35 problems

Let $C$ be the three-punctured sphere and let $\mathcal{F}^{\mathrm{GS}}(\alpha_0,\alpha_1,\alpha_\infty)$ denote the Goncharov–Shen block associated with the configuration shown i…

Let $W_K$ denote the higher-rank free-field algebra associated with the Toda theory under consideration, and consider free-field correlators in the large $N$ limit whose integratio…

The paper studies an extension of the free-field formalism to smooth spectral coverings, using the Maulik–Okounkov R-matrix to incorporate wall-crossing effects in spectral-network…

Let $C$ be the underlying curve, let $\Sigma$ be a real exact Lagrangian in $T^*C$, and let $q_1,\dots,q_\kappa$ be points of $C$. The maps … and … are respectively defined by holo…

Partially wrapped Fukaya categories are related to spectral networks as described above, and Stokes local systems provide the corresponding generalization of local systems for this…

Exponential-network reconstruction conjecture. The underlying quiver structure of $\mathcal{Q}$ is obtained from these calibrated paths: its vertices correspond to the basis states…

Two-matrix Boutroux-curve conjecture. The equilibrium measures and their Stieltjes transforms satisfy

Boutroux-curve conjecture. The large-$N$ equilibrium measure and its Stieltjes transform satisfy

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 $C$ be a surface and let ${\mathrm{Sk}}_{-1}(C, {\mathrm{GL}}(2))={\mathcal O}({\mathcal M}(C,{\mathrm{GL}}(2)))$ be the commutative algebra generated by the Verlinde operators…

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…

Spectral networks are networks of trajectories on a Riemann surface, while clusters are algebraic structures associated with coordinate systems and related moduli spaces. The paper…

Let $\mathbb{L}$ be an embedded $\Lambda_{\mathbb{L}^{\mathrm{trop}}}$-admissible Lagrangian multi-section with simple branching. Existence conjecture. There exists a spectral netw…

Uniqueness conjecture. All spectral network representatives of a given semistable object in $V\left(\mathcal F(S,M;\mathcal E)_{\mathbf K}\right)$ have the same underlying support…

Spectral-network stability conjecture. There is a stability condition on the homotopy category of $\mathcal F(S,M;\mathcal E)_{\mathbf K}$ with central charge $Z$, whose semistable…

Gaiotto–Moore–Neitzke's shortest-saddle-connection conjecture. The exponent $\gamma$ is given by the length of the shortest saddle connection in the metric on $\mathcal{C}$ induced…

Let $\mathcal A$ be the Hitchin base. A basic WKB cameral network associated to $a\in\mathcal A$ consists of a suitable homotopy-equivalent open subset of the curve together with f…

Let $Y\rightarrow \operatorname{Loc}_{G}(X)$ be the spectral-network coordinate chart, let $\chi_{1,a}:\operatorname{Loc}_{G}(X)\rightarrow (\mathbb C^{\times})^{m}$ be the origina…

Let $X$ be the underlying curve, $G$ a reductive group, and let $Y\rightarrow \operatorname{Loc}_{G}(X)$ be a coordinate chart obtained by the spectral-network regluing constructio…

For each topological type $w$ of spectral network, let ${\mathcal V}_w$ be the region where the WKB network has type $w$, let ${\mathcal X}_w$ be the associated coordinates, and le…

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 $A_{N-1}$ theories be associated with Riemann surfaces having full punctures, and let a candidate BPS graph be constructed by replacing each branch point of an $A_1$ BPS graph…

Let $P_0$ be a degree $n$ polynomial, let $\Gamma$ be the charge lattice associated to its spectral curve, and let $\Omega(P_0,\gamma)$ be the BPS count of finite webs of charge…