"Generalised and asymptotic symmetries of hyperkähler manifolds"

”Generalised and asymptotic symmetries of
hyperkähler manifolds”
Roger Bielawski
Fribourg, July 3, 2008

Hyperkähler manifolds
A Riemannian manifold (M,g) is hyperkähler if it possesses 3 almost
complex structures I 1 ,I 2 ,I 3 , parallel for the Levi-Civita connection and
satisfying I 1 I 2 = −I 2 I 1 = I 3 etc.
Holonomy of (M,g) is a subgroup of Sp(n).
I 1 ,I 2 ,I 3 are integrable. In fact a whole 2-sphere of complex
structures: a 1 I 1 + a 2 I 2 + a 3 I 3 , (a 1 ,a 2 ,a 3 ) ∈ S 2 .
Three (or again a 2-sphere) Kähler forms ω i = g(I i·,·).
Complex symplectic forms: ω 2 + √ −1ω 3 is an I 1 -holomorphic
nondegenerate closed form.
Examples: Flat R 4 or T 4 , K 3 surfaces, resolutions of Kleinian
singularities C 2 /Γ, moduli spaces of topological solitons
(instantons, monopoles), moduli spaces of Higgs bundles,
coadjoint orbits of complex reductive Lie groups, quiver varieties,
toric hyperkähler varieties.

Hyperkähler manifolds
A Riemannian manifold (M,g) is hyperkähler if it possesses 3 almost
complex structures I 1 ,I 2 ,I 3 , parallel for the Levi-Civita connection and
satisfying I 1 I 2 = −I 2 I 1 = I 3 etc.
Holonomy of (M,g) is a subgroup of Sp(n).
I 1 ,I 2 ,I 3 are integrable. In fact a whole 2-sphere of complex
structures: a 1 I 1 + a 2 I 2 + a 3 I 3 , (a 1 ,a 2 ,a 3 ) ∈ S 2 .
Three (or again a 2-sphere) Kähler forms ω i = g(I i·,·).
Complex symplectic forms: ω 2 + √ −1ω 3 is an I 1 -holomorphic
nondegenerate closed form.
Examples: Flat R 4 or T 4 , K 3 surfaces, resolutions of Kleinian
singularities C 2 /Γ, moduli spaces of topological solitons
(instantons, monopoles), moduli spaces of Higgs bundles,
coadjoint orbits of complex reductive Lie groups, quiver varieties,
toric hyperkähler varieties.

Hyperkähler manifolds
A Riemannian manifold (M,g) is hyperkähler if it possesses 3 almost
complex structures I 1 ,I 2 ,I 3 , parallel for the Levi-Civita connection and
satisfying I 1 I 2 = −I 2 I 1 = I 3 etc.
Holonomy of (M,g) is a subgroup of Sp(n).
I 1 ,I 2 ,I 3 are integrable. In fact a whole 2-sphere of complex
structures: a 1 I 1 + a 2 I 2 + a 3 I 3 , (a 1 ,a 2 ,a 3 ) ∈ S 2 .
Three (or again a 2-sphere) Kähler forms ω i = g(I i·,·).
Complex symplectic forms: ω 2 + √ −1ω 3 is an I 1 -holomorphic
nondegenerate closed form.
Examples: Flat R 4 or T 4 , K 3 surfaces, resolutions of Kleinian
singularities C 2 /Γ, moduli spaces of topological solitons
(instantons, monopoles), moduli spaces of Higgs bundles,
coadjoint orbits of complex reductive Lie groups, quiver varieties,
toric hyperkähler varieties.

Hyperkähler manifolds
A Riemannian manifold (M,g) is hyperkähler if it possesses 3 almost
complex structures I 1 ,I 2 ,I 3 , parallel for the Levi-Civita connection and
satisfying I 1 I 2 = −I 2 I 1 = I 3 etc.
Holonomy of (M,g) is a subgroup of Sp(n).
I 1 ,I 2 ,I 3 are integrable. In fact a whole 2-sphere of complex
structures: a 1 I 1 + a 2 I 2 + a 3 I 3 , (a 1 ,a 2 ,a 3 ) ∈ S 2 .
Three (or again a 2-sphere) Kähler forms ω i = g(I i·,·).
Complex symplectic forms: ω 2 + √ −1ω 3 is an I 1 -holomorphic
nondegenerate closed form.
Examples: Flat R 4 or T 4 , K 3 surfaces, resolutions of Kleinian
singularities C 2 /Γ, moduli spaces of topological solitons
(instantons, monopoles), moduli spaces of Higgs bundles,
coadjoint orbits of complex reductive Lie groups, quiver varieties,
toric hyperkähler varieties.

