# Talk 3: Dani Kaufman, Cluster varieties, amplitude symbols, and mutation invariants (Tuesday, July 19)

## Cluster Varieties

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Recall: cluster varieties are varieties with a toric atlas generated by some extra combinatorial data.
Encodes unexpected coordinate functions, and automorphisms of the cluster structure encode exotic automorphisms of the variety.
Cluster structure for the Grassmannian: uses Plucker coordinates.
New coordinate functions appear as part of a *symbol alphabet* for scattering amplitudes for $N = 4$ SYM (super Yang-Mills) theory.
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Idea: the same function can be obtained from several different constructions, classifying mutation invariants shows that these must produce the same functions.
Examples of such special functions: traces of monodromy, cluster character, Diophantine equations of Markov numbers, theta functions, Skein modules elements of closed loops, etc.
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Cluster varieties: varieties built out of seed tori glued along birational morphisms called mutations.

![](figures/2022-07-19_15-26-10.png)

Geometric type cluster varieties can be tracked with quivers -- a quiver is used as the initial seed and generated a cluster variety after taking mutations.
Cluster ensembles associated with a quiver $Q$: a pair of cluster varieties $(A, X)$ and a map $p:A\to X$ defined on coordinates on the same seed by $p^*(x_i) = \prod_j a_j^{e_{ij}}$.
Introduced by Fock, Goncharov toward an algebraic theory of higher Teichmüller spaces.
Example: $\Gr_p(p+q)$ and $\Conf_n(\PP^{k-1})$ are an $(A, X)$ pair.
Interestingly, the immutable seeds for $\Gr_3(7)$ look like an $E_6$ diagram!
There are "phase transitions" for values of $p,q$ where there is a jump from finitely many clusters to infinitely many.
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## Amplitudes

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Scattering amplitudes in the planar limit $N=4$ SYM live in $\Conf+n(\PP^3)$, and the amplitudes for up to two loops can be written in terms of polylogarithms.
The arguments of these polylogarithms come from the $X$ cluster variety structure on this configuration space, and constitute the *symbol* of the amplitude which encodes its singularity and branch cut structure.
Goal: understand what happens as cluster ensembles transition from finite type to infinite type.
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## Cluster modular groups

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Idea: automorphism group $\Gamma$ of mutation structures, so similar to a mapping class group.
Can define an action of this on the rational function fields of these varieties.
Mutation invariant functions: those invariant under some subgroup of $\Gamma$.
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Example: take the Markov quiver, which is associated to the punctured torus and the affine root system of type $A_1^{(1, 1)}$ and $\Gamma = \PSL_2(\ZZ)$.
Similarly a 4-punctured sphere is of type elliptic $D_4^{(1, 1)}$.
See Somos sequences.
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## Surface cluster ensembles

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Look at $S$ associated to a marked surface and take $G = \PSL_2(\RR)$, then $A$ corresponds to Penner's decorated Teichmüller spaces and $X$ with Teichmüller space with Fock coordinates parameterizing hyperbolic somethings with an "opening".
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Main theorem of their thesis: classification of invariants for the action of Dehn twists on surface cluster ensembles.
The invariant ring $\RR(X)^{\gens \gamma}$ is generated by traces of monodromy operators of excised closed curves and invariant $X$ coordinates for an excising triangulation of $\delta$ a simple closed curve, where $S$ is a marked surface and $\gamma\in \Gamma_S$ is a Dehn twist about $\delta$.
For $A$, it's generated by curves that don't intersect $\delta$.
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There's some way to obtain a quiver from a triangulation of a surface.
More generally, each cluster modular group has an element that looks like a Dehn twist (a *cluster Dehn twist*), even when the cluster doesn't come from a surface at all.
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Some questions:

- What can we say about cluster invariants from higher Teichmüller spaces? 
  Are these related to higher laminations, tensor webs?

- Are the square root symbols related to polylogarithm relations?

- The collection of invariants for a Dehn twist behave like cluster variables for a limit cluster ensemble.
  How is this limit related to compactifications of Teichmüller space?

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