13.1. Spontaneous supersymmetry breaking Let Q be the generator of a N = 1 supersymmetric theory and Q† its adjoint operator.

With a proper normalization we have

{Q,Q†} = H, where H is the Hamiltonian of the system.

a. Show that the Hamiltonian of a supersymmetric theory contains no negative eigenvalues.

b. Show that any state whose energy is not zero cannot be invariant under a supersymmetry transformation.

c. Show that supersymmetry is spontaneously broken if and only if the energy of the lowest lying state (the vacuum) is not exactly zero.

d. Consider the two-dimensional superconformal models on a cylinder, for which Q = G0 and H = Q2 = L0 −c/24. Show that in the first model of the minimal unitary series, given by the tricritical Ising model, supersymmetry is broken while in the second minimal model, given by the Gaussian model, is exact.