Geometric bulk conductances in even dimensions

Abstract

Let $P$ be a short-range projector on ${\ell }^2({\mathbb{Z}}^{2n},{\mathbb{C}}^m)$ and $U=(U_1,\dots ,U_{2n})$ be a transverse family of open subsets of ${\mathbb{R}}^{2n}$. We give an elementary proof of a recent result of Martinian-Stoiber, which extends a two-dimensional result of Drouot-Zhu to all even dimensions. Specifically, using only Stokes' theorem and the Poincaré lemma, we show that the geometric Kubo formula $\mathrm{Tr}(P[P,1_{U_1}]\wedge \cdots \wedge [P,1_{U_{2n}}])$ is the product of a topological integer ${\mathcal{X}}_U$, which depends only on the relative positions of $U_1,\dots ,U_{2n}$, and of the bulk conductance of $P$. Our proof writes the geometric bulk conductance as the pairing of $P$ with a cochain on ${\mathbb{Z}}^{2n}$ - an idea originally due to Kitaev - and deforms the cochain rather than the sets $(U_1,\dots ,U_{2n})$. The proof presented in this paper is due to Claude Opus 5.5.

arXiv comment15 pages. The proof presented in this paper is due to Claude Opus 5.5. Section 1.4 has some reflections about the state of mathematics