

Thus on $\{\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D}_{1})\geq A^{1/2}\}$ , the desired bound on $|\tilde{\alpha}_{1}-\bar{\alpha}_{1}|$ follows by summing over these estimates over $n$ . It is worth commenting that we expect the exponential decay rate to be sharp. ∎

Remark 3.4.

The periodicity condition is responsible for the exponential decay. Its effect becomes significant when $\eta\sim 1$ , which is compatible with the length scale $|\vec{\mu}|_{a}\sim A^{1/2}$ , or $\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D})\sim A^{1/2}$ . The geometric significance of exponential decay is that the ansatz models the transition from fully quantum into semiflat behaviour (cf. review Section 1.3).

Next we ask for fine asymptote as we move far along $\mathfrak{D}_{i}$ .

Lemma 3.6.

We have the identity

\begin{split}&\tilde{\alpha}_{1}(\mu_{1},\mu_{2},\eta)+\tilde{\alpha}_{1}(-\mu_{1},-\mu_{2},-\eta)\\ =&\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\sum_{n\in\mathbb{Z}\setminus\{0\}}\{\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta+n|^{2}}}-\frac{1}{2\sqrt{a_{22}}|n|}\}\end{split}

where the RHS is recognized as the main part of the complex 2-dimensional Ooguri-Vafa potential. Similarly with $\mathfrak{D}_{i}$ for $i=1,2,3$ .

Proof.

Clear from $\alpha_{1}(\mu_{1},\mu_{2},\eta)+{\alpha}_{1}(-\mu_{1},-\mu_{2},-\eta)=\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}.$ ∎

The utility of this Lemma is that for $\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\lesssim A^{1/2},|\vec{\mu}|_{a}\gtrsim A^{1/2}$ , namely if we move far from the origin along $\mathfrak{D}_{1}$ , then up to exponentially small errors

\begin{split}&\tilde{\alpha}_{1}(-\mu_{1},-\mu_{2},-\eta)\sim\bar{\alpha}_{1}(-\mu_{1},-\mu_{2},-\eta)\\ &=\frac{1}{2\sqrt{a_{22}}}\{\log 2-\gamma_{E}-\log\left(\frac{1}{\sqrt{A}}|(\mu_{1},\mu_{2},y)|_{a}^{\prime}+\frac{\sqrt{a_{22}}}{\sqrt{A}}(\mu_{2}+\frac{a_{12}}{a_{22}}\mu_{1})\right)\}\end{split}

by Proposition 3.5, so the Lemma provides very precise asymptote for $\tilde{\alpha}_{1}(\mu_{1},\mu_{2},\eta)$ along $\mathfrak{D}_{1}$ .

The refined asymptotic behaviour of $\tilde{\alpha}_{i}$ is summarised as

•

Near the origin $\tilde{\alpha}_{i}\sim\alpha_{i}$ . This is designed to match the asymptote of the Taub-NUT type metric on $\mathbb{C}^{3}$ from Chapter 2.
•

Sufficiently far from $\mathfrak{D}_{i}$ , the $\tilde{\alpha}_{i}$ is modelled by an elementary logarithmic function $\bar{\alpha}_{i}$ up to exponentially small fluctuation.
•

Near $\mathfrak{D}_{i}$ and far from the origin, the $\tilde{\alpha}_{i}$ agrees with the 2-dimensional Ooguri-Vafa potential, up to an elementary logarithmic function and some exponentially small fluctuation.

We comment that although $\tilde{\alpha}_{i}$ is defined globally over $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ , the Kähler ansatz is only defined over a finite region and is incomplete, because $A+\tilde{w}$ becomes negative when $\tilde{\alpha}_{i}\sim\bar{\alpha}_{i}\gtrsim A^{1/2}$ , which happens when $\log(A^{-1/2}\varrho)\gtrsim A^{3/4}.$ Conversely for a fixed $0<\epsilon_{0}\ll 1$ independent of $A$ , the Kähler ansatz is positive definite on

(3.9)

M^{+}=\{\log(A^{-1/2}\varrho)<\epsilon_{0}A^{3/4}\}.

3.3. Complex geometric perspective

We now proceed to identify the complex structure $(\tilde{J}^{(1)},\tilde{\Omega}^{(1)})$ on the Kähler ansatz. Our technique is to find a periodic version of the constructions made in Section 2.4 about the Taub-NUT type metric on $\mathbb{C}^{3}$ , in the same way that the Ooguri-Vafa metric is seen as a periodic version of the Taub-NUT metric. The reader is encouraged to warm up by refering to Section 1.3 and 2.4. In this approach algebraic structures will emerge from relations between transcendental integrals of geometric origin. For the converse viewpoint which starts with the algebra, see the review Section 1.1.6.

The generalised Gibbons-Hawking construction provides the $(1,0)$ -forms $\tilde{\zeta}_{i}=\tilde{V}_{(1)}^{ij}d\mu_{j}+\sqrt{-1}\vartheta_{i}$ , and the formula (1.14) computes their differentials. The main idea is to produce holomorphic differentials by adjusting $\tilde{\zeta}_{i}$ . We define the functions

\tilde{\beta}_{i}(\mu_{1},\mu_{2},\eta)=\lim_{N\to\infty}\sum_{n=-N}^{N}\beta_{i}(\mu_{1},\mu_{2},\eta+n),\quad i=0,1,2.

Lemma 3.7.

The series defining $\tilde{\beta}_{i}$ converge for $\eta\notin\mathbb{Z}$ , and $\tilde{\beta}_{i}$ are 1-periodic in $\eta$ . Morever if $|x|\leq\frac{1}{2}$ , then

|\tilde{\beta}_{i}-\beta_{i}|\leq C(|\eta|+\frac{|\mu_{1}|+|\mu_{2}|}{A^{1/4}}).

Proof.

Let $\mu_{1},\mu_{2}$ be fixed. The essential task is to understand the asymptotic behaviour of $\beta_{i}(\mu_{1},\mu_{2},\eta+n)$ as $|n|$ becomes large. We focus on $\beta_{1}$ .

Using the homogeneity property of $\alpha_{1},\alpha_{2},\alpha_{3}$ in the $\mu_{1},\mu_{2}$ and $\eta$ variables, it is easy to see from the integral definition of $\beta_{1}$ that

\beta_{1}(0,0,\eta)=\frac{1}{\eta}\beta_{1}(0,0,1),\quad|\beta_{1}(0,0,1)|\leq C.

By elementary properties of arctan

\alpha_{1}(\mu_{1},\mu_{2},\eta)=\frac{1}{4\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+O(\frac{|\mu_{1}|+|\mu_{2}|}{\mu_{1}^{2}+a_{22}|\eta|^{2}}),

\frac{\partial\alpha_{1}}{\partial\eta}=\frac{-a_{22}\bar{\eta}}{8(\mu_{1}^{2}+a_{22}|\eta|^{2})^{3/2}}\left(1+O(\frac{|\mu_{1}|+|\mu_{2}|}{(\mu_{1}^{2}+a_{22}|\eta|^{2})^{1/2}})\right),

and similarly

\frac{\partial\alpha_{3}}{\partial\eta}=\frac{-(a_{11}+2a_{22}+a_{22})\bar{\eta}}{8((\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{22}+a_{22})|\eta|^{2})^{3/2}}\left(1+O(\frac{|\mu_{1}|+|\mu_{2}|}{((\mu_{1}-\mu_{2})^{2}+A^{1/2}|\eta|^{2})^{1/2}})\right).

After integration

\begin{split}|\beta_{1}(\mu_{1},\mu_{2},\eta)-\beta_{1}(0,0,\eta)|\leq\frac{C(|\mu_{1}|+|\mu_{2}|)}{|\eta|}(\frac{1}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\frac{1}{\sqrt{(\mu_{1}-\mu_{2})^{2}+a_{22}|\eta|^{2}}}).\end{split}

This shows the series

\sum_{n\in\mathbb{Z}}\beta_{1}(\mu_{1},\mu_{2},\eta+n)-\beta_{1}(0,0,\eta+n)

is absolutely convergent if $\eta\notin\mathbb{Z}$ , and if morever $|x|\leq\frac{1}{2}$ then we have the bound

\sum_{n\in\mathbb{Z}\setminus\{0\}}|\beta_{1}(\mu_{1},\mu_{2},\eta+n)-\beta_{1}(0,0,\eta+n)|\leq\frac{C(|\mu_{1}|+|\mu_{2}|)}{A^{1/4}}.

Thus the convergence of the series $\tilde{\beta}_{1}$ is equivalent to the convergence of

\lim_{N\to\infty}\sum_{n=-N}^{N}\beta_{1}(0,0,\eta+n)=\beta_{1}(0,0,1)\lim_{N\to\infty}\sum_{n=-N}^{N}\frac{1}{\eta+n}=\beta_{1}(0,0,1)\pi\cot(\pi\eta),

and similarly for $\tilde{\beta}_{2}$ and $\tilde{\beta}_{0}$ . The periodicity claim follows from standard rearranging theorems for series. The estimate on $\tilde{\beta}_{i}-\beta_{i}$ follows by combining the above discussions. ∎

Lemma 3.8.

Let $0\leq\theta_{1}^{\infty},\theta_{2}^{\infty}\leq 2\pi$ be two real numbers to be determined. The holomorphic 1-forms

\begin{cases}\tilde{\zeta}_{1}^{\prime}=\tilde{\zeta}_{1}+(\tilde{\beta}_{1}+\beta_{1}(0,0,1)\pi\sqrt{-1}-\sqrt{-1}\theta_{1}^{\infty})d\eta,\\ \tilde{\zeta}_{2}^{\prime}=\tilde{\zeta}_{2}+(\tilde{\beta}_{2}+\beta_{2}(0,0,1)\pi\sqrt{-1}-\sqrt{-1}\theta_{2}^{\infty})d\eta,\\ \tilde{\zeta}_{0}^{\prime}=\tilde{\zeta}_{0}+(\tilde{\beta}_{0}+\beta_{0}(0,0,1)\pi\sqrt{-1}+\sqrt{-1}\theta_{1}^{\infty}+\sqrt{-1}\theta_{2}^{\infty})d\eta\end{cases}

are closed, namely they are holomorphic differentials.

Proof.

This is the periodic version of Lemma 2.7. The terms $\beta_{i}(0,0,1)\pi\sqrt{-1}$ and $\theta_{i}^{\infty}$ are added for later convenience. ∎

Lemma 3.9.

The sum $\tilde{\beta}_{1}+\tilde{\beta}_{2}+\tilde{\beta}_{0}=\pi\cot(\pi\eta)$ . Equivalently,

\tilde{\zeta}_{1}+\tilde{\zeta}_{2}+\tilde{\zeta}_{0}=d\log(1-e^{2\pi\sqrt{-1}\eta}).

Proof.

