Lie algebroids, gauge theories, and compatible geometrical structures
Alexei Kotov, Thomas Strobl
TL;DR
The paper develops a framework for gauge theories based on Lie algebroids by introducing Cartan-Lie algebroids, i.e., compatible pairs $(A,\nabla)$ where the $A$-jet splitting is a Lie algebroid morphism, encoded by the tensor $S=0$. It then defines Killing Lie algebroids via ${}^\tau\nabla(g)=0$ and proves that, when $A$ integrates to a proper Lie groupoid, a base metric $g$ can be chosen so that the algebroid data are compatible, yielding a Riemannian foliation on $M$ and a well-behaved quotient geometry. The results extend to settings with compatible symplectic or generalized metric structures, showing how quotients inherit Poisson or generalized Riemannian data under suitable smoothness assumptions. Altogether, the work provides a rigorous geometric foundation for gauged sigma models beyond classical Lie groups, with broad implications for reductions, quotients, and compatibility in mathematical physics.
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.