Multiplicative preprojective algebras are 2-Calabi–Yau

Kaplan, Daniel; Schedler, Travis

doi:10.2140/ant.2023.17.831

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Multiplicative preprojective algebras are 2-Calabi–Yau

Daniel Kaplan and Travis Schedler

Vol. 17 (2023), No. 4, 831–883

DOI: 10.2140/ant.2023.17.831

Abstract

We prove that multiplicative preprojective algebras, defined by Crawley-Boevey and Shaw, are 2-Calabi–Yau algebras, in the case of quivers containing unoriented cycles. If the quiver is not itself a cycle, we show that the center is trivial, and hence the Calabi–Yau structure is unique. If the quiver is a cycle, we show that the algebra is a noncommutative crepant resolution of its center, the ring of functions on the corresponding multiplicative quiver variety with a type A surface singularity. We also prove that the dg versions of these algebras (arising as certain Fukaya categories) are formal. We conjecture that the same properties hold for all non-Dynkin quivers, with respect to any extended Dynkin subquiver (note that the cycle is the type A case). Finally, we prove that multiplicative quiver varieties — for all quivers — are formally locally isomorphic to ordinary quiver varieties. In particular, they are all symplectic singularities (which implies they are normal and have rational Gorenstein singularities). This includes character varieties of Riemann surfaces with punctures and monodromy conditions. We deduce this from a more general statement about 2-Calabi–Yau algebras (following Bocklandt, Galluzzi, and Vaccarino).

Keywords

multiplicative preprojective algebra, Calabi\kern0.05em–\kern-0.1em Yau algebra, NCCR, Ginzburg dg algebra, wrapped Fukaya category, quiver variety, symplectic singularity

Mathematical Subject Classification

Primary: 16G20, 16S38

Secondary: 16E05, 16E65, 53D30

Milestones

Received: 15 October 2020

Revised: 29 September 2021

Accepted: 10 June 2022

Published: 2 May 2023

Authors

Daniel Kaplan
Hasselt University Diepenbeek Belgium

Travis Schedler
Department of Mathematics Imperial College London South Kensington Campus London United Kingdom

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Vol. 17, No. 4, 2023