Hyperkähler manifolds
A Riemannian manifold (M,g) is hyperkähler if it possesses 3 almost
complex structures I 1 ,I 2 ,I 3 , parallel for the Levi-Civita connection and
satisfying I 1 I 2 = −I 2 I 1 = I 3 etc.
Holonomy of (M,g) is a subgroup of Sp(n).
I 1 ,I 2 ,I 3 are integrable. In fact a whole 2-sphere of complex
structures: a 1 I 1 + a 2 I 2 + a 3 I 3 , (a 1 ,a 2 ,a 3 ) ∈ S 2 .
Three (or again a 2-sphere) Kähler forms ω i = g(I i·,·).
Complex symplectic forms: ω 2 + √ −1ω 3 is an I 1 -holomorphic
nondegenerate closed form.
Examples: Flat R 4 or T 4 , K 3 surfaces, resolutions of Kleinian
singularities C 2 /Γ, moduli spaces of topological solitons
(instantons, monopoles), moduli spaces of Higgs bundles,
coadjoint orbits of complex reductive Lie groups, quiver varieties,
toric hyperkähler varieties.

Hyperkähler manifolds
A Riemannian manifold (M,g) is hyperkähler if it possesses 3 almost
complex structures I 1 ,I 2 ,I 3 , parallel for the Levi-Civita connection and
satisfying I 1 I 2 = −I 2 I 1 = I 3 etc.
Holonomy of (M,g) is a subgroup of Sp(n).
I 1 ,I 2 ,I 3 are integrable. In fact a whole 2-sphere of complex
structures: a 1 I 1 + a 2 I 2 + a 3 I 3 , (a 1 ,a 2 ,a 3 ) ∈ S 2 .
Three (or again a 2-sphere) Kähler forms ω i = g(I i·,·).
Complex symplectic forms: ω 2 + √ −1ω 3 is an I 1 -holomorphic
nondegenerate closed form.
Examples: Flat R 4 or T 4 , K 3 surfaces, resolutions of Kleinian
singularities C 2 /Γ, moduli spaces of topological solitons
(instantons, monopoles), moduli spaces of Higgs bundles,
coadjoint orbits of complex reductive Lie groups, quiver varieties,
toric hyperkähler varieties.

Twistor space
(M,g) Z
Z = M × S 2 (S 2 - 2-sphere of complex structures).
Z is a complex manifold with a holomorphic projection
π : Z → CP 1 .
Antiholomorphic involution τ on Z , induced by the antipodal map
on S 2 .
M is recovered as a space of τ-invariant sections of π, with
normal bundle a sum of O(1)-s.
The complex symplectic forms combine to give a twisted
holomorphic symplectic form Ω : Λ 2 T V Z → O(2) on the fibres of
π, taking values in the line bundle O(2) on CP 1 .

Twistor space
(M,g) Z
Z = M × S 2 (S 2 - 2-sphere of complex structures).
Z is a complex manifold with a holomorphic projection
π : Z → CP 1 .
Antiholomorphic involution τ on Z , induced by the antipodal map
on S 2 .
M is recovered as a space of τ-invariant sections of π, with
normal bundle a sum of O(1)-s.
The complex symplectic forms combine to give a twisted
holomorphic symplectic form Ω : Λ 2 T V Z → O(2) on the fibres of
π, taking values in the line bundle O(2) on CP 1 .

Twistor space
(M,g) Z
Z = M × S 2 (S 2 - 2-sphere of complex structures).
Z is a complex manifold with a holomorphic projection
π : Z → CP 1 .
Antiholomorphic involution τ on Z , induced by the antipodal map
on S 2 .
M is recovered as a space of τ-invariant sections of π, with
normal bundle a sum of O(1)-s.
The complex symplectic forms combine to give a twisted
holomorphic symplectic form Ω : Λ 2 T V Z → O(2) on the fibres of
π, taking values in the line bundle O(2) on CP 1 .

