Work

Tautological relations and integrable systems

Year: 2024

Type: article

Abstract: 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 Gu\'er\'e 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 Hern\'andez Iglesias. We also prove all the above mentioned relations in the case $g=0$ and arbitrary $n$. Comment: v3: final journal version, 44 pages

Source: Épijournal de Géométrie Algébrique

DOI: 10.46298/epiga.2024.10382

Authors: Alexandr Buryak, Sergey Shadrin

Institutions: National Research University Higher School of Economics, Skolkovo Institute of Science and Technology, Yaroslavl State University, University of Amsterdam

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Nederlandse Organisatie voor Wetenschappelijk Onderzoek, Russian Science Foundation

Award: G7733117437