Les Houches lectures on non-perturbative Seiberg-Witten geometry
Loïc Bramley, Lotte Hollands, Subrabalan Murugesan
TL;DR
The paper develops a comprehensive framework linking 2d N=(2,2) quantum field theories to exact WKB analysis by introducing a non-perturbative partition function in the Omega-deformed cigar. It systematically builds the formalism from the SUSY algebra and vacuum structure through BPS solitons, spectral networks, and vertex brane realizations, and then applies it to GLSMs and LG models, deriving vortex/Higgs-Coulomb partition functions and their underlying spectral curves. A central theme is the quantization of spectral geometry via d_ε and the emergence of precise wall-crossing behavior encoded in Lefschetz thimbles, spectral networks, and W-abelianization, with connections to Seiberg-Witten geometry in 4d via dimensional reduction. The work provides explicit non-perturbative results for LGs and GLSMs, clarifies the role of the Ω-background in generating quantum curves, and highlights deep ties between 2d non-perturbative data, open M2-branes, and mirror symmetry, offering a unified, geometry-driven approach to BPS spectra and partition functions.
Abstract
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 $ζ= e^{i \vartheta}$, while turning on the two-dimensional $Ω$-background with parameter~$ε$. We show that the resulting partition function $\mathcal{Z}_{\mathrm{2d}}^\vartheta(ε)$ can be characterized as the Borel-summed solution, in the direction $\vartheta$, to an associated Schrödinger equation. The partition function $\mathcal{Z}_{\mathrm{2d}}^\vartheta(ε)$ 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~$ε$, we refer to $Z^\vartheta_\mathrm{2d}(ε)$ as the non-perturbative partition function for the original two-dimensional $\mathcal{N}=(2,2)$ theory. We completely determine this partition function $\mathcal{Z}^\vartheta_\mathrm{2d}(ε)$ in two classes of examples, Landau-Ginzburg models and gauged linear sigma models, and show that $\mathcal{Z}^\vartheta_\mathrm{2d}(ε)$ 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 $\frac{1}{2} Ω$-background.