Laminations from the symplectic double

Abstract

Let $S$ be a compact oriented surface with boundary together with finitely many marked points on the boundary, and let $S^{\circ }$ be the same surface equipped with the opposite orientation. We consider the double $S_{\mathcal{D}}$ obtained by gluing the surfaces $S$ and $S^{\circ }$ along corresponding boundary components. We define a notion of lamination on the double and construct coordinates on the space of all such laminations. We show that this space of laminations is a tropical version of the symplectic double introduced by Fock and Goncharov. There is a canonical pairing between our laminations and the positive real points of the symplectic double. We derive an explicit formula for this pairing using the $F$-polynomials of Fomin and Zelevinsky.

69 pages. Version 2: Added Section 6.4 and various clarifications. Version 3: Revised Section 6

References in corpus (2)

• The quantum dilogarithm and representations quantum cluster varieties
• Bases for cluster algebras from surfaces

Cited by in corpus (3)

• Voros symbols as cluster coordinates
• Categorified canonical bases and framed BPS states
• Quantization of canonical bases and the quantum symplectic double