The quantum integrable hierarchy for the Gromov-Witten theory of elliptic curves
Paolo Rossi, Sergey Shadrin, Ishan Jaztar Singh
TL;DR
The paper constructs a quantum double ramification hierarchy for the Gromov–Witten theory of the elliptic curve, using results of Oberdieck and Pixton to obtain a closed, modular expression that accommodates fermionic fields. It extends the CohFT framework to a $\mathbb{Z}_2$-graded setting and builds the hierarchy via a star product and DR-driven Hamiltonians, proving commutativity of the quantum Hamiltonians. In the elliptic-case, the dynamics couple two bosonic and two fermionic fields with a modular-dependent kernel, producing a quasi-modular dependence on the modular parameter $\tau$ through Eisenstein data and Weierstrass functions, with several limiting regimes like dispersionless and trigonometric kernels. The work also raises questions about quantum Miura-type triviality and potential identifications with known integrable models, highlighting the broader significance of modular structures in quantum integrable hierarchies from CohFTs.
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 $λ_{g-1}$ together with vanishing results for $λ_{g-2}$ 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.