Split Real Form Research Articles

Finiteness of 3D higher spin gravity Landscape

Abstract We give Swampland constraints on the three dimensional Landscape of Anti-de Sitter higher spin gravity in the Chern-Simons formulation with connection valued in various split real forms of Lie algebras. We derive the finiteness conjecture by computing the upper bound on the rank of possible gauge groups then we refine it using the AdS distance conjecture. We discuss the implications of this Swampland constraint on the spectrum of higher spin gravity theories and we compare it with the gravitational exclusion principle required from BTZ black hole consideration to excerpt a constraint on the Chern-Simons level k.

EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS AND VIA CONTACTIFICATIONS

Abstract In Cartan’s PhD thesis, there is a formula defining a certain rank 8 vector distribution in dimension 15, whose algebra of authomorphism is the split real form of the simple exceptional complex Lie algebra $\mathfrak {f}_4$ . Cartan’s formula is written in the standard Cartesian coordinates in $\mathbb {R}^{15}$ . In the present paper, we explain how to find analogous formulae for the flat models of any bracket generating distribution $\mathcal D$ whose symbol algebra $\mathfrak {n}({\mathcal D})$ is constant and 2-step graded, $\mathfrak {n}({\mathcal D})=\mathfrak {n}_{-2}\oplus \mathfrak {n}_{-1}$ . The formula is given in terms of a solution to a certain system of linear algebraic equations determined by two representations $(\rho ,\mathfrak {n}_{-1})$ and $(\tau ,\mathfrak {n}_{-2})$ of a Lie algebra $\mathfrak {n}_{00}$ contained in the $0$ th order Tanaka prolongation $\mathfrak {n}_0$ of $\mathfrak {n}({\mathcal D})$ . Numerous examples are provided, with particular emphasis put on the distributions with symmetries being real forms of simple exceptional Lie algebras $\mathfrak {f}_4$ and $\mathfrak {e}_6$ .

Normalizers of maximal tori and real forms of Lie groups

Given a complex connected reductive Lie group G with a maximal torus Hsubset G, Tits defined an extension W_G^{mathrm{T}} of the corresponding Weyl group W_G. The extended group is supplied with an embedding into the normalizer N_G(H) such that W_G^{mathrm{T}} together with H generate N_G(H). In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form G(mathbb {R})subset G, which leads to a simple topological description of W^{mathrm{T}}_G. We also consider a variation of the Tits construction associated with compact real form U of G. In this case we define an extension W_G^U of the Weyl group W_G, naturally embedded into the group extension widetilde{U}:=U,{rtimes }, Gamma of the compact real form U by the Galois group Gamma ={mathrm{Gal}}(mathbb {C}/mathbb {R}). Generators of W^U_G are squared to identity as in the Weyl group W_G. However, the non-trivial action of Gamma by outer automorphisms requires W^U_G to be a non-trivial extension of W_G. This gives a specific presentation of the maximal torus normalizer of the group extension {widetilde{U}}. Finally, we describe explicitly the adjoint action of W_G^{mathrm{T}} and W^U_G on the Lie algebra of G.

Invariant Einstein metrics on real flag manifolds with two or three isotropy summands

We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.

On $$\mathrm {G}_2$$ and sub-Riemannian model spaces of step and rank three

We give the complete classification of all sub-Riemannian model spaces with both step and rank three. Model spaces in this context refer to spaces where any infinitesimal isometry between horizontal tangent spaces can be integrated to a full isometry. They will be divided into three families based on their nilpotentization. Each family will depend on a different number of parameters, making the result crucially different from the known case of step two model spaces. In particular, there are no nontrivial sub-Riemannian model spaces of step and rank three with free nilpotentization. We also realize both the compact real form $${\mathfrak {g}}_2^c$$ and the split real form $${\mathfrak {g}}_2^s$$ of the exceptional Lie algebra $${\mathfrak {g}}_2$$ as isometry algebras of different model spaces.

