Voros symbols as cluster coordinates

Abstract

We show that the Borel sums of the Voros symbols considered in the theory of exact WKB analysis arise naturally as Fock-Goncharov coordinates of framed $PG{L}_2(\mathbb{C})$-local systems on a marked bordered surface. Using this result, we show that these Borel sums can be meromorphically continued to any point of ${\mathbb{C}}^{\ast }$, and we prove an asymptotic property of the monodromy map introduced in collaboration with Tom Bridgeland.

arXiv comment46 pages. Version 2: Accepted for publication in Journal of Topology. Version 3: Clarifications added to Theorem 1.4 and its proof, superseding the published version

References in corpus (7)

• Four-dimensional wall-crossing via three-dimensional field theory
• Riemann-Hilbert problems from Donaldson-Thomas theory
• The monodromy of meromorphic projective structures
• Stability conditions and cluster varieties from quivers of type $A$
• Laminations from the symplectic double
• The cluster symplectic double and moduli spaces of local systems
• Opers and TBA

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• Stability conditions, cluster varieties, and Riemann-Hilbert problems from surfaces
• A geometric recipe for twisted superpotentials
• Existence and Uniqueness of Exact WKB Solutions for Second-Order Singularly Perturbed Linear ODEs
• Quantum periods and TBA equations for $\mathcal{N}=2SU(2)N_f=2$ SQCD with flavor symmetry
• On the wall-crossing formula for quadratic differentials
• Topological recursion and uncoupled BPS structures II: Voros symbols and the $\tau$-function
• Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies
• Higgs bundles, harmonic maps, and pleated surfaces
• Stability conditions and Teichmüller space
• Nilpotent Higgs bundles and families of flat connections
• Joyce structures on spaces of quadratic differentials