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

TL;DR

This paper investigates partial supersymmetry breaking in $ ext{N}=2$ gauge theories with complex hypermultiplets, extending the ADFT method to include new anomaly contributions and deriving conditions for partial breaking.

Contribution

It generalizes the ADFT formula for partial supersymmetry breaking to include complex hypermultiplet contributions and electric/magnetic charges.

Findings

Derived a generalized symplectic invariant formula for partial breaking.

Extended the anomaly formula to include extra contributions from Killing isometries.

Provided conditions for partial supersymmetry breaking using symplectic symmetry.

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 \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.