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Moduli spaces of residueless meromorphic differentials and the KP hierarchy

Alexandr Buryak, Paolo Rossi and Dimitri Zvonkine

Geometry & Topology 28 (2024) 2793–2824
Abstract

We prove that the cohomology classes of the moduli spaces of residueless meromorphic differentials, ie the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues, form a partial cohomological field theory (CohFT) of infinite rank. To this partial CohFT we apply the double ramification hierarchy construction to produce a Hamiltonian system of evolutionary PDEs. We prove that its reduction to the case of differentials with exactly two zeros and any number of poles coincides with the KP hierarchy up to a change of variables.

Keywords
moduli space of curves, meromorphic differentials, integrable systems, KP hierarchy
Mathematical Subject Classification
Primary: 14H10, 14H70, 37K10
References
Publication
Received: 4 November 2021
Revised: 22 August 2022
Accepted: 19 November 2022
Published: 21 October 2024
Proposed: Yakov Eliashberg
Seconded: Leonid Polterovich, Mark Gross
Authors
Alexandr Buryak
Faculty of Mathematics
National Research University Higher School of Economics
Moscow
Russia
Center for Advanced Studies
Skolkovo Institute of Science and Technology
Moscow
Russia
P G Demidov Yaroslavl State University
Yaroslavl
Russia
Paolo Rossi
Dipartimento di Matematica “Tullio Levi-Civita”
Universita degli Studi di Padova
Padova
Italy
Dimitri Zvonkine
Université de Versailles Saint-Quentin-en-Yvelines
CNRS
Versailles
France

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