Calabi-Yau structures on (quasi-)bisymplectic algebras

TL;DR

This paper establishes a connection between relative Calabi-Yau structures and (quasi-)bisymplectic structures in noncommutative geometry, revealing how fusion processes relate to Calabi-Yau cospans and Poisson structures on moduli spaces.

Contribution

It demonstrates that relative Calabi-Yau structures induce (quasi-)bisymplectic structures and shows how fusion corresponds to composition of Calabi-Yau cospans, preserving dualities.

Findings

Fusion corresponds to composition of Calabi-Yau cospans.

Duality between double quasi-Poisson and quasi-bisymplectic structures is preserved.

Poisson structures on moduli spaces match those from 2-Calabi-Yau structures.

Abstract

We show that relative Calabi--Yau structures on noncommutative moment maps give rise to (quasi-)bisymplectic structures, as introduced by Crawley-Boevey-Etingof-Ginzburg (in the additive case) and Van den Bergh (in the multiplicative case). We prove along the way that the fusion process (a) corresponds to the composition of Calabi-Yau cospans with "pair-of-pants" ones, and (b) preserves the duality between non-degenerate double quasi-Poisson structures and quasi-bisymplectic structures. As an application we obtain that Van den Bergh's Poisson structures on the moduli spaces of representations of deformed multiplicative preprojective algebras coincide with the ones induced by the 2-Calabi-Yau structures on (dg-versions of) these algebras.