# Masha Vlasenko, Integrality of instanton numbers (Friday, July 22)

:::{.remark}
Mirror symmetry: different manifolds give equivalent physics when used to describe particles in string theory.

| A side | B side |
|--------|------------|
| Enumerative geometry (count curves of a fixed degree) | Solve DEs (Picard-Fuchs) for period integrals near MUM |
| GW invariants | Instanton numbers |

The A side is hard!

Beginning of mirror symmetry: Candelas-Ossa-Parkes, exactly soluble superconformal theory from a mirror pair of CYs.
Some polynomial operator $L$ has solutions $Ly=0$ given by $y_0(t)$ a hypergeometric function.
Write $y_1(t) = f_0(t)\log(?) + f_1$ where $f_1\in t\QQ\fps{t}$, guaranteeing a unique solution.
Canonical coordinate: $q(t) = \exp(y_1(t)/y_0(t))$, invert this power series to get the mirror map (near a point of MUM).
Use Frobenius method to solve DE, do some mysterious computation to derive a power series called the Yukawa coupling 
\[
Y(q) = (q\dd{}{q})^2 {y_2\over y_0} = {1\over 5} \sum_{d\geq 0} n_d d^3 {q^d\over 1-q^d}
,\] 
and the $n_d$ are instanton numbers.
These count degree $d$ curves -- in 93, this was shown to compute the number of cubic curves on a general quintic threefold (?).
Perhaps not clear why the $n_d$ are integers.
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:::{.theorem title="2020"}
For quintics, the only primes that divide the denominators of $n_d$ are $p=2,3,5$.
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:::{.remark}
A 4th order differential operator $L\in \QQ\adjoin{t, {d\over dt}}$ is a Calabi-Yau operator if

- Its singularities are regular,
- $t=0$ is a point of MUM,
- $L$ is self-dual,
- $L$ satisfies integrality conditions: 
  - A holomorphic solution $y_0(t)\in \ZZ[t]$,
  - Canonical coordinates $q=\exp(y_1/y_0) \in \ZZ\fps{t}$,
  - Instanton numbers $n_d$ are integral.

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:::{.remark}
Proving integrality goes through \(p\dash \)adic Frobenius structures (originally due to Dwork): equivalence of local systems for $L$ and its pullback under $t\mapsto t^p$ over the field $E_p = \hat{\QQ\rff{t}}$ of \(p\dash \)adic analytic functions.
See *flat sections*.
Can form a matrix $\phi$ representing the Frobenius structure by expanding in a basis of flat sections?
Conjecturally, all Calabi-Yau differential operators have a \(p\dash \)adic Frobenius structure for almost all $p$, as checked against a large database.
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:::{.theorem title="2020"}
If $L$ admits a \(p\dash \)adic Frobenius structure, the previous theorem holds with integrality replaced by $p\dash$integrality, i.e. replace $\ZZ$ with $\ZZpadic$ everywhere.
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:::{.remark}
See \(p\dash \)adic Cartier operation, Dwork crystals.
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:::{.theorem title="?"}
There is a decomposition of \(p\dash \)adic cohomology: fix $1<k<p$ for a fixed prime $p$, assume $R$ is \(p\dash \)adically complete and the $k$th Hasse-Witt condition is satisfied.
Then $\hat\Omega_{f} = \Omega_f(k) \oplus \mcf_k$ where the first term is the free \(R\dash\)module generated by $x^u/f^k$ for $u\in k\Delta \intersect \ZZ^n$ and $\mcf_k$ are forms $\omega = \sum a_u x^u$ where $a_u \in \ell^k R$ where $\ell = \gcd(u_1,\cdots, u_n)$, or alternatively $C_p^s(\omega)\in p^{ks} \hat\Omega_{(f^\circ)^s}$, or $\hat{\Omega}_f$ intersect with the \(R\dash\)module generated by $\prod x_{i_k} \dd{}{x_{i_j}}\sum b_u x_u$, the submodule of $k$th order formal partial derivatives.
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:::{.remark}
Key step in proving integrality: show a modular congruence holds to get a vanishing result?

> See classification of CYs into 14 deformation classes.

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