Geometry of canonical bases and mirror symmetry

Abstract

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 [FG1]. 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 We prove that when S is a disc with n special points on the boundary, the set A (When G = GLm, n = 3, there is a special coordinate system on A G,S [FG1]. We show that it identifies the set A + GLm,S (Z t ) with Knutson-Tao's hives [KT]. Our result generalises a theorem of Kamnitzer [K1], who used hives to parametrise top components of convolution varieties for G = GLm, n = 3. For G = GLm, n > 3, we prove Kamnitzer's conjecture [K1]. Our parametrisation is naturally cyclic invariant. We show that for any G and n = 3 it agrees with Berenstein-Zelevinsky's parametrisation [BZ], 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 of Mirković-Vilonen cycles [MV], and a canonical basis in V λ 1 ⊗ . . . ⊗ V λn , generalizing the Mirković-Vilonen basis in V λ . Our construction comes naturally with a parametrisation of the generalised MV cycles. For the classical MV cycles it is equivalent to the one discovered by Kamnitzer [K].We prove that the set A + G,S (Z t ) parametrises top dimensional components of a new moduli space, surface affine Grasmannian, generalising the fibers of the convolution maps. These components are usually infinite dimensional. We define their dimension being an element of a Z-torsor, rather then an integer. We define a new moduli space Loc G L ,S , which reduces to the moduli spaces of G L -local systems on S if S has no special points. The set A + G,S (Z t ) parametrises a basis in the linear space of regular functions on Loc G L ,S .We suggest that the potential W itself, not only its tropicalization, is important -it should be viewed as the potential for a Landau-Ginzburg model on A G,S . We conjecture that the pair (A G,S , W) is the mirror dual to Loc G L ,S . In a special case, we recover Givental's description of the quantum cohomology connection for flag varieties and its generalisation [GLO2]. [R2]. We formulate equivariant homological mirror symmetry conjectures parallel to our parametrisations of canonical bases.

Canonical bases for cluster algebras

“…The latter work can be viewed as a tropicalization of the descriptions of the potential in terms of holomorphic disks in [CO06], [A07]. Thus our construction explains the emergence of the Landau-Ginzburg potentials in [GS13]. Our potentials are determined by the cluster structure (and conjecturally, just the underlying log Calabi-Yau variety), and in particular are independent of any modular or representation theoretic interpretation of the cluster variety.…”

“…Note that Ξ is convex in our generalized sense. We show, making use of recent results of Magee and Goncharov-Shen, [Ma15], [Ma16], [GS16], that in the representation theoretic examples which were the original motivation for the definition of cluster algebras our polyhedral cones Ξ specialize to the piecewise linear parameterizations of canonical bases of Berenstein and Zelevinsky [BZ01], Knutson and Tao [KT99], and Goncharov and Shen [GS13]:…”

“…Potentials were considered in the work of Goncharov and Shen, [GS13], which in turn built on work of Berenstein and Zelevinsky, [BZ01] and Berenstein and Kazhdan, [BK00], [BK07]. The potential constructed by Goncharov and Shen has a beautiful representation theoretic definition, and was found in many situations to coincide with known constructions of Landau-Ginzburg potentials.…”

Donaldson–Thomas transformations of moduli spaces of G-local systems

“…If the group G has trivial center, then the principal affine space A is the moduli space of pairs pU, χq, where U is a maximal unipotent subgroup of G, and χ : U Ñ A 1 is a non-degenerate character (cf. [GS,Section 1]). For general G, there is a canonical non-degenerate character χ A assigned to a decorated A P A.…”

Towards a cluster structure on trigonometric zastava

“…It would be interesting to study its tropicalization and the corresponding set of positive integral tropical points, cf. [13]. 6.…”