Toric symmetry of $\mathbb{CP}^3$

Abstract

We exhaustively analyze the toric symmetries of $\mathbb{CP}^3$ and its toric blowups. Our motivation is to study toric symmetry as a computational technique in Gromov–Witten (GW) theory and Donaldson–Thomas (DT) theory. We identify all non-trivial toric symmetries. The induced nontrivial isomorphisms lift and provide new symmetries at the level of GW Theory and DT theory. The polytopes of the toric varieties in question include the permutohedron, the cyclohedron, the associahedron, and in fact all graph associahedra, among others.