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Ludwik Dąbrowski

Publications and source records attributed to Ludwik Dąbrowski.

11 recordsLinked to original sources
arXiv2026

On Geometric Spectral Functionals

We investigate spectral functionals associated with Dirac and Laplace-type differential operators on manifolds, defined via the Wodzicki residue, extending classical results for Dirac operators derived from the Levi-Civita connection to geometries with torsion. The local densities of these functionals recover fundamental geometric tensors, including the volume form, Riemannian metric, scalar curvature, Einstein tensor, and torsion tensor. Additionally, we introduce chiral spectral functionals using a grading operator, which yields novel spectral invariants. These constructions offer a richer spectral-geometric characterization of manifolds.

math-ph↗
arXiv2025

Spectral torsion of the internal noncommutative geometry of the Standard Model

We compute the nonvanishing spectral torsion functional of the internal part of the noncommutative geometry behind the Standard Model. We show that with a suitable modification of the usual differential graded calculus it matches an analogous functional constructed in terms of the connection. We study also the impact of the torsion on the other spectral fuctionals, which correspond to geometric invariants such as volume integral, metric and Einstein tensors, and scalar curvature. We discuss the impact of the SM Yukawa couplings and the Majorana mass matrix on our results.

hep-th↗
arXiv2025

Algebraic Versus Spectral Torsion

We relate the recently defined spectral torsion with the algebraic torsion of noncommutative differential calculi on the example of the almost-commutative geometry of the product of a closed oriented Riemannian spin manifold $M$ with the two-point space $\mathcal Z_2$.

math.QA↗
arXiv2025

An impediment to torsion from spectral geometry

Modifications of standard general relativity that bring torsion into a game have a long-standing history. However, no convincing arguments exist for or against its presence in physically acceptable gravity models. In this Letter, we provide an argument based on spectral geometry (using methods of pseudo-differential calculus) that suggests that the torsion shall be excluded from the consideration. We demonstrate that there is no well-defined functional extending to the torsion-full case of the spectral formulation of the Einstein tensor.

gr-qc↗
arXiv2024

Spectral Torsion

We introduce a trilinear functional of differential one-forms for a finitely summable regular spectral triple with a noncommutative residue. We demonstrate that for a canonical spectral triple over a closed spin manifold it recovers the torsion of the linear connection. We examine several spectral triples, including Hodge-de\,Rham, Einstein-Yang-Mills, almost-commutative two-sheeted space, conformally rescaled noncommutative tori, and quantum $SU(2)$ group, showing that the third one has a nonvanishing torsion if nontrivially coupled.

math.QA↗
arXiv2024

Spectral Metric and Einstein Functionals for Hodge-Dirac operator

We examine the metric and Einstein bilinear functionals of differential forms introduced in Adv.Math.,Vol.427,(2023)1091286, for Hodge-Dirac operator $d+δ$ on an oriented even-dimensional Riemannian manifold. We show that they reproduce these functionals for the canonical Dirac operator on a spin manifold up to a numerical factor. Furthermore, we demonstrate that the associated spectral triple is spectrally closed, which implies that it is torsion-free.

math.DG↗
arXiv2023

Spectral Metric and Einstein Functionals

We define bilinear functionals of vector fields and differential forms, the densities of which yield the metric and Einstein tensors on even-dimensional Riemannian manifolds. We generalise these concepts in non-commutative geometry and, in particular, we prove that for the conformally rescaled geometry of the noncommutative two-torus the Einstein functional vanishes.

math.DG↗
arXiv2018

The weak Hilbert-Smith conjecture from a Borsuk-Ulam-type conjecture

We prove a number of results surrounding the Borsuk-Ulam-type conjecture of Baum, Dąbrowski and Hajac (BDH, for short), to the effect that given a free action of a compact group $G$ on a compact space $X$, there are no $G$-equivariant maps $X*G\to X$ (with $*$ denoting the topological join). In particular, we prove the BDH conjecture for locally trivial principal $G$-bundles. The proof relies on the non-existence of $G$-equivariant maps $G^{*(n+1)}\to G^{*n}$, which in turn is a slight strengthening of an unpublished result of M. Bestvina and R. Edwards. Moreover, we show that the BDH conjecture partially settles a conjecture of Ageev. In turn, the latter implies the weak version Hilbert-Smith conjecture stating that no infinite compact zero-dimensional group can act freely on a manifold such that the orbit space is finite-dimensional.

math.GN↗
arXiv2017

Noncommutative Borsuk-Ulam-type conjectures revisited

Let $H$ be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra $A$. It was recently conjectured that there does not exist an equivariant $*$-homomorphism from $A$ (type-I case) or $H$ (type-II case) to the equivariant noncommutative join C*-algebra $A\circledast^δH$. When $A$ is the C*-algebra of functions on a sphere, and $H$ is the C*-algebra of functions on ${\mathbb Z}/2{\mathbb Z}$ acting antipodally on the sphere, then the conjecture of type I becomes the celebrated Borsuk-Ulam theorem. Following recent work of Passer, we prove the conjecture of type I for compact quantum groups admitting a non-trivial torsion character. Next, we prove that, if a compact quantum group admits a representation whose \mbox{$K_1$-class} is non-trivial and $A$ admits a character, then a stronger version of the type-II conjecture holds: the finitely generated projective module associated with $A\circledast^δH$ via this representation is not stably free. In particular, we apply this result to the $q$-deformations of compact connected semisimple Lie groups and to the reduced group C*-algebras of free groups on $n>1$ generators.

math.QA↗
arXiv2016

Twisted reality condition for Dirac operators

Motivated by examples obtained from conformal deformations of spectral triples and a spectral triple construction on quantum cones we propose a new twisted reality condition for the Dirac operator.

math.QA↗