Lectures on AKSZ Sigma Models for Physicists

Abstract

This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analyses of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional topological field theory and explain the ideas of the AKSZ sigma models. This construction is then generalized and leads to a mathematical formulation of a general topological sigma model. We review the mathematical objects, such as algebroids and supergeometry, that are used in the analysis of general gauge structures. The quantization of the Poisson sigma model is presented as an example of a quantization of an AKSZ sigma model.

97 pages, version for proceeding, 'Noncommutative Geometry and Physics 4'. Typos fixed

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Cited by in corpus (30)

• Double Field Theory and Membrane Sigma-Models
• Unified picture of non-geometric fluxes and T-duality in double field theory via graded symplectic manifolds
• Finite deformations from a heterotic superpotential: holomorphic Chern--Simons and an $L_{\infty }$ algebra
• Global Theory of Graded Manifolds
• Double field theory for the A/B-models and topological S-duality in generalized geometry
• Higher Spin Gravities and Presymplectic AKSZ Models
• AKSZ constructions for topological membranes on $G_2$-manifolds
• Momentum sections in Hamiltonian mechanics and sigma models
• Contravariant Gravity on Poisson Manifolds and Einstein Gravity
• Jacobi sigma models
• DFT in supermanifold formulation and group manifold as background geometry
• Presymplectic gauge PDEs and Lagrangian BV formalism beyond jet-bundles
• Gauged sigma-models with nonclosed 3-form and twisted Jacobi structures
• Current algebras from QP-manifolds in general dimensions
• Higher Gauge Theory and Integrability
• Metric algebroid and Dirac generating operator in Double Field Theory
• Geometric BV for twisted Courant sigma models and the BRST power finesse
• The Algebroid Structure of Double Field Theory
• Metric Algebroid and Poisson-Lie T-duality in DFT
• Differential Poisson Sigma Models with Extended Supersymmetry
• Canonical Functions and Differential Graded Symplectic Pairs in Supergeometry and AKSZ Sigma Models with Boundary
• Topological and dynamical aspects of Jacobi sigma models
• Exact renormalization group in Batalin--Vilkovisky theory
• From BFV to BV and spacetime covariance
• AKSZ-type Topological Quantum Field Theories and Rational Homotopy Theory
• Gravity from AKSZ-Manin theories in two, three, and four dimensions
• Homotopy Manin Theories: Generalising Third-Way, Yang-Mills and Integrable Sigma Models
• Higher dimensional Lie algebroid sigma model with WZ term
• Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory
• Instances of Higher Geometry in Field Theory