Theoretical analysis derives a quantum double ramification hierarchy for elliptic curves, uncovering the first explicit quantum integrable hierarchy arising from fields with fermions.
We construct the quantum double ramification (DR) hierarchy associated with the Gromov–Witten theory of elliptic curves. We use results of Oberdieck and Pixton on the intersection numbers of the DR cycle, the Gromov–Witten classes of the elliptic curve, and the Hodge class , together with vanishing results for 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.
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Rossi et al. (2026) studied this question.
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