# String Theory Seminar

### Resurgent transseries and Lefschetz thimble in 3d $\mathcal{N}=2$ supersymmetric Chern-Simons matter theories

We show that a certain class of supersymmetric (SUSY) observables in 3d $\mathcal{N}=2$ SUSY Chern-Simons (CS) matter theories has nontrivial resurgent structures with respect to coupling constants given by inverse CS levels, and that their exact results are expressed as appropriate resummations of weak coupling expansions given by transseries. With a real mass parameter varied, we encounter Stokes phenomena infinitely many times, where the perturbative series gets non-Borel-summable along positive real axis of the Borel plane. We also decompose integral representations of the exact results in terms of Lefschetz thimbles and study how they are related to the resurgent transseries. We further discuss connections between the non-perturbative effects appearing in the transseries and complexified SUSY solutions which formally satisfy SUSY conditions but are not on original path integral contour. We explicitly demonstrate the above for partition functions of rank-1 3d $\mathcal{N}=2$ CS matter theories on sphere. This talk is based on arXiv:1604.08653, 1710.05010, and an on-going collaboration with Toshiaki Fujimori, Syo Kamata, Tatsuhiro Misumi and Norisuke Sakai.​