Abstract
We show that a Calabi–Yau structure of dimension d on a smooth dg category {C} induces a symplectic form of degree 2-d on ‘the moduli space of objects’ {{mathcal {M}}}_{{C}}. We show moreover that a relative Calabi–Yau structure on a dg functor {C}rightarrow { D} compatible with the absolute Calabi–Yau structure on C induces a Lagrangian structure on the corresponding map of moduli {{mathcal {M}}}_{{ D}} rightarrow {{mathcal {M}}}_{{C}}.
Highlights
Given a smooth, proper variety X over a field k, there is a reasonable derived moduli space of perfect complexes MX on X, with the property that at a point in MX corresponding to a perfect complex E on X, the tangent complex at E identifies with the shifted endomorphisms of E: TE (MX ) End(E)[1].For X of dimension d, a trivialisation θ : OX ∧d T ∗(X ) of its canonical bundle gives a trace map tr : End(E) θ Hom(E, E ⊗ ∧d T ∗(X )) → k[−d] such that the Serre pairingTE (MX )[−1]⊗2 End(E)⊗2 →◦ End(E) →tr k[−d] (1.1)is symmetric and non-degenerate
When X is taken to be a compact oriented topological surface, Goldman [10] showed that using Poincaré pairings in place of Serre pairings as above gives a global symplectic form on the moduli space of local systems on X
The main goal of this paper is to establish an analogue of this global symplectic form when a Calabi–Yau variety (X, θ ) is replaced by a ‘non-commutative Calabi–Yau’ in the form of a nice dg category C equipped with some extra structure and the moduli space MX is replaced with a ‘moduli space of objects’ moduli of objects T (MC)
Summary
Proper variety X over a field k, there is a reasonable derived moduli space of perfect complexes MX on X , with the property that at a point in MX corresponding to a perfect complex E on X , the tangent complex at E identifies with the shifted (derived) endomorphisms of E: TE (MX ) End(E)[1]. When X is taken to be a compact oriented topological surface, Goldman [10] showed that using Poincaré pairings in place of Serre pairings as above gives a global symplectic form on the moduli space of local systems on X Such examples motivated Pantev-Toën-Vaquié-Vezzosi [19] to introduce shifted symplectic structures on derived Artin stacks and to show that, in particular, the above pairings (1.1) are induced by a global symplectic form of degree 2 − d on MX. Main theorem Given a non-commutative Calabi–Yau (C, θ ) of dimension d, the moduli space of objects MC has an induced symplectic form of degree 2 − d. Combined with our main theorem, this gives many examples of shifted symplectic moduli spaces and Lagrangians in them coming from non-commutative Calabi–Yaus.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
