  I generalize the theory of phonon topological band structures of isostatic
lattices to frustrated antiferromagnets. I achieve this with a discovery of a
many-body supersymmetry (SUSY) in the phonon problem of balls and springs and
its connection to local constraints satisfied by ground states. The Witten
index of the SUSY model demands the Maxwell-Calladine index of mechanical
structures. "Spontaneous supersymmetry breaking" is identified as the need to
gap all modes in the bulk to create the topological isostatic lattice state.
Since ground states of magnetic systems also satisfy local constraint
conditions (such as the vanishing of the total spin on a triangle) I identify a
similar SUSY structure for many common models of antiferromagnets including the
square, triangluar, kagome, pyrochlore nearest neighbor antiferromagnets, and
the $J_2 = J_1/2$ square lattice antiferromagnet. Remarkably, the kagome family
of antiferromagnets is the analog of topological isostatic lattices among this
collections of models. Thus, a solid state realization of the theory of phonon
topological band structure may be found in frustrated magnetic materials.
