nLab
Courant sigma-model

Context

\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Ingredients

Definition

Examples

Quantum field theory

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

Every symplectic Lie n-algebroid (𝔓,ω)(\mathfrak{P},\omega) carries a specified invariant polynomial ω\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 (𝔓,ω)(\mathfrak{P}, \omega) this is called the Courant σ\sigma-model .

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

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)

References

The action functional was first deduced in

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

Further discussion is in

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

Relations to generalized complex geometry is discussed in

Revised on February 8, 2013 02:09:04 by Urs Schreiber (89.204.155.59)