Webs and tropical coordinates on surfaces

Seminar: 
Clusters and Geometry
Event time: 
Friday, November 20, 2020 - 1:00pm
Speaker: 
Zhe Sun
Speaker affiliation: 
IHES
Event description: 

Kuperberg introduced oriented 3-valent graphs on the surface, called 3-webs, to study the SL_3-invariant tensor products T of irreducible representations of SL_3. Kuntson-Tao found a family of linear inequalities to characterize when T contains an invariant vector. Then Goncharov--Shen identified these inequalities with the positivity of the tropical real A moduli space. On the surface, we identify the space of 3-webs up to homotopy with certain lattice in the positive tropical real A moduli space mapping class group equivariantly. As a consequence, as predicted by Fock--Goncharov duality conjecture, these tropical points parameterize a linear basis of the regular function ring of the dual space explicitly. This is a joint work mostly with Daniel Douglas.