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Comparing tautological relations from the equivariant Gromov-Witten theory of projective spaces and spin structures

Pandharipande-Pixton-Zvonkine's proof of Pixton's generalized Faber-Zagier relations in the tautological ring of $\overline M_{g, n}$ has started the study of tautological relations from semisimple cohomological field theories. In this article we compare the relations obtained in the examples of the equivariant Gromov-Witten theory of projective spaces and of spin structures. We prove an equivalence between the $\mathbb P^1$- and 3-spin relations, and more generally between restricted $\mathbb P^m$-relations and similarly restricted (m + 2)-spin relations. We also show that the general $\mathbb P^m$-relations imply the (m + 2)-spin relations.