Exact WKB analysis and cluster algebras

Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies

“…In this context, the solution was used to construct semi-explicitly the hyperkahler metric on a Hitchin moduli space associated to the given theory. As is mentioned in [Br1,§7], the BPS Riemann-Hilbert problem is also closely related to the Stokes structure of the Voros symbols in the theory of exact WKB analysis of a Schrödinger-type ODE discussed in [IN1,IN2]. See [I1,Al1,Ku] for a relation between Fock-Goncharov coordinates and Voros symbols, and see also [Al2] for the further development in this direction.…”

“…The lattice Γ is a sublattice of the homology group H 1 (Σ; Z) of the spectral cover, and the central charge Z is the period map; i.e., Z(γ) = γ λ = γ √ ϕ for γ ∈ Γ. The BPS invariants are given by a weighted counting of saddle trajectories or ring domains on X associated to ϕ; this is based on the idea of Gaiotto-Moore-Neitzke [GMN2] in which they find a combinatorial algorithm to describe the BPS spectra of certain four-dimensional supersymmetric QFTs through a spectral network (known as a Stokes graph in WKB literature such as [KT,IN1]).…”

Quantum periods and TBA equations for $\mathcal{N}=2\ SU(2)\ N_f=2$ SQCD with flavor symmetry

“…The discontinuity of the Borel resummations of the quantum periods can be captured by the Delabaere-Dillinger-Pham formula (theorem 2.5.1 of [32], and theorem 3.4 of [33]).…”

Constructing Buildings and Harmonic Maps

“…Let us first consider the case of representations into SL 2 . Exact WKB analysis and its relationship with the geometry of quadratic differentials has been considered extensively by Iwaki and Nakanishi [13].…”

“…In the SL 2 case these subjects have been treated recently by Iwaki and Nakanishi [13], and one of the goals in the higher rank case would be to extend the various different aspects of their theory.…”