This is the periodic version of Lemma 2.8, using Euler’s series identity for $\cot(\pi\eta)$ :

\lim_{N\to\infty}\sum_{n=-N}^{N}\frac{1}{\eta+n}=\pi\cot(\pi\eta),

and $\sum_{i}\beta_{i}(0,0,1)=1$ from Lemma 2.8. ∎

To compute the periods of the integrals $\int\tilde{\zeta}_{i}^{\prime}$ , we recall from the topological description (cf. review Section 1.1.3) that there are three $S^{1}$ -cycles generating $H_{1}(T^{3})$ , two of which come from the $T^{2}$ -fibres, and the third comes from lifting the $S^{1}$ on the base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ to the total space, which involves monodromy issues.

Lemma 3.10.

For appropriate choices of $\theta^{\infty}_{1},\theta^{\infty}_{2}$ , the $T^{3}$ -periods of the holomorphic differentials $\tilde{\zeta}_{i}$ take values in $2\pi\sqrt{-1}\mathbb{Z}$ . In particular, the holomorphic functions

Z_{i}=\exp(\int\tilde{\zeta}_{i}),\quad i=0,1,2

are defined without multivalue issues. For a suitable choice of multiplicative normalisation on $Z_{i}$ , we have the functional equation

Z_{0}Z_{1}Z_{2}=1-e^{2\pi\sqrt{-1}\eta}.

Proof.

The periods along the generating cycles in the $T^{2}$ -fibres are straightforward:

\int_{S^{1}}\tilde{\zeta}_{i}^{\prime}=\int_{S^{1}}\sqrt{-1}\vartheta_{i}=2\pi\sqrt{-1},\quad i=1,2,

and $\int_{S^{1}}\tilde{\zeta}_{0}^{\prime}=-\sqrt{-1}\int_{S^{1}}\vartheta_{1}+\vartheta_{2}=-4\pi\sqrt{-1}$ .

Computing the period along the other $S^{1}$ requires a special trick. As a preparatory subtle remark, the Kähler metric is not globally defined over the base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ due to incompleteness issues, but the quantities $\tilde{v}^{ij},\tilde{w},\vartheta_{i}$ make sense globally. Consider the $S^{1}$ on the base defined by $\{\mu_{1}=\mu_{2}=0,y=\text{const}\}$ . If we attempt to lift this $S^{1}$ by parallel transport, in general we cannot get a closed loop, and this failure is measured by the holonomy of the $T^{2}$ -connection $\vartheta=(\vartheta_{1},\vartheta_{2})$ along the $S^{1}$ . When $y\to+\infty$ , due to the exponential decay of the $x$ -dependent part of $\tilde{v}^{ij},\tilde{w},\vartheta_{i}$ , this holonomy converges to two real numbers $(\theta_{1}^{\infty},\theta_{2}^{\infty})$ modulo $2\pi\mathbb{Z}$ . In particular, if we twist $\vartheta$ by a flat $T^{2}$ -connection, then $\theta_{1}^{\infty},\theta_{2}^{\infty}$ receive a corresponding twist so that $\tilde{\zeta}_{i}$ is unaffected. Thus we can assume without loss of generality that $\theta_{i}^{\infty}=0$ , namely the asymptotic holonomy of $\vartheta$ is zero, so in the limit the $S^{1}$ cycle lifts to a closed loop, on which we can evaluate the period asymptotically.

By construction $\int_{S^{1}}\vartheta_{i}=0$ , and using $\tilde{\beta}_{i}(0,0,\eta)=\beta_{i}(0,0,1)\pi\cot(\pi\eta)$ from the proof of Lemma 3.7, we compute

\begin{split}\int_{S^{1}}\tilde{\zeta}_{i}^{\prime}&=\int_{S^{1}}(\tilde{\beta}_{i}+\beta_{i}(0,0,1)\pi\sqrt{-1})d\zeta\\ &=\lim_{y\to\infty}\int_{S^{1}}\beta_{i}(0,0,1)(\pi\cot(\pi\eta)+\pi\sqrt{-1})d\eta\\ &=\lim_{y\to\infty}\beta_{i}(0,0,1)\int_{S^{1}}d\log(1-e^{2\pi\sqrt{-1}\eta})=0.\end{split}

From this we see the integrality condition on the periods, so the holomorphic functions $Z_{i}$ are well defined without multivalue issues.

Notice the definition of $Z_{i}$ for $i=0,1,2$ involve three unspecified multiplicative constants; by prescribing their product appropriately, the functional equation follows from Lemma 3.9. The remaining two free multiplicative constants will be fixed in later Sections. ∎

We denote $Z_{3}=\exp(2\pi\sqrt{-1}\eta)\in\mathbb{C}^{*}$ . The functional equation gives a map

M^{+}\to\{Z_{0}Z_{1}Z_{2}=1-Z_{3}\}\subset\mathbb{C}^{3}_{Z_{1},Z_{2},Z_{0}}\times\mathbb{C}^{*}_{Z_{3}}.

By the same argument as Section 2.4, this is a holomorphic map on $M^{+}\setminus\{0\}$ and extends continuously at the origin.

Proposition 3.11.

The map $M^{+}\to\{Z_{0}Z_{1}Z_{2}=1-Z_{3}\}$ is a holomorphic open embedding. The $T^{2}$ -action is identified as

e^{i\theta_{1}}\cdot(Z_{0},Z_{1},Z_{2})=(e^{-i\theta_{1}}Z_{0},e^{i\theta_{1}}Z_{1},Z_{2}),\quad e^{i\theta_{2}}\cdot(Z_{0},Z_{1},Z_{2})=(e^{-i\theta_{2}}Z_{0},Z_{1},e^{i\theta_{2}}Z_{2}),

and the holomorphic volume form is $\tilde{\Omega}^{(1)}=-\frac{\sqrt{-1}}{2\pi Z_{3}}dZ_{0}\wedge dZ_{1}\wedge dZ_{2}$ .

Proof.

The $T^{2}$ -action follows the same argument as Proposition 2.11. The holomorphic volume form is characterised by $\tilde{\Omega}^{(1)}(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\cdot)=d\eta.$ Notice also

-dZ_{0}\wedge dZ_{1}\wedge dZ_{2}(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\cdot)=d(Z_{0}Z_{1}Z_{2})=-dZ_{3}=-2\pi\sqrt{-1}Z_{3}d\eta,

so $\tilde{\Omega}^{(1)}=-\frac{\sqrt{-1}}{2\pi Z_{3}}dZ_{0}\wedge dZ_{1}\wedge dZ_{2}$ . This formula in particular implies the map $M^{+}\to\{Z_{0}Z_{1}Z_{2}=1-Z_{3}\}$ is a local biholomorphism. We finally need to show this map is injective. Since both $M^{+}$ and $\{Z_{0}Z_{1}Z_{2}=1-Z_{3}\}$ fibre over the $\mathbb{C}^{*}$ coordinate $\eta$ in a compatible way, it suffices to compare the fibres, which have compatible $T^{2}$ -actions, so boils down to the injectivity of $(\mu_{1},\mu_{2})\mapsto(\log|Z_{0}|,\log|Z_{1}|,\log|Z_{2}|)$ for fixed $\eta$ . ∎

Remark 3.5.

Section 2.5 shows that the algebraic structure on Taub-NUT type $\mathbb{C}^{3}$ emerges from holomorphic functions with controlled growth at infinity. Since our Kähler ansatz is incomplete, it makes no literal sense to speak of spatial infinity. Instead growth rate is thought in terms of effective estimates. For a holomorphic function $f$ on $M^{+}$ normalised to $\left\lVert f\right\rVert_{L^{2}}=1$ , if we decompose $f$ according to the weights of the $T^{2}$ -action, then in a smaller metric ball around the origin only Fourier components with small $T^{2}$ -weights contribute significantly to $|f|$ . The intuition is that $T^{2}$ -weights are related to an effective filtration of local holomorphic functions.

3.4. Weighted Hölder norms and initial error estimates

The central analytic difficulty comes from three sources:

•

The metric ansatz behaves very differently in various characteristic regions, and for different Fourier modes. In short, the geometry is multi-scaled.
•

The volume form error becomes larger at large distance, a problem closely related to the incompleteness of the metric.
•

We wish to treat the error estimates with relatively high precision, incorporating features such as exponential decay of higher Fourier modes.

These difficulties require us to introduce some weighted Hölder norms which are more complicated than the ones used in a standard gluing problem. The purpose of this Section is to give precise estimates on the volume form errors of the ansatz, in the complement of a small ball near the origin in $M^{+}$ ; the small ball itself will be later replaced in our gluing construction by a region in $\mathbb{C}^{3}$ equipped with the Taub-NUT type metric. The task of developing the requisite linear analysis will be deferred to later Sections.

There are 3 useful weight parameters or characteristic length scales:

•

The $g_{a}^{\prime}$ -distance to the origin is $\varrho=|(\mu_{1},\mu_{2},y)|_{a}^{\prime}$ .
•

The regularity scale is controlled by the parameter $\ell\sim A^{-1/4}+\text{dist}_{g_{a}}(\cdot,\mathfrak{D}).$
•

The parameter $\tilde{\ell}=2\pi A^{-1/2}\text{dist}_{g_{a}^{\prime}}(\cdot,\mathfrak{D})$ is useful for measuring the rate of exponential decay of higher Fourier modes.

The key quantity to understand is the volume form error:

\tilde{E}^{(1)}=\frac{\tilde{W}_{(1)}}{\det(\tilde{V}^{ij}_{(1)})}-1=\frac{A+\tilde{w}}{A+Aa^{ij}\tilde{v}^{ij}+\det(\tilde{v}^{ij})}-1=-\frac{\det(\tilde{v}^{ij})}{A+\tilde{w}+\det(\tilde{v}^{ij})},

where $\det(\tilde{v}^{ij})=\tilde{\alpha}_{1}\tilde{\alpha}_{2}+\tilde{\alpha}_{1}\tilde{\alpha}_{3}+\tilde{\alpha}_{2}\tilde{\alpha}_{3}$ and $\tilde{w}=a_{22}\tilde{\alpha}_{1}+a_{11}\tilde{\alpha}_{2}+(a_{11}+2a_{12}+a_{22})\tilde{\alpha}_{3}$ . The weighted Hölder norms will be taylor made for the volume form error. Familiarity with Section 2.2, 2.3 and 2.6 will be assumed.

Let $\delta\leq 0$ . We shall define the weighted Hölder norms $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ for $T^{2}$ -invariant tensor fields $T$ on $M^{+}\cap\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}$ , by prescribing the norm on a number of overlapping regions up to uniform equivalence.

