Explicit metrics with holonomy G_{2}

Simon M. Salamon Department of Mathematics, Imperial College

The aim of this talk is to construct explicit metrics with holonomy equal to G_{2} on 7-manifolds. Nowadays this is elementary, but it wasn’t some 20 years ago. These are all non-compact examples, as no compact example is explicitly known.

G_{2} is a subgroup of SO (7), so a G_{2}-structure determines an orientation and a Riemannian metric g on a 7-manifold. G_{2} has two important subgroups:

• SU(3); indeed we have G_{2}/SU(3) = S^{6}, where S^{6} written this way is not symmetric but 3-symmetric.

• SO(4); indeed we have G_{2}/SO(4) = W^{8}, where W^{8} is one of Wolf’s quaternion-Kähler symmetric spaces.

I shall build a G_{2}-structure from an SO(4)-structure. The existence of a G_{2}-struc-ture is really a topological condition, but we want Hol(g) = G_{2}, which is a strong integrability condition implying the vanishing of the Ricci tensor, akin to the Calabi-Yau condition on a 6-manifold, i.e. Hol(g) = SU(3). Just as Calabi-Yau manifolds lead to 10-dimensional string theories, G_{2} manifolds lead to an 11-di-mensional string theory on a manifold M^{7} × ℝ^{1,3}, where M^{7} is G_{2} and ℝ^{1,3} is Minkowski space. The point here is that there exists a parallel spinor (unique up to multiples) on a manifold with holonomy G_{2}.

1 Self-duality

Let M^{4} be an oriented Riemannian 4-manifold. Let e^{1}, e^{2}, e^{3}, e^{4} be an oriented orthonormal basis of T_{x}^{∗}M = Λ^{1}. There is a well-known SO(4)-invariant decomposition Λ^{2}T_{x}^{∗}M = Λ_{+}^{2} ⊕ Λ_{−}^{2}. As a matter of taste I’ll fix my attention on Λ_{−}^{2} = ⟨ω^{1}, ω^{2}, ω^{3}⟩, where

\begin{aligned}ω^{1} & = e^{1} ∧ e^{2} − e^{3} ∧ e^{4} \\ ω^{2} & = e^{1} ∧ e^{3} − e^{4} ∧ e^{2} \\ ω^{3} & = e^{1} ∧ e^{4} − e^{2} ∧ e^{3} \end{aligned}  (1)

is the standard basis of the space of anti-selfdual 2-forms. One can do that pointwise or locally on any manifold, but in some cases one can do this globally.

If ω^{1}, ω^{2}, ω^{3} exist globally, then the 4-manifold M has an SU(2)_{+}-structure (where in the SU(2) × SU(2) → SO(4) double covering SU(2)_{+} is the copy preserving everything in Λ_{−}^{2}). There are topological obstructions to finding such a basis. If dω^{i} = 0 for all i, then Hol(g) ⊆ SU(2)_{+} and (M^{4}, g) is hyperkähler. Hyperkähler means that in addition to the ω^{i}’s we have 3 complex structures J_{1}, J_{2}, J_{3}, such that (M, J_{i}, g) is Kähler and ω^{2}+iω^{3} is a holomorphic symplectic form relative to J_{1}. This implies Ric(g) = 0.

This is the simplest example of a differential geometric structure defined by closed differential forms.

The second aim of this talk is to estabish links between hyperkähler 4manifolds and metrics with holonomy G_{2}.

2 Definition of G_{2}

Relative to the inclusions

SO(7) ⊃ G_{2} ⊃ SO(4) ,

we can write

ℝ^{7} = H^{4} ⊕ V^{3}

The group SO(7) acts on ℝ^{7}, which is irreducible for the G_{2} action, but splits up for the SO(4) action as H^{4} ⊕ V^{3}. H^{4} is the standard representation of SO(4), i.e. Λ^{1}, but V^{∗} ≅ Λ_{−}^{2}; both are irreducible representations. Let b_{1}, b_{2}, b_{3} be a basis of V^{∗} corresponding to ω^{1}, ω^{2}, ω^{3}. We have split a 7-dimensional space into a 4-dimensional H (think of H as horizontal) and a 3-dimensional V (think of V as vertical). The 3-form

b^{1} ∧ ω^{1} + b^{2} ∧ ω^{2} + b^{3} ∧ ω^{3},

where ω^{1}, ω^{2}, ω^{3} ∈ Λ^{2}H^{∗}, is invariant by SO(4). The 3-form b^{123} (short for b^{1} ∧ b^{2} ∧ b^{3}) on V is preserved by SO(4). So

