On the structure of graded symplectic supermanifolds and Courant algebroids

By Dmitry Roytenberg. 2002

387 citations recorded by the source corpus.

This paper is devoted to a study of geometric structures expressible in terms of graded symplectic supermanifolds. We extend the classical BRST formalism to arbitrary pseudo-Euclidean vector bundles (E\to M_{0}) by canonically associating to such a bundle a graded symplectic supermanifold ((M,\Omega)), with (\textrm{deg}(\Omega)=2). Conversely, every such manifold arises in this way. We describe the algebra of functions on (M) in terms of (E) and show that ``BRST charges'' on (M) correspond to Courant algebroid structures on (E), thereby constructing the standard complex for the latter as a generalization of the classical BRST complex. As an application of these ideas, we prove the acyclicity of ``higher de Rham complexes'', a generalization of a classic result of Fr\"{o}hlicher-Nijenhuis, and derive several easy but useful corollaries.

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/0b00468af5c63b83cfc9c0780e84f8825294163b

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Citations in the corpus

Supergroupoids, double structures, and equivariant cohomology (2006, methodology): “[33], [63], [68], where the notation E[−j] has been used for the supermanifold which we denote by [j]E.”
Topological strings on noncommutative manifolds (2003, neutral): “The Courant bracket is a bilinear operation on TX ⊕ TX defined as follows [15]:”
Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, background): “In [49], Roytenberg showed that the non-negatively graded symplectic manifolds of degree 2 are the pseudo-euclidian vector bundles, and that the Courant algebroids are defined by an additional structure, that of a homological vector field, associated to a cubic hamiltonian Θ of Poisson square 0, pre”
Membrane sigma-models and quantization of non-geometric flux backgrounds (2012, methodology): “The corresponding AKSZ sigma-model has target space comprising a symplectic Lie 2-algebroid with a “degree 2 symplectic form”, which is the same thing as a Courant algebroid E over a manifold M [71].”
Topological sigma-models with H-flux and twisted generalized complex manifolds (2004, background): “metric of signature (n, n), which we call q. For a detailed discussion of the Dorfman and Courant brackets and their geometric meaning, see Ref. [ 16 ].”
Maximally homogeneous para-CR manifolds of semisimple type (2006, neutral): “[13] or also [32]): for any graded manifold Z and a morphism ψ : Z ×M1 → M2 there exists a morphism ψ̃ : Z → Map(M1,M2) such that ψ = ẽv ◦ (ψ̃× Id), where the evaluation map ẽv is (formally) defined in the obvious way:”
Higher derived brackets and homotopy algebras (2003, background): “Another interesting direction of study should be derived brackets and homotopy algebras arising from graded manifolds [35] (see also[29]).”
Graded bundles and homogeneity structures (2011, methodology): “This notion is equivalent to the notion of an N-manifold in the terminology of Ševera and Roytenberg [11, 9].”
L∞‐Algebras of Classical Field Theories and the Batalin–Vilkovisky Formalism (2018, background): “ipped with a graded symplectic structure !of degree kfor which Qis symplectic, that is, L Q!0. It is rather straightforward to see that !must be exact for k˘0 and QHamiltonian for k˘ 1, respectively [38]. Indeed, k! L ! d! dp !q dp !q øæ ! dp1 k !q: (2.21) Likewise, to verify that Qis Hamiltonia”
On weak Lie 2‐algebras (2007, neutral): “Furthermore, writing Sx,y = ([x,y],−〈x,y〉) (8) Jx,y,z = ([x, [y,z]],−〈x,y,z〉) (9)”
AKSZ–BV Formalism and Courant Algebroid-Induced Topological Field Theories (2006, positive): “We have shown [9] that such structures are in canonical 1-1 correspondence with what is known as Courant algebroids.”
Q-algebroids and their cohomology (2007, background): “For example, Roytenberg [32] generalized the homological approach to Lie bialgebras [14, 17] to give a homological compatibility condition for Lie bialgebroids.”