The Critical Groups of Adinkras up to 2-Rank of Cayley Graphs

TL;DR

This paper determines the critical groups of Adinkras, which are graphs modeling supersymmetry algebras, based on the 2-rank of their Laplacian, and shows their independence from the signature.

Contribution

It extends previous results by explicitly linking the critical group to the 2-rank of the Laplacian and proves signature independence for Adinkras.

Findings

Critical groups are determined by the 2-rank of the Laplacian.

Critical groups are independent of the Adinkra's signature.

The proof employs the monodromy pairing on critical groups.

Abstract

Adinkras are graphical gadgets introduced by physicists to study supersymmetry, which can be thought of as the Cayley graphs for supersymmetry algebras. Improving the result of Iga et al., we determine the critical group of an Adinkra given the 2-rank of the Laplacian of the underlying Cayley graph. As a corollary, we show that the critical group is independent of the signature of the Adinkra. The proof uses the monodromy pairing on these critical groups.