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Andrea Santi

Publications and source records attributed to Andrea Santi.

At least 19 recordsLinked to original sources
arXiv2026

Classification of nilpotent and semisimple fourvectors of a real eight-dimensional space

In 1981 Antonyan classified the orbits of SL$(8,\mathbb{C})$ on $\bigwedge^4 \mathbb{C}^8$. This is an example of a $θ$-group action as introduced and studied by Vinberg. The orbits of a $θ$-group are divided into three classes: nilpotent, semisimple and mixed. We consider the action of SL$(8,\mathbb{R})$ on $\bigwedge^4 \mathbb{R}^8$ and classify the nilpotent and semisimple orbits as well as the Cartan subspaces. The semisimple orbits are divided into 1441 parametrized classes. Due to this high number a classification of the mixed orbits does not seem feasible. Our methods are based on Galois cohomology.

math.RT↗
arXiv2026

Some rigidity results for supergravity backgrounds in 11 dimensions

This paper is a contribution to the supersymmetry gap problem for supergravity backgrounds $(M,g,F)$ in $11$ dimensions. We study restrictions on the curvature of $(M,g,F)$ and, using the bijective correspondence between the space of certain filtered deformations of Lie superalgebras and the space of highly supersymmetric supergravity backgrounds, we establish the following general rigidity result: if the $4$-form $F$ has rank $\operatorname{rk}(F)\leq 6$, Euclidean support, and the space $\mathfrak{k}_{\bar 1}$ of Killing spinors has dimension $\dim\mathfrak{k}_{\bar 1}> 26$ then $(M,g,F)$ is locally isometric to the maximally supersymmetric Minkowski spacetime or Freund Rubin background $\mathrm{AdS}_7\times\mathrm{S}^4$. The same rigidity result but with finer estimates on $\dim\mathfrak{k}_{\bar 1}$ is provided for certain types of $\mathfrak k_{\bar 1}$ and specific orbits of the $4$-form under the action of the Lorentz group.

hep-th↗
arXiv2026

Hyperplane arrangements and Vinberg's $θ$-groups

Let $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$ be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of $\mathfrak{g}$ to a Cartan subspace $\mathfrak{c} \subset \mathfrak{g}_1$ coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of $\mathfrak{g} = \bigoplus_{i \in \mathbb{Z} /m \mathbb{Z}} \mathfrak{g}_i$.

math.RT↗
arXiv2025

On 3-nondegenerate CR manifolds in dimension 7 (II): the intransitive case

We investigate 3-nondegenerate CR structures in the lowest possible dimension 7 and show that 8 is the maximal dimension for the Lie algebra of symmetries of such structures. The next possible symmetry dimension is 6, and for the automorphism groups the dimension 7 is also realizable. This part (II) is devoted to the case where the symmetry algebra acts intransitively. We use various methods to bound its dimension and demonstrate the existence of infinitely many non-equivalent submaximally symmetric models. Summarizing, we get a stronger form of Beloshapka's conjecture on the symmetry dimension of hypersurfaces in $\mathbb{C}^4$.

math.CV↗
arXiv2024

On 3-nondegenerate CR manifolds in dimension 7 (I): the transitive case

We investigate 3-nondegenerate CR structures in the lowest possible dimension 7, and one of our goals is to prove Beloshapka's conjecture on the symmetry dimension bound for hypersurfaces in $\mathbb{C}^4$. We claim that 8 is the maximal symmetry dimension of 3-nondegenerate CR structures in dimension 7, which is achieved on the homogeneous model. This part (I) is devoted to the homogeneous case: we prove that the model is locally the only homogeneous 3-nondegenerate CR structure in dimension 7.

math.CV↗
arXiv2024

Exceptionally simple super-PDE for $F(4)$

For the largest exceptional simple Lie superalgebra $F(4)$, having dimension $(24|16)$, we provide two explicit geometric realizations as supersymmetries, namely as the symmetry superalgebra of super-PDE systems of second and third order respectively.

