\(\mathfrak g\)-symplectic orbits and a solution of the BRST consistency condition. (English) Zbl 1072.53030

Schmid, Rudolf

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 4th international conference on geometry, integrability and quantization, Sts. Constantine and Elena, Bulgaria, June 6–15, 2002. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-4-1/pbk). 284-295 (2003).

For a principal \(G\)-bundle \((P,\pi, M)\), let \(\Omega^k(P,{\mathfrak g})\) be the space of \({\mathfrak g}\)-valued \(k\)-forms on \(P\) (shortly: \({\mathfrak g}\)-forms). Then a closed and nondegenerate \({\mathfrak g}\)-form \(\Omega\in \Omega^2(P,{\mathfrak g})\) is called a \({\mathfrak g}\)-symplectic structure. Let \(\varphi\) be the right invariant Maurer-Cartan form on \(G\) and \(R: P\times G\to P\) the right action of \(G\) on \(P\). Then every \(G\)-orbit \(O_p\subset P\) through a point \(p\in P\) carries a natural \({\mathfrak g}\)-symplectic form \(\Omega_p= dR_{p^*}\vartheta\) and if \(G\) is a semisimple group, then \(O_p\) becomes a \({\mathfrak g}\)-symplectic manifold. The author deals with the canonical momentum map on \(O_p\), recalls the variational bicomplex on \(J^\infty(\pi)\) and connections to the BRST bicomplex (consisting of \({\mathfrak g}\)-valued forms with Chevalley-Eilenberg coboundary operators, see R. Schmid [Differ. Geom. Appl. 4, 107–116 (1994; Zbl 0798.58011)]). The associated momentum map is a solution of the Wess-Zumino consistency condition for the anomaly (the form \(R_{p^*}\vartheta\)).
For the entire collection see [Zbl 1008.00022].

Reviewer: Jan Chrastina (Brno)

MSC:

53D20	Momentum maps; symplectic reduction
58A10	Differential forms in global analysis

Keywords:

principal \(G\)-bundle; \({\mathfrak g}\)-symplectic structure; momentum map; BRST bicomplex

Citations:

Zbl 0798.58011