LAUSR

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 L. Andrianopoli et al., Phys. Lett. B 744, 116 (2015), which we refer to as the ADFT method. We derive the generalization of the symplectic invariant ADFT formula $\zeta _{a}=\frac{1}{2}\varepsilon _{abc}(\mathcal{P}^{bM}\mathcal{C}_{MN}\mathcal{P}^{cN}) $, capturing information on partial breaking. Our extension of this anomaly is expressed as $d_{a}=\frac{1}{2}\varepsilon _{abc}\mathbb{P}^{bM}\mathcal{C}_{MN}\mathbb{P}^{cN}+\mathcal{J}_{a}$. The generalized moment maps $\mathbb{P}^{aM}$ contain $\mathcal{P}^{aM}$ and also depend on electric/magnetic coupling charges $G^{M}=( \eta ^{i},g_{i}) $; the $\mathcal{J}_{a}$ is an extra contribution induced by Killing isometries in the 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 I. Antoniadis et al., Nucl. Phys. B 863, 471 (2012) by a particular realization of the $d_{a}$ anomaly.