Different representations for the action principle in 4D N = 2 supergravity

TL;DR

This work delivers two complementary, fully covariant routes to reduce the curved ${ m N}=2$ supergravity action from superspace to components. First, it exploits the projective invariance of the covariant projective multiplets to perform a gauge-preserving component reduction, yielding the complete ${ m N}=2$ component action and enabling applications such as the curved c-map. Second, it develops a general normal-coordinate framework around a submanifold, deriving differential relations for the vielbein, connection, and super-determinant to reconstruct the full supergeometry order by order in transverse directions, and then specializes to chiral subspace reductions. Together, these methods provide both a robust, gauge-free path to component actions and a versatile technique for reducing full superspace integrals to chiral or other subspaces, with explicit results for the chiral density and a new chiral projector representation. The paper also demonstrates key applications, including the gauge-invariance of vector-tensor couplings and a curved-space formulation of the ${ m N}=2$ c-map, highlighting the practical impact of these representations for ${ m N}=2$ supergravity-matter systems.

Abstract

Within the superspace formulation for four-dimensional N = 2 matter-coupled supergravity developed in arXiv:0805.4683, we elaborate two approaches to reduce the superfield action to components. One of them is based on the principle of projective invariance which is a purely N = 2 concept having no analogue in simple supergravity. In this approach, the component reduction of the action is performed without imposing any Wess-Zumino gauge condition, that is by keeping intact all the gauge symmetries of the superfield action, including the super-Weyl invariance. As a simple application, the c-map is derived for the first time from superfield supergravity. Our second approach to component reduction is based on the method of normal coordinates around a submanifold in a curved superspace, which we develop in detail. We derive differential equations which are obeyed by the vielbein and the connection in normal coordinates, and which can be used to reconstruct these objects, in principle in closed form. A separate equation is found for the super-determinant of the vielbein, which allows one to reconstruct it without a detailed knowledge of the vielbein. This approach is applicable to any supergravity theory in any number of space-time dimensions. As a simple application of this construction, we reduce an integral over the curved N = 2 superspace to that over the chiral subspace of the full superspace. We also give a new representation for the curved projective-superspace action principle as a chiral integral.