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 is supposed to analyze 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 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 Riemannian structure.

The quasilocal degrees of freedom of Yang-Mills theory

“…14 This condition implies (19) as well as the bundle-like condition mentioned in the previous footnote. 15 A finite dimensional analogue of this condition was recently studied and generalized to more general Lie algebroids A KS than the action Lie algebroid featuring studied here, by Kotov and Strobl [34,39]. They named Lie algebroids satisfying such a generalized condition Killing Lie algebroids, and related their properties to the ability of "gauging" a Poisson-sigma-model with a A KS -symmetry.…”

“…In non-Abelian theories, reducible configurations form a meager set, 39 in the same way as those spacetime metrics which admit non-trivial Killing vector fields are "extremely rare" (i.e. form a meager set).…”

Momentum Sections in Hamiltonian Mechanics and Sigma Models

“…for all sections e 1 , e 2 ∈ Γ(E). We note these conditions have already appeared in [17] as compatibility conditions of geometric quantities as a metric and a closed differential form with a Lie algebroid structure.…”

“…Blohmann and Weinstein [1] have proposed a generalization of a momentum map and a Hamiltonian G-space on a Lie algebra (a Lie group) to Lie algebroid setting, based on analysis of the general relativity [2]. It is called a momentum section and a Hamiltonian Lie algebroid This structure is also regarded as reinterpretation of compatibility conditions of geometric quantities such as a metric g and a closed differential form H with a Lie algebroid structure, which was analyzed by Kotov and Strobl [17].…”

Transverse generalized metrics and 2d sigma models

“…While every isometric G-action equips (M, g) with an-in general only singular-Riemannian foliation, where the leaves are given by the G-orbits, not every such a foliation results from a Gaction. According to [8,9] 2 , it is not necessary to restrict gauging to eventual isometries of g, it is sufficient that (M, g) defines (a somewhat controlled form of) a singular Riemannian foliation. While the general theory of such gaugings is not worked out yet in the presence of a Wess-Zumino twist for arbitrary dimensions d, it was done so in [3] for d = 2.…”