# Talk 1: Tyler Kelly, Open Saito-Givental Invariants and Mirror Symmetry (Tuesday, July 19)

:::{.remark}
Realizations of cycles and motives play a role in the physics of amplitudes in string theory and the study of Feynman integrals.
Today's focus: mirror symmetry, with a focus on certain integrals over cycles.
Goal of mirror symmetry: link

- Enumerative data for a symplectic manifold, algebraic variety, or complex function, or
- Periods of a so-called mirror

The ur-example is a quintic threefold.
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How this example works: count degree $d$ rational curves in $X \subset \PP^4$ a quintic threefold.
The mirror is a crepant resolution of 
\[
Z(\sum x_i^5 - 5\psi \prod x_i ) \subseteq \PP^4/(C_5)^3
,\]
a 1-parameter deformation of the quintic.
Compute as some integral $\int_\gamma \Omega_\psi$.
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How to construct the mirror? And how to distill the enumerative information?
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Today's goal: answer these questions for Landau-Ginzberg (LG) models $(X,G,W)$ where $X$ is a quasi-affine variety $G\actson X$ is a finite group, $W = ?$.
These give mirrors to Fanos, and noncommutative symplectic deformations of complete intersections in toric varieties.
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Example LG model: $(\CC^n, \prod \mu_{r_i}, \sum x_i^{r_i}) \mapstofrom (\CC^n, 1, \sum x_i^{r_i})$.
Answers to these questions for LG models: the mirror is built via a combinatorial construction, which annoyingly isn't geometric.
For enumerative data: oscillatory integrals associated to the RHS using Saito-Givental theory the encapsulate FJRW invariants.
Issue: deformation must be built perturbatively term-by-term.
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## Saito-Givental Theory in dimension 1

:::{.remark}
Start with $W:\CC^n\to \CC$ given by $\sum x_i^{r_i}$.
Build a twisted de Rham complex where $\delta = d + h\inv dW \wedgeprod (\wait)$ to get $0 \to \Omega^0 \mapsvia{\delta} \cdots$, then take hypercohomology.
For e.g. $W = x^r$, a basis can be computed explicitly for $H^1$.
One can cook up a perfect pairing between a certain relative homology $H_n(X, \Re(W/H)\ll 0, \CC)$ and this hypercohomology, roughly 
\[
(\psi, \omega) \mapsto \int_\xi e^{W\over h} \omega
\]
where $\Re(W/h) \ll 0$ so that the exponential term is small.
Now take deformations to distill enumerative info from this periods.
First approximation: take the versal deformation 
\[
W^s = \sum x_i^{r_i} + \sum s_{a_1,\cdots, a_n} \prod x_i^{a_i}
.\]
Integrate to get a power series whose coefficients are $\psi_k \in \CC\fps{s_{a_1,\cdots, a_n}}$.
Compute this using a dual basis and integration by parts.

Can try $W = x^4 + s_2 s^2 + s_1 x + s_0$. Problem: lack of linearity!
But this can be fixed to obtain flat coordinates and a primitive form $\omega$.
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Theorem: there is a deformed potential $W^t$ which has primitive form $\omega = \dx$ and flat coordinates $t_0, \cdots, t_{r-2}$ for the Frobenius manifold construction via Saito-Givental theory for the LG model $W = x^r$.
This is a generating function for open enumerative invariants.
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Strategy: start with $X$, where we want to understand its enumerative geometry.
Build the mirror $\tilde X$ using some type of *open* enumerative geometry, along with a deformation.
Look at integrals of forms over pieces of $\tilde X$, and that should recover the enumerative counts.
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Build a moduli of orbidisks $\mcm$ by taking configurations of marked points on $\PP^1$, where points off of $\RR$ come in pairs of complex conjugates. 
The local picture at marked points: $[\CC/\mu_r]$.
Compactify by considering degenerations of configurations -- internal nodes, boundary nodes, contracted boundaries:

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![](figures/2022-07-19_10-37-48.png)

The compactified moduli space is a real manifold with corners.
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Take the universal line bundle $\mcl \to \mcm$ and define the Witten bundle $R^1(\pi_* \mcl)^*$.
One needs to specify a canonical multisection $s_\can$ to control what happens on the boundary in order to integrate over $\mcm$.
Theorem: in dimension one, the open $r\dash$spin invariants exist and don't depend on choice of this multisection, enjoy topological recursion, and there are closed form solutions of certain invariants.
Parts of this theorem fail in dimension 2, although analogous statements can be made.
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:::{.theorem title="Gross-K-Tessler"}
Open FJRW invariants exist for $(\CC^2, \mu_r \times \mu_s, x^r+y^2)$ and do depend on the choice of $s_\can$.
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## Dimension 2

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Proceed as before, but run into issues with extra terms involving Gamma functions in $h^0$, as opposed to terms attached to $h\inv$ previously.
A solution: insert a hypergeometric function to cancel it, as Li-Li-Saito-Shen do.

What's the picture?

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![](figures/2022-07-19_10-56-16.png)

There are some cases with mild singularities (ADE and elliptic type), outside of these things become unmanageable.
Key foothold in the proof: no wall-crossing to deal with, which is advantageous.
Proof uses topological recursion: compute something in two different ways, set them equal, and play the two formulae off of each other.
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