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

Supersymmetry "protected" topological phases of isostatic lattices and kagome antiferromagnets