nLab Courant sigma-model

Idea

Every symplectic Lie n-algebroid $(\mathfrak{P},\omega)$ carries a specified invariant polynomial $\omega$. The action functional induced by the corresponding Chern-Simons element in ∞-Chern-Simons theory defined the corresponding AKSZ theory sigma model.

For a Courant algebroid $(\mathfrak{P}, \omega)$ this is called the Courant $\sigma$-model .

Related concepts

• ∞-Chern-Simons theory

• sigma-model

  • AKSZ sigma-model

    • Poisson sigma-model

      • A-model, B-model

    • Courant sigma-model

      • Chern-Simons theory

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

References

The action functional was first deduced in

• Noriaki Ikeda, Chern-Simons Gauge Theory coupled with BF Theory Int. J. Mod. Phys. A18 (2003) 2689-2702 (arXiv:hep-th/0203043)

As an example of an AKSZ sigma-model it was later re-derived in

• Dmitry Roytenberg, AKSZ-BV Formalism and Courant Algebroid-induced Topological Field Theories (2006) (arXiv:hep-th/0608150).

Further discussion is in

• Alberto Cattaneo, Jian Qiu, Maxim Zabzine, 2D and 3D topological field theories for generalized complex geometry (pdf)

The interpretation in terms of infinity-Chern-Weil theory is discussed in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory (2011)

Relations to generalized complex geometry is discussed in

• Alberto Cattaneo, Jian Qiub, Maxim Zabzine, 2D and 3D topological field theories for generalized complex geometry (pdf)