Tropical cycles and Chow polytopes

The Chow polytope of an algebraic cycle in a torus depends only on its tropicalisation. Generalising this, we associate a Chow polytope to any abstract tropical variety in a tropicalised toric variety. Several significant polyhedra associated to tropical varieties are special cases of our Chow polytope. The Chow polytope of a tropical variety $X$ is given by a simple combinatorial construction: its normal subdivision is the Minkowski sum of $X$ and a reflected skeleton of the fan of the ambient toric variety.Comment: 22 pp, 3 figs. Added discussion of arbitrary ambient toric varieties; several improvements suggested by Eric Katz; some rearrangemen

Mesonic Chiral Rings in Calabi-Yau Cones from Field Theory

We study the half-BPS mesonic chiral ring of the N=1 superconformal quiver theories arising from N D3-branes stacked at Y^pq and L^abc Calabi-Yau conical singularities. We map each gauge invariant operator represented on the quiver as an irreducible loop adjoint at some node, to an invariant monomial, modulo relations, in the gauged linear sigma model describing the corresponding bulk geometry. This map enables us to write a partition function at finite N over mesonic half-BPS states. It agrees with the bulk gravity interpretation of chiral ring states as cohomologically trivial giant gravitons. The quiver theories for L^aba, which have singular base geometries, contain extra operators not counted by the naive bulk partition function. These extra operators have a natural interpretation in terms of twisted states localized at the orbifold-like singularities in the bulk.Comment: Latex, 25pgs, 12 figs, v2: minor clarification