•

In the region close to $\mathfrak{D}$ characterised by $\{\ell\lesssim A^{1/2}\}$ , the ansatz metric is approximated by $g_{\text{Taub}}$ (cf. Section 2.3), and $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is uniformly equivalent to the norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ for $g_{\text{Taub}}$ introduced in Section 2.3.
•

The region $\{\ell\gtrsim A^{1/2}\}$ can be covered by subregions of diameter $\sim\ell$ , where the $T^{2}$ -bundle is topologically trivial. Over each subregion the metric is approximated by the periodic version of the constant solution $g_{\text{flat}}$ (cf. Section 2.2). The $x$ -variable defines an $S^{1}$ direction. We decompose $T$ into the part $\bar{T}$ independent of $x$ (the ‘zeroth Fourier mode’) and the oscillatory part $T-\bar{T}$ (the ‘higher Fourier mode’), and define the weighted Hölder norm separately on the two parts.
•

On the zeroth Fourier mode, the norm $\left\lVert\bar{T}\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is equivalent to

$A^{-3\delta/4}(\sum_{j=0}^{k}\left\lVert\ell^{j}\nabla^{j}\bar{T}\right\rVert_{L^{\infty}}+[\ell^{k}\nabla^{k}\bar{T}]_{\alpha}),$

where $[]_{\alpha}$ denotes the appropriately normalised Hölder seminorm. Here the $\ell$ -dependence is inserted to reflect the regularity scale.

•

On the higher Fourier modes we build in the exponential decay. Fix a parameter $0<\kappa<1$ . The norm $\left\lVert T-\bar{T}\right\rVert_{C^{k,\alpha}_{\delta,0}}$ in this region is equivalent to

A^{-3\delta/4}\sup_{\ell(p)\gtrsim A^{1/2}}e^{\kappa\tilde{\ell}}(\sum_{j=0}^{k}\left\lVert A^{j/2}\nabla^{j}(T-\bar{T})\right\rVert_{L^{\infty}}+A^{k/2}[\nabla^{k}(T-\bar{T})]_{\alpha}).

An estimate in this norm is the higher order version of $|T-\bar{T}|\leq CA^{3\delta/4}e^{-\kappa\tilde{\ell}}.$

Notation.

The norm $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta,0}}$ can refer to any type of tensors depending on the context, such as functions, 1-forms, symmetric 2-tensors, and in some cases can refer to the norm computed in a subregion. Strictly speaking this norm depends on $\kappa$ , but we suppress this to avoid cluttering the notation.

We will also need a variant weighted Hölder norm $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta}}$ . The only difference from $\left\lVert T\right\rVert_{C^{k,\alpha}_{\delta,0}}$ is that in the region $\{\ell\gtrsim A^{1/2}\}$ on the zeroth Fourier mode, $\left\lVert\bar{T}\right\rVert_{C^{k,\alpha}_{\delta}}$ is equivalent to

A^{-\delta/4}(\sum_{j=0}^{k}\left\lVert\ell^{j-\delta}\nabla^{j}\bar{T}\right\rVert_{L^{\infty}}+[\ell^{k-\delta}\nabla^{k}\bar{T}]_{\alpha}),

so an estimate in this norm is the higher order version of $|\bar{T}|=O(A^{3\delta/4}(A^{-1/2}\ell)^{\delta})$ . We have inserted an extra decay factor $(A^{-1/2}\ell)^{\delta}$ .

Notation.

For a parameter $\nu$ with $1\ll\nu<\epsilon_{0}A^{3/4}$ , define the subregion of $M^{+}$

M_{\nu}^{+}=\{A^{-1/2}\varrho<e^{\nu}\}\subset M^{+}.

Its base is $\mathcal{B}^{+}_{\nu}=\{A^{-1/2}\varrho<e^{\nu}\}\subset\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ .

The following Lemmas are simple consequences of asymptotes in Section 3.1 and 3.2. The higher order estimates are taken care by $\Delta_{a}$ -harmonicity of $\tilde{\alpha}_{i}$ .

Lemma 3.12.

In the region $M^{+}_{\nu}\setminus\{\vec{\mu}|_{a}\lesssim A^{1/2}\}$ ,

\left\lVert\tilde{\alpha}_{i}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq CA^{1/2}\nu,\quad i=1,2,3.

Lemma 3.13.

In the region $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\lesssim A^{1/2}\lesssim|\vec{\mu}|_{a}\}\subset M^{+}_{\nu}$ , which is far away from $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$ ,

\begin{cases}\left\lVert\tilde{\alpha}_{1}-\frac{1}{2\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-1/4}\nu,\\ \left\lVert\tilde{\alpha}_{2}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-1/4}\nu,\\ \left\lVert\tilde{\alpha}_{3}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-1/4}\nu.\end{cases}

Likewise with the neighbourhood of $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$ .

Lemma 3.14.

In the region $\{\text{dist}_{g_{a}}(\cdot,\mathfrak{D}_{1})\lesssim A^{1/2}\lesssim|\vec{\mu}|_{a}\}\subset M^{+}_{\nu}$ , the Kähler ansatz is approximated by the suitably gauge fixed metric model $g_{\text{Taub}}$ , with metric deviation estimate

\left\lVert\tilde{g}^{(1)}-g_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-3/4}\nu,\quad\left\lVert\tilde{\Omega}^{(1)}-\Omega_{\text{Taub}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-3/4}\nu.

Similarly with the neighbourhood of $\mathfrak{D}_{2}$ and $\mathfrak{D}_{3}$ .

Lemma 3.15.

The region $\{\ell\gtrsim A^{1/2}\}\subset M^{+}_{\nu}$ is covered by subregions of diameter $\sim\ell$ where the Kähler ansatz is approximated by suitably gauge fixed flat models $g_{\text{flat}}$ , with metric deviation estimate

\left\lVert\tilde{g}^{(1)}-g_{\text{flat}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-3/4}\nu,\quad\left\lVert\tilde{\Omega}^{(1)}-\Omega_{\text{flat}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq CA^{-3/4}\nu.

Finally, multiplication property for the weighted Hölder norms implies

Lemma 3.16.

In the region $M^{+}_{\nu}\setminus\{|\vec{\mu}|_{a}\lesssim A^{1/2}\}$ , the volume form error is estimated by

\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq CA^{-3/4}\nu^{2}.

3.5. Harmonic analysis I: periodic Euclidean region

This Section obtains refined mapping properties of the Euclidean Green operator $\Delta_{a}^{-1}$ on $\mathbb{R}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ , which will be used to correct volume form error away from $\mathfrak{D}$ . The method is similar to Lemma 2.18, and the new technical difficulties are the exponential decay estimate and the growth of the error at large distance. We shall identify $T^{2}$ -invariant functions with functions on the base.

As a preliminary observation, the periodic Newtonian potential on $\mathbb{R}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ equipped with the Euclidean metric $g_{a}$ is given by

G_{a}(\mu_{1},\mu_{2},\eta)=\sum_{n\in\mathbb{Z}}\frac{-1}{4\pi^{2}|(\mu_{1},\mu_{2},\eta+n)|_{a}^{2}}.

Its zeroth Fourier mode is

\bar{G}_{a}(\mu_{1},\mu_{2},y)=\int_{0}^{1}G_{a}dx=-\frac{1}{4\pi A^{1/2}\varrho}.

Up to a factor $A^{1/2}$ this agrees with the Newtonian potential for $g_{a}^{\prime}$ on $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times\mathbb{R}_{y}$ . Our real emphasis will be on the second derivatives $\nabla^{2}_{g_{a}}G_{a}$ . Since higher order estimates follow from easy bootstrap arguments, we will focus on absolute estimates.

Lemma 3.17.

For $\varrho\gtrsim A^{1/2}$ ,

|\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a}|_{g_{a}}\leq CA^{1/2}\varrho^{-5}.

Proof.

By the mean value inequality

|\int_{\eta-\frac{1}{2}}^{\eta+\frac{1}{2}}|(\mu_{1},\mu_{2},s+\sqrt{-1}y)|_{a}^{-2}ds-|(\mu_{1},\mu_{2},\eta)|_{a}^{-2}|\leq CA|(\mu_{1},\mu_{2},\eta)|_{a}^{-4},

changing $\eta$ to $\eta+n$ and summing over $n\in\mathbb{Z}$ , we obtain for $\varrho\gtrsim A^{1/2}$ that

\begin{split}&|G_{a}(\mu_{1},\mu_{2},\eta)-\bar{G}_{a}(\mu_{1},\mu_{2},y)|\leq\sum_{n}CA|(\mu_{1},\mu_{2},\eta+n)|_{a}^{-4}\\ \leq&CA\int_{-\infty}^{\infty}|(\mu_{1},\mu_{2},s+\sqrt{-1}y)|_{a}^{-4}ds\\ \leq&CA^{1/2}\varrho^{-3}.\end{split}

The claim follows from $\Delta_{a}$ -harmonicity and bootstrap arguments. ∎

We can improve this to an exponential decay estimate:

Lemma 3.18.

For $\varrho\gtrsim A^{1/2}$ ,

|\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a}|_{g_{a}}\leq CA^{-2}e^{-2\pi A^{-1/2}\varrho}.

Proof.

The basic idea is Fourier analysis in the $x$ -variable combined with $\Delta_{a}$ -harmonicity. The argument is a simpler version of Proposition 3.5, using the barrier method. ∎

Lemma 3.19.

Let $-3<\delta<0$ . Let a function $f$ be compactly supported in $\{\ell>2A^{-1/4}\}\subset\mathcal{B}^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq 1$ . Then $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f$ is estimated on $\mathcal{B}^{+}_{\nu}$ by

|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{g_{a}}\leq C\nu\begin{cases}A^{\delta/4}\ell^{\delta}\quad&\ell\lesssim A^{1/2},\\ A^{3\delta/4}\quad&\ell\gtrsim A^{1/2}.\end{cases}

Remark 3.6.

The support cutoff condition $\varrho<A^{1/2}e^{\nu}$ is needed because sources located at exponentially large distance drives up the elliptic constants; this suggests the metric ansatz destabilizes at exponentially large distance (cf. Section 3.10).

Remark 3.7.

The Green operator will propagate the effects out of $\text{supp}(f)$ into the tail region $\{\varrho\geq A^{1/2}e^{\nu}\}$ and the neighbourhood $\{\ell\leq 2A^{-1/4}\}$ of $\mathfrak{D}$ .

Proof.

The basic idea is similar to Proposition 2.18. We analyse the contribution of the source located at $q$ to the convolution integral $\nabla^{2}_{g_{a}}G_{a}*f(p)$ , depending on the spatial separation between $p$ and $q$ . We write $|q|_{a}^{\prime}=\varrho(q),|p|_{a}^{\prime}=\varrho(p)$ .

Suppose $p$ and $q$ do not belong to the same dyadic scale, namely $|p|_{a}^{\prime}\geq A^{1/2}+2|q|_{a}^{\prime}$ or $|q|_{a}^{\prime}\geq A^{1/2}+2|p|_{a}^{\prime}$ . From Lemma 3.17 we easily deduce

|\nabla^{2}_{g_{a}}G_{a}|_{g_{a}}\leq A^{-1/2}\min(|q|_{a}^{\prime-3},|p|_{a}^{\prime-3}),

so the contribution from all such dyadic scales on $\text{supp}(f)$ is bounded by

\begin{split}&CA^{3\delta/4-1/2}(\int_{2|p|_{a}<\varrho<A^{1/2}e^{\nu}}\varrho^{-3}d\text{Vol}_{a}+\int_{A^{1/2}\lesssim\varrho<|p|_{a}^{\prime}/2}|p|_{a}^{\prime-3}d\text{Vol}_{a})\leq CA^{3\delta/4}\nu,\end{split}

where we use $\delta>-3$ to control the source $f=O(A^{\delta/4}\ell^{\delta})$ in $L^{1}$ .

