Logarithmic Gromov-Witten theory and double ramification cycles

TL;DR

This paper demonstrates that all logarithmic Gromov-Witten pushforwards for maps to a toric variety relative to its boundary are contained within the tautological ring of the moduli space of curves, using a novel approach that avoids virtual localization.

Contribution

It introduces a new technique relating rigid and rubber geometries in logarithmic Gromov-Witten theory, and shows all such pushforwards lie in the tautological ring, generalizing previous work.

Findings

All logarithmic Gromov-Witten pushforwards lie in the tautological ring.

A new method relates rigid and rubber geometries via maps to the logarithmic algebraic torus.

The approach bypasses the undeveloped virtual localization formula.

Abstract

We examine the logarithmic Gromov-Witten cycles of a toric variety relative to its full toric boundary. The cycles are expressed as products of double ramification cycles and natural tautological classes in the logarithmic Chow ring of the moduli space of curves. We introduce a simple new technique that relates the Gromov-Witten cycles of rigid and rubber geometries; the technique is based on a study of maps to the logarithmic algebraic torus. By combining this with recent work on logarithmic double ramification cycles, we deduce that all logarithmic Gromov-Witten pushforwards, for maps to a toric variety relative to its full toric boundary, lie in the tautological ring of the moduli space of curves. A feature of the approach is that it avoids the as yet undeveloped logarithmic virtual localization formula, instead relying directly on piecewise polynomial functions to capture the structure that would be provided by such a formula. The results give a common generalization of work of Faber-Pandharipande, and more recent work of Holmes-Schwarz and Molcho-Ranganathan. The proof passes through general structure results on the space of stable maps to the logarithmic algebraic torus, which may be of independent interest.