[Paper Review] Symmetric Spaces with Conformal Symmetry

André Wehner|ArXiv.org|Jul 1, 2001

Advanced Algebra and Geometry4 references4 citations

TL;DR

This paper introduces a new symmetric space, homogeneous biconformal space $B(n)$, constructed as a coset $C(n)/W(n)$ of the conformal group $C(n)$ modulo a subgroup $W(n)$, which exhibits intrinsic conformal symmetry, a group-invariant metric, a canonical torsion-free connection, and a natural symplectic and Kähler structure. The key contribution is the construction of a model geometry for a generalized conformal gauge theory with full scale and Lorentz invariance, realized through a nonlinear action of the conformal group that treats translations and special conformal transformations symmetrically.

ABSTRACT

We consider an involutive automorphism of the conformal algebra and the resulting symmetric space. We display a new action of the conformal group which gives rise to this space. The space has an intrinsic symplectic structure, a group-invariant metric and connection, and serves as the model space for a new conformal gauge theory.

Motivation & Objective

• To construct a new symmetric space based on the conformal group that supports intrinsic conformal symmetry and rich geometric structures.
• To develop a nonlinear action of the conformal group on this space that treats translations and special conformal transformations symmetrically.
• To establish a geometric framework—homogeneous biconformal space $B(n)$—that supports a conformally invariant connection, a symplectic form, and a Kähler structure.
• To provide a model geometry for biconformal space, a generalized curved conformal geometry, which allows for scale-invariant Lagrangians and reproduces general relativity on subspaces.
• To demonstrate that $B(n)$ admits a canonical connection with covariantly constant curvature and vanishing torsion, compatible with a non-degenerate metric.

Proposed method

• The construction uses an involutive automorphism of the conformal algebra to define a symmetric space structure on the coset $C(n)/W(n)$, where $W(n)$ is the isotropy subgroup of the origin under the new action.
• The tangent space at the origin is spanned by the translation generators $P_{ u}$ and special conformal generators $K^{ u}$, which are treated symmetrically in the nonlinear group action.
• The canonical connection on the principal bundle $C(n) \to B(n)$ is derived from the Maurer-Cartan form $\mathbf{\omega}$, with the $\mathfrak{h}$-component $\mathbf{\tilde{\omega}} = M_b^a \mathbf{\omega}_a^b + D \mathbf{\omega}^0$ defining the invariant connection.
• The symplectic structure $\mathbf{\Omega} = \mathbf{\omega}^a \mathbf{\omega}_a$ is shown to be non-degenerate and closed via the Maurer-Cartan equation $\mathbf{d}\omega^0 = -\mathbf{\omega}^a \mathbf{\omega}_a$.
• The Kähler structure is established by defining an almost complex structure $J$ compatible with $\mathbf{\Omega}$, satisfying $\mathbf{\Omega}(Ju,Jv) = \mathbf{\Omega}(u,v)$, making $B(n)$ a complex manifold with Kähler metric $g(u,v) = \mathbf{\Omega}(Ju,v)$.
• The geometry is compared to Minkowski, Weyl, and conformal spaces, highlighting that only $B(n)$ supports a symplectic and Kähler structure intrinsically.

Experimental results

Research questions

• RQ1 Can a symmetric space be constructed from the conformal group that supports both conformal symmetry and a natural symplectic structure?
• RQ2 How can the conformal group be nonlinearly realized on a space where translations and special conformal transformations are treated symmetrically?
• RQ3 Does the resulting space admit a canonical, torsion-free, group-invariant connection with covariantly constant curvature?
• RQ4 What geometric structures (symplectic, Kähler, metric) emerge intrinsically in this new symmetric space?
• RQ5 Can this space serve as a model geometry for a generalized conformal gauge theory with scale invariance and consistent dynamics?

Key findings

• The homogeneous biconformal space $B(n) = C(n)/W(n)$ is a $2n$-dimensional symmetric space with intrinsic conformal symmetry and a group-invariant indefinite Riemannian metric.
• The space admits a canonical, torsion-free, and covariantly constant curvature connection derived from the $\mathfrak{h}$-component of the Maurer-Cartan form.
• A non-degenerate, closed $2$-form $\mathbf{\Omega} = \mathbf{\omega}^a \mathbf{\omega}_a$ endows $B(n)$ with a natural symplectic structure.
• The symplectic structure is compatible with an integrable almost complex structure $J$, yielding a Kähler geometry on the $2n$-dimensional real space, equivalent to an $n$-dimensional complex Kähler manifold.
• Unlike Minkowski, Weyl, or standard conformal spaces, $B(n)$ uniquely supports all three structures—symplectic, Kähler, and conformally invariant metric—simultaneously and intrinsically.
• The construction provides a model geometry for biconformal space, enabling linear scale-invariant Lagrangians in any dimension and reproducing general relativity on specific subspaces without unphysical size changes.