The paper is about log Gromov-Witten theory of a toric variety relative to its toric boundary. Log Gromov-Witten theory, developed by Abramovich-Chen and Gross-Siebert is a generalization of Gromov-Witten theory, which concerns counts of stable log maps, which in brief terms are stable maps satisfying given tangency conditions relative to a (not necessarily smooth) divisor in the target.
Gromov-Witten cycles are cycles in the Chow ring of the Deligne-Mumford moduli space of stable curves \(\overline{\mathcal{M}}_{g,n}\), obtained by pushing-forward cycles on moduli spaces of stable maps defined in terms of the virtual fundamental class. Log Gromov-Witten cycles are defined analogously, considering moduli spaces of stable log maps.
The main result of the paper shows that log Gromov-Witten cycles of a toric variety are contained in the tautological ring of \(\overline{\mathcal{M}}_{g,n}\), which is a natural subring of the Chow ring that is closed under natural operations, such as push-forwards and pull-backs along the inclusion of the boundary strata of \(\overline{\mathcal{M}}_{g,n}\). To show this, the authors express log Gromov-Witten cycles in terms of log toric contact cycles, which are generalizations of the double ramification cycle. The proof also systematically uses the logarithmic Chow ring of \(\overline{\mathcal{M}}_{g,n}\), defined as a limit of Chow rings of blow-ups of \(\overline{\mathcal{M}}_{g,n}\).