D0-branes in two-dimensional ${\cal N}=(2,2)$ Landau-Ginzburg models at critical points of the superpotential

Question

My question concerns Section 6.1 of Hori's 'Linear Models of Supersymmetric D-branes' (http://arxiv.org/abs/hep-th/0012179).

Firstly, some background. The quantum field theory in question is a 2d ${\cal N}=(2,2)$ Landau-Ginzburg (LG) model on a worldsheet with boundaries (e.g. the infinite strip, or a disk). Boundary conditions for the fields ought to be chosen at the boundaries. The possible boundary conditions can only preserve a subset of the supersymmetries at the boundaries, in this case only B-type supersymmetry (see Section 2.2.2 for definition) is chosen to be preserved, and the corresponding boundary conditions are called B-branes.

Now, on to my question. The B-brane studied in Section 6.1 is a D0-brane, and Hori argues that the D0-brane must be located at a critical point ($\partial_i W=0$) of the superpotential. Unfortunately, I cannot grasp his argument completely. He presents the conserved supercharge,

\begin{equation} Q={1\over 2\pi}\int d x^1\left\{ g_{i\bar{\jmath}}(\overline{\psi}_-^{\bar{\jmath}}+\overline{\psi}_+^{\bar{\jmath}})\partial_0\phi^i +g_{i\bar{\jmath}}(\overline{\psi}_-^{\bar{\jmath}}-\bar{\psi}_+^{\bar{\jmath}})\partial_1\phi^i +(\psi_-^i-\psi_+^i)\partial_iW\right\}. \end{equation}

and from there says that 'Since the boundary point, say at $x^1 = π$ is locked at that point, we see that the supersymmetry is indeed broken for any configuration. Thus, we will not consider such a D-brane. In other words, D0-branes must be located at one of the critical points of W.'

I suspect that his argument has something to do with the SUSY algebra $\{Q,Q^\dagger\}=H$, where $H$ is the Hamiltonian, and when the Hamiltonian has non-zero vacuum expectation value, there is spontaneous supersymmetry breaking. But what exactly does he mean by 'locked', and why does it imply breaking of supersymmetry? Aren't all the other fields also 'locked' at the boundary due to their boundary conditions?

This post imported from StackExchange Physics at 2016-09-28 07:45 (UTC), posted by SE-user Mtheorist