VIRTUAL SEMINAR SERIES -- INTEGRABLE SYSTEMS: Givental theory for F-manifolds and non-Hamiltonian integrable PDEs
Paolo Rossi (University of Padova)
Wednesday 13th May, 2020 14:00-15:00 Zoom seminar
With A. Buryak, we have introduced a higher genus theory for flat F-manifolds, encoded in the notion of F-cohomological field theory. F-CohFTs are weaker analogues of CohFTs, i.e. families of cohomology classes on the moduli space of stable marked curves well beahved with respect to natural maps between moduli spaces. In the same way as CohFTs reduce to Frobenius manifolds for genus 0, F-CohFTs reduce to flat F-manifolds. Employing intersection theory with the double ramification (DR) cycle we have then associated to any F-CohFT an integrable hierarchy of (generically non Hamiltonian) evolutionary dispersive PDEs called the DR hierarchy (generalizing our previous construction for CohFTs). The dispersionless limit of the DR hierarchy coincides with the principal hierarchy of the underlying flat F-manifold. In a joint work with A. Arsie, A. Buryak and P. Lorenzoni we have developed a Givental-type theory for flat F-manifolds and F-CohFTs proving in particular that to any semisimple flat F-manifold one can associate an all genera F-CohFT. We then studied in detail the properties of the corresponding DR hierarchy. In particular this means that we can find a nontrivial full dispersive perturbation of the principal hierarchy for any semisimple flat F-manifold. The perturbation in non-unique, but depends on a number of parameters equal to the dimension of the flat F-manifold. This theory subsumes several known integrable hierarchies, including some celebrated examples who wouldn't fit in the traditional Frobenius manifold theory. We completely classify all semisimple flat F-manifolds, F-CohFTs and resulting hierarchies.
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