[Paper Review] Geometry of canonical bases and mirror symmetry

A. B. Goncharov, Linhui Shen|arXiv (Cornell University)|Sep 23, 2013

Algebraic structures and combinatorial models62 references4 citations

TL;DR

This paper establishes a geometric framework linking canonical bases in representation theory to mirror symmetry via decorated surfaces and positive structures on moduli spaces of G-local systems. It introduces a potential function W on the moduli space A_{G,S}, whose tropicalization parametrizes top-dimensional components of convolution morphism fibers and provides a canonical basis for tensor product invariants of the Langlands dual group G^L, generalizing Kamnitzer's hive model and extending Mirković-Vilonen cycles to mixed flag configurations.

ABSTRACT

A decorated surface S is a surface with a finite set of special points on the boundary, considered modulo isotopy. Let G be a split reductive group. A pair (G, S) gives rise to a moduli space A(G, S), closely related to the space of G-local systems on S. It has a positive structure. So the set of its integral tropical points is defined. We introduce a rational positive function W on A(G, S), the potential. The condition that its tropicalisation is non-negative determines its subset. For SL(2), we recover the set of integral laminations on S. We prove that when S is a disc with n special points on the boundary, this set parametrises top dimensional components of the convolution varieties. Thus, via geometric Satake correspondence, they provide a canonical basis in tensor product invariants of irreducible modules for the Langlands dual group. When G=GL(m), n=3, there is a special coordinate system on A(G,S). We show that it identifies our set with the set of with Knutson-Tao's hives. Our result generalises a theorem of Kamnitzer, who used hives to parametrise top components of convolution varieties for GL(m), n=3. For n>3, we prove Kamnitzer's conjecture. We define canonical bases in tensor products, generalizing the Mirkovic-Vilonen basis in a single representation. We prove that for any S, the set of positive integral tropical points of A(G, S) parametrise top components in a new space, surface affine Grasmannian. We view W as a potential for Landau-Ginzburg model on A(G,S). We conjecture that the pair (A(G,S), W) is the mirror dual to the moduli space of local systems on S for the Langlands dual group. In a special case, we recover Givental's description of the quantum cohomology connection for flag varieties.

Motivation & Objective

• To establish a geometric correspondence between canonical bases in tensor product invariants of Langlands dual groups and moduli spaces of G-local systems on decorated surfaces.
• To generalize the Mirković-Vilonen basis and its parametrization to mixed configurations of flags and higher n-tuples of representations.
• To define a canonical basis in V_{λ₁} ⊗ ⋯ ⊗ V_{λₙ}^{G^L} using tropical integral points of a positive moduli space with a potential function W.
• To formulate a Landau-Ginzburg mirror symmetry duality between the space (A_{G,S}, W) and the moduli space of G^L-local systems on S.

Proposed method

• Introduces a positive structure on the moduli space A_{G,S} of G-local systems on a decorated surface S, enabling the definition of integral tropical points A_{G,S}(Z^t).
• Defines a rational positive function W on A_{G,S}, whose tropicalization W^t maps to Z, and defines the subset A^+_{G,S}(Z^t) where W^t ≥ 0.
• Uses the geometric Satake correspondence to identify top-dimensional components of fibers of convolution morphisms with A^+_{G,S}(Z^t), thus parametrizing canonical bases in tensor product invariants.
• For GL_m and n=3, shows that A^+_{GL_m,S}(Z^t) is in bijection with Knutson-Tao hives, proving Kamnitzer’s conjecture for n>3.
• Constructs generalized Mirković-Vilonen cycles as components of intersections S_e^λ ∩ S_w^μ, parametrized by tropical points of mixed configuration spaces.
• Defines a new moduli space Loc_{G^L,S} of G^L-local systems on S, and shows that A^+_{G,S}(Z^t) parametrizes a canonical basis in the space of regular functions on Loc_{G^L,S}.

Experimental results

Research questions

• RQ1 How can canonical bases in tensor product invariants of irreducible representations of G^L be geometrically parametrized using moduli spaces of G-local systems?
• RQ2 What is the role of the potential function W on A_{G,S}, and how does its tropicalization relate to integrality and positivity in representation theory?
• RQ3 How does the parametrization of canonical bases via A^+_{G,S}(Z^t) generalize Kamnitzer’s hive model and extend to n>3?
• RQ4 Can generalized Mirković-Vilonen cycles be constructed and parametrized using mixed configurations of flags and positive structures?
• RQ5 Is there a mirror symmetry duality between (A_{G,S}, W) and the moduli space Loc_{G^L,S} of G^L-local systems?

Key findings

• For G=SL₂, the set A^+_{G,S}(Z^t) recovers the set of positive integral A-laminations from [FG1], establishing a geometric link to surface topology.
• For G=GL_m and n=3, the set A^+_{GL_m,S}(Z^t) is in bijection with Knutson-Tao hives, proving Kamnitzer’s conjecture for n>3.
• The parametrization of canonical bases via A^+_{G,S}(Z^t) is cyclically invariant, resolving an obscurity in Berenstein-Zelevinsky’s original parametrization.
• The generalized Mirković-Vilonen cycles are parametrized by A^+_{G,S}(Z^t), and their construction avoids explicit parametrizations via a canonical map κ.
• The potential W on A_{G,S} is proposed as the Landau-Ginzburg superpotential, with (A_{G,S}, W) conjectured to be mirror dual to Loc_{G^L,S}.
• The construction yields a canonical basis in the space of regular functions on Loc_{G^L,S}, with the basis elements indexed by A^+_{G,S}(Z^t).