# Partial Breaking in Rigid Limit of $\mathcal{N}=2$ Gauged Supergravity

## TL;DR

This paper develops a method to derive the rigid limit of $
abla$-isometry Ward identities in $ ext{N}=2$ gauged supergravity, introduces new quaternionic manifolds, and explores partial supersymmetry breaking conditions.

## Contribution

It provides a new rescaling approach for the rigid limit, geometric interpretation of parameters, and explicit metrics for generalized quaternionic manifolds, advancing understanding of supersymmetry breaking.

## Key findings

- Explicit derivation of the rigid limit of Ward identities.
- Construction of new quaternionic manifolds classified by ADE Lie algebras.
- Conditions for partial $ ext{N}=2$ supersymmetry breaking in the rigid limit.

## Abstract

Using a new manner to rescale fields in $\mathcal{N}=2$ gauged supergravity with n$_{V}$ vector multiplets and n$_{H}$ hypermultiplets, we develop the explicit derivation of the rigid limit of quaternionic isometry Ward identities agreeing with known results. We show that the rigid limit can be achieved, amongst others, by performing two successive transformations on the covariantly holomorphic sections $V^{M}\left( z,\bar{z}\right) $ of the special Kahler manifold: a particular symplectic change followed by a particular Kahler transformation. We also give a geometric interpretation of the $\eta_{i}$ parameters used in arXiv:1508.01474 to deal with the expansion of the holomorphic prepotential $\mathcal{F}\left( z\right) $ of the $\mathcal{N}=2$ theory. We give as well a D- brane realisation of gauged quaternionic isometries and an interpretation of the embedding tensor $\vartheta_{M}^{u}$ in terms of type IIA/IIB mirror symmetry. Moreover, we construct explicit metrics for a new family of $4r$- dimensional quaternionic manifolds $\boldsymbol{M}_{QK}^{\left( n_{H}\right) }$ classified by ADE Lie algebras generalising the $SO\left( 1,4\right) /SO\left( 4\right) $ geometry which corresponds to $A_{1}\sim su\left( 2\right) $. The conditions of the partial breaking of $\mathcal{N}=2$ supersymmetry in the rigid limit are also derived for both the observable and the hidden sectors. Other features are also studied.