φ = b^{1} ∧ ω^{1} + b^{2} ∧ ω^{2} + b^{3} ∧ ω^{3} + b^{123} ∈ Λ^{3}(ℝ^{7})^{∗}

is preserved by SO(4). Setting b^{1} = e^{5}, b^{2} = e^{6}, b^{3} = e^{7}, we can also write

φ = e^{125} − e^{345} + e^{136} − e^{426} + e^{147} − e^{237} + e^{567}.

The following is well known:

Proposition 1. {g ∈ GL(7, ℝ) : g · φ = φ} ≅ G_{2} .

Since dim G_{2} = 14, the orbit O = {g · φ : g ∈ GL(7, ℝ)} has dimension

dim(GL(7, ℝ)) − dim G_{2} = 49 − 14 = 35 = \binom{7}{3} = dim Λ^{3}(ℝ^{7})^{∗} ,

so O is open in Λ^{3}(ℝ^{7})^{∗}. The 3-form φ is the analogue of a non degenerate 2-form; it is generic in some sense. If I change just one sign, the stabilizer becomes the non-compact form G_{2}^{∗} (there are two open orbits). Forms with stabilizer G_{2} we call positive as opposed to forms with stabilizer G_{2}^{∗}. So φ is “positive non-degenerate”. What does that mean geometrically? It means that to define a G_{2}-structure on a 7-manifold it suffices to find a global 3-form φ such that φ_{x} ∈ O_{x} for all x. Note that φ determines a G_{2}-structure and therefore a metric g and an orientation, so one can define ∗φ ∈ Λ^{4}T_{x}^{∗} (beware though, ∗ depends on φ).

Proposition 2. Hol(g) ⊆ G_{2} ⟺ dφ = 0, d ∗ φ = 0

The condition Hol(g) ⊆ G_{2} is the same thing as saying ∇φ = 0 which easily implies

dφ = 0 and d ∗ φ = 0 .

The converse is a matter of decomposing ∇φ = 0 into components, and comparing with components of dφ and d ∗ φ. Also,

Hol(g) ⊆ G_{2} ⟹ Ric(g) = 0 .

Later, we shall construct forms φ which satisfy dφ = 0 and d ∗ φ = 0. First though, I want to do something well known.

3 A hyperkähler metric on T^{∗}S^{2}

Let M^{4} = T^{∗}S^{2}. I’ll do this in detail because then we can copy the construction and get a manifold with holonomy G_{2}. Fix an oriented orthonormal basis e^{1}, e^{2} of 1-forms on an open set of S^{2}. I could do this explicitly, but let’s pretend for a moment S^{2} is an arbitrary 2-manifold. Thus, there exists a 1-form σ such that

\{ \begin{matrix}de^{1} & = σ ∧ e^{2} \\ de^{2} & = −σ ∧ e^{1} \end{matrix} , dσ = −e^{1} ∧ e^{2} ,

if the scalar curvature is 1. To work this out explicitly, take e^{i} = 2dx^{i}/ (1 + r^{2}).

The total space M has a tautological 1-form τ; in terms of fibre coordinates a^{1}, a^{2} corresponding to our basis

τ = a^{1}π^{∗}e^{1} + a^{2}π^{∗}e^{2} = ∑ a^{i}e^{i}.

