Supersymmetry in Singular Spaces

TL;DR

This paper extends supersymmetry to singular spaces by introducing a SUSY singlet field $G(x)$ and a $(D-1)$-form, enabling consistent SUSY on $S^1/\mathbb{Z}_2$ orbifolds with branes that act as flux sinks. The bulk theory is built from a modified 5D gauged supergravity, and a 4-form together with brane delta-function sources ensure the flux changes sign across walls while preserving SUSY separately in bulk and brane sectors. The energy of static configurations vanishes locally ($E=0$), and BPS domain-wall solutions are constructed in very special geometry, including fixed-scalar doubling and non-constant scalar walls described by harmonic functions $H_I(y)$, with explicit STU and Calabi–Yau wall examples. The framework also discusses 8-branes in 10D and provides a platform for exploring RS-type brane-worlds without fine-tuning, offering a bridge between flux stabilization, domain-wall physics, and higher-dimensional supergravity. Overall, the work clarifies how to formulate and analyze supersymmetry in the presence of singular branes and fluxes, with potential implications for string/M-theory constructions and warped extra dimensions.

Abstract

We develop the concept of supersymmetry in singular spaces, apply it in an example for 3-branes in D=5 and comment on 8-branes in D=10. The new construction has an interpretation that the brane is a sink for the flux and requires adding to the standard supergravity a (D-1)-form field and a supersymmetry singlet field. This allows a consistent definition of supersymmetry on a S_1/Z_2 orbifold, the bulk and the brane actions being separately supersymmetric. Randall-Sundrum brane-worlds can be reproduced in this framework without fine tuning. For fixed scalars, the doubling of unbroken supersymmetries takes place and the negative tension brane can be pushed to infinity. In more general BPS domain walls with 1/2 of unbroken supersymmetries, the distance between branes in some cases may be restricted by the collapsing cycles of the Calabi-Yau manifold. The energy of any static x^5-dependent bosonic configuration vanishes, E=0, in analogy with the vanishing of the Hamiltonian in a closed universe.