Massless four-dimensional $N=2$ multiplet counting conventions

For the adopted two-derivative four-dimensional $N=2$ on-shell framework, fix a four-dimensional Minkowski vacuum preserving all eight real supercharges and a linearized sector consisting of one massless gravity multiplet, $n_V$ massless vector multiplets, and $n_H$ massless hypermultiplets. Here a Minkowski vacuum means a solution of the adopted field equations on ${\mathbb{R}}^4$ with metric $\eta =\mathrm{diag}(-1,1,1,1)$, constant scalars, and zero gauge field strengths and fermion fields; preservation of all eight supercharges means that every constant supersymmetry parameter in the adopted eight-real-dimensional parameter space gives zero fermion variations on this background. The linearized sector consists of the stipulated massless multiplet solutions of the field equations to first order about this vacuum. Here $n_V,n_H$ are nonnegative integers counting these massless representations about this vacuum, after any mass generation; they need not equal the matter-field counts of a gauged action before expansion about the vacuum. Count each vector multiplet as one massless vector and one complex scalar (two real scalar coordinates), each hypermultiplet as four real scalar coordinates, and the gravity multiplet as containing one graviphoton. The graviphoton belongs to the gravity multiplet and is excluded from the number $n_V$ of vector multiplets. These are the component-counting conventions used in the conditional reductions on this A page.