An element of the fibre stands for a form on the base so I just pull it back. Then

dτ = ∑ b^{i} ∧ e^{i} where \{ \begin{matrix}b^{1} & = da^{1} − a^{2}σ \\ b^{2} & = da^{2} + a^{1}σ \end{matrix}

So we have e^{1}, e^{2} horizontally and b^{1}, b^{2} vertically. So we get the a^{i}’s combined with the e^{i}’s, the b^{i}’s combined with the e^{i}’s, what about the a^{i}’s combined with the b^{i}’s? In fact,

∑ a^{i}b^{i} = ∑ a^{i}da^{i} = rdr where r^{2} = ∑(a^{i})^{2} .

We now need the following lemma:

Lemma 3. d(b^{1} ∧ b^{2}) = rdr ∧ e^{1} ∧ e^{2} .

Set s = (λ+r^{2})^{1/4} with λ constant and let g be the riemannian metric on the total space of the cotangent bundle for which se^{1}, se^{2}, \frac{1}{s}b^{1}, \frac{1}{s}b^{2} is an orthonormal basis, i.e.

g = s^{2}g_{ij} + \frac{1}{s^{2}} g_{ℝ^{2}} ,

where g_{ij} is the standard metric on S^{2}, g_{ℝ^{2}} is the standard metric on the fibre ℝ^{2} and + is defined with respect to the Levi-Civita connection. Think of se^{1}, se^{2}, \frac{1}{s}b^{1}, \frac{1}{s}b^{2} as the e^{1}, e^{2}, e^{3}, e^{4} in (1). Define

\begin{aligned}ω^{1} & = s^{2}e^{1} ∧ e^{2} − \frac{1}{s^{2}} b^{1} ∧ b^{2} \\ ω^{2} & = e^{1} ∧ b^{1} + e^{2} ∧ b^{2} = −dτ \\ ω^{3} & = e^{1} ∧ b^{2} − e^{2} ∧ b^{1}. \end{aligned}  (2)

These 2-forms are all closed: ω^{1} is closed by the lemma, ω^{2} is closed since exact, and ω^{3} is closed because ω^{3} = dτ′ where

τ′ = a^{1}e^{2} − a^{2}e^{1}

is another canonical form, obtained by rotating by π/2 (in fact, by the underlying complex structure) before combining the a^{i}’s with the e^{i}’s.

Corollary 4. Hol(g) ⊆ SU(2)_{+} and (T^{∗}S^{2}, g) is hyperkähler.

If λ = 1, Hol(g) = SU(2) and g = g_{EH} is a complete metric known as the Eguchi-Hanson metric.

Remark 1. If λ = 0, s^{2} = r and it is easy to see that g is flat. In fact g_{EH} is asymptotic to the flat metric on the quotient R^{+} × S^{3}/ℤ_{2}; it’s one of a well known family of “ALE” metrics.

Remark 2. Since S^{2} = ℂℙ^{1}, we have a natural complex structure on T^{∗}S^{2}, and we have

ω^{2} − iω^{3} = d (τ^{1,0})

where τ^{1,0} = τ − iτ′.

4 A metric with holonomy G_{2} on Λ_{−}^{2}S^{4}

Let Y^{7} denote the total space of the rank 3 vector bundle Λ_{−}^{2}S^{4} of anti-selfdual 2-forms over S^{4}. Choose a local oriented orthonormal basis e^{1}, e^{2}, e^{3}, e^{4} of 1forms on S^{4} with its standard metric. Then Y has a tautological 2-form

τ = ∑_{i=1}^{3} a^{i}π^{∗}ω^{i} ,

where the ω^{i}’s are defined as in (1), and

dτ = ∑_{1}^{3} b^{i} ∧ ω^{i} ,  (3)

where b^{i} = da^{i} + a^{i} ∑ γ_{j}^{i} ∧ ω^{j}, where γ_{j}^{i} are 1-forms such that ∇ω^{i} = γ_{j}^{i} ⊗ ω^{j}. So we have b_{1}, b_{2}, b_{3} vertically and e^{1}, e^{2}, e^{3}, e^{4} horizontally.

Lemma 5. If the scalar curvature of S^{4} is 1, d(b^{123}) = −\frac{1}{12}rdr ∧ dτ.