math.DG↗
arXiv2021

On Jordan classes for Vinberg's theta-groups

Popov has recently introduced an analogue of Jordan classes (packets, or decomposition classes) for the action of a theta-group (G_0,V), showing that they are finitely-many, locally-closed, irreducible unions of G_0-orbits of constant dimension partitioning V. We carry out a local study of their closures showing that Jordan classes are smooth and that their closure is a union of Jordan classes. We parametrize Jordan classes and G_0-orbits in a given class in terms of the action of subgroups of Vinberg's little Weyl group, and include several examples and counterexamples underlying the differences with the symmetric case and the critical issues arising in the theta-situation.

math.RT↗
arXiv2020

G(3)-supergeometry and a supersymmetric extension of the Hilbert-Cartan equation

We realize the simple Lie superalgebra G(3) as supersymmetry of various geometric structures, most importantly super-versions of the Hilbert-Cartan equation (SHC) and Cartan's involutive PDE system that exhibit G(2) symmetry. We provide the symmetries explicitly and compute, via the first Spencer cohomology groups, the Tanaka-Weisfeiler prolongation of the negatively graded Lie superalgebras associated with two particular choices of parabolics. We discuss non-holonomic superdistributions with growth vector (2|4,1|2,2|0) deforming the flat model SHC, and prove that the second Spencer cohomology group gives a binary quadratic form, thereby providing a "square-root" of Cartan's classical binary quartic invariant for generic rank 2 distributions in a 5-dimensional space. Finally, we obtain super-extensions of Cartan's classical submaximally symmetric models, compute their symmetries and observe a supersymmetry dimension gap phenomenon.

math.DG↗
arXiv2020

Remarks on highly supersymmetric backgrounds of 11-dimensional supergravity

This note focuses on some properties and uses of filtered deformations in the context of D=11 supergravity. We define the concept of abstract symbol and give a strong version of the Reconstruction Theorem, namely a bijective correspondence from the space of highly supersymmetric supergravity backgrounds to the space of abstract symbols. We propose a general strategy to construct highly supersymmetric supergravity backgrounds and present an example in detail, which includes the computation of the ideal generated by the Killing spinors of two known pp-wave backgrounds with N=24 supersymmetry. Finally, we give an alternative proof, based on the isotropy algebra of a supergravity background, of a classical supersymmetry gap result of Gran, Gutowski, Papadopoulos and Roest.

hep-th↗
arXiv2020

On the algebraic structure of Killing superalgebras

We study the algebraic structure of the Killing superalgebra of a supersymmetric background of $11$-dimensional supergravity and show that it is isomorphic to a filtered deformation of a $\mathbb Z$-graded subalgebra of the Poincaré superalgebra. We are able to map the classification problem for highly supersymmetric backgrounds (i.e., those which preserve more than half the supersymmetry) to the classification problem of a certain class of filtered deformations of graded subalgebras of the Poincaré superalgebra. We show that one can reconstruct a highly supersymmetric background from its Killing superalgebra; in so doing, we relate the bosonic field equations of $11$-dimensional supergravity to the Jacobi identity of the Killing superalgebra and show in this way that preserving more than half the supersymmetry implies the bosonic field equations.

hep-th↗
arXiv2018

Killing superalgebras for Lorentzian six-manifolds

We calculate the Spencer cohomology of the $(1,0)$ Poincaré superalgebras in six dimensions: with and without R-symmetry. As the cases of four and eleven dimensions taught us, we may read off from this calculation a Killing spinor equation which allows the determination of which geometries admit rigidly supersymmetric theories in this dimension. We prove that the resulting Killing spinors generate a Lie superalgebra and determine the geometries admitting the maximal number of such Killing spinors. They are divided in two branches. One branch consists of the lorentzian Lie groups with bi-invariant metrics and, as a special case, it includes the lorentzian Lie groups with a self-dual Cartan three-form which define the maximally supersymmetric backgrounds of $(1,0)$ Poincaré supergravity in six dimensions. The notion of Killing spinor on the other branch does not depend on the choice of a three-form but rather on a one-form valued in the R-symmetry algebra. In this case, we obtain three different (up to local isometry) maximally supersymmetric backgrounds, which are distinguished by the causal type of the one-form.

