  A key result in equivariant symplectic geometry is Delzant's classification
of compact connected symplectic toric manifolds. The moment map induces an
embedding of the quotient of the manifold by the torus action into the dual of
the Lie algebra of the torus; its image is a unimodular ("Delzant") polytope;
this gives a bijection between unimodular polytopes and isomorphism classes of
compact connected symplectic toric manifolds. In this paper we extend Delzant's
classification to non-compact symplectic toric manifolds. For a non-compact
symplectic toric manifold the image of the moment map need not be convex and
the induced map on the quotient need not be an embedding. Moreover, even when
the map on the quotient is an embedding, its image no longer determines the
symplectic toric manifold; a degree two characteristic class on the quotient
makes an appearance. Nevertheless, the quotient is a manifold with corners, and
the induced map from the quotient to the dual of the Lie algebra is what we
call a unimodular local embedding. We classify non-compact symplectic toric
manifolds in terms of manifolds with corners equipped with degree two
cohomology classes and unimodular local embeddings into the dual of the Lie
algebra of the corresponding torus. The main new ingredient is the construction
of a symplectic toric manifold from such data. The proof passes through an
equivalence of categories between symplectic toric manifolds and symplectic
toric bundles over a fixed unimodular local embedding. This equivalence also
gives a geometric interpretation of the degree two cohomology class.

Symplectic Geometry;Algebraic Geometry