Courant sigma-model


\infty-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory




Quantum field theory


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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)


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 (