We are left with one dyadic scale $|q|_{a}^{\prime}\sim|p|_{a}^{\prime}\lesssim A^{1/2}e^{\nu}$ . By a similar argument, the contribution from sources at $A^{1/2}\lesssim|p-q|_{a}\lesssim 2|p|_{a}^{\prime}$ is bounded by $CA^{3\delta/4}\nu.$ If $\ell(p)>\frac{1}{2}A^{1/2}$ , then the contribution from sources at $|p-q|_{a}\leq\frac{1}{4}A^{1/2}$ is controlled by $CA^{3\delta/4}$ using standard Schauder theory.

If $\ell(p)\leq\frac{1}{2}A^{1/2}$ and $|p-q|_{a}^{\prime}\lesssim A^{1/2}$ , then for the purpose of estimating the convolution integral we can simply replace the Green kernel $G_{a}$ by $\frac{-1}{4\pi^{2}|(\mu_{1},\mu_{2},\eta)|_{a}^{2}}$ , and correspondingly for their second derivatives. The point is that at this length scale the periodicity effect is secondary, and we are essentially in the same situation as Lemma 2.18 with $-3<\delta<0$ and $\tau=0$ . A careful examination of that argument there, restoring the $A$ -dependence, shows that the contribution of sources inside this region towards $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f$ is bounded by $C(A^{1/4}\ell)^{\delta}.$

Combining the above shows the claim. ∎

Lemma 3.20.

(Exponential decay of higher Fourier modes) In the situation of Lemma 3.19, the higher Fourier modes of $\nabla_{g_{a}}^{2}\Delta_{a}^{-1}f$ admit estimate in the region $\{\ell>A^{1/2}\}\subset\mathcal{B}^{+}_{\nu}$ ,

|\nabla^{2}_{g_{a}}\Delta_{a}^{-1}(f-\bar{f})|_{g_{a}}\leq CA^{3\delta/4}e^{-\kappa\tilde{\ell}}.

Proof.

The key observation is that if without loss of generality $f$ has no zeroth Fourier modes, then the convolution integral

\nabla^{2}_{g_{a}}G_{a}*f=(\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a})*f,

but Lemma 3.18 says the integral kernel $\nabla^{2}_{g_{a}}G_{a}-\nabla^{2}_{g_{a}}\bar{G}_{a}$ has exponential decay, at a rate faster than the exponential decay rate of $f$ itself. Thus at any point $p$ in the region $\{\ell>A^{1/2}\}\cap\mathcal{B}^{+}_{\nu}$ , the contribution to $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f|_{p}$ from sources outside the ball $\{|p-q|_{a}^{\prime}\lesssim A^{1/2}\}$ is negligible. The contribution from sources inside the ball is treated by standard Schauder theory, and inherits the same exponential decay factor $e^{-\kappa\tilde{\ell}}$ as $f$ itself. ∎

Combining the Lemmas shows the main result of this Section after bootstrap.

Proposition 3.21.

(Periodic Euclidean region) In the situation of Lemma 3.19, in the region $\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}\cap\mathcal{B}^{+}_{\nu}$

\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq C\nu.

The constant only depends on $k,\alpha,\delta,\kappa$ and the scale-invariant uniform ellipticity bound on $a_{ij}$ .

We also record the following variant (cf. Section 3.4 for definition of norm).

Proposition 3.22.

Let $-3<\delta<0$ . Let a function $f$ be compactly supported in $\{\ell>2A^{-1/4}\}\subset\mathcal{B}^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}\leq 1$ . Then in the region $\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}\cap\mathcal{B}^{+}_{\nu}$

\left\lVert\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f\right\rVert_{C^{k,\alpha}_{\delta}}\leq C.

We do not need the extra log factor $\nu$ in the RHS because $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}\leq 1$ implies power law decay on $f$ in the generic region, wheras $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta,0}}\leq 1$ implies no decay.

Remark 3.8.

In the small ball $\{|\vec{\mu}|_{a}<A^{-1/4}\}$ the norms are not defined yet, but the regularity of $\nabla^{2}_{g_{a}}\Delta_{a}^{-1}f$ is well controlled by $\Delta_{a}$ -harmonicity, since here $f=0$ by assumption.

3.6. Perturbation in the Euclidean region

This Section corrects the volume form error sufficiently away from $\mathfrak{D}$ , by perturbatively solving the generalised Gibbons-Hawking equation. We will circumvent the problem caused by metric incompleteness by a trick from [27] called extension norm. From now on $1\ll\nu\ll A^{3/8}$ .

Proposition 3.23.

Let $1\ll\nu\ll A^{3/8}$ . Then there is a real valued function $\varphi_{1}$ on $\mathcal{B}^{+}_{\nu}$ , solving the generalised Gibbons-Hawking equation on $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$

\tilde{V}^{ij}_{(2)}=\tilde{V}^{ij}_{(1)}+\frac{\partial^{2}\varphi_{1}}{\partial\mu_{i}\partial\mu_{j}},\quad\tilde{W}_{(2)}=\tilde{W}_{(1)}-4\frac{\partial^{2}\varphi_{1}}{\partial\eta\partial\bar{\eta}},\quad\det(\tilde{V}^{ij}_{(2)})=\tilde{W}_{(2)}.

Morever $\varphi_{1}$ is $\Delta_{a}$ -harmonic on $\mathcal{B}^{+}_{\nu}\cap\{\ell<A^{1/2}\}$ , and

\left\lVert\nabla^{2}_{g_{a}}\varphi_{1}\right\rVert_{C^{k,\alpha}_{-1,0}(\mathcal{B}^{+}_{\nu}\cap\{\ell\gtrsim A^{1/2}\})}\leq C\nu^{3}A^{-3/4},

and $|\nabla^{2}_{g_{a}}\varphi_{1}|_{g_{a}}\leq C\nu^{3}A^{-3/2}$ on $\mathcal{B}^{+}_{\nu}$ . In particular the matrix $(\tilde{V}^{ij}_{(2)})$ is positive definite and $\tilde{W}_{(2)}$ is positive on $\mathcal{B}^{+}_{\nu}$ .

Proof.

The method is to set up a Banach iteration scheme to correct the volume form error. The generalised Gibbons-Hawking equation can be rewritten in the linearised form

\mathcal{L}\varphi_{1}+\frac{1}{\tilde{W}_{(1)}}\det(\frac{\partial^{2}\varphi_{1}}{\partial\mu_{i}\partial\mu_{j}})=\tilde{E}^{(1)}.

where the linearised operator

\mathcal{L}=\frac{1}{\tilde{W}_{(1)}}(\tilde{V}_{(1)}^{11}\frac{\partial^{2}}{\partial\mu_{2}\partial\mu_{2}}+\tilde{V}_{(1)}^{22}\frac{\partial^{2}}{\partial\mu_{1}\partial\mu_{1}}-2\tilde{V}^{12}_{(1)}\frac{\partial^{2}}{\partial\mu_{1}\partial\mu_{2}}+4\frac{\partial^{2}}{\partial\eta\partial\bar{\eta}}).

The key point below is that in $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$ the quadratic term is small while $\mathcal{L}$ is approximately $\Delta_{a}$ .

•

Start with the initial volume form error $\tilde{E}^{(1)}$ on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell\geq A^{1/2}\}$ , where $\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq CA^{-3/4}\nu^{2}$ according to Lemma 3.16. We will only need the precise value of $\tilde{E}^{(1)}$ in the shrinked region $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$ .
•

Define the extension norm for a function $f$ on $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$ as the infimum of the $C^{k,\alpha}_{-1,0}$ -norms for all functions $f^{\prime}$ extending $f$ with compact support inside $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$ . The extension norm of $\tilde{E}^{(1)}$ is bounded by $CA^{-3/4}\nu^{2}$ , since we can find an appropriate cutoff function $\chi$ such that $\chi\tilde{E}^{(1)}$ provides a required extension.

•

Apply Proposition 3.21 to produce $u_{1}=\Delta_{a}^{-1}(\chi\tilde{E}^{(1)}),$ with second derivative bound on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$ ,

\left\lVert\nabla^{2}_{g_{a}}u_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C\nu\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C\nu^{3}A^{-3/4}.

In particular on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$ ,

|\nabla^{2}_{g_{a}}u_{1}|_{g_{a}}\leq C\nu^{3}A^{-3/2},

which in fact holds on the entire $\mathcal{B}^{+}_{\nu+1}$ using $\Delta_{a}$ -harmonicity in $\{\ell<A^{1/2}\}$ . Whence the quadratic term is bounded on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$ by

\begin{split}\left\lVert\frac{1}{\tilde{W}_{(1)}}\det(\frac{\partial^{2}u_{1}}{\partial\mu_{i}\partial\mu_{j}})\right\rVert_{C^{k,\alpha}_{-1,0}}\leq&C\nu^{3}A^{-3/2}\left\lVert\nabla^{2}_{g_{a}}u_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\\ \leq&C\nu^{4}A^{-3/2}\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}\\ \ll&\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}.\end{split}

The last inequality uses the condition $\nu\ll A^{3/8}$ .

The linearised equation is approximately satisfied on $\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\}$ :

\begin{split}&\left\lVert\mathcal{L}u_{1}-\chi\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}=\left\lVert\mathcal{L}u_{1}-\Delta_{a}u_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\\ \leq&A^{-3/4}\nu\left\lVert\nabla^{2}_{g_{a}}u_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\ll\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}}.\end{split}

where we used the metric deviation estimate in Lemma 3.12.

Elementary algebra shows that inside $\mathcal{B}^{+}_{\nu}\cap\{\ell>A^{1/2}\}$ , the volume form error is improved:

\tilde{E}^{(1)}_{1}=\det(W^{p\bar{q}}_{(1)}-4\frac{\partial^{2}u_{1}}{\partial\eta_{p}\partial\bar{\eta}_{q}})(V_{(1)}+\frac{\partial^{2}u_{1}}{\partial\mu\partial\mu})^{-1}-1,

\left\lVert\tilde{E}^{(1)}_{1}\right\rVert_{C^{k,\alpha}_{-1}(\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\})}\ll\left\lVert\tilde{E}^{(1)}\right\rVert_{C^{k,\alpha}_{-1,0}(\mathcal{B}^{+}_{\nu+1}\cap\{\ell>A^{1/2}\})}.

More formally the extension norm of $E^{(1)}$ is far smaller than that of $E^{(1)}$ , after taking into account the cutoff procedures.

