# Talk 2: Man Wai Cheung, Cluster dualities and superpotentials for Grassmannian (Tuesday, July 19)

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Goal: Marsh-Rietsch potential write down a potential for Grassmannians.
Gross-Hacking-Keel-Kontsevich do as well, we'll construct a map $P$ going from the former to the latter.
Makes use of cluster structure on $\Gr$, cluster varieties, and a GHKK construction.
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Cluster algebras: start with a seed $s = (\ts{A_1,\cdots, A_n}, B = (b_{ij}))$ where $A_i$ are variables and $B = (b_{ij})$ is a skew-symmetrizable matrix.
Start with an initial seed, allow "mutations" taking $s\to s'$ where the new $A_i$ are functions of the old, and take all such possible iterations to generate the algebra.
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Example: $s = \qty{\ts{A_1, A_2}, \matt 0 {-1} 1 0}$.
Mutate at $1$ to get $A_1 A_1' = 1 + A_2$; write $A_3 \da A_1'$ to get $A_3 = {1+A_2\over A_1}$.
The next seed is $(\ts{A_3, A_2}, B')$.
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Can express cluster structures in terms of triangulations of $(n+3)\dash$gons:

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![](figures/2022-07-19_11-30-13.png)

This flip corresponds to the mutation relation $A_{24} A_{13} = A_{12} A_{34} + A_{14} A_{23}$.
Since we can not flip the boundary, some variables are frozen -- namely those corresponding to adjacent nodes, so $A_{12}, A_{23}, A_{34}, A_{41}$.
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The homogeneous coordinate ring of $\CC[\Gr_k(n)]$ carries a cluster structure.
On the geometry side, cluster varieties are log Calabi-Yau whose rings of regular functions carry cluster structures.
For seeds, variables correspond to algebraic tori $(\CC\units)^n$, and mutations are birational maps between tori and defines a gluing between them.
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Formally defining a cluster variety: start with a lattice $N$ and skew-symmetric form $\ts{\wait, \wait}$ on $N$, and integers $d_i$, and a seed given by a basis for $N$.
Take a dual basis $M$ and scale to $M^\circ$ by taking a basis ${1\over d_i} e_i^*$.
Take the corresponding tori $A, X$ and glue by mutations to get the corresponding cluster variety.
The form gives a cluster ensemble lattice map $p^*: N\to M^\circ$ where $n \mapsto \ts{n, \wait}$.
Since these commute with mutation, and produce a map of varieties $A\to X$.
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There is a Langlands dual given by sending $B\mapsto B^{-t}$.
A conjecture (Bardwellcot-Evans-C-Hong-Lin) of mirror symmetry: $A \mapstofrom X^L$ and $X\mapstofrom A^L$.
For today, the exchange matrices are skew-symmetric, so the expectation is that $A \mapstofrom X$ are mirrors.
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Cluster varieties can be described by scattering diagrams: defining $N_\RR, M_\RR$, this is a collection of walls with finiteness and consistency conditions.
A wall is a pair $(d, f_d)$ where $d \subseteq M_\RR$ is the support of walls, a convex rational polyhedral cone of codimension 1.

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![](figures/2022-07-19_11-51-51.png)

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Crossing walls: $p_\gamma: z^m \mapsto z^m f_d^{\pm \ip{n_0}{m} }$ where $n_0$ is a primitive normal to a wall and $\gamma$ is a path between chambers.
A motivating example for theta functions: $(\CCstar)^2$ satisfies $H^0((\CCstar)^2; \OO) = \bigoplus _{m_1, m_2\in \ZZ} \CC z_1^{m_1} z_2^{m_2}$.
To each nonzero point in $M^\circ$ assign a theta function $\theta_m$, defined from a collection of broken lines with initial slope $m$ and endpoint $Q$ in the positive chamber.
Identify points in the scattering diagram with tropical points of Langlands dual cluster varieties.
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Cluster structures for $\Gr$: have a distinguished anticanonical divisor $D_{\mathrm{ac} } = \sum _{i=1}^n D_i$ where $D_k = D_{[i+1, i+k]} = \ts{p_{[i+1, i+k]} = 0}$.
Let $\Gr_k(\CC^n)^\circ = \Gr_k(\CC^n) / D$, the positroid... something.
Compactify using boundary divisors $D_k = D_{[i+1, i+(n-k)]}$.
Each irreducible component yields a term in the potential of the mirror (something about LG models).
Can somehow get valuations on the coordinate ring and write the potential directly.
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Marsh-Rietsch define an isomorphism from the Jacobi ring of $(\Gr_k(\CC^n)^\circ, W)$ to the equivariant quantum cohomology ring of $\Gr_{n-k}(\CC^n)$.
Rietsch-Williams: there is a Newton-Okounkov body on one side $\Delta(D)$, identifies points in this as tropical points on functions extending to the boundary of... a potential $\Gamma$?
Use theta reciprocity to get $\Delta(D) \cong \Gamma$ via $p^*$.
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