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

TL;DR

This paper establishes upper bounds on the Witten index for supersymmetric lattice models using discrete Morse theory, linking topological invariants of independence complexes to graph properties, and extends existing methods to higher dimensions.

Contribution

It proves a general theorem on independence complexes, providing new bounds on the Witten index and generalizing the 3-rule to arbitrary dimensions.

Findings

Upper bounds on the Witten index for various lattice classes.

Confirmation of previous computational results for small lattices.

Generalization of the 3-rule to higher-dimensional lattices.

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.