Twistor space
(M,g) Z
Z = M × S 2 (S 2 - 2-sphere of complex structures).
Z is a complex manifold with a holomorphic projection
π : Z → CP 1 .
Antiholomorphic involution τ on Z , induced by the antipodal map
on S 2 .
M is recovered as a space of τ-invariant sections of π, with
normal bundle a sum of O(1)-s.
The complex symplectic forms combine to give a twisted
holomorphic symplectic form Ω : Λ 2 T V Z → O(2) on the fibres of
π, taking values in the line bundle O(2) on CP 1 .

Twistor space
(M,g) Z
Z = M × S 2 (S 2 - 2-sphere of complex structures).
Z is a complex manifold with a holomorphic projection
π : Z → CP 1 .
Antiholomorphic involution τ on Z , induced by the antipodal map
on S 2 .
M is recovered as a space of τ-invariant sections of π, with
normal bundle a sum of O(1)-s.
The complex symplectic forms combine to give a twisted
holomorphic symplectic form Ω : Λ 2 T V Z → O(2) on the fibres of
π, taking values in the line bundle O(2) on CP 1 .

ALE gravitational instantons
ALE metrics on deformations of Kleinian singularities. The twistor
spaces are (almost):
A k : Z = { (x,y,z) ∈ O(k) ⊕O(k) ⊕O(2); xy − z k =
a 1 z k−2 + ··· + a k−2
}
Note: A 1 is the flat space; Z ≃ O(1) ⊕O(1).
D k : Z = { (x,y,z) ∈ O(k + 1) ⊕O(k) ⊕O(4); x 2 + y 2 z + z k+1 =
a 1 z k−1 + ··· + a k−1 + b 1 y 2 + b 2 y }
E 6 : Z = { (x,y,z) ∈ O(12) ⊕O(8) ⊕O(6); x 2 + y 3 + z 4 =
a 1 z 2 + a 2 z + a 3 + y(b 1 z 2 + b 2 z + b 3 ) }
E 7 : Z = { (x,y,z) ∈ O(18) ⊕O(12) ⊕O(8); x 2 + y 3 + yz 3 =
a 1 z 2 + a 2 z + a 3 + y(b 1 z + b 2 ) + y 2 (c 1 z + c 2 ) }
E 8 : Z = { (x,y,z) ∈ O(30) ⊕O(20) ⊕O(12); x 2 + y 3 + z 5 =
a 1 z 3 + a 2 z 2 + a 3 z + a 4 + +y(b 1 z 3 + b 2 z 2 + b 3 z + b 4 ) }

ALE gravitational instantons
ALE metrics on deformations of Kleinian singularities. The twistor
spaces are (almost):
A k : Z = { (x,y,z) ∈ O(k) ⊕O(k) ⊕O(2); xy − z k =
a 1 z k−2 + ··· + a k−2
}
Note: A 1 is the flat space; Z ≃ O(1) ⊕O(1).
D k : Z = { (x,y,z) ∈ O(k + 1) ⊕O(k) ⊕O(4); x 2 + y 2 z + z k+1 =
a 1 z k−1 + ··· + a k−1 + b 1 y 2 + b 2 y }
E 6 : Z = { (x,y,z) ∈ O(12) ⊕O(8) ⊕O(6); x 2 + y 3 + z 4 =
a 1 z 2 + a 2 z + a 3 + y(b 1 z 2 + b 2 z + b 3 ) }
E 7 : Z = { (x,y,z) ∈ O(18) ⊕O(12) ⊕O(8); x 2 + y 3 + yz 3 =
a 1 z 2 + a 2 z + a 3 + y(b 1 z + b 2 ) + y 2 (c 1 z + c 2 ) }
E 8 : Z = { (x,y,z) ∈ O(30) ⊕O(20) ⊕O(12); x 2 + y 3 + z 5 =
a 1 z 3 + a 2 z 2 + a 3 z + a 4 + +y(b 1 z 3 + b 2 z 2 + b 3 z + b 4 ) }

(E-H)-formalism
An alternative to the twistor language:
The structure group of a hyperkähler manifold reduces to Sp(n).
Let E be the vector bundle on M associated to the the standard
representation of Sp(n). Then T C M ≃ E ⊗ H, where H is the
trivial bundle of rank 2 (representation of Sp(1)).
In dimension 4, E and H are spinor bundles.
The representations of Sp(n) and Sp(1) are quaternionic, so
there are anti-holomorphic involutions on E,H.
Both E and H are symplectic vector bundles, with symplectic
forms ω E ,ω H . The metric is given by g = ω E ⊗ ω H .

