A higher Chern-Weil derivation of AKSZ sigma-models

Details A higher Chern-Weil derivation of AKSZ sigma-models A higher Chern-Weil derivation of AKSZ sigma-models

2011-08-22

arxiv.org/abs/1108.4378v2

Physics

Mathematical Physics

39 pages, further example added

Scientific paper

Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. We observe that in the context of higher Chern-Weil theory in smooth infinity-groupoids this statement generalizes from Lie algebras to L-infinity-algebras and further to L-infinity-algebroids. It turns out that the symplectic form on a symplectic higher Lie algebroid (for instance a Poisson Lie algebroid or a Courant Lie 2-algebroid) is infinity-Lie-theoretically an invariant polynomial. We show that the higher Chern-Simons action functional associated to this by higher Chern-Weil theory is the action functional of the AKSZ sigma-model whose target space is the given L-infinity-algebroid (for instance the Poisson sigma-model or the Courant-sigma-model including ordinary Chern-Simons theory, or higher dimensional abelian Chern-Simons theory).

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