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 N=(2,2) 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 = ei ϑ ζ = e i ϑ , while turning on the two-dimensional Ω Ω -background with complex parameter ε ϵ . We show that the resulting partition function Z2d^ϑ(ε) 𝒵 2 d ϑ ( ϵ ) can be characterized as the Borel-summed solution, in the direction ϑ ϑ , to an associated Schrödinger equation. The partition function Z2d^ϑ(ε) 𝒵 2 d ϑ ( ϵ ) is locally constant in the phase ϑ ϑ and jumps across phases ϑ_BPS ϑ BPS associated with the BPS solitons. Since these jumps are non-perturbative in the parameter ε ϵ , we refer to Z^ϑ_2d(ε) Z 2 d ϑ ( ϵ ) as the non-perturbative partition function for the original two-dimensional N=(2,2) 𝒩 = ( 2 , 2 )