Alexandr Buryak ; Sergey Shadrin - Tautological relations and integrable systems

epiga:10382 - Épijournal de Géométrie Algébrique, August 23, 2024, Volume 8 - https://doi.org/10.46298/epiga.2024.10382
Tautological relations and integrable systemsArticle

Authors: Alexandr Buryak ; Sergey Shadrin

    We present a family of conjectural relations in the tautological cohomology of the moduli spaces of stable algebraic curves of genus $g$ with $n$ marked points. A large part of these relations has a surprisingly simple form: the tautological classes involved in the relations are given by stable graphs that are trees and that are decorated only by powers of the psi-classes at half-edges. We show that the proposed conjectural relations imply certain fundamental properties of the Dubrovin-Zhang (DZ) and the double ramification (DR) hierarchies associated to F-cohomological field theories. Our relations naturally extend a similar system of conjectural relations, which were proposed in an earlier work of the first author together with Guéré and Rossi and which are responsible for the normal Miura equivalence of the DZ and the DR hierarchy associated to an arbitrary cohomological field theory. Finally, we prove all the above mentioned relations in the case $n=1$ and arbitrary $g$ using a variation of the method from a paper by Liu and Pandharipande, this can be of independent interest. In particular, this proves the main conjecture from our previous joined work together with Hernández Iglesias. We also prove all the above mentioned relations in the case $g=0$ and arbitrary $n$.


    Volume: Volume 8
    Published on: August 23, 2024
    Accepted on: February 27, 2024
    Submitted on: November 29, 2022
    Keywords: Mathematics - Algebraic Geometry,Mathematical Physics

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