Comments on fake supersymmetry

Flat domain walls and spherical black holes are solutions to coupled second-order ODE's of the Hamiltonian form. Hamilton-Jacobi theory then implies that first-order flow equations always exist (possibly up to isolated submanifolds). If the first-order equations factorise in a specific way, they take a form that has been named fake supersymmetry. We point out that this factorisation is always possible at zero temperature. We therefore propose a less generic definition of fake supersymmetry, which involves the boundary conditions in a non-trivial way, and we analyse its physical relevance. For instance attractor flows are necessarily fake supersymmetric in our restricted sense. To illustrate the definition we provide new analytic solutions for axion-dilaton domain walls with fake superpotentials that were argued not to exist.