26 problems

Let $\hbar$ be the semiclassical parameter, let $\theta_{\mathrm{B}}$ be the Bloch angle, and let $A_{\mathrm{ZJJ}}$ and $B_{\mathrm{ZJJ}}$ denote the functions entering the Zinn-J…

Let $Z(t,\nu,\hbar)$ be the perturbative partition function for the Painlevé I system, and let $\mathfrak S$ denote the Stokes automorphism associated with the $B$-periods. Write…

Let $(\nu^\pm,\rho^\pm)$ be two parameter pairs, and let $\mathscr T^\pm(t,\nu^\pm,\rho^\pm,\hbar)$ be analytic $\tau$-functions of $(P_{\rm I})$ defined on domains containing…

The Borel-summability and connection-formula conjecture. If the Stokes graph is saddle-free, then $Z$ is Borel summable, the displayed discrete Fourier series converges and defines…

Let $\nabla$ be a WKB-regular $\hbar$-flat connection with strongly GMN spectral curve, let $\Psi$ be a WKB formal solution, and let $\cL\Psi$ be its Borel–Laplace dual. Let…

Non-abelianization convergence conjecture. The resulting local system is convergent under the substitution $T=e^{-1/\hbar}$ for $0<\hbar\ll 1$. This predicts that the formal Noviko…

Let $\Gamma$ be a spectral curve over a punctured base, let $\vartheta$ be a phase, and let $H(\Gamma,\vartheta)=H_1(\widehat\Gamma;L(\vartheta))$ be the extended homology of the r…

Let $\nabla(\varepsilon)$ be a family of flat $\mathrm{GL}(n)$-connections whose rescaled limit is a Higgs bundle with spectral curve $\Gamma$. Fix a phase $\vartheta$ and let…

The quantization condition gives $\widetilde R(x)=\nu$, with $\nu=\pi\left(n+\frac{1}{2}\right)$, and perturbative inversion produces…

Let $\cQ$ be an $\hbar$-differential operator, lifted to an $\hbar$-connection after fixing a spin structure, and assume WKB-regularity. Let $L$ be the characteristic variety of…

Let $\cQ$ be an $\hbar$-differential operator, lifted to an $\hbar$-connection after fixing a spin structure, and assume WKB-regularity. Let $L$ be the characteristic variety of…

Consider conventional 2d-4d systems with local solutions of the associated differential equations, and let the Borel plane denote the plane in which the Borel singularities of thes…

Let ${\mathcal X}$ be the coordinates assigned to saddle-free WKB networks, including networks of FN type, by the construction using canonical bases of solutions to…

Let $C=C_{0,2}$ and restrict the coordinate function $\sigma$ on the space of quantum curves to the subspace represented by opers. Consider the Voros symbol associated to this rest…

Consider the non-perturbatively defined partition functions associated with the quantum curves studied in the paper, together with exact WKB and topological recursion. Canonical de…

Let ${\mathcal X}_\gamma(\hbar)$ be a spectral coordinate and define $\widetilde Z_\gamma(\hbar)=\hbar\log{\mathcal X}_\gamma(\hbar)$. Let ${\mathcal X}_\gamma^{{\rm RH}}(\hbar)$ b…

Let $u>0$ and $\hbar>0$, and consider the $T_3$ equation and its associated ${\rm SL}(3)$-connection. Apply the higher-rank exact WKB construction with phase $\vartheta=0$. Exact W…

Consider the cubic equation with spectral covering $\Sigma=\{y^3+p_2(z)y+p_3(z)=0\}$, with simple branch points, and let $\psi_i^\vartheta$ be the local WKB solution associated wit…

Choose a function $\vartheta(\arg\hbar)$ satisfying $\lvert\vartheta(\arg\hbar)-\arg\hbar\rvert\leq\frac{\pi}{2}$, and define…

Let ${\rm e}^{{\rm i}\vartheta}\mathbb R_+$ be the ray associated with the phase $\vartheta$, and let ${\mathcal X}_\gamma^\vartheta$ have the asymptotic expansion given by. On the…

Fix $(t,\nu)\in D_{t_\ast}\times D_{\nu_\ast}$ so that the Stokes graph has no Stokes segment. Let $C$ be a Stokes curve connecting a branch point $e$ and $\infty$, forming a commo…

Fix $(t,\nu)\in D_{t_\ast}\times D_{\nu_\ast}$ and suppose that the Stokes graph for $(t,\nu)$ has no Stokes segment. Let $U$ be a Stokes region, and let…

Consider a difference equation with a potential having simple turning points, and study the transition behavior of its WKB solutions near those turning points. Simple-turning-point…

The partition function of pure ${\cal N}=2$ $epsilon_i$-deformed gauge theory in four dimensions is studied using exact WKB methods, with corrections of the form … to the expansion…

Stokes triangulation–seed correspondence conjecture. We have $T=T'$ if and only if $(B',x',y')=(B,x,y)$. This conjecture asserts that the seed obtained from signed mutations record…