On the partial breaking of N=2 rigid supersymmetry with a complex hypermultiplet

Abstract

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.