arXiv subjects

Luca Accornero

Publications and source records attributed to Luca Accornero.

6 recordsLinked to original sources

arXiv2025

A note on Chern-Weil classes of Cartan connections

We present a construction of Chern-Weil characteristic classes for pairs Cartan geometries sharing the same underlying data. Given any model Cartan geometry $(Q,\omega)$ with underlying data $(G,V)$ and a second Cartan geometry $(P,\theta)$ with the same underlying data, we define a subalgebra of polynomials on the Atiyah algebroid of $Q$ together with a characteristic map that recovers the classical Chern-Weil map of a Cartan connection when $(Q,\omega)$ arises from a Klein pair modeling $(P,\theta)$.

math.DG↗

arXiv2023

Haefliger's differentiable cohomology

We review Haefliger's differentiable cohomology for the pseudogroup of diffeomorphisms of $\mathbb{R}^q$. We investigate the structure needed to define such a cohomology, which, remarkably, is related to the so called Cartan distribution underlying the geometric study of PDE. We define an analogue of Haefliger differentiable cohomology for flat Cartan groupoids, investigate its infinitesimal counterpart and relate the two by a Van Est-like map. Finally, we define a characteristic map for geometric structures on manifolds $M$ associated to flat Cartan groupoids. The outcome generalizes the existing approaches to characteristic classes for foliations.

math.DG↗

arXiv2022

Pseudogroups of symmetries and Morita equivalences

This work is a spin-off of an on-going programme which aims at revisiting the original studies of Lie and Cartan on pseudogroups and geometric structures from a modern perspective. Within the framework of Lie groupoids equipped with a special multiplicative form - called Pfaffian groupoids - we focus on principal bibundles and Morita equivalences. In particular, we discuss in details the notion of Pfaffian Morita equivalence, its relation to the gauge construction in the Pfaffian setting, and its interactions with principal actions. We briefly present some examples and applications to transitive pseudogroups of symmetries, which we explored in great detail in arXiv:2211.16639.

math.DG↗

arXiv2022

A groupoid approach to transitive differential geometry

This work is a spin-off of an on-going programme which aims at revisiting the original studies of Lie and Cartan on pseudogroups and geometric structures from a modern perspective. We encode geometric structures induced by transitive Lie pseudogroups into principal $G$-bundles equipped with a transversally parallelisable foliation generated by a subalgebra of $\mathfrak{g}$, called Cartan bundles. Our approach is complementary to arXiv:1911.13147 and is based on Morita equivalence of Lie groupoids. After identifying the main examples and properties, we develop a notion of flatness with respect to a Lie algebra, which encompasses the classical integrability of $G$-structures, the flatness of Cartan geometries, as well as the integrability of contact structures.

math.DG↗

arXiv2021

Symmetry transformations of extremals and higher conserved quantities: invariant Yang--Mills connections

We characterize symmetry transformations of Lagrangian extremals generating `on shell' conservation laws. We relate symmetry transformations of extremals to Jacobi fields and study symmetries of higher variations by proving that a pair given by a symmetry of the $l$-th variation of a Lagrangian and by a Jacobi field of the $s$-th variation of the same Lagrangian (with $s<l$) is associated with an `off shell' conserved current. The conserved current associated with two symmetry transformations is constructed and, as a case of study, its expression for invariant Yang--Mills connections on Minkowski space-times is obtained.

math-ph↗

arXiv2020

The Jacobi morphism and the Hessian in higher order field theory; with applications to a Yang-Mills theory on a Minkowskian background

We characterize the second variation of an higher order Lagrangian by a Jacobi morphism and by currents strictly related to the geometric structure of the variational problem. We discuss the relation between the Jacobi morphism and the Hessian at an arbitrary order. Furthermore, we prove that a pair of Jacobi fields always generates a (weakly) conserved current. An explicit example is provided for a Yang-Mills theory on a Minkowskian background.

math-ph↗