Derived smooth stacks and prequantum categories
Abstract
The WeilKostant integrality theorem states that given a smooth manifold endowed with an integral complex closed 2form, then there exists a line bundle with connection on this manifold with curvature the given 2form. It also characterises the moduli space of line bundles with connection that arise in this way. This theorem was extended to the case of pforms by Gajer in [Ga]. In this paper we provide a generalization of this theorem where we replace the original manifold by a derived smooth Artin stack. Our derived Artin stacks are geometric stacks on the étale (\infty,1)site of affine derived smooth manifolds. We introduce the notion of a nshifted ppreplectic derived smooth Artin stack in analogy with the algebraic case constructed by PantevToënVaquiéVezzosi in [PTVV]. This is a derived smooth Artin stack endowed with a complex closed (p+1)form which has been cohomologically shifted by degree n. It is a far reaching generalization of a ppreplectic manifold which includes orbifolds and other highly singular objects. We then show that when its nshifted ppreplectic form is integral, then there exists a (p+n1)gerbe with pconnection data and curvature corresponding to the original ppreplectic form. We also provide the characterization of the moduli stack of gerbes with connections arising in this context. We construct a canonical functor from the (\infty,1)category of integral nshifted ppreplectic derived smooth Artin stacks to the (\infty,1)category of linear (\infty,p+n1)categories. When n=0 and p=1, this functor can be thought of like a cohomology functor in that it associates to a derived presymplectic smooth Artin stack a linear invariant in the form of a differential graded module. In the general case we obtain higher prequantum categories which requires the machinery of linear (\infty,n)categories.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 arXiv:
 arXiv:1610.00441
 Bibcode:
 2016arXiv161000441W
 Keywords:

 Mathematics  Symplectic Geometry;
 Mathematics  Algebraic Topology
 EPrint:
 V2, 67 pages, minor corrections