(E-H)-formalism
An alternative to the twistor language:
The structure group of a hyperkähler manifold reduces to Sp(n).
Let E be the vector bundle on M associated to the the standard
representation of Sp(n). Then T C M ≃ E ⊗ H, where H is the
trivial bundle of rank 2 (representation of Sp(1)).
In dimension 4, E and H are spinor bundles.
The representations of Sp(n) and Sp(1) are quaternionic, so
there are anti-holomorphic involutions on E,H.
Both E and H are symplectic vector bundles, with symplectic
forms ω E ,ω H . The metric is given by g = ω E ⊗ ω H .

(E-H)-formalism
An alternative to the twistor language:
The structure group of a hyperkähler manifold reduces to Sp(n).
Let E be the vector bundle on M associated to the the standard
representation of Sp(n). Then T C M ≃ E ⊗ H, where H is the
trivial bundle of rank 2 (representation of Sp(1)).
In dimension 4, E and H are spinor bundles.
The representations of Sp(n) and Sp(1) are quaternionic, so
there are anti-holomorphic involutions on E,H.
Both E and H are symplectic vector bundles, with symplectic
forms ω E ,ω H . The metric is given by g = ω E ⊗ ω H .

(E-H)-formalism
An alternative to the twistor language:
The structure group of a hyperkähler manifold reduces to Sp(n).
Let E be the vector bundle on M associated to the the standard
representation of Sp(n). Then T C M ≃ E ⊗ H, where H is the
trivial bundle of rank 2 (representation of Sp(1)).
In dimension 4, E and H are spinor bundles.
The representations of Sp(n) and Sp(1) are quaternionic, so
there are anti-holomorphic involutions on E,H.
Both E and H are symplectic vector bundles, with symplectic
forms ω E ,ω H . The metric is given by g = ω E ⊗ ω H .

(E-H)-formalism
An alternative to the twistor language:
The structure group of a hyperkähler manifold reduces to Sp(n).
Let E be the vector bundle on M associated to the the standard
representation of Sp(n). Then T C M ≃ E ⊗ H, where H is the
trivial bundle of rank 2 (representation of Sp(1)).
In dimension 4, E and H are spinor bundles.
The representations of Sp(n) and Sp(1) are quaternionic, so
there are anti-holomorphic involutions on E,H.
Both E and H are symplectic vector bundles, with symplectic
forms ω E ,ω H . The metric is given by g = ω E ⊗ ω H .

Symmetries
A symmetry of a hyperkähler manifold is a vector field X which is
Killing and tri-holomorphic.
X is tri-Hamiltonian, if there are moment maps µ X 1 ,µX 2 ,µX 3 for the
three Kähler forms.
µ X = (µ X 1 ,µX 2 ,µX 3 ) : M → R3
tri-Hamiltonian =⇒ Killing and tri-holomorphic.
X is Hamiltonian w.r.t. to complex-symplectic forms, hence w.r.t.
the twisted complex-symplectic form Ω on Z and induces a
moment map µ : Z → O(2).
X - Killing is equivalent to X in the kernel of
TM ∇ → TM ⊗ T ∗ M g → TM ⊗ TM ≃ Λ 2 TM ⊕ S 2 TM → S 2 TM.
X is tri-Hamiltonian if it is in the image under the second
projection of the kernel of the first projection in:
S 2 H ∇ → S 2 H ⊗(E ⊗H) ≃ E ⊗(S 3 H ⊕H) → (E ⊗S 3 H)⊕(E ⊗H).

Symmetries
A symmetry of a hyperkähler manifold is a vector field X which is
Killing and tri-holomorphic.
X is tri-Hamiltonian, if there are moment maps µ X 1 ,µX 2 ,µX 3 for the
three Kähler forms.
µ X = (µ X 1 ,µX 2 ,µX 3 ) : M → R3
tri-Hamiltonian =⇒ Killing and tri-holomorphic.
X is Hamiltonian w.r.t. to complex-symplectic forms, hence w.r.t.
the twisted complex-symplectic form Ω on Z and induces a
moment map µ : Z → O(2).
X - Killing is equivalent to X in the kernel of
TM ∇ → TM ⊗ T ∗ M g → TM ⊗ TM ≃ Λ 2 TM ⊕ S 2 TM → S 2 TM.
X is tri-Hamiltonian if it is in the image under the second
projection of the kernel of the first projection in:
S 2 H ∇ → S 2 H ⊗(E ⊗H) ≃ E ⊗(S 3 H ⊕H) → (E ⊗S 3 H)⊕(E ⊗H).

