Logarithmic Gromov-Witten theory and double ramification cycles

Abstract

This thesis contains two research papers, whose main topics concern the logarithmic enumerative geometry of toric varieties and its interactions with the moduli space of curves. In the first paper (joint work with Dhruv Ranganathan), we prove that the logarithmic Gromov-Witten cycles of a toric variety endowed with its toric log structure lie in the tautological subring of the moduli space of stable curves. The result gives a common generalization of work of Faber-Pandharipande, and more recent work of Holmes-Schwarz and Molcho-Ranganathan. The crucial idea we use is a method to relate these logarithmic Gromov-Witten cycles with the higher double ramification cycles. In the second paper (joint work with Patrick Kennedy-Hunt and Qaasim Shafi) we prove a refined tropical correspondence theorem for higher genus descendant logarithmic Gromov-Witten invariants of toric surfaces with a top lambda class insertion. This generalizes a result due to Pierrick Bousseau. The proof uses the aforementioned idea from the first paper to reduce the calculation to an intersection problem on the moduli space of curves. The intersection problem against the higher double ramification cycle is then solved using integrable systems methods due to Buryak-Rossi.