Les Houches lectures on non-perturbative Seiberg-Witten geometry

In these lectures we detail the interplay between the low-energy dynamics of quantum field theories with four supercharges and the exact WKB analysis. This exposition may be the first comprehensive account of this connection and includes new arguments and results. The lectures start with the introduction of massive two-dimensional $\mathcal{N}=(2,2)$ theories and their spectra of BPS solitons. We place these theories in a two-dimensional cigar background with supersymmetric boundary conditions labelled by a phase $\zeta = e^{i \vartheta}$, while turning on the two-dimensional $\Omega$-background with parameter~$\epsilon$. We show that the resulting partition function $\mathcal{Z}_{\mathrm{2d}}^\vartheta(\epsilon)$ can be characterized as the Borel-summed solution, in the direction $\vartheta$, to an associated Schr\"odinger equation. The partition function $\mathcal{Z}_{\mathrm{2d}}^\vartheta(\epsilon)$ is locally constant in the phase $\vartheta$ and jumps across phases $\vartheta_\textrm{BPS}$ associated with the BPS solitons. Since these jumps are non-perturbative in the parameter~$\epsilon$, we refer to $Z^\vartheta_\mathrm{2d}(\epsilon)$ as the non-perturbative partition function for the original two-dimensional $\mathcal{N}=(2,2)$ theory. We completely determine this partition function $\mathcal{Z}^\vartheta_\mathrm{2d}(\epsilon)$ in two classes of examples, Landau-Ginzburg models and gauged linear sigma models, and show that $\mathcal{Z}^\vartheta_\mathrm{2d}(\epsilon)$ encodes the well-known vortex partition function at a special phase $\vartheta_\textrm{FN}$ associated with the presence of self-solitons. This analysis generalizes to four-dimensional $\mathcal{N}=2$ theories in the $\frac{1}{2} \Omega$-background.