The cluster symplectic double and moduli spaces of local systems

Abstract

We prove a conjecture of Fock and Goncharov which provides a birational equivalence of a cluster variety called the cluster symplectic double and a certain moduli space of local systems associated to a surface.

Voros symbols as cluster coordinates

“…In 1983, Voros [26] described a reformulation of this method which can be used to construct exact solutions. In this reformulation, one begins by constructing formal series solutions of (1). These formal series are generally divergent, and genuine analytic solutions are obtained by taking Borel sums.…”

“…Then the Voros symbol associated to a class γ ∈ H 1 (Σ φ \ π −1 Crit(φ)) is a formal series in . It is defined as e Vγ where V γ is the period of a certain meromorphic 1-form integrated around the cycle γ. Voros symbols play an important role in WKB analysis [21], where they appear in the explicit calculation of the monodromy group of equation (1). If φ is a saddle-free differential, then there is a canonical cycle γ j associated to each arc j of the WKB triangulation, and hence we have the associated Voros symbol e Vγ j .…”

“…Such a moduli space would have twice the dimension of and would be birational to the principal cluster variety . Closely related spaces associated to the symplectic double cluster variety were described in .…”

Quantization of canonical bases and the quantum symplectic double

“…As explained in the introduction, these parametrize points of A and D valued in the semifield Z t of tropical integers. Further explanation can be found in [2,3,8,11]. 1.…”

“…This cluster variety was introduced by Fock and Goncharov and plays a key role in their work on quantization of cluster Poisson varieties [10]. As before, one can associate, to a marked bordered surface , a certain moduli space of local systems, and this moduli space is birationally equivalent to a symplectic double cluster variety, which we denote by D [2,11].…”