On the partial breaking of $\mathcal{N}=2$ rigid supersymmetry with complex hypermultiplet

We study partial supersymmetry breaking in effective $\mathcal{N}=2$ U${\left(1\right)}^n$ gauge theory coupled to complex hypermultiplets by using the method of \textrm{“{\small arXiv:1501.07842}” to which we refer to as ADFT method}. We derive the generalisation of the symplectic invariant ADFT formula $\zeta {a}=\frac{1}{2}\varepsilon {abc}\left($\mathcal{P}$^{bM}$\mathcal{C}${MN}$\mathcal{P}$^{cN}\right) $ capturing information on partial breaking. Our extension of this anomaly is expressed like $d{a}=\frac{1}{2}\varepsilon {abc}$\mathbb{P}$^{bM}$\mathcal{C}${MN}$\mathbb{P}$^{cN}+$\mathcal{J}${a}$; the generalized moment maps ${\mathbb{P}}^{aM}$ contain ${\mathcal{P}}^{aM}$ and depend as well on electric/magnetic coupling charges $G^{M}=\left( \eta ^{i},g{i}\right) $,the$$\mathcal{J}$_{a}$ is an extra contribution \textrm{
induced by Killing isometries in complex hypermatter sector}. Using SP$(2n,\mathbb{R})$ symplectic symmetry, we also give the $\mathcal{N}=2$ partial breaking condition and derive the model of “{\small arXiv: 1204.2141}” by a particular realisation of $d_a$ anomaly.