On certain classes of mathbf{Sp}(4,mathbb{R}) symmetric G_2 structures

We find two different families of mathbf{Sp}(4,mathbb{R}) symmetric G_2 structures in seven dimensions. These are G_2 structures with G_2 being the split real form of the simple exceptional complex Lie group G_2. The first family has tau _2equiv 0, while the second family has tau _1equiv tau _2equiv 0, where tau _1, tau _2 are the celebrated G_2-invariant parts of the intrinsic torsion of the G_2 structure. The families are different in the sense that the first one lives on a homogeneous space mathbf{Sp}(4,mathbb{R})/mathbf{SL}(2,mathbb{R})_l, and the second one lives on a homogeneous space mathbf{Sp}(4,mathbb{R})/mathbf{SL}(2,mathbb{R})_s. Here mathbf{SL}(2,mathbb{R})_l is an mathbf{SL}(2,mathbb{R}) corresponding to the mathfrak{sl}(2,mathbb{R}) related to the long roots in the root diagram of mathfrak{sp}(4,mathbb{R}), and mathbf{SL}(2,mathbb{R})_s is an mathbf{SL}(2,mathbb{R}) corresponding to the mathfrak{sl}(2,mathbb{R}) related to the short roots in the root diagram of mathfrak{sp}(4,mathbb{R}).

Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. This generalizes Simpson's results for GL(n,C) to arbitrary complex and real reductive Lie groups. Three interesting features are the relation between the parabolic degree and the Tits geometry of the boundary at infinity of the symmetric space, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant–Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.

Topological invariants of parabolic G-Higgs bundles

For a semisimple real Lie group $G$, we study topological properties of moduli spaces of polystable parabolic $G$-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichm{\"u}ller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic $G$-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit an alternative explanation of fundamental results on counting components in the absence of a parabolic structure.

Aerobatics of Flying Saucers

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer’s velocity. These restrictions define certain ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘$$G_2$$ mode’ manœuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manœuvre corresponds to the flat Legendrean contact structure, the landing manœuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1, 1), and the most complicated $$G_2$$ manœuvre corresponds to the contact Engel structure (Engel in C R Acad Sci 116:786–788, 1893; Mano et al. in The geometry of marked contact twisted cubic structures, 2018, arXiv:1809.06455) with split real form of the exceptional Lie group $$G_2$$ as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic $$G_2$$ geometries is used to construct a ‘$$G_2$$ joystick,’ consisting of two balls of radii in ratio $$1\!:\!3$$ that transforms the difficult $$G_2$$ manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting.

Multiplet Classification of Reducible Verma Modules over the G2 Algebra

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra G2(2) which is split real form of G2. We give the classification of reducible Verma modules G2. We give also the singular vectors between these modules, thus setting the stage for construction of the invariant differential operators over G2(2).

Exotic components of SO(p,q) surface group representations, and their Higgs bundle avatars

For semisimple Lie groups, moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. In many cases, natural topological invariants label connected components of the moduli spaces. Hitchin representations into split real forms, and maximal representations into Hermitian Lie groups, are the only previously know cases where natural invariants do not fully distinguish connected components. In this note we announce the existence of new such exotic components in the moduli spaces for the groups SO(p,q) with 2<p<q. These groups lie outside formerly know classes of groups associated with exotic components.

Invariant almost complex structures on real flag manifolds

In this work, we study the existence of invariant almost complex structures on real flag manifolds associated to split real forms of complex simple Lie algebras. We show that, contrary to the complex case where the invariant almost complex structures are well known, some real flag manifolds do not admit such structures. We check which invariant almost complex structures are integrable and prove that only some flag manifolds of the Lie algebra $$ C_{l}$$ admit complex structures.

