Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2302.05891

Dirac operator associated to a quantum metric

Shahn Majid
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct a canonical geometrically realised Connes spectral triple or `Dirac operator' $D\!\!\!/$ from the data of a quantum metric $g\in \Omega^1\otimes_A\Omega^1$ and quantum Levi-Civita bimodule connection, at the pre-Hilbert space level. Here $A$ is a possibly noncommutative coordinate algebra, $\Omega^1$ a bimodule of 1-forms and the spinor bundle is $S=A\oplus\Omega^1$. When applied to graphs or lattices, we essentially recover a Dirac operator previously proposed by Bianconi but now as a geometrically realised spectral triple. We also apply the construction to the fuzzy sphere and to $2\times 2$ matrices with their standard quantum Riemannian geometries. We also propose how $D\!\!\!/$ can be minimally coupled to an external potential.

Explore related subjects

math.QAgr-qc

Keep this discovery

BibTeXRIS

Shahn Majid. 2023-02-12. Dirac operator associated to a quantum metric. https://arxiv.org/abs/2302.05891

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Diagrammatic Categorification of Verma Modules I: Braiding

In this paper, we study the extensions of KLRW algebras to tensor products of Verma module representations of $\mathfrak{sl}_2$. Our motivation is to construct a theory of Khovanov homology for knot complements in $S^3$ (and which also categorifies the Gukov-Manolescu two-variable series for knot complements), which will be done in the second part of this work. We construct the categorification of R-matrices for Verma modules as functors given by derived tensor products with diagrammatic bimodules and explicitly compute their projective resolutions. We also prove these braiding functors induce an action of the braid group on the relevant categories. Then, we describe how to incorporate strands in finite-dimensional representations of $\mathfrak{sl}_2$, thereby establishing functors that serve as the Khovanov homology on a braid complement. In the case of the unknot, this gives knot homologies in $S^1\times D^2$, which we compare to Annular Khovanov Homology through several examples and show they are very closely related, conjecturing they are of the same dimension. We conclude with a proposal for the categorification of the cups and caps of Verma module colored strands, which we build upon in the next paper.

math.QA↗
arXiv2026

Some finite dimensional representations of shifted quantum affine algebras of type A

In this paper, we study finite dimensional representations of shifted quantum affine algebras of type A. We give an explicit description of the tensor product of simple evaluation modules of the quantum loop algebra and a one-dimensional representation of the shifted quantum affine algebra under the separation condition. As a consequence, we give the q-characters of some finite dimensional simple modules of the shifted quantum affine algebra.

math.QA↗