•

Iterate this procedure to produce $u_{1},u_{2},\ldots$ , each time improving the extension norm by a factor say $10^{-1}$ . The second derivative estimate

$\left\lVert\nabla^{2}_{g_{a}}u_{j}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C10^{-j}\nu^{3}A^{-3/4}$

implies that the series $\sum_{j}\nabla^{2}_{g_{a}}u_{j}$ converges. The series $\varphi_{1}=\sum_{j}u_{j}$ also converges after possibly adjusting $u_{j}$ by some affine linear functions, and satisfies the Hessian estimate $\left\lVert\nabla^{2}_{g_{a}}\varphi_{1}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C\nu^{3}A^{-3/4}.$ By construction the generalised Gibbons-Hawking equation holds on $\mathcal{B}^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$ .

∎

Applying the generalised Gibbons-Hawking ansatz, we obtain a second Kähler ansatz $(\tilde{g}^{(2)},\tilde{\omega}^{(2)},\tilde{J}^{(2)},\tilde{\Omega}^{(2)})$ associated to the data $\tilde{V}^{ij}_{(2)}$ and $\tilde{W}_{(2)}$ . The new $T^{2}$ -connection is (cf. (1.13))

\vartheta_{i}^{(2)}=\vartheta_{i}+\sqrt{-1}\frac{\partial^{2}\varphi_{1}}{\partial\eta\partial\mu_{i}}d\eta-\sqrt{-1}\frac{\partial^{2}\varphi_{1}}{\partial\bar{\eta}\partial\mu_{i}}d\bar{\eta}.

Corollary 3.24.

The volume form error $\tilde{E}^{(2)}$ of $(\tilde{g}^{(2)},\tilde{\omega}^{(2)},\tilde{J}^{(2)},\tilde{\Omega}^{(2)})$ is zero on $M^{+}_{\nu}\cap\{\ell>2A^{1/2}\}$ and satisfies the bound on $M^{+}_{\nu}\cap\{|\vec{\mu}|_{a}\gtrsim A^{1/2}\}$

\left\lVert\tilde{E}^{(2)}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq CA^{-3/4}\nu^{2}.

3.7. Glue in the Taub-NUT type metric on $\mathbb{C}^{3}$

The Ooguri-Vafa type Kähler metric ansatz is designed as a periodic version of the Taub-NUT type metric on $\mathbb{C}^{3}$ , the latter having the correct topology and metric asymptote to glue in as a metric bubble inside the former. We shall produce the gluing ansatz while maintaining control on the complex structure. This will be divided into a number of steps.

3.7.1. Relative Gibbons-Hawking potential

We plan to exhibit a $T^{2}$ -bundle preserving diffeomorphism $\Psi_{1}$ between the Taub-NUT type $\mathbb{C}^{3}$ and the positive vertex space $M^{+}_{\nu}$ over the common base $\{0<|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$ , with good estimates on the deviations between both Kähler structures. Since $\frac{1}{3}A^{1/2}<\frac{1}{2}A^{1/2}$ , the $\eta$ -periodic copies of such punctured discs do not overlap. The topology of the $T^{2}$ -bundle structures on both spaces agree by construction. The remaining degrees of freedom in defining $\Psi_{1}$ amounts to a gauge choice, which is the same as a prescription of $\vartheta_{i}^{\mathbb{C}^{3}}-\Psi_{1}^{*}\vartheta_{i}$ .

As a general guideline, the corresponding quantities on $\mathbb{C}^{3}$ and $M^{+}_{\nu}$ have the same singularity, so their difference are smooth quantities. We use superscripts for quantities on $\mathbb{C}^{3}$ to disambiguate from quantities on $M^{+}_{\nu}$ .

Lemma 3.25.

Over the region $\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$ ,

|\Psi_{1}^{-1*}\alpha_{i}-\tilde{\alpha}_{i}|\leq CA^{-3/4}|\vec{\mu}|_{a},\quad|\nabla_{g_{a}}^{k}(\Psi_{1}^{-1*}\alpha_{i}-\tilde{\alpha}_{i})|_{g_{a}}\leq CA^{-1/4-k/2},\quad i=1,2,3.

Proof.

The absolute estimate follows from Lemma 3.2. The higher order estimate follows from $\Delta_{a}$ -harmonicity. ∎

Lemma 3.26.

Over the disc $\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$

|\tilde{\beta}_{i}-\Psi_{1}^{-1*}\beta_{i}|\leq CA^{-1/2}|\vec{\mu}|_{a},\quad|\nabla^{k}_{g_{a}}(\tilde{\beta}_{i}-\Psi_{1}^{-1*}\beta_{i})|_{g_{a}}\leq CA^{-k/2}.

Proof.

The absolute estimate is contained in Lemma 3.7. The higher order estimates follow from the differential relations between $\tilde{\beta}_{i}$ and $\tilde{\alpha}_{i}$ , vis-a-vis ${\beta}_{i}$ and ${\alpha}_{i}$ (cf. Lemma 2.7). ∎

Corollary 3.27.

There is a real-valued relative Gibbons-Hawking potential $\varphi_{2}$ on the disc $\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$ , such that its second derivatives are given by

\begin{cases}\frac{\partial^{2}\varphi_{2}}{\partial\mu_{i}\partial\mu_{j}}=V_{(1)}^{ij}-\tilde{V}_{(1)}^{ij}-\frac{\partial^{2}\varphi_{1}}{\partial\mu_{i}\partial\mu_{j}},\quad i,j=1,2,\\ \frac{\partial^{2}\varphi_{2}}{\partial\eta\partial\bar{\eta}}=-\frac{1}{4}(W_{(1)}-\tilde{W}_{(1)})-\frac{\partial^{2}\varphi_{1}}{\partial\eta\partial\bar{\eta}},\\ \frac{\partial^{2}\varphi_{2}}{\partial\eta\partial\mu_{i}}=\frac{1}{2}(\beta_{i}-\tilde{\beta}_{i})-\frac{\partial^{2}\varphi_{1}}{\partial\eta\partial\mu_{i}},\quad i=1,2.\end{cases}

We can demand the estimates in $\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$ :

|\nabla^{k}_{g_{a}}\varphi_{2}|_{g_{a}}\leq C\nu A^{1/4-k/2},\quad k\geq 0.

Proof.

The existence of $\varphi_{2}$ with presecribed second order derivatives is a consequence of integrability, notably Lemma 2.7 and its counterpart for $M^{+}_{\nu}$ . If we impose that $\varphi$ and its first order derivatives vanish at the origin, then the estimates follow immediately from the Lemmas above and Proposition 3.23. ∎

3.7.2. Modifying the Kähler ansatz I

We now modify $(\tilde{g}^{(2)},\tilde{\omega}^{(2)},\tilde{\Omega}^{(2)})$ to an intermediate Kähler ansatz $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ designed to match up exactly with $(g^{(1)},\omega^{(1)},\Omega_{\mathbb{C}^{3}})$ over $\{|\vec{\mu}|_{a}\leq\frac{1}{6}A^{1/2}\}$ . This will be constructed using the generalised Gibbons-Hawking ansatz.

Take a standard cutoff function $\chi$ on $\mathbb{R}$ with

\chi(s)=\begin{cases}1\quad s\leq 1,\\ 0\quad s\geq 2,\end{cases}

and let $\varphi_{3}=\chi(\frac{|\vec{\mu}|_{a}}{\frac{1}{6}A^{1/2}})\varphi_{2}$ with $\varphi_{2}$ from Corollary 3.27,

\tilde{V}_{(3)}^{ij}=\tilde{V}_{(2)}^{ij}+\frac{\partial^{2}\varphi_{3}}{\partial\mu_{i}\partial\mu_{j}},\quad\tilde{W}_{(3)}^{ij}=\tilde{W}_{(2)}^{ij}-4\frac{\partial^{2}\varphi_{3}}{\partial\eta\partial\bar{\eta}}.

The perturbations are sufficiently small so that positive definiteness is not affected. The generalised Gibbons-Hawking construction produces the intermediate Kähler ansatz $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ . We identify $M^{+}_{\nu}$ with the underlying space of $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ . The $T^{2}$ -connection for $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ is identified as (cf. (1.13))

\vartheta_{i}^{(3)}=\vartheta_{i}^{(2)}+\sqrt{-1}\frac{\partial^{2}\varphi_{3}}{\partial\eta\partial\mu_{i}}d\eta-\sqrt{-1}\frac{\partial^{2}\varphi_{3}}{\partial\bar{\eta}\partial\mu_{i}}d\bar{\eta}.

This amounts to making a gauge choice.

By construction $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ agrees identically with $(\tilde{g}^{(2)},\tilde{\omega}^{(2)},\tilde{\Omega}^{(2)})$ over $\{|\vec{\mu}|_{a}\geq\frac{1}{3}A^{1/2}\}$ , and modulo diffeomorphism agrees identically with $({g}^{(1)},{\omega}^{(1)},\Omega_{\mathbb{C}^{3}})$ over $\{|\vec{\mu}|_{a}\leq\frac{1}{6}A^{1/2}\}$ . By Corollary 3.27,

Lemma 3.28.

Over the region $\{\frac{1}{6}A^{1/2}\leq|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$ ,

\left\lVert\tilde{g}^{(3)}-g^{(1)}\right\rVert_{C^{k,\alpha}_{0,0}}\leq C\nu A^{-3/4},\quad\left\lVert\Omega-\Omega_{\mathbb{C}^{3}}\right\rVert_{C^{k,\alpha}_{0,0}}\leq C\nu A^{-3/4},

and the volume form error $\tilde{E}^{(3)}$ of $\tilde{g}^{(3)}$ satisfies

\left\lVert\tilde{E}^{(3)}-\Delta_{a}\varphi_{3}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C\nu^{2}A^{-3/4}.

Henceforth the complex structure will be fixed, and can be identified as follows. The new holomorphic differentials are

(3.10)

\begin{cases}d\log\tilde{Z}_{i}=d\log Z_{i}+d(\frac{\partial(\varphi_{3}+\varphi_{1})}{\partial\mu_{i}}),\quad i=1,2,\\ d\log\tilde{Z}_{0}=d\log Z_{0}-d(\frac{\partial(\varphi_{3}+\varphi_{1})}{\partial\mu_{1}}+\frac{\partial(\varphi_{3}+\varphi_{1})}{\partial\mu_{2}}).\end{cases}

These have the same $T^{3}$ -periods as $d\log Z_{i}$ , which lie inside $2\pi\sqrt{-1}\mathbb{Z}$ , so the new holomorphic functions $\tilde{Z}_{0},\tilde{Z}_{1},\tilde{Z}_{2}$ are defined without multivalue issues. The functional equation

\tilde{Z}_{0}\tilde{Z}_{1}\tilde{Z}_{2}=1-e^{2\pi\sqrt{-1}\eta}=1-Z_{3}

persists from Lemma 3.10. The results in Proposition 3.11 hold verbatim:

Proposition 3.29.