Symmetries
A symmetry of a hyperkähler manifold is a vector field X which is
Killing and tri-holomorphic.
X is tri-Hamiltonian, if there are moment maps µ X 1 ,µX 2 ,µX 3 for the
three Kähler forms.
µ X = (µ X 1 ,µX 2 ,µX 3 ) : M → R3
tri-Hamiltonian =⇒ Killing and tri-holomorphic.
X is Hamiltonian w.r.t. to complex-symplectic forms, hence w.r.t.
the twisted complex-symplectic form Ω on Z and induces a
moment map µ : Z → O(2).
X - Killing is equivalent to X in the kernel of
TM ∇ → TM ⊗ T ∗ M g → TM ⊗ TM ≃ Λ 2 TM ⊕ S 2 TM → S 2 TM.
X is tri-Hamiltonian if it is in the image under the second
projection of the kernel of the first projection in:
S 2 H ∇ → S 2 H ⊗(E ⊗H) ≃ E ⊗(S 3 H ⊕H) → (E ⊗S 3 H)⊕(E ⊗H).

Symmetries
A symmetry of a hyperkähler manifold is a vector field X which is
Killing and tri-holomorphic.
X is tri-Hamiltonian, if there are moment maps µ X 1 ,µX 2 ,µX 3 for the
three Kähler forms.
µ X = (µ X 1 ,µX 2 ,µX 3 ) : M → R3
tri-Hamiltonian =⇒ Killing and tri-holomorphic.
X is Hamiltonian w.r.t. to complex-symplectic forms, hence w.r.t.
the twisted complex-symplectic form Ω on Z and induces a
moment map µ : Z → O(2).
X - Killing is equivalent to X in the kernel of
TM ∇ → TM ⊗ T ∗ M g → TM ⊗ TM ≃ Λ 2 TM ⊕ S 2 TM → S 2 TM.
X is tri-Hamiltonian if it is in the image under the second
projection of the kernel of the first projection in:
S 2 H ∇ → S 2 H ⊗(E ⊗H) ≃ E ⊗(S 3 H ⊕H) → (E ⊗S 3 H)⊕(E ⊗H).

Symmetries
A symmetry of a hyperkähler manifold is a vector field X which is
Killing and tri-holomorphic.
X is tri-Hamiltonian, if there are moment maps µ X 1 ,µX 2 ,µX 3 for the
three Kähler forms.
µ X = (µ X 1 ,µX 2 ,µX 3 ) : M → R3
tri-Hamiltonian =⇒ Killing and tri-holomorphic.
X is Hamiltonian w.r.t. to complex-symplectic forms, hence w.r.t.
the twisted complex-symplectic form Ω on Z and induces a
moment map µ : Z → O(2).
X - Killing is equivalent to X in the kernel of
TM ∇ → TM ⊗ T ∗ M g → TM ⊗ TM ≃ Λ 2 TM ⊕ S 2 TM → S 2 TM.
X is tri-Hamiltonian if it is in the image under the second
projection of the kernel of the first projection in:
S 2 H ∇ → S 2 H ⊗(E ⊗H) ≃ E ⊗(S 3 H ⊕H) → (E ⊗S 3 H)⊕(E ⊗H).

Symmetries
A symmetry of a hyperkähler manifold is a vector field X which is
Killing and tri-holomorphic.
X is tri-Hamiltonian, if there are moment maps µ X 1 ,µX 2 ,µX 3 for the
three Kähler forms.
µ X = (µ X 1 ,µX 2 ,µX 3 ) : M → R3
tri-Hamiltonian =⇒ Killing and tri-holomorphic.
X is Hamiltonian w.r.t. to complex-symplectic forms, hence w.r.t.
the twisted complex-symplectic form Ω on Z and induces a
moment map µ : Z → O(2).
X - Killing is equivalent to X in the kernel of
TM ∇ → TM ⊗ T ∗ M g → TM ⊗ TM ≃ Λ 2 TM ⊕ S 2 TM → S 2 TM.
X is tri-Hamiltonian if it is in the image under the second
projection of the kernel of the first projection in:
S 2 H ∇ → S 2 H ⊗(E ⊗H) ≃ E ⊗(S 3 H ⊕H) → (E ⊗S 3 H)⊕(E ⊗H).

