Generalizing Geometry - Algebroids and Sigma Models

Abstract

In this contribution we review some of the interplay between sigma models in theoretical physics and novel geometrical structures such as Lie (n-)algebroids. The first part of the article contains the mathematical background, the definition of various algebroids as well as of Dirac structures, a joint generalization of Poisson, presymplectic, but also complex structures. Proofs are given in detail. The second part deals with sigma models. Topological ones, in particular the AKSZ and the Dirac sigma models, as generalizations of the Poisson sigma models to higher dimensions and to Dirac structures, respectively, but also physical ones, that reduce to standard Yang Mills theories for the "flat" choice of a Lie algebra: Lie algebroid Yang Mills theories and possible action functionals for nonabelian gerbes and general higher gauge theories. Characteristic classes associated to Dirac structures and to higher principal bundles are also mentioned.

52 pages, Contribution to the Handbook on Pseudo-Riemannian Geometry and Supersymmetry, ed. V. Cortes

References in corpus (15)

• The Geometry of the Master Equation and Topological Quantum Field Theory
• Two-Dimensional Gravity and Nonlinear Gauge Theory
• Derived brackets
• WZW-Poisson manifolds
• Semiclassical approximation in Batalin-Vilkovisky formalism
• Current Algebras and Differential Geometry
• Chern-Simons Gauge Theory coupled with BF Theory
• Lie algebroid morphisms, Poisson Sigma Models, and off-shell closed gauge symmetries
• Algebroid Yang-Mills Theories
• Lectures on Generalized Complex Geometry and Supersymmetry
• Generalized complex structures and Lie brackets
• Lie Algebroid Yang Mills with Matter Fields
• Gravity from Lie algebroid morphisms
• Characteristic classes associated to Q-bundles
• Lie algebroids, Lie groupoids and TFT

Cited by in corpus (16)

• A higher stacky perspective on Chern-Simons theory
• A higher Chern-Weil derivation of AKSZ sigma-models
• Six-Dimensional (1,0) Superconformal Models and Higher Gauge Theory
• Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions
• Dirac Sigma Models from Gauging
• The Non-Abelian Self-Dual String and the (2,0)-Theory
• p-Brane Actions and Higher Roytenberg Brackets
• Extended generalized geometry and a DBI-type effective action for branes ending on branes
• Automorphisms of Strong Homotopy Lie Algebras of Local Observables
• Graded geometry in gauge theories and beyond
• On higher-dimensional Courant algebroids
• Dissections and automorphisms of regular Courant algebroids
• Pure Spinor String and Generalized Geometry
• The Free Courant Algebroid
• Geometry of Membrane Sigma Models
• A heterotic sigma model with novel target geometry