Hitchin systems, spectral networks and noncommutative clusters
Hot Topics: Cluster algebras and wall-crossing March 28, 2016 - April 01, 2016
Location: SLMath:
algebraic combinatorics
cluster algebras
Hitchin representation
Hitchin system
local systems
moduli spaces
hyperkahler
scattering diagram
parallel transport
14K22 - Complex multiplication and abelian varieties [See also 11G15]
14J50 - Automorphisms of surfaces and higher-dimensional varieties
13P10 - Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14483
Hitchin's integrable system is one of the fundamental examples of mirror symmetry in the sense of Strominger-Yau-Zaslow.
Many of the structures described in this workshop can be seen very concretely in this example. This example has moreover some extra structure, coming from the fact that it is actually a hyperkahler space.
I will review an approach to these moduli spaces which arose in my joint work with Davide Gaiotto and Greg Moore. The key player in the story is a set of cluster-type coordinate systems, very closely related to those appearing in the work of Fock-Goncharov. We will also see scattering diagrams very similar to those in the work of Gross-Hacking-Keel.
In the end I will describe a new point: when one tries to extend the cluster description over certain singular loci of the moduli space, the most natural description seems to involve not an ordinary cluster algebra, but rather a noncommutative version thereof. One special case of this story appears closely related to the "noncommutative marked surfaces" recently introduced by Berenstein-Retakh; there is also related work of Goncharov-Kontsevich.
Neitzke Notes
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14483
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