arXiv subjects

Aren Martinian

Publications and source records attributed to Aren Martinian.

2 recordsLinked to original sources

arXiv2026

Kubo Formulas and Bulk-Edge Correspondence for Curved Boundaries

Strong topological insulators are classified by integer invariants which admit different real-space expressions. Inspired by recent work studying these invariants for spaces with curved boundaries, we revisit the problem of equivalence of the various expressions, including Fredholm index pairings and Kubo formulas involving half-space projections. Using Roe algebras and a variant of KK-theory suitable for non-separable $C^*$-algebras, we prove a general form of bulk-edge correspondence for the index pairing with a general position-space Dirac operator in the presence of arbitrary boundaries. For spaces coarsely equivalent to $\mathbb{R}^d$, we show that up to multiplicity, these pairings are equal to those obtained with the standard dual Dirac operator, with the multiplicity explicitly given as the topological degree of the symbol function. In addition, we prove a generalized Kubo formula which computes the general index pairing by a real-space formula.

math-ph↗

arXiv2022

On the critical group of hinge graphs

Let $G$ be a finite, connected, simple graph. The critical group $K(G)$, also known as the sandpile group, is the torsion subgroup of the cokernel of the graph Laplacian $\operatorname{cok}(L)$. We investigate a family of graphs with relatively simple non-cyclic critical group with an end goal of understanding whether multiple divisors, i.e., formal linear combinations of vertices of $G$, generate $K(G)$. These graphs, referred to as hinge graphs, can be intuitively understood by taking multiple base shapes and ``gluing" them together by a single shared edge and two corresponding shared vertices. In the case where all base shapes are identical, we compute the explicit structure of the critical group. Additionally, we compute the order of three special divisors. We prove the structure of the critical group of hinge graphs when variance in the number of vertices of each base shape is allowed, generalizing many of the aforementioned results.

math.CO↗