hep-th↗
arXiv2017

Homogeneous models for Levi-degenerate CR manifolds

We extend the notion of a fundamental negatively $\mathbb Z$-graded Lie algebra $\mathfrak{m}_x=\bigoplus_{p\leq -1}\mathfrak{m}_x^p$ associated to any point of a Levi nondegenerate CR manifold to the class of $k$-nondegenerate CR manifolds $(M,\mathcal D,\mathcal J)$ for all $k\geq 2$ and call this invariant the core at $x\in M$. It consists of a $\mathbb Z$-graded vector space $\mathfrak{m}_x=\bigoplus_{p\leq k-2}\mathfrak{m}_x^p$ of height $k-2$ endowed with the natural algebraic structure induced by the Tanaka and Freeman sequences of $(M,\mathcal D,\mathcal J)$ and the Levi forms of higher order. In the case of CR manifolds of hypersurface type we propose a definition of a homogeneous model of type $\mathfrak m$, that is, a homogeneous $k$-nondegenerate CR manifold $M=G/G_o$ with core $\mathfrak m$ associated with an appropriate $\mathbb Z$-graded Lie algebra $Lie(G)=\mathfrak g=\bigoplus\mathfrak g^p$ and subalgebra $Lie(G_o)=\mathfrak g_o=\bigoplus\mathfrak g_o^p$ of the nonnegative part $\bigoplus_{p\geq 0}\mathfrak g^p$. It generalizes the classical notion of Tanaka of homogeneous model for Levi nondegenerate CR manifolds and the tube over the future light cone, the unique (up to local CR diffeomorphisms) maximally homogeneous $5$-dimensional $2$-nondegenerate CR manifold. We investigate the basic properties of cores and models and study the $7$-dimensional CR manifolds of hypersurface type from this perspective. We first classify cores of $7$-dimensional $2$-nondegenerate CR manifolds up to isomorphism and then construct homogeneous models for seven of these classes. We finally show that there exists a unique core and homogeneous model in the $3$-nondegenerate class.

math.DG↗
arXiv2016

Almost CR quaternionic manifolds and their immersibility in HP^n

We apply the general theory of codimension one integrability conditions for $G$-structures developed in arXiv:1306.6817v3 [math.DG] to the case of quaternionic CR geometry. We obtain necessary and sufficient conditions for an almost CR quaternionic manifold to admit local immersions as an hypersurface of the quaternionic projective space. We construct a deformation of the standard quaternionic contact structure on the quaternionic Heisenberg group which does not admit local immersions in any quaternionic manifold.

math.DG↗
arXiv2016

Killing superalgebras for Lorentzian four-manifolds

We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of $\mathbb{Z}$-graded subalgebras with maximum odd dimension of the $N{=}1$ Poincaré superalgebra in four dimensions. Part of this calculation involves computing a Spencer cohomology group which, by analogy with a similar result in eleven dimensions, prescribes a notion of Killing spinor, which we identify with the defining condition for bosonic supersymmetric backgrounds of minimal off-shell supergravity in four dimensions. We prove that such Killing spinors always generate a Lie superalgebra, and that this Lie superalgebra is a filtered deformation of a subalgebra of the $N{=}1$ Poincaré superalgebra in four dimensions. Demanding the flatness of the connection defining the Killing spinors, we obtain equations satisfied by the maximally supersymmetric backgrounds. We solve these equations, arriving at the classification of maximally supersymmetric backgrounds whose associated Killing superalgebras are precisely the filtered deformations we classify in this paper.

hep-th↗
arXiv2016

Spencer cohomology and eleven-dimensional supergravity

We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the $\mathbb Z$-graded subalgebras $\mathfrak{h}=\mathfrak{h}_{-2}\oplus\mathfrak{h}_{-1}\oplus\mathfrak{h}_{0}$ of the Poincaré superalgebra $\mathfrak{g}=\mathfrak{g}_{-2}\oplus\mathfrak{g}_{-1}\oplus\mathfrak{g}_{0}=V\oplus S\oplus \mathfrak{so}(V)$ which differ only in zero degree, that is $\mathfrak{h}_0\subset\mathfrak{g}_0$ and $\mathfrak{h}_j=\mathfrak{g}_j$ for $j<0$. Aside from the Poincaré superalgebra itself and its $\mathbb Z$-graded subalgebras, there are only three other Lie superalgebras, which are the symmetry superalgebras of the non-flat maximally supersymmetric backgrounds. In passing we identify the gravitino variation with (a component of) a Spencer cocycle.

hep-th↗