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$\left(2n,\mathbb{R}\right)$ 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.