On the structure of graded symplectic supermanifolds and Courant algebroids. (English) Zbl 1036.53057

Voronov, Theodore (ed.), Quantization, Poisson brackets and beyond. London Mathematical Society regional meeting and workshop on quantization, deformations, and new homological and categorical methods in mathematical physics, Manchester, UK, July 6–13, 2001. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3201-8). Contemp. Math. 315, 169-185 (2002).
Graded symplectic supermanifolds have been known to physicists since the 70’s providing the framework for the so-called BRST formalism. The purpose of this paper is to generalize the BRST formalism to the case of an arbitrary pseudo-Euclidean vector bundle \(E\) over a manifold \(M_0\), and to study the resulting geometric structures, Courant algebroids. The special case \(M_0=\) point was treated in [B. Kostant and S. Sternberg, Ann. Phys. 176, 49–113 (1987; Zbl 0642.17003)]. The author uses the supermanifold approach, which offers significant advantages when we study geometric structures on vector bundles. In particular, pseudo-Euclidean vector bundles are in 1-1 correspondence with graded symplectic supermanifolds \((M,\Omega )\) with \(\deg (\Omega )\)=2. Under this correspondence, we get that “BRST charges” on \(M\), i.e., cubic Hamiltonians \(\Theta\) satisfying the structure equation \(\{ \Theta ,\Theta \}=0\), induce Courant algebroid structures on \(E\).
The notion of a Courant algebroid was first introduced in [Z. Liu, A. Weinstein and P. Xu, J. Differ. Geom. 45, 547–574 (1997; Zbl 0885.58030)] and some of its properties were developed in [D. Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, PhD thesis, UC Berkeley, 1999, math.DG/9910078].
Finally, given a Courant algebroid with BRST charge \(\Theta\), the author introduces the differential complex \(({\mathcal A}=\bigoplus _{k\geq 0}{\mathcal A}^k ,D=\{ \Theta ,\cdot \})\), where \({\mathcal A}^k\) is the algebra of polynomial functions on the supermanifold \(M\). This complex is called the standard complex. In the particular case of \(E=TM_0\oplus T^\ast M_0\), the author introduces certain differential subcomplexes called higher de Rham complexes, proving the acyclicity of these complexes.
For the entire collection see [Zbl 1007.53002].


53D05 Symplectic manifolds (general theory)
81T70 Quantization in field theory; cohomological methods
58A50 Supermanifolds and graded manifolds
81T45 Topological field theories in quantum mechanics
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