Geometric bulk conductances in even dimensions
Abstract
Let P be a short-range projector on ℓ2(ℤ2n,ℂm) and U=(U1,…,U2n) be a transverse family of open subsets of ℝ2n. 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 Tr(P[P,1U1]∧⋯∧[P,1U2n]) is the product of a topological integer 𝒳U, which depends only on the relative positions of U1, …, U2n, and of the bulk conductance of P. Our proof writes the geometric bulk conductance as the pairing of P with a cochain on ℤ2n - an idea originally due to Kitaev - and deforms the cochain rather than the sets (U1, …, U2n). The proof presented in this paper is due to Claude Opus 5.5.
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BibTeX
@misc{drouot2026geometric,
title = {{Geometric bulk conductances in even dimensions}},
author = {Alexis Drouot},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16208},
eprint = {2610.16208},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16208},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometric bulk conductances in even dimensions
AU - Alexis Drouot
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16208
UR - https://arxiv.org/abs/2610.16208
AB - Let P be a short-range projector on ℓ^2(ℤ^2n,ℂ^m) and U=(U_1,…,U_2n) be a transverse family of open subsets of 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 Tr(P[P,1_U_1]∧⋯∧[P,1_U_2n]) is the product of a topological integer 𝒳_U, which depends only on the relative positions of U_1, …, 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 ℤ^2n - an idea originally due to Kitaev - and deforms the cochain rather than the sets (U_1, …, U_2n). The proof presented in this paper is due to Claude Opus 5.5.
ER -