Actually, it will simplify subsequent formulae to change the coefficient 1/12 in the lemma to 1/2. We therefore assume that the metric on S^{4} is re-scaled so as to have scalar curvature equal to 6! Set s = (λ + r^{2})^{1/4} with λ constant. Let k be the metric on Y with orthonormal basis se^{1}, se^{2}, se^{3}, se^{4}, \frac{1}{s}b^{1}, \frac{1}{s}b^{2}, \frac{1}{s}b^{3}, i.e.

k = s^{2}g_{base} + \frac{1}{s^{2}} g_{fibre} .

This metric is compatible with the G_{2}-structure defined by

φ = s ∑ b^{i} ∧ ω^{i} + \frac{1}{s^{3}} b^{123} = sdτ + \frac{1}{s^{3}} b^{123}

using (3). Then φ is closed because of Lemma 5, and so is

∗φ = −s (b^{12} ∧ ω^{3} + b^{23} ∧ ω^{1} + b^{31} ∧ ω^{2}) + s^{4}e^{1234} .

Corollary 6. If λ > 0, k is a complete Ricci-flat metric with holonomy G_{2} on the total space of Λ_{−}^{2}S^{4}.

Observe that S^{2} is totally geodesic in S^{4}, and we have the following diagram:

\xymatrix{T^{∗}S^{2} \ar[d] \ar@{^{(}->}[r] & Λ_{−}^{2} \ar[d] \\ S^{2} \ar@{^{(}->}[r] & S^{4}}

Restricting the G_{2} metric on T^{∗}S^{2} we get the Eguchi-Hanson metric on T^{∗}S^{2} (see [Han78]).

The second lecture and subsequent sections regard hypersurfaces and quotients of G_{2}-manifolds, i.e. manifolds with an explicit metric with holonomy G_{2}, typically non-compact.

5 Hypersurfaces

We constructed a metric of holonomy G_{2} on Λ_{−}^{2}S^{4}. Explicitly, there was a one-parameter family of metrics

k_{λ} = (λ + r^{2})^{1/2}g_{base} + (λ + r^{2})^{−1/2}g_{fibre} .

We have a diagram:

\xymatrix{ℂℙ^{3} \ar[d]^{S^{2}} \ar@{^{(}->}[r] & Λ_{−}^{2} \ar[d]^{ℝ^{3}} \\ _{ℍℙ_{1}} \ar[r]^{≅} & S^{4}}

where the induced map on the fibres is the inclusion of the 2-sphere in ℝ^{3}. For λ = 0 the metric is

k_{0} = rg_{S^{4}} + \frac{1}{r} (dr^{2} + r^{2}g_{S^{2}}) = r (g_{S^{4}} + g_{S^{2}}) + \frac{1}{r^{2}} dr^{2} .

Put r = ρ^{2}; then k_{0} = ρ^{2}g_{ℂℙ}^{3} + 4dρ^{2} is a conical metric on ℂℙ^{3} × ℝ^{+}, where g_{ℂℙ}^{3} is not the standard metric, but a 3-symmetric Einstein, non-Kähler metric. Hol(k_{0}) = G_{2}, but the metric is not complete because of the singularity at 0.

Theorem 7 ([Sal89]). If N^{6} is any of ℂℙ^{3}, SU(3)/T^{2}, S^{3} × S^{3}, then N × ℝ^{+} has a holonomy G_{2} metric that can be deformed to a complete one.

Remark 1. SU(3)/T^{2} is the “twistor space” that fibres over ℂℙ^{2}, and can be treated by considering Λ_{−}^{2}ℂℙ^{2} in place of Λ_{−}^{2}S^{4}. On the other hand, S^{3} × S^{3} is considered as the sphere bundle in the (topologically trivial) spin bundle over S^{3}.

Remark 2. Even at the present day, there are only a handful of other holonomy G_{2} metrics known explicitly.

