From $3$-spin to Negative $r$-spin: Limits of Cohomological Field Theories and Tautological Relations

We prove that the negative $r$-spin tautological relations of Chidambaram–Garcia-Failde–Giacchetto are a consequence of Pixton’s $3$-spin relations, for every $r\ge 2$, and deduce that they hold in the tautological Chow ring of ${\bar{\mathcal{M}}}_{g,n}$. In particular, our result for $r=2$ implies that Kazarian–Norbury $K$-relations hold in Chow. The proof factors through two essential parts. The first part shows that the tautological relations arising from Chiodo’s class imply the negative $r$-spin relations. This requires an intricate study of limiting behavior of the Givental–Teleman graph contributions. In the second part, we show that the Chiodo relations are a consequence of Pixton’s $3$-spin relations as an application of a generalization of Janda’s work on tautological relations from CohFTs.