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Abstract

This thesis studies the Gromov-Witten and stable pair invariants of local curves. In particular, we give a closed formula for the full descendent stable pair theory of all (non-relative) local curves in terms of the Bethe roots of the quantum intermediate long wave system. In the process, we derive a new explicit description of these Bethe roots, which may be of independent interest. We further deduce rationality, functional equation and a pole restriction for the descendent stable pair theory of local curves as conjectured by Pandharipande and Pixton. Furthermore, we show how the Bethe roots can be used to diagonalize the descedent invariants of the tube and give explicit formulas for the first few descendents. On the Gromov-Witten side, we conjecture that the Gromov-Witten theory of the local elliptic curve is governed by quasi-Jacobi forms. Finally, we compute an infinite series of special cases, which provides evidence for our conjecture.