(complex structure) The map $M^{+}\to\{\tilde{Z}_{0}\tilde{Z}_{1}\tilde{Z}_{2}=1-Z_{3}\}$ is a holomorphic open embedding. The $T^{2}$ -action is identified as

e^{i\theta_{1}}\cdot(\tilde{Z}_{0},\tilde{Z}_{1},\tilde{Z}_{2})=(e^{-i\theta_{1}}\tilde{Z}_{0},e^{i\theta_{1}}\tilde{Z}_{1},\tilde{Z}_{2}),\quad e^{i\theta_{2}}\cdot(\tilde{Z}_{0},\tilde{Z}_{1},\tilde{Z}_{2})=(e^{-i\theta_{2}}\tilde{Z}_{0},\tilde{Z}_{1},e^{i\theta_{2}}\tilde{Z}_{2}),

and the holomorphic volume form is $\Omega=-\frac{\sqrt{-1}}{2\pi Z_{3}}d\tilde{Z}_{0}\wedge d\tilde{Z}_{1}\wedge d\tilde{Z}_{2}$ . We shall identify $M^{+}$ with its image.

Over $\{|\vec{\mu}|_{a}\leq\frac{1}{6}A^{1/2}\}$ the ansatz $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ is identified with $({g}^{(1)},\omega^{(1)},\Omega_{\mathbb{C}^{3}})$ after suitable diffeomorphism. An identification of complex coordinates compatible with the holomorphic differential formula (3.10) is

\begin{cases}\tilde{Z}_{1}=z_{1}\exp\{(\pi\sqrt{-1}\beta_{1}(0,0,1)-\sqrt{-1}\theta_{1}^{\infty})\eta\},\\ \tilde{Z}_{2}=z_{2}\exp\{(\pi\sqrt{-1}\beta_{2}(0,0,1)-\sqrt{-1}\theta_{2}^{\infty})\eta\},\\ \tilde{Z}_{0}=\frac{-2\sqrt{-1}\sin(\pi\eta)}{\eta}z_{0}\exp\{(\pi\sqrt{-1}\beta_{0}(0,0,1)+\sqrt{-1}\theta_{1}^{\infty}+\sqrt{-1}\theta_{2}^{\infty})\eta\}.\end{cases}

This fixes the normalisation for the multiplicative constants of $\tilde{Z}_{i}$ .

3.7.3. Modifying the Kähler ansatz II

We make a second modification from $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ to another new Kähler ansatz $(\tilde{g}^{(4)},\tilde{\omega}^{(4)},\Omega)$ designed to match up with the Taub-NUT type metric $(g_{\mathbb{C}^{3}},\omega_{\mathbb{C}^{3}},\Omega_{\mathbb{C}^{3}})$ in Chapter 2.

Recall from Theorem 2.26 that there is a Kähler potential $\phi^{\mathbb{C}^{3}}$ such that

\omega_{\mathbb{C}^{3}}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}=\omega^{(1)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}},\quad\text{for }|\vec{\mu}|_{a}\gtrsim A^{-1/4},

with bound $\left\lVert d\phi^{\mathbb{C}^{3}}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3})}\leq CA^{-1/4}.$ We can impose a normalisation such that $|\phi^{\mathbb{C}^{3}}|\leq CA^{-1/2+\frac{3}{4}\epsilon}$ for $\frac{1}{100}A^{1/2}\lesssim|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}.$

We then define a modified Kähler metric ansatz $\tilde{\omega}^{(4)}$ on $M^{+}_{\nu}$ . Take a standard cutoff function

\chi(s)=\begin{cases}1\quad s\leq 1,\\ 0\quad s\geq 2,\end{cases}

and define

\tilde{\omega}^{(4)}=\tilde{\omega}^{(3)}+\sqrt{-1}\partial\bar{\partial}\phi_{4},\quad\phi_{4}=\chi(\frac{|\vec{\mu}|_{a}}{\frac{1}{12}A^{1/2}})\phi^{\mathbb{C}^{3}}-2\varphi_{3}.

In particular

\begin{cases}\tilde{\omega}^{(4)}=\omega_{\mathbb{C}^{3}}-2\sqrt{-1}\partial\bar{\partial}\varphi_{3},\quad&|\vec{\mu}|_{a}\leq\frac{1}{12}A^{1/2},\\ \tilde{\omega}^{(4)}=\tilde{\omega}^{(3)},\quad&|\vec{\mu}|_{a}\geq\frac{1}{3}A^{1/2}.\end{cases}

The positive definiteness of $\tilde{\omega}^{(4)}$ follows from the metric deviation estimate:

\begin{cases}\left\lVert\partial\bar{\partial}\varphi_{3}\right\rVert_{C^{k,\alpha}_{0,0}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\})}\leq C\nu A^{-3/4},\\ \left\lVert\partial\bar{\partial}\{\chi(\frac{|\vec{\mu}|_{a}}{\frac{1}{12}A^{1/2}})\phi^{\mathbb{C}^{3}}\}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,0}(|\vec{\mu}|_{a}\sim A^{1/2})}\leq CA^{3/4(-1+\epsilon)}.\end{cases}

Here $\varphi_{3}$ is inserted to approximately cancel the cutoff error $\Delta_{a}\varphi_{3}$ in the volume form error $\tilde{E}^{(3)}$ (cf. Lemma 3.28).

Lemma 3.30.

The volume form error for $\tilde{g}^{(4)}$ admits bound in $\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$ :

\tilde{E}^{(4)}=\frac{4}{3}\frac{(\tilde{\omega}^{(4)})^{3}}{\sqrt{-1}\Omega\wedge\overline{\Omega}}-1,\quad\left\lVert\tilde{E}^{(4)}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,0}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\})}\leq C\nu^{2}A^{3/4(-1+\epsilon)}.

3.7.4. Global weighted Hölder norms and error estimates

Now we introduce the global weighted Hölder norms $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta}}$ on $M^{+}_{\nu}$ by demanding that up to uniform equivalence the norm is

•

$\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta}}$ on $M^{+}\cap\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}$ , as defined in Section 3.4.
•

$\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta,0}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\})}$ on $\mathbb{C}^{3}$ for $|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}$ .

On overlapping regions the definitions are equivalent.

Proposition 3.31.

On $M^{+}_{\nu}$ the volume form error satisfies the estimate

(3.11)

\left\lVert\tilde{E}^{(4)}\right\rVert_{C^{k,\alpha}_{-1-\epsilon}}\leq CA^{3/4(-1+\epsilon)}\nu^{2}.

Proof.

Combine Lemma 3.30 with Corollary 3.24. ∎

3.8. Harmonic analysis II: perturbation to Calabi-Yau metric

We now shift to the complex geometric viewpoint and solve the complex Monge-Ampère equation by perturbative methods. The main result of the linear theory is (Compare Proposition 2.23):

Proposition 3.32.

Let $-3<\delta<-1$ and $1\ll\nu\ll A^{3/8}$ . Let $f$ be a $T^{2}$ -invariant function compactly supported in $M^{+}_{\nu}$ with $\left\lVert f\right\rVert_{C^{k,\alpha}_{\delta}}=1$ . Then there is a $T^{2}$ -invariant function $u$ such that the Poisson equation is approximately solved on $M^{+}_{\nu}$ :

\left\lVert\Delta_{\tilde{g}^{(4)}}u-f\right\rVert_{C^{k,\alpha}_{\delta}}\ll 1,

with the Hessian bound

\left\lVert\nabla^{2}_{\tilde{g}^{(2)}}u\right\rVert_{C^{k,\alpha}_{\delta}}\leq C,\quad\left\lVert du\right\rVert_{C^{k+1,\alpha}_{\delta+1}(M^{+}_{\nu})}\leq CA^{-1/4}.

The constants depend only on $k,\alpha,\delta,\kappa$ and the scale invariant uniform ellipticity constant of $a_{ij}$ .

Proof.

(Sketch) The method is the decomposition and patching argument of Section 2.8, using Proposition 3.22, Lemma 2.20 and Proposition 2.23 as ingredients to provide local parametrices. ∎

Theorem 3.33.

(Ooguri-Vafa type metric on the positive vertex) Fix $k,\alpha,\kappa$ and $0<\epsilon\ll 1$ , and let $1\ll\nu\ll A^{3/8}$ . Then there is a $T^{2}$ -invariant Calabi-Yau metric on $M^{+}_{\nu}$ given by a $T^{2}$ -invariant Kähler potential $\phi^{+}$ ,

\omega_{+}=\tilde{\omega}^{(4)}+\sqrt{-1}\partial\bar{\partial}\phi^{+},\quad\omega_{+}^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega},

satisfying the metric deviation estimate

(3.12)

\left\lVert\omega_{+}-\tilde{\omega}^{(4)}\right\rVert_{C^{k,\alpha}_{-1-\epsilon}(M^{+}_{\nu})}\leq C\nu^{2}A^{3/4(-1+\epsilon)},\quad\left\lVert d\phi^{+}\right\rVert_{C^{k+1,\alpha}_{-\epsilon}(M^{+}_{\nu})}\leq C\nu^{2}A^{-1+3\epsilon/4}.

The constants depend only on $k,\alpha,\epsilon,\kappa$ and the scale invariant ellipticity bound on $a_{ij}$ .

Proof.

(Sketch) Given Proposition 3.32, one can set up a Banach iteration scheme to correct the volume form error. A subtlety caused by metric incompleteness is that the parametrix $P_{\nu}$ can only invert sources with compact supports. This problem can be circumvented using the extension norm trick as in Proposition 3.23, and we obtain a Calabi-Yau metric on a shrinked domain $M^{+}_{\nu-1}$ . Changing $\nu$ to $\nu+1$ gives the statement. ∎

Remark 3.9.

The metric lives over a region $M^{+}_{\nu}$ with exponential neck length, but the exponent $\nu$ is not expected to be optimal. For existence results over longer necks one should allow the coupling constants $a_{ij}$ to drift according to the renormalisation flow equation (cf. Section 3.10 and 4.13 for discussions).

3.9. Ooguri-Vafa type metric on the positive vertex

We discuss geometric aspects of the Ooguri-Vafa type metric $\omega_{+}$ .

Corollary 3.34.

(Exponential decay to semiflat metric away from $\mathfrak{D}$ ) Assume the setup of Theorem 3.33. In the subregion $\{\tilde{\ell}\gtrsim 1\}\subset M^{+}_{\nu}$ , the deviation of $\omega_{+}$ from its zeroth Fourier mode $\bar{\omega}_{+}$ decays exponentially:

(3.13)

|\omega_{+}-\bar{\omega}_{+}|\leq CA^{-3/4}\nu e^{-\kappa\tilde{\ell}}.

The constant depends only on $\epsilon,\kappa$ and the scale invariant ellipticity bound on $a_{ij}$ . The decay rate $0<\kappa<1$ can be chosen arbitrarily close to 1.

Corollary 3.35.

