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Towards a description of the double ramification hierarchy for Witten’s \(r\)-spin class. (English. French summary) Zbl 1352.37176

The double ramification hierarchy is a new integrable hierarchy of Hamiltonian PDEs introduced recently by the first author. It is associated to a given cohomological field theory. The aim of this paper is to study the double ramification hierarchy associated to the cohomological field theory formed by Witten’s \(r\)-spin classes. Using the formula for the product of the top Chern class of the Hodge bundle with Witten’s class, found by the second author, the authors present an effective method for a computation of the double ramification hierarchy. They do explicit computations for \(r=3,4,5\) and prove that the double ramification hierarchy is Miura equivalent to the corresponding Dubrovin-Zhang hierarchy. As an application, this result together with a recent work of the first author with Paolo Rossi gives a quantization of the \(r\)-th Gelfand-Dickey hierarchy for \(r=3,4,5\).

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
35S35 Topological aspects for pseudodifferential operators in context of PDEs: intersection cohomology, stratified sets, etc.
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