Symmetries
A symmetry of a hyperkähler manifold is a vector field X which is
Killing and tri-holomorphic.
X is tri-Hamiltonian, if there are moment maps µ X 1 ,µX 2 ,µX 3 for the
three Kähler forms.
µ X = (µ X 1 ,µX 2 ,µX 3 ) : M → R3
tri-Hamiltonian =⇒ Killing and tri-holomorphic.
X is Hamiltonian w.r.t. to complex-symplectic forms, hence w.r.t.
the twisted complex-symplectic form Ω on Z and induces a
moment map µ : Z → O(2).
X - Killing is equivalent to X in the kernel of
TM ∇ → TM ⊗ T ∗ M g → TM ⊗ TM ≃ Λ 2 TM ⊕ S 2 TM → S 2 TM.
X is tri-Hamiltonian if it is in the image under the second
projection of the kernel of the first projection in:
S 2 H ∇ → S 2 H ⊗(E ⊗H) ≃ E ⊗(S 3 H ⊕H) → (E ⊗S 3 H)⊕(E ⊗H).

Generalised symmetries
Notion goes back to [Lindström and Roček, 1988], but it has been
formalised by [-,1999] (in a twistor language) and by [Dunajski, 1999],
[Dunajski & Mason, 2000, 2003] as follows:
A (real) section of E ⊗ S k H (k -odd) is a generalised symmetry if it is
in the kernel of:
E ⊗S k H ∇ → (E ⊗S k H)⊗(E ∗ ⊗H ∗ ) → (S 2 E ⊗S k+1 H)⊕(Λ 2 E ⊗S k−1 H).
Similarly, such a section is a tri-Hamiltonian generalised symmetry if it
is in the image of the kernel of
S k+1 H ∇ → S k+1 H ⊗ (E ⊗ H) ≃ E ⊗ (S k+2 H ⊕ S k H) → E ⊗ S k+2 H
under the map
S k+1 H ∇ → S k+1 H ⊗ (E ⊗ H) ≃ E ⊗ (S k+2 H ⊕ S k H) → E ⊗ S k H.

Generalised symmetries
Notion goes back to [Lindström and Roček, 1988], but it has been
formalised by [-,1999] (in a twistor language) and by [Dunajski, 1999],
[Dunajski & Mason, 2000, 2003] as follows:
A (real) section of E ⊗ S k H (k -odd) is a generalised symmetry if it is
in the kernel of:
E ⊗S k H ∇ → (E ⊗S k H)⊗(E ∗ ⊗H ∗ ) → (S 2 E ⊗S k+1 H)⊕(Λ 2 E ⊗S k−1 H).
Similarly, such a section is a tri-Hamiltonian generalised symmetry if it
is in the image of the kernel of
S k+1 H ∇ → S k+1 H ⊗ (E ⊗ H) ≃ E ⊗ (S k+2 H ⊕ S k H) → E ⊗ S k+2 H
under the map
S k+1 H ∇ → S k+1 H ⊗ (E ⊗ H) ≃ E ⊗ (S k+2 H ⊕ S k H) → E ⊗ S k H.

Generalised symmetries
Notion goes back to [Lindström and Roček, 1988], but it has been
formalised by [-,1999] (in a twistor language) and by [Dunajski, 1999],
[Dunajski & Mason, 2000, 2003] as follows:
A (real) section of E ⊗ S k H (k -odd) is a generalised symmetry if it is
in the kernel of:
E ⊗S k H ∇ → (E ⊗S k H)⊗(E ∗ ⊗H ∗ ) → (S 2 E ⊗S k+1 H)⊕(Λ 2 E ⊗S k−1 H).
Similarly, such a section is a tri-Hamiltonian generalised symmetry if it
is in the image of the kernel of
S k+1 H ∇ → S k+1 H ⊗ (E ⊗ H) ≃ E ⊗ (S k+2 H ⊕ S k H) → E ⊗ S k+2 H
under the map
S k+1 H ∇ → S k+1 H ⊗ (E ⊗ H) ≃ E ⊗ (S k+2 H ⊕ S k H) → E ⊗ S k H.

