Reasons we use the rank 2 subtorus \(T\subset (\mathbb C^ * )^3\) instead of the whole rank 3 torus
- the rank \(2\) torus is specifically chosen to preserve the volume form, it’s the subtorus defined by the equation \(t_1t_2t_3 = 1\), or written additively as weights, \(w_1 + w_2 + w_3 = 0\).
- the \(T\)-equivariant 1st Chern class of \(TY\) vanishes but its \((\mathbb C^*)^3\)-equivariant class does not. I believe this vanishing of the first equivariant Chern class is of vital importance for the constancy of the defined GW invariant.