BFV-Complex and Higher Homotopy Structures

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L ∞ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instea

The A∞ de Rham Theorem and Integration of Representations up to Homotopy

We use Chen's iterated integrals to integrate representations up to homotopy. That is, we construct an functor from the representations up to homotopy of a Lie algebroid A to those of its infinity groupoid. This construction extends the usual integration of representations in Lie theory. We discuss several examples including Lie algebras and Poisson manifolds. The construction is based on an version of de Rham's theorem due to Gugenheim [15]. The integration procedure we explain here amounts to extending the construction of parallel transport for superconnections, introduced by Igusa [17] and Block-Smith [6], to the case of certain differential graded manifold

Deformations of Lie brackets and representations up to homotopy

We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.Comment: 28 page

Deformations of coisotropic submanifolds for fibrewise entire Poisson structures

We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the $L_\infty$-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo-Felder. In the symplectic case, we recover results previously obtained by Oh-Park. Moreover we consider the extended deformation problem and prove its obstructedness

BFV-complex and higher homotopy structures

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the so-called strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are $L_{\infty}$ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.Comment: 50 pages, 6 figures; version 4 is heavily revised and extende

Verified System Development with the AutoFocus Tool Chain

This work presents a model-based development methodology for verified software systems as well as a tool support for it: an applied AutoFocus tool chain and its basic principles emphasizing the verification of the system under development as well as the check mechanisms we used to raise the level of confidence in the correctness of the implementation of the automatic generators.Comment: In Proceedings WS-FMDS 2012, arXiv:1207.184

Coisotropic submanifolds and the BFV-complex

Koisotrope Untermannigfaltigkeiten bilden eine wichtige Klasse von Unterobjekten in Poisson Mannigfaltigkeiten. In dieser Dissertation werden jeder koisotrope Untermannigfaltigkeit zwei algebraische Strukturen zuge- ordnet und deren Eigenschaften untersucht. Zunächst wird die Konstruktion des ”homotopy Lie algebroid” erläutert. Hierbei folgen wir Oh und Park beziehungsweise Cattaneo und Felder. Die Invarianz des ”homotopy Lie algebroid” wird bewiesen. Dieses Resultat basiert auf einer gemeinsamen Arbeit mit Cattaneo. Dann wird der BFV-Komplex eingeführt. Dabei geben wir eine neue, konzeptuelle Konstruktion der BFV-Klammer. Anschliessend wird die Ab- hängigkeit des BFV-Komplexes von gewissen Wahlen geklärt. Ausserdem wird ein L∞ Quasi-Isomorphismus zwischen dem ”homotopy Lie algberoid” und dem BFV-Komplex konstruiert. Schlussendlich stellen wir eine Verbindung zwischen dem BFV-Komplex und der lokalen Deformationstheorie koisotroper Untermannigfaltigkeiten her. Es stellt sich heraus, dass man mit Hilfe des BFV-Komplexes ein Groupoid konstruieren kann, welches zum Deformationsgroupoid der koisotropen Untermannigfaltigkeit isomorph ist. Coisotropic submanifolds form an important class of subobjects of Poisson manifolds. In this thesis two algebraic structures associated to coisotropic submanifolds are constructed and their properties are investigated. First we explain the construction of the homotopy Lie algebroid – following Oh, Park and Cattaneo, Felder, respectively. The invariance of the homotopy Lie algberoid is established. This part of the thesis relies on joint work with Cattaneo. Next the BFV-complex is introduced. To this end we give a new conceptual construction of the BFV-bracket. The dependence of the BFV-complex on certain input data is clarified. Moreover an L∞ quasi-isomorphism be- tween the homotopy Lie algebroid and the BFV-complex is constructed. Finally we connect the BFV-complex with the local deformation theory of coisotropic submanifolds. It turns out that the BFV-complex allows to construct a groupoid which is isomorphic to the deformation groupoid of the coisotropic submanifold