(Special Lagrangian fibration) There exist moment coordinates $\tilde{\mu}_{1},\tilde{\mu}_{2}$ for the $T^{2}$ action on $\omega^{+}$ . The special Lagrangian fibration

(3.14)

M^{+}_{\nu}\xrightarrow{(\tilde{\mu}_{1},\tilde{\mu}_{2},\text{Im}\eta)}\mathbb{R}^{3}

is proper over $\{|(\tilde{\mu}_{1},\tilde{\mu}_{2},\text{Im}\eta)|_{a}^{\prime}\leq\frac{1}{2}A^{1/2}e^{\nu}\}\subset\mathbb{R}^{3}$ where the generic fibre is topologically $T^{3}$ . The critical point set is $\bigcup_{i,j\in\{0,1,2\}}\{\tilde{Z}_{i}=\tilde{Z}_{j}=0\}$ and the discriminant locus is contained in $\mathfrak{D}$ . The monodromy of the fibration and the topology of the central singular fibre agrees with the Gross-Ruan prediction in Section 1.1.3.

Proof.

By similar calculations as in Corollary 2.29, the moment map on $M^{+}_{\nu}$ is expressed as

\tilde{\mu}_{i}=\begin{cases}\mu_{i}+\iota_{\frac{\partial}{\partial\theta_{i}}}d^{c}\phi^{+},\quad&|\vec{\mu}|_{a}>\frac{1}{3}A^{1/2}\\ \mu_{i}+\iota_{\frac{\partial}{\partial\theta_{i}}}d^{c}(\phi^{+}+\phi_{4}),\quad&\frac{1}{12}A^{1/2}<|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2},\\ \tilde{\mu}_{i}^{\mathbb{C}^{3}}+\iota_{\frac{\partial}{\partial\theta_{i}}}d^{c}(\phi^{+}-2\varphi_{3}),\quad&|\vec{\mu}|_{a}\leq\frac{1}{12}A^{1/2}.\end{cases}

where $\phi_{4}$ is the Kähler potential between $\tilde{\omega}^{(4)}$ and $\tilde{\omega}^{(3)}$ (cf. Section Section 3.7), and $\tilde{\mu}_{i}^{\mathbb{C}^{3}}$ are the moment coordinates for $\omega_{\mathbb{C}^{3}}$ (cf. Corollary 2.29). By construction $\tilde{\mu}_{1},\tilde{\mu}_{2},\tilde{\mu}_{1}-\tilde{\mu}_{2}$ vanish respectively along $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$ , due to the respective vanishing of the circle generators $\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial\theta_{1}}-\frac{\partial}{\partial\theta_{2}}$ . This fixes the additive normalisation on the moment coordinates.

The gradient estimates on Kähler potentials and Corollary 2.29 imply on $M^{+}_{\nu}$

(3.15)

|\mu_{i}-\tilde{\mu}_{i}|\leq\begin{cases}C\nu^{2}A^{-5/4+3\epsilon/4},\quad&|\vec{\mu}|_{a}\geq\frac{1}{3}A^{1/2},\\ CA^{-3/4}\ell^{-\epsilon}|\vec{\mu}|_{a}^{-1+\epsilon}+C\nu A^{-1/2},\quad&|\vec{\mu}|_{a}<\frac{1}{12}A^{1/2}.\end{cases}

In particular, if $|(\tilde{\mu}_{1},\tilde{\mu}_{2},y)|_{a}^{\prime}\leq\frac{1}{2}A^{1/2}e^{\nu}$ , then $\varrho\leq\frac{3}{4}A^{1/2}e^{\nu}$ , so the map (3.14) is proper over $\{|(\tilde{\mu}_{1},\tilde{\mu}_{2},y)|_{a}^{\prime}\leq\frac{1}{2}A^{1/2}e^{\nu}\}\subset\mathbb{R}^{3}$ . By the same argument in Corollary 2.29, the fibres of (3.14) are special Lagrangians of phase angle zero, the critical point set is $\bigcup_{i,j\in\{0,1,2\}}\{\tilde{Z}_{i}=\tilde{Z}_{j}=0\}$ and the discriminant locus is contained in $\mathfrak{D}$ .

Next we consider the map $(\mu_{1},\mu_{2},\eta)\mapsto(\tilde{\mu}_{1},\tilde{\mu}_{2},\eta)$ on the region $\{\varrho<A^{1/2}e^{\nu}\}$ . Using (3.15) and the implicit function theorem, this map restricted to the region $\{\ell\geq A^{1/2},\varrho<\frac{3}{4}A^{1/2}e^{\nu}\}$ is an approximate identity, and in particular a diffeomorphism onto its image. Morever by (3.15) no points elsewhere can map into $\text{Image}(\{\ell\geq A^{1/2},\varrho<\frac{3}{4}A^{1/2}e^{\nu}\})$ . Interpreted geometrically, this implies that the special Lagrangian fibres of (3.14) lying over the region $\{|(\tilde{\mu}_{1},\tilde{\mu}_{2},y)|_{a}^{\prime}\leq\frac{1}{2}A^{1/2}e^{\nu}\}$ and suitably away from $\mathfrak{D}$ , must be small perturbations of the $T^{3}$ -fibres of the map $M^{+}_{\nu}\xrightarrow{(\mu_{1},\mu_{2},y)}\mathbb{R}^{3}$ . This shows the generic fibre of (3.14) is topologically $T^{3}$ , and the monodromy data of (3.14) is the same as for $M^{+}\xrightarrow{(\mu_{1},\mu_{2},y)}\mathbb{R}^{3}$ , which by construction agrees with the Gross-Ruan prediction in Section 1.1.3.

Finally we need to determine the topology of the central singular fibre, defined as the set $X_{0}=\{\tilde{\mu}_{1}=\tilde{\mu}_{2}=0,y=0\}$ , which is invariant under the $T^{2}$ -action. From our knowledge of the critical point set, the only singular point on the central fibre is $\tilde{Z}_{0}=\tilde{Z}_{1}=\tilde{Z}_{2}=0$ . Thus the quotient $X_{0}/T^{2}$ must be a compact 1-dimensional manifold with possibly one singular point. But there is also a homological constraint

\text{Vol}_{g_{+}}(X_{0})=\int_{X_{0}}\Omega=4\pi^{2}\int_{X_{0}/T^{2}}d\eta=\int_{T^{3}}\Omega=4\pi^{2},

so $X_{0}/T^{2}$ is connected and must in fact be a circle. Therefore $X_{0}$ has the topology of $T^{3}$ with a copy of $T^{2}$ collapsed to a point, in accordance with the Gross-Ruan prediction on the positive vertex. ∎

Remark 3.10.

The singular fibres over $\mathfrak{D}\setminus\{0\}$ have non-isolated singularities by $T^{2}$ -invariance. This does not contradict Joyce’s critique, since the Ooguri-Vafa type metric on the positive vertex is not a generic metric (cf. review Section 1.1.5). But when we glue the Ooguri-Vafa type metric into the global Calabi-Yau metric on a degenerating 3-fold (cf. review Section 1.1.6), the exponentially small corrections to the complex structure will destroy $T^{2}$ -invariance. We then expect the singularity structure of the SYZ fibration to be drastically changed, and in particular its discriminant locus thickens into a ribbon around $\mathfrak{D}$ as predicted by Joyce [14].

3.10. Incompleteness and running coupling

We now give a deeper perspective on the incompleteness of the metric, and a semi-heuristic discussion about how to partially overcome one of the main limitations of the perturbation method: the final metric one constructs is by necessity $C^{0}$ -close to the metric ansatz one starts with.

The main insights are as follows. The Ooguri-Vafa type metric is intended as an effective local description below a certain distance scale for the collapsing family of Calabi-Yau metrics on compact manifolds near the large complex structure limit. Our starting assumption is that the metric is a perturbation of a constant solution after incorporating topology. These constant solutions come naturally in a family parametrised by the coupling constants $a_{ij}$ , which have a geometric meaning in terms of the size and shape of the generic $T^{2}$ -fibres in the local region. The nontrivial topology manifests itself in a distributional equation which dictates the first order corrections $\tilde{\alpha}_{i}$ to the constant solutions, and after Fourier analysis we see the dominant correction terms $\bar{\alpha}_{i}$ depend logarithmically on $\mu_{i},\eta$ . The slow growth of $\log$ means it can be treated as a perturbation term in an exponentially long region, but once we attempt to go beyond, the correction will have a perceptible effect on the size and shape of the average $T^{2}$ -fibres, which would break down our initial effective description via the original constant solution. This suggests that the coupling constants in the effective description drift slowly as we move up the logarithmic scale, a phenomenon we call running coupling. Morever, the precise formula of these log corrections dictate how these coupling constants change as a function of the logarithmic scale, which we will discuss under the name of renormalisation flow equation. Geometrically, the ansatz metrics naturally come in families, and each time we move up a log scale, we really should glue a different ansatz with slightly changed coupling constants to the previous ansatz. The fact that at very large distance scales the original ansatz should be replaced by another ansatz within the same family, is the deep reason why the ansatz metric is incomplete.

The terminologies are based on the following analogy. According to my rudimentary understanding of high energy physics, Quantum Electrodynamics (QED) is intended as an effective description below a certain energy scale for some more sophisticated theories. The starting assumption of Feynman diagram calculations in QED is that the scattering amplitudes are perturbations of the free field theory, after adding new interaction terms in the Lagrangian. These interaction terms come naturally in a family parametrised by the coupling constants, whose physical meaning is related to the observed charges in low energy experiments. Loop calculations in Feynmann diagrams suggest that the coupling constants depend on the energy scale at which one conducts the experiments, a phenomenon known as running coupling. The equation which governs how the coupling constants change as a function of the cutoff energy scale is known as the renormalisation flow equation.

We now flesh out the ideas in the setting of our Ooguri-Vafa type metrics on the positive vertices. We begin by recalling some main features about the family of ansatz metrics. The construction begins with the choice of parameters $a_{ij}$ , and outputs a generalised Gibbons-Hawking metric associated to the data

\begin{cases}V^{11}_{a}=a_{11}+\tilde{\alpha}_{1,a}+\tilde{\alpha}_{3,a}+\text{constant},\\ V^{12}_{a}=a_{12}-\tilde{\alpha}_{3,a}+\text{constant},\\ V^{22}_{a}=a_{22}+\tilde{\alpha}_{2,a}+\tilde{\alpha}_{3,a}+\text{constant},\\ W_{a}=A+a_{22}\tilde{\alpha}_{1}+a_{11}\tilde{\alpha}_{2}+(a_{11}+2a_{12}+a_{22})\tilde{\alpha}_{3}+\text{constant}.\end{cases}

Here the subscript is to emphasize the dependence on $a_{ij}$ . The ambiguity of twisting by a flat connection is not important for the discussions below. The functions $\tilde{\alpha}_{i}$ generically behave like the logarithmic functions $\bar{\alpha}_{i}$ . The constants above refer to numbers independent of $\mu_{1},\mu_{2},\eta$ which are up to our choice (cf. Remark 3.3). The significance of this extra freedom is that if we are interested only in the ansatz at one particular logarithmic scale, then we can always adjust the constants to cancel some log factors in $\tilde{\alpha}_{i}$ so that $a_{ij}$ is the average value of $V^{ij}_{a}$ over this log scale. The QFT analogue of these constants are called counterterms. This step is needed to back up the idea that the constant solution defined by $a_{ij}$ really offers an effective description at the given log scale of the metric, suitably away from the discriminant locus $\mathfrak{D}$ . This issue did not appear previously, because when $A^{-1/2}\varrho\lesssim 1$ these constants were essentially zero.

