arXiv subjects
Publications and source records attributed to Andrea Santi.
We extend the link between Einstein Sasakian manifolds and Killing spinors to a class of $η$-Einstein Sasakian manifolds, both in Riemannian and Lorentzian settings, characterising them in terms of generalised Killing spinors. We propose a definition of supersymmetric M-theory backgrounds on such a geometry and find a new class of such backgrounds, extending previous work of Haupt, Lukas and Stelle.
Given an $\widetilde n$-dimensional manifold $\widetilde M$ equipped with a $\widetilde G$-structure $\widetildeπ:\widetilde P\rightarrow \widetilde M$, there is a naturally induced $G$-structure $π: P\rightarrow M$ on any submanifold $M\subset\widetilde M$ that satisfies appropriate regularity conditions. We study generalized integrability problems for a given $G$-structure $π: P\rightarrow M$, namely the questions of whether it is locally equivalent to induced $G$-structures on regular submanifolds of homogeneous $\widetilde G$-structures $\widetildeπ:\widetilde P\to \widetilde{H}/\widetilde{K}$. If $\widetildeπ:\widetilde P\to \widetilde{H}/\widetilde{K}$ is flat $k$-reductive we introduce a sequence of generalized curvatures taking values in appropriate cohomology groups and prove that the vanishing of these curvatures are necessary and sufficient conditions for the solution of the corresponding generalized integrability problems.
Let $V$ be a complex vector space with a non-degenerate symmetric bilinear form and $\mathbb S$ an irreducible module over the Clifford algebra $Cl(V)$ determined by this form. A supertranslation algebra is a $\mathbb Z$-graded Lie superalgebra $\mathfrak m=\mathfrak{m}_{-2}\oplus\mathfrak{m}_{-1}$, where $\mathfrak{m}_{-2}=V$ and $\mathfrak{m}_{-1}=\mathbb S\oplus\cdots\oplus\mathbb{S}$ is the direct sum of an arbitrary number $N\geq 1$ of copies of $\mathbb S$, whose bracket $[\cdot,\cdot]|_{\mathfrak{m}_{-1}\otimes \mathfrak{m}_{-1}}:\mathfrak{m}_{-1}\otimes\mathfrak{m}_{-1}\rightarrow\mathfrak{m}_{-2}$ is symmetric, $\mathfrak{so}(V)$-equivariant and non-degenerate (that is the condition "$s\in\mathfrak{m}_{-1}, [s,\mathfrak{m}_{-1}]=0$" implies $s=0$). We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finite-dimensional for $\dim V\geq3$ and classify them in terms of super-Poincaré algebras and appropriate $\mathbb Z$-gradings of simple Lie superalgebras.
Let (V,(.,.)) be a pseudo-Euclidean vector space and S an irreducible Cl(V)-module. An extended translation algebra is a graded Lie algebra m = m_{-2}+m_{-1} = V+S with bracket given by ([s,t],v) = b(v.s,t) for some nondegenerate so(V)-invariant reflexive bilinear form b on S. An extended Poincaré structure on a manifold M is a regular distribution D of depth 2 whose Levi form L_x: D_x\wedge D_x\rightarrow T_xM/D_x at any point x\in M is identifiable with the bracket [.,.]: S\wedge S\rightarrow V of a fixed extended translation algebra m. The classification of the standard maximally homogeneous manifolds with an extended Poincaré structure is given, in terms of Tanaka prolongations of extended translation algebras and of appropriate gradations of real simple Lie algebras.
The presentation of supergravity theories of our previous paper "Super-Poincare' algebras, space-times and supergravities (I)" is re-formulated in the language of Berezin-Leites-Kostant theory of supermanifolds. It is also shown that the equations of Cremmer, Julia and Scherk's theory of 11D-supergravity are equivalent to manifestly covariant equations on a supermanifold.
Let M_0=G_0/H be a (pseudo)-Riemannian homogeneous spin manifold, with reductive decomposition g_0=h+m and let S(M_0) be the spin bundle defined by the spin representation Ad:H->\GL_R(S) of the stabilizer H. This article studies the superizations of M_0, i.e. its extensions to a homogeneous supermanifold M=G/H whose sheaf of superfunctions is isomorphic to Lambda(S^*(M_0)). Here G is the Lie supergroup associated with a certain extension of the Lie algebra of symmetry g_0 to an algebra of supersymmetry g=g_0+g_1=g_0+S via the Kostant-Koszul construction. Each algebra of supersymmetry naturally determines a flat connection nabla^{S} in the spin bundle S(M_0). Killing vectors together with generalized Killing spinors (i.e. nabla^{S}-parallel spinors) are interpreted as the values of appropriate geometric symmetries of M, namely even and odd Killing fields. An explicit formula for the Killing representation of the algebra of supersymmetry is obtained, generalizing some results of Koszul. The generalized spin connection nabla^{S} defines a superconnection on M, via the super-version of a theorem of Wang.
Let M_0=G_0/H be a (n+1)-dimensional Cahen-Wallach Lorentzian symmetric space associated with a symmetric decomposition g_0=h+m and let S(M_0) be the spin bundle defined by the spin representation r:H->GL_R(S) of the stabilizer H. This article studies the superizations of M_0, i.e. its extensions to a homogeneous supermanifold M=G/H whose sheaf of superfunctions is isomorphic to Lambda(S^*(M_0)). Here G is a Lie supergroup which is the superization of the Lie group G_0 associated with a certain extension of the Lie algebra g_0 to a Lie superalgebra g=g_0+g_1=g_0+S, via the Kostant construction. The construction of the superization g consists of two steps: extending the spin representation r:h->gl_R(S) to a representation r:g_0->gl_R(S) and constructing appropriate r(g_0)-equivariant bilinear maps on S. Since the Heisenberg algebra heis is a codimension one ideal of the Cahen-Wallach Lie algebra g_0, first we describe spin representations of gheis and then determine their extensions to g_0. There are two large classes of spin representations of gheis and g_0: the zero charge and the non-zero charge ones. The description strongly depends on the dimension n+1 (mod 8). Some general results about superizations g=g_0+g_1 are stated and examples are constructed.