Supersymmetric backgrounds from 5d N=1 supergravity

TL;DR

The paper addresses constructing rigid supersymmetric backgrounds on curved 5d Euclidean manifolds by employing 5d ${\cal N}=1$ off-shell Poincar\'e supergravity. It solves the fermionic SUSY variations $\delta_Q\psi_\mu=0$ and $\delta_Q\eta=0$, expressing the backgrounds in terms of independent base fields and showing that local partition functions are invariant under $Q$-exact background deformations. It then extends the setup to background vector multiplets, derives explicit realizations on geometries such as ${\bm S}^5$, ${\bm S}^4\times\mathbb{R}$, and ${\bm S}^3\times\Sigma$, and discusses global issues and the connection to conformal supergravity. The work provides a systematic framework linking 5d rigid SUSY, curved backgrounds, and partition-function computations, with implications for understanding dualities and higher-dimensional SUSY field theories.

Abstract

We construct curved backgrounds with Euclidean signature admitting rigid supersymmetry by using a 5d N=1 off-shell Poincare supergravity. We solve the conditions for the background Weyl multiplet and vector multiplets that preserve at least one supersymmetry parameterized by a symplectic Majorana spinor, and represent the solution in terms of several independent fields. We also show that the partition function does not depends on the local degrees of freedom of the background fields. Namely, as far as we focus on a single coordinate patch, we can freely change the independent fields by combining Q-exact deformations and gauge transformations. We also discuss realization of several known examples of supersymmetric theories in curved backgrounds by using the supergravity.