Higgs bundles for real groups and the Hitchin–Kostant–Rallis section

We consider the moduli space of polystable L L -twisted G G -Higgs bundles over a compact Riemann surface X X , where G G is a real reductive Lie group and L L is a holomorphic line bundle over X X . Evaluating the Higgs field on a basis of the ring of polynomial invariants of the isotropy representation defines the Hitchin map. This is a map to an affine space whose dimension is determined by L L and the degrees of the polynomials in the basis. In this paper, we construct a section of this map and identify the connected components of the moduli space containing the image. This section factors through the moduli space for G s p l i t G_{\mathrm {split}} , a split real subgroup of G G . Our results generalize those by Hitchin, who considered the case when L L is the canonical line bundle of X X and G G is complex. In this case, the image of the section is related to the Hitchin–Teichmüller components of the moduli space of representations of the fundamental group of X X in G s p l i t G_{\mathrm {split}} , a split real form of G G . The construction involves the notion of a maximal split subgroup of a real reductive Lie group and builds on results by Kostant and Rallis.

Higgs bundles and exceptional isogenies

We explore relations between Higgs bundles that result from isogenies between low-dimensional Lie groups, with special attention to the spectral data for the Higgs bundles. We focus on isogenies onto $$\mathrm {SO}(4,\mathbb {C})$$ and $$\mathrm {SO}(6,\mathbb {C})$$ and their split real forms. Using fiber products of spectral curves, we obtain directly the desingularizations of the (necessarily singular) spectral curves associated with orthogonal Higgs bundles. In the case of $$\mathrm {SO}(6,\mathbb {C})$$ , our construction can be interpreted as a new description of Recillas’ trigonal construction.

Cellular homology of real maximal isotropic grassmannians

Abstract In this paper we obtain a formula for the boundary map of the cellular ℤ-homology of the real maximal isotropic Grassmannians of type B, C and D which are obtained as minimal flag manifolds of split real forms of semi-simple Lie algebras of the respective type. We use the model of shifted Young diagrams presented in [9] to describe the Schubert varieties and the cellular decompositions of such flag manifolds. The results are applied to determine the orientability of the real maximal isotropic Grassmannians.

Adjoint Orbits of sl ( 2 , ℝ ) $sl(2,\mathbb {R})$ on Real Simple Lie Algebras and Controllability

This paper shows that for any Lie group G whose Lie algebra L is the split real form of a complex simple Lie algebra, and for any arbitrary root α, there exists a Cartan decomposition of L, related to α, which characterizes some controllability properties by using the adjoint orbits of $\text {sl}(2, \mathbb {R})$ . For a class of invariant control systems evolving on G, it is proved that the necessary full rank condition for controllability is also sufficient.

The isotropy representation of a real flag manifold: Split real forms

We study the isotropy representation of real flag manifolds associated to simple Lie algebras that are split real forms of complex simple Lie algebras. For each Dynkin diagram the invariant irreducible subspaces for the compact part of the isotropy subgroup are described. Contrary to the complex flag manifolds the decomposition into irreducible components is not unique in general. In other words there are cases with infinitely many invariant subspaces.

Cyclic Higgs bundles and the affine Toda equations

We introduce a class of Higgs bundles called cyclic which lie in the Hitchin component of representations of a compact Riemann surface into the split real form of a simple Lie group. We then prove that such Higgs bundles correspond to a class of solutions to the affine Toda equations. This relationship is further explained in terms of lifts of harmonic maps.

A quaternionic construction of 𝐸₇

We give an explicit construction of the simply-connected compact real form of the Lie group of type E 7 E_7 , as a group of 28 × 28 28\times 28 matrices over quaternions, acting on a 28 28 -dimensional left quaternion vector space. This leads to a description of the simply-connected split real form, acting on a 56 56 -dimensional real vector space, and thence to the finite quasi-simple groups of type E 7 E_7 . The sign problems usually associated with constructing exceptional Lie groups are almost entirely absent from this approach.

Isotropic foliations of coadjoint orbits from the Iwasawa decomposition

Let G be a noncompact real semisimple Lie group. The set of regular coadjoint orbits of G can be partitioned according to a finite set of types. We show that on each regular orbit, the Iwasawa decomposition induces a left-invariant foliation which is isotropic with respect to the Kirillov symplectic form. Moreover, the leaves are affine subspaces of the dual of the Lie algebra, and the dimension of the leaves depends only on the type of the orbit. When G is a split real form, the foliations induced from the Iwasawa decomposition are actually Lagrangian fibrations with a global transverse Lagrangian section.