arXiv subjects

A. Santi

Publications and source records attributed to A. Santi.

4 recordsLinked to original sources

arXiv2024

Fermionic Spencer Cohomologies of D=11 Supergravity

We combine the theory of Cartan-Tanaka prolongations with the Molien-Weyl integral formula and Hilbert-Poincar\'e series to compute the Spencer cohomology groups of the $D=11$ Poincar\'e superalgebra $\mathfrak p$, relevant for superspace formulations of $11$-dimensional supergravity in terms of nonholonomic superstructures. This includes novel fermionic Spencer groups, providing with new cohomology classes of $\mathbb Z$-grading $1$ and form number $2$. Using the Hilbert-Poincar\'e series and the Euler characteristic, we also explore Spencer cohomology contributions in higher form numbers. We then propose a new general definition of filtered deformations of graded Lie superalgebras along first-order fermionic directions and investigate such deformations of $\mathfrak p$ that are maximally supersymmetric. In particular, we establish a no-go type theorem for maximally supersymmetric filtered subdeformations of $\mathfrak p$ along timelike (i.e., generic) first-order fermionic directions.

hep-th↗

arXiv2019

Homogeneous symplectic 4-manifolds and finite dimensional Lie algebras of symplectic vector fields on the symplectic 4-space

We classify the finite type (in the sense of E. Cartan theory of prolongations) subalgebras $\mathfrak{h}\subset\mathfrak{sp}(V)$, where $V$ is the symplectic 4-dimensional space, and show that they satisfy $\mathfrak{h}^{(k)}=0$ for all $k>0$. Using this result, we reduce the problem of classification of graded transitive finite-dimensional Lie algebras $\mathfrak{g}$ of symplectic vector fields on $V$ to the description of graded transitive finite-dimensional subalgebras of the full prolongations $\mathfrak{p}_1^{(\infty)}$ and $\mathfrak{p}_2^{(\infty)}$, where $\mathfrak{p}_1$ and $\mathfrak{p}_2$ are the maximal parabolic subalgebras of $\mathfrak{sp}(V)$. We then classify all such $\mathfrak{g}\subset\mathfrak{p}_i^{(\infty)}$, $i=1,2$, under some assumptions and describe the associated homogeneous symplectic 4-manifolds $(M=G/K,ω)$. We prove that any reductive homogeneous symplectic manifold (of any dimension) admits an invariant torsion free symplectic connection, i.e., it is a homogeneous Fedosov manifold, and give conditions for uniqueness of the Fedosov structure. Finally, we show that any nilpotent symplectic Lie group (of any dimension) admits a natural invariant Fedosov structure which is Ricci-flat.

math.DG↗

arXiv2016

Homogeneous irreducible supermanifolds and graded Lie superalgebras

A depth one grading $\mathfrak{g}= \mathfrak{g}^{-1}\oplus \mathfrak{g}^0 \oplus \mathfrak{g}^1 \oplus \cdots \oplus \mathfrak{g}^{\ell}$ of a finite dimensional Lie superalgebra $\mathfrak{g}$ is called nonlinear irreducible if the isotropy representation $\mathrm{ad}_{\mathfrak{g}^0}|_{\mathfrak{g}^{-1}}$ is irreducible and $\mathfrak{g}^1 \neq (0)$. An example is the full prolongation of an irreducible linear Lie superalgebra $\mathfrak{g}^0 \subset \mathfrak{gl}(\mathfrak{g}^{-1})$ of finite type with non-trivial first prolongation. We prove that a complex Lie superalgebra $\mathfrak{g}$ which admits a depth one transitive nonlinear irreducible grading is a semisimple Lie superalgebra with the socle $\mathfrak{s}\otimes Λ(\mathbb{C}^n)$, where $\mathfrak{s}$ is a simple Lie superalgebra, and we describe such gradings. The graded Lie superalgebra $\mathfrak{g}$ defines an isotropy irreducible homogeneous supermanifold $M=G/G_0$ where $G$, $G_0$ are Lie supergroups respectively associated with the Lie superalgebras $\mathfrak{g}$ and $\mathfrak{g}_0 := \bigoplus_{p\geq 0} \mathfrak{g}^p$.

math.RT↗

arXiv2012

Super-Poincare' algebras, space-times and supergravities (I)

A new formulation of theories of supergravity as theories satisfying a generalized Principle of General Covariance is given. It is a generalization of the superspace formulation of simple 4D-supergravity of Wess and Zumino and it is designed to obtain geometric descriptions for the supergravities that correspond to the super Poincare' algebras of Alekseevsky and Cortes' classification.

math-ph↗