# Michel van Garrel, N. Takahashi's 3 Conjectures (Friday, July 22)

## The first two conjectures

:::{.remark}
This talk: 3 conjectures from a 2000 paper of Takahashi.
Main topic for today: mirror theorem within the Gross-Siebert mirror construction.
Outline:

- The 3 conjectures
- Describe the GS mirror
- Deduce one of the conjectures.

:::

:::{.remark}
Takahashi's 96 paper: let $E \subseteq \PP^2$ be an elliptic curve, choose $0\in E$ a flex point, a point $P_d$ of order $3d$ for $d\geq 1$, and consider the curve count
\[
m_d \da \size \ts{\PP^1 \mapsvia{f} \PP^2 \st f_*[\PP^1] = dH , f(\PP^1) \intersect E = 3d\cdot P_d}
\]
where $H \in c_1(\OO(1))$.
Takahashi computes $m_d$ for $d=1,\cdots 6$.
:::

:::{.conjecture}
Consider $\mcm_d$ the moduli of stable sheaves $F$ on $\PP^2$, $\chi(F)$, and $[F] = dH$.
Define $n_d = (-1)^{d-1} \chi(\mcm_d)$.
Then there is a conjectured formula
\[
n_d = (-1)^{d-1} 3d m_d
,\]
so this moduli space knows about the enumerative counts from before.
:::

:::{.remark}
This was proved around 2019, complicated and involves Bridgeland stability conditions.
Is there an invariant that is independent of choice of the point $P$?
The answer is yes, look at a moduli space $\bar{\mcm_d^{\log}}$ of stable log maps $f: C\to \PP^2$ (stable: tree of copies of $\PP^1$ in this case) where $f_*[C] = d H$ is degree $d$, and $f(C) \intersect E = 3d P$ where $P$ is not a particular specified point.
:::

:::{.example title="?"}
An example of $\bar{\mcm_f^{\log} }$:

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![](figures/2022-07-22_15-06-54.png)

:::

:::{.conjecture}
By some virtual intersection theory magic, there is a log GW invariant $N_d^{\log} = \deg [ \bar{ \mcm_d^{\log} } ]^{\vir} \in \QQ$.
A conjectured formula:
\[
N_d^{\log} = (-1)^{d-1} 3d \sum _{k\divides d} k^{-3} n_{d\over k }
.\]
This was proved by Gathmann in 2000.
:::

## The mirror symmetry conjecture 

:::{.remark}
There is a mirror family
\[
\hat E_\phi = Z(xy - \phi(x^3+y^3 + 1)) \subseteq (\CCstar)^2
,\]
which determines an affine elliptic fibration whose fibers are 3-punctured elliptic curves, where there are 3 singular fibers.
Tracing out the vanishing cycles horizontally yield *Lefschetz thimbles*:

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![](figures/2022-07-22_15-18-00.png)

He studies $I(\phi) = \int_{\Gamma_\phi} {\dx\over x}\wedge {\dy\over y}$ and finds a basis for solutions of the Picard-Fuchs equations $I_0(\phi^3) = 1, I_1(\phi^3) = \log(\phi^3) + \cdots, I_2(\phi^3) = {1\over 2}\log^2(\phi^3) + \log(\phi^3) + \cdots$ where the omitted terms form holomorphic functions.
:::

:::{.conjecture title="Mirror symmetry"}
There is a transcendental coordinate change (the mirror map) $\phi\to Q$, and after this change,
\[
I_3(Q) = {\log^2(-Q) \over 2} + \sum_{d\geq 1} N_d^{\log} (-Q)^d
.\]
:::

:::{.remark}
Why these integrals know about the enumerative counts: VHS on one side and extract Yukawa coupling, VHS on the other side with Gauss-Manin, and they match by mirror symmetry.
Can we extract this somehow from the GS mirror construction?
Upshot of the talk: yes.
:::

:::{.remark}
Will use the GS construction and the A-model correspondence:
\[
(\PP^2, E) \mapstofrom (\mcx_t, E\dual_t)
\]
where $\mcx_t\to \AA^1$ is an elliptic fibration and $t$ is a GS parameter.
:::

:::{.remark}
Some heuristics: there is a tropicalization functor $(\PP^2, E) \to T$ the infinite Toblerone with a cross-sectional triangle $E$ mirror to the original $E$.
There is a moment map $\mathrm{mm} (\mcx_t, E\dual_t)\to T$?
$N_d^{\log}$ for $(\PP^2, E)$ gets sent to some tropical curves, and the flex line to an infinite line starting at the vertex.

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![](figures/2022-07-22_15-35-37.png)

:::

:::{.remark}
The situation:

\begin{tikzcd}
	{[v_i]\in H^{i, i}(\PP^2, E)} &&& {} & {[L_i] \in H^{i, 2-i}(\mcx_t, E\dual_t)} \\
	\\
	&& {[\mathrm{trop}(v_i)] = \mathrm{mm}(L_i)}
	\arrow[from=1-1, to=3-3]
	\arrow[from=3-3, to=1-5]
	\arrow["{\text{duality of Hodge diamond}}", from=1-1, to=1-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJbdl9pXVxcaW4gSF57aSwgaX0oXFxQUF4yLCBFKSJdLFszLDBdLFs0LDAsIltMX2ldIFxcaW4gSF57aSwgMi1pfShcXG1jeF90LCBFXFxkdWFsX3QpIl0sWzIsMiwiW1xcbWF0aHJte3Ryb3B9KHZfaSldID0gXFxtYXRocm17bW19KExfaSkiXSxbMCwzXSxbMywyXSxbMCwyLCJcXHRleHR7ZHVhbGl0eSBvZiBIb2RnZSBkaWFtb25kfSJdXQ==)

Here $L_i$ is a Lagrangian fibred over a tropical $i\dash$cycle, and it turns out that integrating over tropical cycles recovers the conjectural equation by looking at periods.

Note that for $[\pt]\in H^{0, 0}(\PP^2, E)$ one obtains ${1\over (2\pi i)^2} \int {\dx\over x} \wedge {\dy\over y} = 1$, recovering $I_0(\phi^3) = 1$:

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![](figures/2022-07-22_15-42-52.png)

Ruddat-Siebert show that for a curve $[M] \in H^{1, 1}(\PP^2, E)$:

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![](figures/2022-07-22_15-45-12.png)

RJ show that 
\[
\exp\qty{{1\over 2\pi i} \int_\Sigma {\dx\over x}\wedge {\dy\over y}} = -t^3 = Q
.\]
:::

:::{.theorem title="?"}
In the case $[\PP^2] \in H^{2, 2}$ yields a cycle $\Sigma$ determined by GS over $\RR$, and 
\[
Q\partial_Q \int_\Sigma {\dx\over x} \wedge {\dy\over y} = \log(-Q) = \sum_d dN_d^{\log}(-Q)^d
.\]

The novelty here is that this falls out of the GS construction by a simple calculation.
:::