# Is there a stacky definition of irreducible symplectic manifold?

I am now interested in studying symplectic structures in the field of stacks.

In particular, is there a stacky definition of irreducible symplectic manifold ? I'm also interested in similar things in derived algebraic geometry.

Any comment welcome! Thank you!

• Have a look at arxiv.org/pdf/1903.05632.pdf. In page number 7, they define Symplectic stacks. It might be of some use.. You can also look at marle.perso.math.cnrs.fr/publications/Liegroupoids.pdf This define/discuss symplectic Lie groupoids. It is not necessary to remind that Lie groupoids and differentible stacks are closely related.. :) May 10, 2020 at 14:49
• There is a relatively large body of recent literature on derived (shifted) sympectic structures - look in works of a subset of Calanque, Pantev, Toen, Vaquie, Vezzosi, etc. A review is in arxiv.org/pdf/1603.02753.pdf May 10, 2020 at 15:53
• Thank you very much! And what is the difference between the definition on the third page of arxiv.org/abs/1111.6294v1 and the shifted Symplectic structures? May 10, 2020 at 16:06
• The definition you mention in arxiv.org/abs/1111.6294v1 looks like an attempt at a $0$-shifted symplectic structure, but seems to omit any sort of closure condition. Beware that the survey arxiv.org/pdf/1603.02753.pdf which Balazs mentions really just summarises CPTVV, neglecting to mention much related literature. For differentiable stacks, a summary on shifted structures is arxiv.org/pdf/1804.07622 , including references to related mathematical physics papers. Some of Safronov's papers are also a good place to look if you want to see where these structures arise in the wild. May 10, 2020 at 17:01
• yes, by throwing away the higher structure May 10, 2020 at 18:26