Geometry of canonical bases and mirror symmetry

A decorated surface S is an oriented surface with boundary and a finite, possibly empty, set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group over Q . A pair ( G , S ) gives rise to a moduli space A G , S , closely related to the moduli space of G -local systems on S . It is equipped with a positive structure (Fock and Goncharov, Publ Math IHES 103:1–212, 2006 ). So a set A G , S ( Z t ) of its integral tropical points is defined. We introduce a rational positive function W on the space A G , S , called the potential . Its tropicalisation is a function W t : A G , S ( Z t ) → Z . The condition W t ≥ 0 defines a subset of positive integral tropical points A G , S + ( Z t ) . For G = SL 2 , we recover the set of positive integral A -laminations on S from Fock and Goncharov (Publ Math IHES 103:1–212, 2006 ). We prove that when S is a disc with n special points on the boundary, the set A G , S + ( Z t ) parametrises top dimensional components of the fibers of the convolution maps. Therefore, via the geometric Satake correspondence (Lusztig, Astérisque 101–102:208–229, 1983 ; Ginzburg, 1995 ; Mirkovic and Vilonen, Ann Math (2) 166(1):95–143, 2007 ; Beilinson and Drinfeld, Chiral algebras. American Mathematical Society Colloquium Publications, vol. 51, 2004 ) they provide a canonical basis in the tensor product invariants of irreducible modules of the Langlands dual group G L : 1 ( V λ 1 ⊗ … ⊗ V λ n ) G L . When G = GL m , n = 3 , there is a special coordinate system on A G , S (Fock and Goncharov, Publ Math IHES 103:1–212, 2006 ). We show that it identifies the set A GL m , S + ( Z t ) with Knutson–Tao’s hives (Knutson and Tao, The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture, 1998 ). Our result generalises a theorem of Kamnitzer (Hives and the fibres of the convolution morphism, 2007 ), who used hives to parametrise top components of convolution varieties for G = GL m , n = 3 . For G = GL m , n > 3 , we prove Kamnitzer’s conjecture (Kamnitzer, Hives and the fibres of the convolution morphism, 2012 ). Our parametrisation is naturally cyclic invariant. We show that for any G and n = 3 it agrees with Berenstein–Zelevinsky’s parametrisation (Berenstein and Zelevinsky, Invent Math 143(1):77–128, 2001 ), whose cyclic invariance is obscure. We define more general positive spaces with potentials ( A , W ) , parametrising mixed configurations of flags. Using them, we define a generalization