Consistent reduction of N=2 -> N=1 four dimensional supergravity coupled to matter

TL;DR

This work addresses the consistent four-dimensional reduction of general matter-coupled N=2 supergravity to N=1 by analyzing the fermionic sector. The authors perform a gravitino truncation ($\psi_{2\mu}=0$) and enforce the integrability of the corresponding supersymmetry variation, deriving explicit constraints that fix the geometry (hypermultiplets live on a K"ahler-Hodge submanifold; surviving vector multiplets form a reduced special-K"ahler sector) and the gauging data. They obtain explicit expressions for the N=1 D-terms and a holomorphic gravitino mass superpotential $L(z,w)$, along with precise conditions on prepotentials and Killing vectors that survive the reduction; they also show that the reduced theory’s holomorphic gauge couplings arise as antiholomorphic functions of the chiral scalars. A central result is the demonstration that all Yukawa terms mixing retained and truncated fermions vanish, ensuring that the reduced Lagrangian matches the standard N=1 form with the potential ${\cal V}^{N=2\rightarrow N=1}=4[-3L\bar L+g^{i\bar j}\nabla_i L\nabla_{\bar j}\bar L+g^{s\bar s}\nabla_s L\nabla_{\bar s}\bar L+\tfrac{1}{16}\mathrm{Im}f_{\Lambda\Sigma}D^{\Lambda}D^{\Sigma}]$, thereby providing a general framework for consistent partial supersymmetry breaking in string/M-theory contexts.

Abstract

We analyze the constraints which follow both on the geometry and on the gauge sector for a consistent supergravity reduction of a general matter-coupled N=2 supergravity theory in four dimensions. These constraints can be derived in an elegant way by looking at the fermionic sector of the theory.