AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories

Abstract

We give a detailed exposition of the Alexandrov-Kontsevich-Schwarz- Zaboronsky superfield formalism using the language of graded manifolds. As a main illustarting example, to every Courant algebroid structure we associate canonically a three-dimensional topological sigma-model. Using the AKSZ formalism, we construct the Batalin-Vilkovisky master action for the model.

arXiv comment13 pages, based on lectures at Rencontres mathematiques de Glanon 2003

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Cited by in corpus (83)

• Higher vector bundles and multi-graded symplectic manifolds
• Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds
• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• Classical BV theories on manifolds with boundary
• Introduction to supergeometry
• Double Field Theory and Membrane Sigma-Models
• General Yang-Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical framework OR From Bianchi identities to twisted Courant algebroids
• A higher stacky perspective on Chern-Simons theory
• A minimal BV action for Vasiliev's four-dimensional higher spin gravity
• A higher Chern-Weil derivation of AKSZ sigma-models
• An action principle for Vasiliev's four-dimensional higher-spin gravity
• Poisson-Lie T-duality as a boundary phenomenon of Chern-Simons theory
• Sigma models for genuinely non-geometric backgrounds
• Courant-Dorfman algebras and their cohomology
• Courant Algebroids. A Short History
• Odd Scalar Curvature in Anti-Poisson Geometry
• QP-Structures of Degree 3 and 4D Topological Field Theory
• The Hitchin Model, Poisson-quasi-Nijenhuis Geometry and Symmetry Reduction
• Finite dimensional AKSZ-BV theories
• BV-BFV approach to General Relativity, Einstein-Hilbert action
• Pre-Courant Algebroids
• Double field theory for the A/B-models and topological S-duality in generalized geometry
• Higher Spin Gravities and Presymplectic AKSZ Models
• $L_{\infty }$-Algebra Models and Higher Chern-Simons Theories
• On time
• Knot Invariants and New Weight Systems from General 3D TFTs
• Symplectic structures on the integration of exact Courant algebroids
• Equivariant Rozansky-Witten classes and TFTs
• Topological Field Theories induced by twisted R-Poisson structure in any dimension
• Fluxes in Exceptional Field Theory and Threebrane Sigma-Models
• Courant sigma model and $L_{\infty }$-algebras
• On the AKSZ formulation of the Rozansky-Witten theory and beyond
• E-Courant algebroids
• Twisted Courant algebroids and coisotropic Cartan geometries
• Global Aspects of Doubled Geometry and Pre-rackoid
• Equivariant Batalin-Vilkovisky formalism
• Courant cohomology, Cartan calculus, connections, curvature, characteristic classes
• AKSZ constructions for topological membranes on $G_2$-manifolds
• Graded Bundles in the Category of Lie Groupoids
• Chern-Simons Theory with Wilson Lines and Boundary in the BV-BFV Formalism
• Momentum section on Courant algebroid and constrained Hamiltonian mechanics
• Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids
• The geometry of graded cotangent bundles
• The BV action of 3D twisted R-Poisson sigma models
• On the relation of Lie algebroids to constrained systems and their BV/BFV formulation
• Odd Chern-Simons Theory, Lie Algebra Cohomology and Characteristic Classes
• Algebraic formulation of higher gauge theory
• BV equivalence with boundary
• Renormalization group flow of Chern-Simons boundary conditions and generalized Ricci tensor
• Comparing Poisson Sigma Model with A-model
• Higher Gauge Theory
• Topological A-Type Models with Flux
• Geometric BV for twisted Courant sigma models and the BRST power finesse
• AKSZ construction from reduction data
• An Alternative Topological Field Theory of Generalized Complex Geometry
• Basic curvature and the Atiyah cocycle in gauge theory
• Formal Global AKSZ Gauge Observables and Generalized Wilson Surfaces
• Boundary coupling of Lie algebroid Poisson sigma models and representations up to homotopy
• Topology, rigid cosymmetries and linearization instability in higher gauge theories
• On Quantum Obstruction Spaces and Higher Codimension Gauge Theories
• Ricci tensor in graded geometry
• Observables in the equivariant A-model
• Transitive Courant algebroids and double symplectic groupoids
• Fluxes of Courant bracket twisted at the same time by $B$ and $\theta$
• From BFV to BV and spacetime covariance
• Quillen superconnections and connections on supermanifolds
• BV Quantization
• Normal forms of $\mathbb{Z}$-graded $Q$-manifolds
• Wilson Lines from Representations of NQ-Manifolds
• Super AKSZ construction, integral forms, and the 2-dimensional $\mathcal{N}=(1,1)$ sigma model
• Dimensional reduction of Courant sigma models and Lie theory of Poisson groupoids
• Comomentum sections and Poisson maps in Hamiltonian Lie algebroids
• Gravity from AKSZ-Manin theories in two, three, and four dimensions
• Homotopy Manin Theories: Generalising Third-Way, Yang-Mills and Integrable Sigma Models
• Pre-NQ Manifolds and Correspondence Spaces: the Nilmanifold Example
• Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds
• Quantizing Derived Mapping Stacks
• The standard cohomology of regular Courant algebroids
• Instances of Higher Geometry in Field Theory
• A topological quantum field theory for $\mathrm{Spin}(7)$-instantons
• Supersymmetric Poisson and Poisson-supersymmetric sigma models
• Higher Courant-Dorfman algebras and associated higher Poisson vertex algebras
• Categorification of VB-Lie algebroids and VB-Courant algebroids