Let us describe the general theory of constructing a G_{2} manifold from a 6manifold. Suppose (Y^{7}, φ) has holonomy G_{2}, i.e. dφ = 0, d ∗ φ = 0 (note that the latter is a non-linear equation since the ∗ operator depends on φ). Let N^{6} be a hypersurface of Y. Then

\{ \begin{matrix}φ & = σ ∧ η + ψ^{+} \\ ∗φ & = ψ^{−} ∧ η − \frac{1}{2}σ ∧ σ \end{matrix}  (4)

where η is the unit one-form annihilating T_{x}N, σ is a 2-form on N defining an almost complex structure J on N, and ψ^{+} + iψ^{−} is a (3, 0)-form on N with respect to J. The pair (σ, ψ^{+}) gives N an SU(3)-structure with

dψ^{+} = 0 and d(σ ∧ σ) = 0 .  (5)

In fact the intrinsic torsion of the SU(3)-structure has 42 components; equations (5) tell us that 21 are zero, so it’s exactly half-way to being Calabi-Yau. I call it a “half-flat” or “\frac{1}{2}CY ” SU(3)-structure.

6 A nilpotent example

Let

G = \{(\begin{matrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) : x, y, z ∈ ℂ\} , Γ = \{(\begin{matrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1 \end{matrix}) : x, y, z ∈ ℤ[i]\} .

Let N^{6} = Γ\G be the space of right cosets. It is a compact space, called the Iwasawa manifold. It is a principal bundle over T^{4} with fibre T^{2} (the generic element is mapped to (x, y) + ℤ^{2}). Define real forms by

dx = e^{1} + ie^{4}
dy = e^{3} + ie^{2}
−dz + xdy = e^{5} − ie^{6} ;

the e^{i}’s are then left-invariant forms on G, so they pass to N. So N is a nilmanifold with a global basis of 1-forms such that

de^{i} = \begin{cases}0 & i = 1, 2, 3, 4, \\ e^{13} − e^{42} = ω^{2} & i = 5, \\ −e^{12} + e^{34} = −ω^{1} & i = 6. \end{cases}

I want to define a metric with holonomy G_{2}. Let k be the metric on N^{6} ×ℝ^{+} (where ℝ^{+} is parametrized by s) with orthonormal basis

(se^{1}, se^{2}, se^{3}, se^{4}, \frac{1}{s}e^{5}, \frac{1}{s}e^{6}, s^{2}ds) ;

using (4) this basis determines a G_{2}-structure with

\{ \begin{matrix}η & = s^{2}ds \\ σ & = s^{2}ω^{3} + \frac{1}{s^{2}} e^{56} \\ ψ^{+} + iψ^{−} & = s(e^{5} − ie^{6}) ∧ (ω^{1} + iω^{2}) . \end{matrix}

Lemma 8. For fixed s, (σ, ψ^{±}) defines an SU(3)-structure on N^{6} such that dσ = \frac{1}{s^{3}} ψ^{+}, and dψ^{−} = −4se^{1234}.

Remark 1. The first equation implies dψ^{+} = 0 and d(σ∧σ) = s^{−3}(ψ^{+}∧σ) = 0.

The nilmanifold N is close to what we call a nearly-Kähler manifold (that would correspond to dψ^{−} being a constant times σ ∧ σ). Moreover, on N × ℝ^{+}, it is now easy to check that dφ = 0, d ∗ φ = 0. So Hol(k) ⊆ G_{2}. One can check directly that k is Ricci-flat (i.e. g^{ik}R_{ijkl} = 0), and that the rank of R viewed as a symmetric bilinear form on Λ^{2}(N × ℝ^{+}) is 14 = dim G_{2}, so Hol(k) = G_{2}.

7 S^{1} quotients

Let

S^{1} = \{(\begin{matrix}1 & 0 & z \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}) : z ∈ ℝ/ℤ\} .

