Title: Non-perturbative partition functions for N=2 theories of class S

Abstract: The beautiful works by Gaiotto, Moore and Neitzke have taught us (in particular) how to understand various kinds of BPS spectra for 4d N=2 theories in terms of certain configurations of curves on a Riemann surface, called spectral networks. In this talk I'll review how to generate a spectral network for each point of the 4d Coulomb branch together with the choice of a phase. I'll point out that this data corresponds to a boundary condition for the 4d N=2 theory in the half-Omega background, and associate to it a new kind of N=2 partition function. This partition function is locally constant in the phase, while its jumps are associated to 4d BPS particles in the corresponding Coulomb vacuum. The well-known Nekrasov partition function (in the half-Omega background) may be recovered in a weakly coupled region of the Coulomb branch (if the theory admits any), at a special phase corresponding to a Fenchel-Nielsen spectral network. When lifted to five dimensions, this new partition function becomes the non-perturbative topological string partition function.

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