Teorie di campo coomologiche, gerarchie quantistiche e forme quasimodulari

This thesis investigates the Quantum Double Ramification (qDR) hierarchy of Buryak–Rossi in the Gromov–Witten theory of elliptic curves, with emphasis on its explicit quasimodular dependence and its relation to the Dubrovin–Zhang (DZ) hierarchy. Defined via Kontsevich’s deformation of the Poisson structure, the qDR hierarchy extends the classical Double Ramification hierarchy to a quantized setting. In addition to the rich quasimodular behaviour observed in the elliptic case, a further motivation is to understand the role of odd cohomology in non-semisimple theories, which endows Frobenius manifolds and their associated hierarchies with natural super-structures; these appear nontrivially in the qDR recursion for elliptic curves. A central contribution is a refinement of results by Oberdieck and Pixton on quasimodular forms and holomorphic anomaly equations in CohFTs. We derive explicit cyclic expressions for Gromov–Witten classes of elliptic curves paired with Hodge classes, leading to new tautological relations and providing an alternative proof of Faber’s socle intersection formula. This approach is further extended to quadratic Hodge integrals, where we conjecture new tautological relations via the holomorphic anomaly equation. Building on these results, the main achievement of this thesis is the advancement of a dispersive quantum hierarchy for the Gromov–Witten theory of elliptic curves, which exhibits explicit quasimodular dependence and provides a new example of a hierarchy arising from a non-semisimple theory. Its construction relies on geometric tools including the Pixton–Zagier formula for the double ramification cycle, the BSSZ splitting formula, cycle-graph relations, and quasimodular techniques. A conjectural algorithm for obtaining the Dubrovin–Zhang hierarchy associated with elliptic curves in this framework is also proposed.