# Extended $r$-spin theory in all genera and the discrete KdV hierarchy

Abstract : In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to the extended $r$-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the $r$-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus $0$ it corresponds to a flat F-manifold. For $r=2$ we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.
Type de document :
Pré-publication, Document de travail
2018

https://hal.archives-ouvertes.fr/hal-01945149
Contributeur : Inspire Hep <>
Soumis le : mercredi 5 décembre 2018 - 10:49:03
Dernière modification le : mardi 19 février 2019 - 13:19:56

### Citation

Alexandr Buryak, Paolo Rossi. Extended $r$-spin theory in all genera and the discrete KdV hierarchy. 2018. 〈hal-01945149〉

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