The quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves

Abstract

We construct the quantum double ramification hierarchy associated with the Gromov-Witten theory of elliptic curves. We use results of Oberdieck and Pixton on the intersection numbers of the double ramification cycle, the Gromov-Witten classes of the elliptic curve and the Hodge class ${\lambda }_{g-1}$ together with vanishing results for ${\lambda }_{g-2}$ to produce a closed, modular expression for the resulting integrable hierarchy. It is the first explicit nontrivial example of a quantum integrable hierarchy from a cohomological field theory containing fermionic fields, which correspond to the odd classes in the cohomology of the elliptic curve.

arXiv comment11 pages

References in corpus (7)

• Gromov-Witten classes, quantum cohomology, and enumerative geometry
• A polynomial bracket for the Dubrovin--Zhang hierarchies
• Double ramification cycles and integrable hierarchies
• Holomorphic anomaly equations and the Igusa cusp form conjecture
• Tau-structure for the Double Ramification Hierarchies
• Integrable systems of double ramification type
• Gromov-Witten theory of target curves and the tautological ring