A note on 4-forms in 8-dimensions

Abstract: We present a note on two related questions on 4-forms in 8-dimensions. In particular, we clarify that the Cayley form is not unique in determining a metric via the formula of Karigiannis. Additionally, we discuss and prove part of a conjecture of Salamon and Walpuski.

• The paper demonstrates that the Cayley form is not unique in defining a non-degenerate Riemannian metric via Karigiannis's formula, identifying additional 4-forms with this property.
• The Salamon-Walpuski conjecture on non-degenerate 4-forms is partially confirmed, showing that 4-forms with $H_\alpha$ strongly non-degenerate do not have a vanishing self-wedge.
• The paper introduces a new concept, strong non-degeneracy, which is more restrictive but implies ordinary non-degeneracy, and suggests future research into the equivalence of the two notions.

Background and context

The geometry of 4-forms in 8 dimensions occupies an unusual position in the theory of stable forms. Algebraically, they are the first case — scanning all pairs $(p,n)$ of form degree and dimension in increasing $n$ — for which the space $\Lambda^4 V^*$ carries infinitely many $GL(V)$-orbits [Ryvkin]. Orbit classification was initiated by Antonyan over $\mathbb{C}$ and recently completed over $\mathbb{R}$, building on earlier proposals. The dominant object in this landscape is the Cayley 4-form $\Phi$, stabilised by $Spin(7)$, first identified by Bonan in 1966 and since studied intensively: it determines both an orientation and a Riemannian metric via a nontrivial formula due to Karigiannis, and admits comass-based characterisations.

This note by Sam Close addresses two related questions. First, it shows that the folklore belief that the Cayley form is the 4-form whose self-wedge structure recovers a metric through Karigiannis's construction is false. Second, it makes partial progress on a conjecture attributed to Salamon–Walpuski concerning non-degenerate 4-forms. Throughout, $V$ denotes a real 8-dimensional vector space, and a "Cayley form" means any 4-form with $GL(V)$-stabiliser exactly $n$0.

Metric 4-forms and the failure of uniqueness

Karigiannis gave an explicit reconstruction of the metric determined by a Cayley form. Given a basis $n$1 of $n$2 and a 4-form $n$3, define

$n$4

$n$5

and the scalar function

$n$6

The paper includes an absolute value absent from Karigiannis's original definition, accounting for possible orientation discrepancy between $n$7 and $n$8. Two structural facts hold generally: $n$9 is absolutely homogeneous of order 1 in $\Lambda^4 V^*$0 and homogeneous of order 4 in $\Lambda^4 V^*$1; and for a Cayley form, $\Lambda^4 V^*$2 is proportional to $\Lambda^4 V^*$3, so that polarisation of $\Lambda^4 V^*$4 yields the metric,

$\Lambda^4 V^*$5

Nothing about this expression demands special properties of $\Lambda^4 V^*$6 beyond producing a top-form, and the note exploits this observation to prove its main result:

Theorem (metric non-uniqueness). The Cayley form is not unique among 4-forms defining a non-degenerate Riemannian metric via the Karigiannis formula.

The proof constructs a one-parameter family built from $\Lambda^4 V^*$7-structure data on $\Lambda^4 V^*$8: the Kähler form $\Lambda^4 V^*$9 and holomorphic volume form $GL(V)$0, setting

$GL(V)$1

Since $GL(V)$2 and direct computation gives

$GL(V)$3

every member of this family defines the same underlying Riemannian norm, up to scale. The family intersects the $GL(V)$4 orbit at $GL(V)$5 (where $GL(V)$6), while at $GL(V)$7 the stabiliser enlarges to $GL(V)$8 and the form ceases to be metric. A second example uses a three-parameter family $GL(V)$9 assembled from self-dual triples of 2-forms; for generic parameters its stabiliser is $\mathbb{C}$0 (the hyper-Kähler 4-form), and for $\mathbb{C}$1 up to permutation and sign it reproduces the Cayley form, while $\mathbb{C}$2 yields the Kraines/quaternion-Kähler form stabilised by $\mathbb{C}$3. In general,

$\mathbb{C}$4

The author notes honestly that these examples are constructed from auxiliary data ($\mathbb{C}$5 structures, or pairs of 2-form triplets satisfying an Urbantke-like relation $\mathbb{C}$6) that already encode metrics, so their metric characterisation is unsurprising given orthogonal stabilisers. The substantive point stands nonetheless: starting abstractly from an orbit representative, the Karigiannis formula recovers the same metric more widely than appreciated, in contrast with dimension 7 where only the two $\mathbb{C}$7 orbits define a non-degenerate metric. The precise necessary-and-sufficient condition for a 4-form to be metric remains open; orthogonality of the stabiliser is conjectured to be the relevant criterion, with compactness required only for Riemannian signature.

