Transitive Lie algebroids and Q-manifolds.

Abstract

We will start by giving an outline of the fundamentals ofsupergeometry and Q-manifolds. Then, we will give local descriptionof the Atiyah sequence of a principal bundle. We will constructlocal charts for the manifolds involved and write the expressions explicitly in local form. The Atiyah sequence encodes the different notions in connection theory in a compact way. For example a section of the Atiyah algebroid will give a connection in the principal bundle and curvature will be the failure of this section to be a Lie algebroid morphism.We will describe the Lie brackets for the Atiyah algebroid$\frac{TP}{G}$ and the adjoint bundle $\frac{P\tm \li{g}}{G}$ inlocal coordinates. After that, we will describe the Lie algebroid ofderivations $\li{D}(E)$. We will see how the curvature of somesection gives the curvature on the vector bundle $E$ and whenexpressed locally gives the corresponding local connection forms. Onthe other hand we will give an explicit expression of the morphismfrom $\frac{TP}{G}$ to $\li{D}\lf(\frac{P\tm V}{G}\rt)$ where $V$ issome vector space on which $G$ acts. As a corollary we will get anisomorphism between $\frac{TFE}{G}$ and $\li{D}(E)$ where $FE$ isthe frame bundle of $E$ and $G$ is the general linear group of thefibre $V$. We will establish an explicit equivalence betweencurvature and field strength in a more general sense. We will recallconstructions from the paper of Kotov and Strobl~\cite{kot1} thatdescribe the construction of characteristic classes associated witha section (connection=gauge field) of a $Q$-bundle$\cl{E}(\cl{M},\cl{F},\pi)$. Finally, we state and prove thenon-abelian Poincar\'{e} lemma in the case when $G=\diff(F)$, thediffeomorphism group of some supermanifold $F$, which has the spaceof vector fields on $F$, $\li{X}(F)$, as its super Lie algebra. Thediffeomorphism group is generally infinite dimensional. It is thisthat makes the non-abelian Poincar\'{e} lemma more interesting toapplications. Then we will show how it is applied to prove thatevery transitive Lie algebroid is locally trivial.

Date of Award	1 Aug 2011
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Theodore Voronov (Supervisor) & Hovhannes Khudaverdyan (Supervisor)