# Elana Kalashnikov, Unwinding the Gelfand—Cetlin toric degeneration on the mirror (Thursday, July 21)

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Batyrev mirror symmetry: anticanonical CYs have mirrors which are Fanos in toric varieties, where $Z \subseteq X_p$ and $Z\dual \subseteq X_{p\dual}$ for $p$ a polytope and $p\dual$ its dual polytope.
How to extend from mirror hypersurfaces in a Calabi-Yau to mirrors in a Grassmannian?
Gelfand-Cetlin degenerate $\Gr_n(m)\leadsto X_p$ a toric variety and proceed as before (?), how do we leave the toric world?
It turns out that there is a Grassmannian on the mirror, and a natural generalization of the Greene-Plesser quintic mirror.
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Generalizing Greene-Plesser mirrors: mirror pairs of the form $Z \subset \PP^{n-1} \mapstofrom Z\dual \subseteq \PP^{n-1}/ H$ for $H$ a finite group constructed as follows.
Write $\PP^{n-1} = \CC^n \gitquot \CCstar$ as a GIT quotient.
Think of this as $\Mat_{1\times n}(\CC) \gitquot \GL_1(\CC)$.
Define $\tilde H_n = \ts{(\zeta_1, \cdots, \zeta_n) \st \zeta_i^n=1, \prod \zeta_i = 1}$, then $\tilde H_n \actson \Mat_{1\times n}(\CC)$, and finally $H_n = \tilde H_n / \tilde H_n \intersect \GL_1(\CC)$.
Replacing $\GL_1(\CC)$ with $\GL_r(\CC)$ yields $\Gr_n(r) = \Mat_{r\times n}(\CC) \gitquot \GL_r(\CC)$, and one defines $H_{n, r}$.
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The CYs $Z \subseteq \Gr_n(r)$ and $\tilde Z \in \Gr_n(r)/ H_{n, r}$ are related.
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Take a toric degeneration $Z \subseteq \Gr_n(r)\leadsto Z_p \subseteq X_p$ a Fano toric, apply BB mirror symmetry to get $Z_p\dual \subseteq X_{p\dual}$, take a "geometric transition" (blow up, change the stability conditions, variation of GIT) to get $\tilde Z_p \subseteq \tilde X_p$, and smooth to get $\tilde Z$.

![](figures/2022-07-21_16-18-56.png)
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Degenerate toric variety to one associated to a certain reflexive polytope coming from a quiver moduli spaces $\mcm_\theta(LQ, (1,1,\cdots, 1))$ which is a Fano where $\theta$ is a stability condition.
Here $LQ$ is a ladder quiver.
There are correspondences between arrows in $LQ$ and coordinates on $X_{P_{n, r}}$, paths $0\to 1$ and partitions $\lambda \vdash r\times (n-r)$, subsets of $[n]$ of size $r$, and sections of a very ample line bundle $\OO(D)$ where $S_\lambda = \prod_{a\in \lambda} x_a$ yielding $X_{P_{n, r}} \injects \PP^{{n\choose r} - 1}$ with Plucker coordinates $p_\lambda$.
What equations cut out $X_{P_{n, r}}$ in this projective space?
Can define operations $\sigma \wedge \lambda, \sigma \vee \lambda$ for two "incomparable" grid paths (pairs of paths which cross) through an $r\times (n-r)$ grid by taking maximal or minimal parts of each, and these generate the correct equations.
This yields the toric degeneration described as step 1 in the image above.
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Step 2: use a new version of polytope duality. Let $P \subseteq N_\RR$ be a reflexive polytope, let $Q$ denote the **primitive dual** of $P$.
For fixed $P, Q$, BB mirror symmetry looks like $X_P\mapstofrom X_Q/G$ and $X_Q \mapstofrom X_P/G$, where $G = M/\bar M \cong \bar{M}\dual/ N$.
See Gelfand-Cetlin polytope, toric variation of GIT.
What geometric transition means in this context:


![](figures/2022-07-21_16-37-56.png)

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## Toric un-degenerating

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We get $H_{n, r}$ from mirror symmetry and $G$ from the Gelfand-Cetlin polytope, and it turns out that $H_{n, r} \leq G$ is the maximal subgroup for which $\Gr_n(r)$ is an equivariant subvariety.
Constructing the toric degeneration as an equivariant family: let $Z \subseteq X\da \PP^{{n\choose r} - 1} \times \CC^k$ be the family cut out by Plucker relations with extra parameters tracked by the $\CC^k$ part.
Extend the action of $G$ to $\CC^k$ so that $Z$ is $G\dash$equivariant.
Then $Z/G \subseteq X/G \to \CC^k/G$ is a family with special fiber $X_{P_{n, r}}/G$ and general fiber $\Gr_n(r)/H_{n, r}$, so $X_{P_{n, r}}/ G$ smooths to the Grassmannian analogy of Greene-Plesser.
This more or less comes from the orbit-stabilizer theorem and carefully tracking the stabilizer.
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The family of hypersurfaces $\sum_\lambda p_\lambda^n + \psi \prod_{\lambda } p_\lambda = 0$ in $\Bl \Gr_n(r)/H_{n, r}$ is a Calabi-Yau compactification of a Laurent polynomial mirror for $\Gr_n(r)$.
This is a candidate mirror to $Z \subseteq \Gr_n(r)$.
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