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Modular Classes of Q-Manifolds, Part II: Riemannian Structures & Odd Killing Vectors Fields Bruce, Andrew in Archivum Mathematicum (2020) We define and make an initial study of (even) Riemannian supermanifolds equipped with a homological vector field that is also a Killing vector field. We refer to such supermanifolds as Riemannian Q ... [more ▼] We define and make an initial study of (even) Riemannian supermanifolds equipped with a homological vector field that is also a Killing vector field. We refer to such supermanifolds as Riemannian Q-manifolds. We show that such Q-manifolds are unimodular, i.e., come equipped with a Q-invariant Berezin volume. [less ▲] Detailed reference viewed: 807 (0 UL)The graded differential geometry of mixed symmetry tensors Bruce, Andrew ; Ibarguengoytia, Eduardo in Archivum Mathematicum (2019), 55(2), 123-137 We show how the theory of $\mathbb{Z}_2^n$-manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual ... [more ▼] We show how the theory of $\mathbb{Z}_2^n$-manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed. [less ▲] Detailed reference viewed: 93 (8 UL)Modular classes of Q-manifolds: a review and some applications Bruce, Andrew in Archivum Mathematicum (2017) A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold – which is viewed as the obstruction to the existence of a Q ... [more ▼] A Q-manifold is a supermanifold equipped with an odd vector field that squares to zero. The notion of the modular class of a Q-manifold – which is viewed as the obstruction to the existence of a Q-invariant Berezin volume – is not well know. We review the basic ideas and then apply this technology to various examples, including $L_{\infty}$-algebroids and higher Poisson manifolds. [less ▲] Detailed reference viewed: 87 (4 UL)Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications Qiu, Jian ; in Archivum Mathematicum (2011), 47 These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded ... [more ▼] These notes are intended to provide a self-contained introduction to the basic ideas of finite dimensional Batalin-Vilkovisky (BV) formalism and its applications. A brief exposition of super- and graded geometries is also given. The BV-formalism is introduced through an odd Fourier transform and the algebraic aspects of integration theory are stressed. As a main application we consider the perturbation theory for certain finite dimensional integrals within BV-formalism. As an illustration we present a proof of the isomorphism between the graph complex and the Chevalley-Eilenberg complex of formal Hamiltonian vectors fields. We briefly discuss how these ideas can be extended to the infinite dimensional setting. These notes should be accessible to both physicists and mathematicians. [less ▲] Detailed reference viewed: 40 (0 UL) |
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