Precise zero energy bound for supersymmetry Usually we can shift the energy $E$ by any amount $\delta$ to redefine the lowest energy as
$$
E + \delta.
$$
However, in supersymmetry, there is a precise $E=0$ must be true, so that the supercharge $Q$ annihilates some state $|\psi_{min}\rangle$ to give the minimal energy
$$
Q |\psi_{min}\rangle =0
$$
and also
$$
H|\psi_{min}\rangle =Q^2|\psi_{min}\rangle=0.
$$
Question

This raises the question that we have a precise zero energy bound for supersymmetry theory.


Does it mean that we cannot shift the energy $E$ to $E + \delta$ in supersymmetry theory? What is the  deep reason behind it?

 A: I will offer two arguments to try to exhibit what the problem with ground states with non zero energy in a supersymmetric theory is.
The case of gauged supersymmetry: Energy backreacts on the geometry.
If the energy of the ground states of a supersymmetric theory were non exactly zero the underlying geometry of the background would be allowed to be curved in arbitrary ways. The latter is imposible in a supersymmetric theory because supersymmetry enforce at least an spin structure of the underlying background geometry or possibly a trivial canonical bundle or a special holonomy (as in the Calabi-Yau, $G_{2}$ or $Spin(7)$ cases) on the target manifold.
Supersymmetry in quantum mechanics: The hamiltonian in a $(0+1)$-supersymmetric theory can be schematically written (see Supersymmetry and Morse theory) as $$H=\frac{1}{2}(Q_{1}^{2}+Q_{2}^{2}).$$
If for some ground state $\psi$ $$H\psi \neq 0,$$ it would follow that $\psi$ wouldn't be annihilated by the (positive definite) squares of the supercharges; then $\psi$ wouldn't preserve some supersymmetry.
It is illustrative to notice that in the case of a Riemannian manifold $\mathcal{M}$ the hamiltonian coincides with the laplacian of $\mathcal{M}$ (see chapter 2 in Supersymmetry and Morse theory); if the laplacian was not zero, then you can make an analogy with electrodynamics, you can't be possibly describing the vaccum of the theory, sources must be present.