S^{1} acts on N, in fact N is a principal T^{2} bundle over T^{4} and I am choosing one of the S^{1} factors of T^{2}. Let Q^{5} = N/S^{1}: we get the following:

\xymatrix{N^{6} \ar@{^{(}->}[r] \ar[d]^{S^{1}} & N × ℝ^{+} \ar[d] \\ Q \ar@{^{(}->}[r] \ar[d]^{S^{1}} & Q × ℝ^{+} \ar[dl]^{ℂ^{∗}} \\ T^{2}}

N^{6} is half-flat, not integrable and compact. For both the inclusion N ×ℝ^{+} ⊃ N and the quotient N × ℝ^{+} → Q × ℝ^{+}, we are losing one dimension. In both cases we have a pointwise splitting ℝ^{7} = ℝ^{6} ⊕ ℝ; so in the quotient case we can still use equations (4) where η = \frac{1}{s}e^{5}. Doing the calculations (since we know that dφ = 0, d ∗ φ = 0), we get

σ = s(e^{6} ∧ ds + sω^{1}) = sd(−se^{6}) ,

where d(−se^{6}) is a symplectic form, and

ψ^{+} + iψ^{−} = s(e^{6} − is^{3}ds) ∧ (ω^{2} + iω^{3}) = sΨ

with dΨ = 0.

Corollary 9. With the natural induced structure, Q × ℝ^{+} is Kähler.

The Kähler form is just σ̂ = d(−se^{6}). Moreover, ψ^{+} + iψ^{−} is a multiple of a closed (3, 0) form, so the almost complex structure J is integrable. Since Je^{6} = s^{3}ds we have

d(−se^{6}) = dJd(\frac{1}{5}s^{5}) = −2i∂\overline{∂} \frac{1}{5}s^{5} ;

the circumference of the fibres is 2π/s and in fact the Ricci form is −∂\overline{∂}(log s). It turns out that there are a lot of other examples where the quotient is Kähler.

8 More generally

Suppose (Y^{7}, φ) is a 7-manifold with a G_{2} holonomy metric invariant by some S^{1} action with Killing vector field V and ∥V ∥ = \frac{1}{s}. Then L_{v}φ = 0 and σ̂ = V ⌟φ is a symplectic form on Y/S^{1}. Let U be the Hamiltonian vector field on Y/S^{1} corresponding to s, i.e. σ̂(U, ·) = ds.

Theorem 10 ([Sal04]). Assume (Y/S^{1}, σ̂, J) is Kähler. Then L_{U}J = 0 (indeed U preserves the SU(3)-structure), and there is a Kähler quotient

M^{4} = \frac{\{s = constant\}}{S^{1}} = \frac{Y/S^{1}}{ℂ^{∗}}

since we have a holomorphic vector field.

M^{4} admits:

• a complex structure J

• a holomorphic symplectic 2-form ω^{2} + iω^{3}

• a one-parameter family of real 2-forms ω̃^{1}(s)

One can compute the volume using the complex forms, but one can also compute the volume using the symplectic form and the two won’t be equal in general. So I define a function f(s) on M as

sω̃^{1} ∧ ω̃^{1} = f(s)ω^{2} ∧ ω^{2} ;

then one shows

Proposition 11. ω̃^{1} and f satisfy the second order evolution equation

\frac{1}{2}ω̃^{1}(s) = i∂_{M} \overline{∂_{M}}f(s) .

So the idea is to start from this data on M^{4} and reconstruct (Y^{7}, φ) with G_{2} holonomy. There are 2 classes of examples:

1. If f is constant on M, ω̃^{1} is linear in s (as in Iwasawa’s case).

2. More generally, if f is non-constant, the Monge-Ampère equation can be solved in some cases.

Further details can be found in [Sal04] and [Sal03].

References

[Han78] T. Eguchi; A.J. Hanson. Asymptotically flat selfdual solutions to Euclidean gravity. Phys. Lett. B, 74(249), 1978.

[Sal89] R. Bryant; S. Salamon. On the construction of some complete metrics with exceptional holonomy. Duke Math. J., 58:829–850, 1989.

[Sal03] S. Salamon. A tour of exceptional geometry. Milan j. Math, 71:59–94, 2003.

[Sal04] V. Apostolov; S. Salamon. Kähler reduction of metrics with holonomy G_{2}. Comm. Math. Phys., 246:43–61, 2004.