Upper bounds on the Witten index for supersymmetric lattice models by discrete Morse theory

Abstract

The Witten index for certain supersymmetric lattice models treated by de Boer, van Eerten, Fendley, and Schoutens, can be formulated as a topological invariant of simplicial complexes arising as independence complexes of graphs. We prove a general theorem on independence complexes using discrete Morse theory: If G is a graph and D a subset of its vertex set such that G\D is a forest, then $sum_i \dim H_i(Ind(G);Q) \leq |Ind}(G[D])|$. We use the theorem to calculate upper bounds on the Witten index for several classes of lattices. These bounds confirm some of the computer calculations by van Eerten on small lattices. The cohomological method and the 3-rule of Fendley et al. is a special case of when G\D lacks edges. We prove a generalized 3-rule and introduce lattices in arbitrary dimensions satisfying it.