The central question is how these effective coupling constants $a_{ij}$ vary as a function of the log scale $\lambda$ . Moving up to the next log scale means

\lambda\mapsto\lambda+1,\quad\mu_{i}\mapsto e\mu_{i},\quad y\mapsto ey.

Since $a_{ij}$ drifts very slowly, to zeroth order we can treat them as constants. Now $\bar{\alpha}_{i}$ are explicit functions given by formula (3.6), whose values receive a small increment as we move up the log scale:

\begin{cases}\bar{\alpha}_{1}\mapsto\bar{\alpha}_{1}-\frac{1}{2\sqrt{a_{22}}},\\ \bar{\alpha}_{2}\mapsto\bar{\alpha}_{2}-\frac{1}{2\sqrt{a_{11}}},\\ \bar{\alpha}_{3}\mapsto\bar{\alpha}_{3}-\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}}.\end{cases}

Since $\tilde{\alpha}_{i}$ are generically almost the same as $\bar{\alpha}_{i}$ , this means as we move up a log scale, the average value of $V^{ij}_{a}$ drift by

\begin{cases}V^{11}_{a}\mapsto V^{11}_{a}-\frac{1}{2\sqrt{a_{22}}}-\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}},\\ V^{12}_{a}\mapsto V^{12}_{a}+\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}},\\ V^{22}_{a}\mapsto V^{22}_{a}-\frac{1}{2\sqrt{a_{11}}}-\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}}.\end{cases}

In our viewpoint, it means the first order change of the coupling constants when we move up a log scale is

\begin{cases}a_{11}\mapsto a_{11}-\frac{1}{2\sqrt{a_{22}}}-\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}},\\ a_{12}\mapsto a_{12}+\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}},\\ a_{22}\mapsto a_{22}-\frac{1}{2\sqrt{a_{11}}}-\frac{1}{2\sqrt{a_{11}+2a_{12}+a_{22}}}.\end{cases}

We now denote

p_{1}=\sqrt{a_{22}},\quad p_{2}=\sqrt{a_{11}},\quad p_{3}=\sqrt{a_{11}+2a_{12}+a_{22}}.

As we move up a log scale,

\begin{cases}\lambda\mapsto\lambda+1,\\ p_{1}^{2}\mapsto p_{1}^{2}-\frac{1}{2p_{2}}-\frac{1}{2p_{3}},\\ p_{2}^{2}\mapsto p_{2}^{2}-\frac{1}{2p_{1}}-\frac{1}{2p_{3}},\\ p_{3}^{2}\mapsto p_{3}^{2}-\frac{1}{2p_{1}}-\frac{1}{2p_{2}}.\end{cases}

We have presented this discussion from a discretized viewpoint, which the author thinks is conceptually simpler. The continuum version is the renormalisation flow equation

(3.16)

\begin{cases}\frac{d}{d\lambda}p_{1}^{2}=-\frac{1}{2p_{2}}-\frac{1}{2p_{3}},\\ \frac{d}{d\lambda}p_{2}^{2}=-\frac{1}{2p_{1}}-\frac{1}{2p_{3}},\\ \frac{d}{d\lambda}p_{3}^{2}=-\frac{1}{2p_{1}}-\frac{1}{2p_{2}}.\end{cases}

The remarkable fact is that this ODE system is exactly solvable.

Proposition 3.36.

There exist constants $K_{1},K_{2},K_{3}$ such that the solution to the renormalisation flow equation admits the parametrised representation

\begin{cases}p_{1}=t^{2/3}+\frac{1}{3}(K_{1}+K_{2})t^{-1/3},\\ p_{2}=t^{2/3}+\frac{1}{3}(K_{2}-2K_{1})t^{-1/3},\\ p_{3}=t^{2/3}+\frac{1}{3}(K_{1}-2K_{2})t^{-1/3},\\ \lambda=-\frac{2}{3}t^{2}+\frac{4}{9}(K_{1}^{2}+K_{2}^{2}-K_{1}K_{2})\log t+\frac{4}{81}(K_{1}+K_{2})(K_{2}-2K_{1})(K_{1}-2K_{2})\frac{1}{t}+K_{3}.\end{cases}

Proof.

The renormalisation flow equation is equivalent to

\begin{cases}\frac{d}{d\lambda}p_{1}=-\frac{p_{2}+p_{3}}{4p_{1}p_{2}p_{3}},\\ \frac{d}{d\lambda}p_{2}=-\frac{p_{1}+p_{3}}{4p_{1}p_{2}p_{3}},\\ \frac{d}{d\lambda}p_{3}=-\frac{p_{1}+p_{2}}{4p_{1}p_{2}p_{3}}.\end{cases}

Summing over the three equations,

\frac{d}{d\lambda}(p_{1}+p_{2}+p_{3})=-\frac{p_{1}+p_{2}+p_{3}}{2p_{1}p_{2}p_{3}},

and taking the differences give

\frac{d}{d\lambda}(p_{1}-p_{2})=\frac{p_{1}-p_{2}}{4p_{1}p_{2}p_{3}},\quad\frac{d}{d\lambda}(p_{1}-p_{3})=\frac{p_{1}-p_{3}}{4p_{1}p_{2}p_{3}}.

Without loss of generality $p_{1}\geq p_{2}\geq p_{3}$ , then

d\log(p_{1}+p_{2}+p_{3})=-2d\log(p_{1}-p_{2})=-2\log(p_{1}-p_{3})=-\frac{d\lambda}{2p_{1}p_{2}p_{3}}.

In the degenerate case where $p_{1}=p_{2}$ say, it is understood that $p_{1}=p_{2}$ identically. Denote $p_{1}+p_{2}+p_{3}=3t^{2/3}$ for some new parameter $t$ , then after integration

p_{1}-p_{2}=K_{1}t^{-1/3},\quad p_{1}-p_{3}=K_{2}t^{-1/3},

for some constants $0\leq K_{1}\leq K_{2}$ . Rewriting these equations give

\begin{cases}p_{1}=t^{2/3}+\frac{1}{3}(K_{1}+K_{2})t^{-1/3},\\ p_{2}=t^{2/3}+\frac{1}{3}(K_{2}-2K_{1})t^{-1/3},\\ p_{3}=t^{2/3}+\frac{1}{3}(K_{1}-2K_{2})t^{-1/3}.\end{cases}

Now

d\lambda=-\frac{4}{3}p_{1}p_{2}p_{3}\frac{dt}{t}=\frac{-4}{3t^{2}}(t+\frac{1}{3}(K_{1}+K_{2}))(t+\frac{1}{3}(K_{1}-2K_{2}))(t+\frac{1}{3}(K_{2}-2K_{1}))dt.

Increasing $\lambda$ corresponds to decreasing $t$ . We integrate to obtain

\lambda=-\frac{2}{3}t^{2}+\frac{4}{9}(K_{1}^{2}+K_{2}^{2}-K_{1}K_{2})\log t+\frac{4}{81}(K_{1}+K_{2})(K_{2}-2K_{1})(K_{1}-2K_{2})\frac{1}{t}+K_{3},

where $K_{3}$ is an integration constant. ∎

The rest of the Section offers a heuristic interpretation of the renormalisation flow, whose power is to predict effective metric behvaiour up to a very large distance scale. It is helpful to keep in mind the Gross-Wilson K3 metric [11]. The positive vertex is best understood as part of a global SYZ $T^{3}$ -fibration on a Calabi-Yau 3-fold near the large complex structure limit consisting of a finite number of overlapping pieces with simple complex geometric descriptions (cf. review Section 1.1.6). The renormalisation flow breaks down when $\min_{i}p_{i}$ becomes negative, which indicates a metric transition into a different piece in the 3-fold.

In our normalisation convention $\text{Vol}(T^{3})=4\pi^{2}\sim 1$ . In the generic region of the 3-fold, it is reasonable to expect the 3 circle factors of the SYZ $T^{3}$ -fibres to have comparable length scales, so $\text{diam}(T^{3})\sim 1$ . In contrast, our starting point for constructing the Ooguri-Vafa type metric on the positive vertex is that a $T^{2}$ factor inside $T^{3}$ has much smaller diameter compared to $\text{diam}(T^{3}/T^{2})$ . In order for the Ooguri-Vafa type metric to smoothly transition into the generic region of the SYZ fibration, we require an exponentially long neck region, modelled by the renormalisation flow.

The renormalisation flow has the curious feature that at smaller distance scales $\sum p_{i}$ becomes larger but $|p_{i}-p_{j}|$ becomes smaller, so the scale invariant ellipticity bound $CA^{1/2}\delta_{ij}\leq a_{ij}\leq CA^{1/2}\delta_{ij}$ works better at smaller distance scales. Suppose this bound holds throughout the renormalisation flow until $A$ decreases to $A\sim 1$ where the metric transitions into the generic region, then in Proposition 3.36 the constants $K_{1},K_{2}=O(1)$ . Consequently at smaller distance scales, where $t$ is large, the terms $K_{1},K_{2}$ are neglegible, and the solution of the renormalisation flow is approximated by the special solution

\begin{cases}p_{1}=p_{2}=p_{3}=t^{2/3},\\ \lambda=-\frac{2}{3}t^{2}+K_{3},\quad t\gg 1.\end{cases}

This special solution is invariant under the $S_{3}$ -discrete symmetry interchanging the 3 edges $\mathfrak{D}_{i}$ . The insight is that the most symmetric configuration of $a_{ij}$ is the attractive fixed point of the renormalisation flow.

We can also use the special solution to approximately count the number of log scales involved in the neck region. At the innermost log scale

\lambda\sim 0,\quad t\sim\sqrt{\frac{3}{2}K_{3}},\quad p_{1}\sim p_{2}\sim p_{3}\sim t^{2/3}\sim(\frac{3}{2}K_{3})^{1/3},\quad A_{\text{max}}\sim\frac{3}{4}p_{1}^{4}\sim\frac{3}{4}(\frac{3}{2}K_{3})^{4/3},

and at the outermost log scale

p_{1},p_{2},p_{3}\sim 1,\quad A\sim 1,\quad t\sim 1,\quad\lambda\sim K_{3}.

The total number of log scales is roughly $K_{3}\sim\frac{2}{3}(\frac{4A_{\text{max}}}{3})^{3/4}$ . The diameter of the neck region is of the order

\int_{0}^{K_{3}}p_{1}(\lambda)e^{\lambda}d\lambda\sim e^{K_{3}}\sim\exp(\frac{2}{3}(\frac{4A_{\text{max}}}{3})^{3/4}).

Chapter 3 The Positive Vertex [041X]