Lie algebroids, gauge theories, and compatible geometrical structures

Abstract

The construction of gauge theories beyond the realm of Lie groups and algebras leads one to consider Lie groupoids and algebroids equipped with additional geometrical structures which, for gauge invariance of the construction, need to satisfy particular compatibility conditions. This paper analyzes these compatibilities from a mathematical perspective. In particular, we show that the compatibility of a connection with a Lie algebroid that one finds is the Cartan condition, introduced previously by A. Blaom. For the metric on the base M of a Lie algebroid equipped with any connection, we show that the compatibility suggested from gauge theories implies that the (possibly singular) foliation induced by the Lie algebroid becomes a Riemannian foliation. Building upon a result of del Hoyo and Fernandes, we prove furthermore that every Lie algebroid integrating to a proper Lie groupoid admits a compatible Riemannian base. We also consider the case where the base is equipped with a compatible symplectic or generalized metric structure.

25 pages. This is the first part of the original preprint that was split into two parts for publication, with a new title, abstract, and introduction. The second, somewhat extended part, entitled 'Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries' is published at Journal of Geometry and Physics 135 (2019) 1-6 and can be found under arXiv:1904.05809

References in corpus (4)

• Lie Algebroid Yang Mills with Matter Fields
• 2d Gauge Theories and Generalized Geometry
• Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids
• Universal Cartan-Lie algebroid of an anchored bundle with connection and compatible geometries

Cited by in corpus (14)

• Universal Corner Symmetry and the Orbit Method for Gravity
• Lie Algebroids and the Geometry of Off-shell BRST
• Born Sigma-Models for Para-Hermitian Manifolds and Generalized T-Duality
• Momentum sections in Hamiltonian mechanics and sigma models
• Momentum section on Courant algebroid and constrained Hamiltonian mechanics
• Transverse generalized metrics and 2d sigma models
• The BV action of 3D twisted R-Poisson sigma models
• Homotopy momentum sections on multisymplectic manifolds
• BRST Cohomology is Lie Algebroid Cohomology
• The quasilocal degrees of freedom of Yang-Mills theory
• Geometric BV for twisted Courant sigma models and the BRST power finesse
• Basic curvature and the Atiyah cocycle in gauge theory
• Singular Riemannian foliations and $\mathcal{I}$-Poisson manifolds
• Instances of Higher Geometry in Field Theory