Non-degeneracy of 4-forms

In dimension 7, a 3-form is non-degenerate precisely when it determines a metric via $\mathbb{C}$8, and there are exactly two such orbits, $\mathbb{C}$9 and its split form. No equally clean picture exists in 8 dimensions. Salamon and Walpuski proposed a notion: $\mathbb{R}$0 is non-degenerate if for every linearly independent triple $\mathbb{R}$1 there exists $\mathbb{R}$2 with $\mathbb{R}$3. This implies but is not equivalent to multisymplectic non-degeneracy (injectivity of $\mathbb{R}$4): the split volume form $\mathbb{R}$5, stabilised by $\mathbb{R}$6, is multisymplectically non-degenerate yet degenerate in the Salamon–Walpuski sense, witnessed by $\mathbb{R}$7.

The conjecture under study is:

Conjecture (Salamon–Walpuski). If $\mathbb{R}$8, then $\mathbb{R}$9 is degenerate.

The converse fails trivially (degenerate forms with non-vanishing self-wedge exist, e.g. $\Phi$0 above). Non-degeneracy admits several equivalent reformulations: $\Phi$1 is symplectic on $\Phi$2; equivalently $\Phi$3 for independent $\Phi$4. A further lemma reduces this condition to requiring that, for every $\Phi$5, the restriction $\Phi$6 be the standard $\Phi$7 3-form.

Area metrics, strong non-degeneracy, and partial results

The technical engine of the second half is an area-metric formalism. Since no metric is given a priori, index raising uses the quasi-Hodge isomorphism built from Levi-Civita tensor densities, which densitises expressions. Two canonical scalar densities of weight $\Phi$8 and degree 2 in $\Phi$9 are available: $Spin(7)$0, and $Spin(7)$1, where $Spin(7)$2 is viewed as a matrix. The latter fact follows from the Sylvester–Franke theorem applied to $Spin(7)$3. These densities are a priori independent; the paper writes $Spin(7)$4 for either choice.

Define the area metric

$Spin(7)$5

Two structural results govern it. First, $Spin(7)$6 is congruent to $Spin(7)$7, so by Sylvester's law of inertia $Spin(7)$8 is always indefinite; moreover, non-degeneracy of $Spin(7)$9 is equivalent to $V$0. Second, under the canonical splitting $V$1 (the algebraic Bianchi decomposition),

$V$2

Because $V$3 is totally antisymmetric, evaluating $V$4 — the quantity controlling non-degeneracy per the equivalent characterisations — sees only the Bianchi part $V$5. This motivates the central new definition: $V$6 is strongly non-degenerate if $V$7 is definite. Strong non-degeneracy evidently implies non-degeneracy, and the indefinite-signature result immediately yields the paper's main partial answer:

Corollary. There are no strongly non-degenerate 4-forms with vanishing self-wedge.

Indeed, if $V$8 then $V$9, and indefiniteness of $GL(V)$0 forces $GL(V)$1 indefinite. Thus the implication chain

$GL(V)$2

is established, and the Salamon–Walpuski conjecture would follow if strong non-degeneracy were equivalent to non-degeneracy. The author could not locate any 4-form that is non-degenerate without being strongly non-degenerate, and elevates this equivalence itself to a conjecture.

The relationship between the two notions studied in the paper is subtle. Since $GL(V)$3 appears in the denominator of $GL(V)$4, being metric forces non-vanishing self-wedge. However, the family $GL(V)$5 demonstrates that the implications do not close neatly: choosing $GL(V)$6,

$GL(V)$7

which is strongly non-degenerate only for $GL(V)$8, whereas $GL(V)$9 is a metric 4-form for all $n$00. Hence strong non-degeneracy does not imply metricality; whether the converse holds is unproved.

Limitations and open questions

Several caveats qualify the results. The key computation of $n$01 was performed symbolically in Mathematica rather than by hand, and the theorem is proved by exhibiting a single one-parameter family rather than a classification of all metric 4-forms. The exact criterion for a 4-form to be metric — presumably orthogonality of the stabiliser — is asserted as intuition, not proven. On the non-degeneracy side, the equivalence of strong and ordinary non-degeneracy is conjectural, so the Salamon–Walpuski conjecture is reduced but not resolved. The choice between the two scalar densities $n$02 and $n$03 is left somewhat ambiguous: the determinant is natural for non-degeneracy of $n$04, while the self-wedge removes the coefficient in the Bianchi decomposition. Forms for which both densities vanish are explicitly excluded from consideration. Finally, the paper does not assume the algebraic Bianchi identity on the area metric, leaving open how imposing it would interact with the decomposition $n$05.

Conclusion

This note makes two contributions to the structure theory of 4-forms in 8 dimensions. It establishes, via an explicit $n$06-based family and a self-dual three-parameter family, that metric recovery through the Karigiannis formula is a property shared by many orbits beyond $n$07 — including hyper-Kähler and broken-symmetry representatives — correcting a piece of folklore. It also reframes the Salamon–Walpuski non-degeneracy conjecture through an area-metric decomposition, proving that definiteness of the Bianchi component $n$08 excludes vanishing self-wedge, thereby reducing the conjecture to the question of whether strong non-degeneracy coincides with non-degeneracy. Both the full classification of metric 4-forms and the resolution of either remaining conjecture remain open.