The algebraic modular functor conjecture of Fock and Goncharov, that cutting a surface yields a canonical gluing isomorphism of the associated quantum algebras, is proven for type A_n (G = PGL(n+1)).
Cluster structure on genus 2 spherical DAHA: seven-colored flower
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abstract
We construct an embedding of the Arthamonov-Shakirov algebra of genus 2 knot operators into the quantized coordinate ring of the cluster Poisson variety of exceptional finite mutation type $X_7$. The embedding is equivariant with respect to the action of the mapping class group of the closed surface of genus 2. The cluster realization of the mapping class group action leads to a formula for the coefficient of each monomial in the genus 2 Macdonald polynomial of type $A_1$ as sum over lattice points in a convex polyhedron in 7-dimensional space.
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The algebraic modular functor conjecture in type $A_n$ quantum Teichm\"uller theory
The algebraic modular functor conjecture of Fock and Goncharov, that cutting a surface yields a canonical gluing isomorphism of the associated quantum algebras, is proven for type A_n (G = PGL(n+1)).