The tautological ring of ${\mathcal{M}}_{g,n}$ via Pandharipande-Pixton-Zvonkine $r$-spin relations

Abstract

We use relations in the tautological ring of the moduli spaces ${\overline{\mathcal{M}}}_{g,n}$ derived by Pandharipande, Pixton, and Zvonkine from the Givental formula for the $r$-spin Witten class in order to obtain some restrictions on the dimensions of the tautological rings of the open moduli spaces ${\mathcal{M}}_{g,n}$. In particular, we give a new proof for the result of Looijenga (for $n=1$) and Buryak et al. (for $n\ge 2$) that $\dim R^{g-1}({\mathcal{M}}_{g,n})\le n$. We also give a new proof of the result of Looijenga (for $n=1$) and Ionel (for arbitrary $n\ge 1$) that $R^i({\mathcal{M}}_{g,n})=0$ for $i\ge g$ and give some estimates for the dimension of $R^i({\mathcal{M}}_{g,n})$ for $i\le g-2$.

24 pages

References in corpus (7)

• Three questions in Gromov-Witten theory
• The equivariant Gromov-Witten theory of P^1
• Dubrovin's superpotential as a global spectral curve
• The hypergeometric functions of the Faber-Zagier and Pixton relations
• Powers of the theta divisor and relations in the tautological ring
• A calculus for the moduli space of curves
• The moduli space of curves and its invariants

Cited by in corpus (4)

• Topological recursion relations from Pixton's formula
• Half-Spin Tautological Relations and Faber's Proportionalities of Kappa Classes
• Shifted Witten classes and topological recursion
• Punctures and p-spin curves from matrix models II