[Paper Review] Parallel tractor extension and ambient metrics of holonomy split G_2

C. Robin Graham, Travis Willse|arXiv (Cornell University)|Sep 15, 2011

Geometric Analysis and Curvature Flows28 references4 citations

TL;DR

This paper establishes that the ambient metric construction applied to real-analytic generic 2-plane fields on 5-manifolds yields a Ricci-flat metric of signature (3,4) whose holonomy is contained in the split real form $G_2$. It proves that holonomy equals $G_2$ if two pointwise nondegeneracy conditions—on the Weyl-Cotton map $L_x$ and the 4-form invariant $A$—are satisfied, generalizing earlier explicit constructions to a broad class of geometric structures.

ABSTRACT

The holonomy of the ambient metrics of Nurowski's conformal structures associated to generic real-analytic 2-plane fields on 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G_2 of the exceptional Lie group, and is equal to G_2 for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinite-dimensional family of metrics of holonomy equal to split G_2. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G_2 is established by proving an ambient extension theorem for parallel tractors for conformal structures in general signature and dimension, which is expected to be of independent interest. Parallel extension beyond the critical order in even dimensions is considered in certain cases.

Motivation & Objective

• To extend the holonomy containment $\operatorname{Hol}(\widetilde{g}) \subset G_2$ from explicit 8-parameter families to all real-analytic generic 2-plane fields on 5-manifolds.
• To identify sufficient pointwise conditions under which the ambient metric $\widetilde{g}$ has full $G_2$ holonomy.
• To characterize the geometric conditions—specifically the rank of the Weyl-Cotton map $L_x$ and 3-nondegeneracy of the Cartan invariant $A$—that ensure $\operatorname{Hol}(\widetilde{g}) = G_2$.
• To show that these conditions are generic in the space of local 2-plane fields, confirming that $G_2$ holonomy is prevalent in the real-analytic category.

Proposed method

• Utilizes the ambient metric construction of Fefferman-Graham to lift a real-analytic conformal structure of signature (2,3) associated to a generic 2-plane field $\mathcal{D}$ to a Ricci-flat metric $\widetilde{g}$ of signature (3,4) on $\mathbb{R}_+ \times M \times \mathbb{R}$.
• Applies Cartan's method of equivalence and normal Cartan connections to express the Weyl and Cotton tensors of the conformal structure in terms of Cartan's scalar invariants $A, B, C, D, E$.
• Defines the linear map $L_x: T_xM \times \mathbb{R} \to \otimes^3 T_x^*M$ via $L_x(v,\lambda) = W_{ijkl}v^i + C_{jkl}\lambda$, whose rank is conformally invariant and used as a nondegeneracy condition.
• Introduces the 3-nondegeneracy condition on the symmetric 4-form $A \in S^4\mathcal{D}^*$, defined as the absence of nonzero $X \in \mathcal{D}_x$ with $A(Y,X,X,X) = 0$ for all $Y \in \mathcal{D}_x$.
• Uses the normal Cartan connection and curvature components $\Omega_{ijkl}$ to express the Weyl and Cotton tensors explicitly in terms of the invariants, showing $W_{ijkl} = \Omega_{ijkl} \circ \sigma$ and $C_{jkl} = \Omega_{jkl} \circ \sigma$.
• Analyzes the $SL(2,\mathbb{R})$-representation decomposition of the Weyl tensor space, identifying the 15-dimensional subspace corresponding to generic 2-plane fields, and isolates the $S^4$-component as the invariant $A$.

Experimental results

Research questions

• RQ1 Does the ambient metric of any real-analytic generic 2-plane field on a 5-manifold have holonomy contained in $G_2$?
• RQ2 What pointwise geometric conditions on the Weyl and Cotton tensors ensure that the ambient metric has full $G_2$ holonomy?
• RQ3 Is the condition that the map $L_x$ is injective (rank 6) generic among real-analytic 2-plane fields?
• RQ4 Is the 3-nondegeneracy of the Cartan invariant $A$ generic at a point for generic 2-plane fields?
• RQ5 How do the Cartan invariants $A, B, C, D, E$ decompose under the $SL(2,\mathbb{R})$-action, and which components correspond to the Weyl and Cotton tensors?

Key findings

• The ambient metric $\widetilde{g}$ associated to any real-analytic generic 2-plane field $\mathcal{D}$ on a 5-manifold satisfies $\operatorname{Hol}(\widetilde{g}) \subset G_2$, extending the result of Leistner-Nurowski beyond explicit families.
• If at some point $x \in M$ the map $L_x$ has rank 6 (i.e., is injective), and at some point $y \in M$ the 4-form $A_y$ is 3-nondegenerate, then $\operatorname{Hol}(\widetilde{g}) = G_2$.
• The set of 2-plane fields for which $L_x$ has rank less than 6 or $A_x$ is 3-degenerate is contained in a proper algebraic subvariety of the jet space, implying that the $G_2$ holonomy condition is generic.
• In a normal form $\mathcal{D} = \operatorname{span}\{\partial_q, \partial_x + p\partial_y + q\partial_p + F\partial_z\}$ with $F_{qq} \neq 0$, the components of the Weyl and Cotton tensors are rational functions of $F$ and its derivatives up to order 6 and 7, respectively.
• The Weyl tensor components $W_{ijkl}$ and Cotton tensor components $C_{jkl}$ are explicitly expressed via curvature components $\Omega_{ijkl}$ and $\Omega_{jkl}$ of the normal Cartan connection, with $W_{ijkl} = \Omega_{ijkl} \circ \sigma$ and $C_{jkl} = \Omega_{jkl} \circ \sigma$.
• The Cartan invariant $A$ is symmetric and corresponds to the $S^4$-component of the Weyl tensor under the $SL(2,\mathbb{R})$-action, and its 3-nondegeneracy is a necessary and sufficient condition (along with $L_x$ injectivity) for full $G_2$ holonomy.