Roček).
Examples include all toric hyperkähler manifolds (all r i are 1),
monopole metrics, asymptotic monopole metrics, gravitational
instantons, and various quiver varieties, in particular, hyperkähler
metrics on adjoint orbits of GL(n,C) and the natural metrics on
Calogero-Moser spaces.
In the twistor language, a generalised symmetry in Γ(E ⊗ S 2r−1 H)
corresponds to a map Z → O(2r), Hamiltonian for the twisted
symplectic form Ω. Examples: ALE spaces.
hK manifolds of dim 4n with an abelian group of symmetries of
dimension n correspond to polyharmonic functions on R n ⊗ R 3
(harmonic on every v ⊗ R 3 ) (Lindström-Roček).E.g.
A k ALE space is given by ∑ k i=1
1
|x−x i | ,
1
|x−x i |
A k ALF space is given by c + ∑ k i=1
R n ⊗ R 3 is the image of the moment map (a space of τ-invariant
sections of O(2) × C n );
Similarly, hK manifolds of dimension 4n, with n commuting
generalised symmetries (Z → L n
i=1 O(2r i )) correspond to
solutions of a system of linear PDEs on a space of sections of
L n
i=1 O(2r i ) (generalised Legendre transform of Lindström and

Roček).
Examples include all toric hyperkähler manifolds (all r i are 1),
monopole metrics, asymptotic monopole metrics, gravitational
instantons, and various quiver varieties, in particular, hyperkähler
metrics on adjoint orbits of GL(n,C) and the natural metrics on
Calogero-Moser spaces.
In the twistor language, a generalised symmetry in Γ(E ⊗ S 2r−1 H)
corresponds to a map Z → O(2r), Hamiltonian for the twisted
symplectic form Ω. Examples: ALE spaces.
hK manifolds of dim 4n with an abelian group of symmetries of
dimension n correspond to polyharmonic functions on R n ⊗ R 3
(harmonic on every v ⊗ R 3 ) (Lindström-Roček).E.g.
A k ALE space is given by ∑ k i=1
1
|x−x i | ,
1
|x−x i |
A k ALF space is given by c + ∑ k i=1
R n ⊗ R 3 is the image of the moment map (a space of τ-invariant
sections of O(2) × C n );
Similarly, hK manifolds of dimension 4n, with n commuting
generalised symmetries (Z → L n
i=1 O(2r i )) correspond to
solutions of a system of linear PDEs on a space of sections of
L n
i=1 O(2r i ) (generalised Legendre transform of Lindström and

Roček).
Examples include all toric hyperkähler manifolds (all r i are 1),
monopole metrics, asymptotic monopole metrics, gravitational
instantons, and various quiver varieties, in particular, hyperkähler
metrics on adjoint orbits of GL(n,C) and the natural metrics on
Calogero-Moser spaces.
In the twistor language, a generalised symmetry in Γ(E ⊗ S 2r−1 H)
corresponds to a map Z → O(2r), Hamiltonian for the twisted
symplectic form Ω. Examples: ALE spaces.
hK manifolds of dim 4n with an abelian group of symmetries of
dimension n correspond to polyharmonic functions on R n ⊗ R 3
(harmonic on every v ⊗ R 3 ) (Lindström-Roček).E.g.
A k ALE space is given by ∑ k i=1
1
|x−x i | ,
1
|x−x i |
A k ALF space is given by c + ∑ k i=1
R n ⊗ R 3 is the image of the moment map (a space of τ-invariant
sections of O(2) × C n );
Similarly, hK manifolds of dimension 4n, with n commuting
generalised symmetries (Z → L n
i=1 O(2r i )) correspond to
solutions of a system of linear PDEs on a space of sections of
L n
i=1 O(2r i ) (generalised Legendre transform of Lindström and

Roček).
Examples include all toric hyperkähler manifolds (all r i are 1),
monopole metrics, asymptotic monopole metrics, gravitational
instantons, and various quiver varieties, in particular, hyperkähler
metrics on adjoint orbits of GL(n,C) and the natural metrics on
Calogero-Moser spaces.
In the twistor language, a generalised symmetry in Γ(E ⊗ S 2r−1 H)
corresponds to a map Z → O(2r), Hamiltonian for the twisted
symplectic form Ω. Examples: ALE spaces.
hK manifolds of dim 4n with an abelian group of symmetries of
dimension n correspond to polyharmonic functions on R n ⊗ R 3
(harmonic on every v ⊗ R 3 ) (Lindström-Roček).E.g.
A k ALE space is given by ∑ k i=1
1
|x−x i | ,
1
|x−x i |
A k ALF space is given by c + ∑ k i=1
R n ⊗ R 3 is the image of the moment map (a space of τ-invariant
sections of O(2) × C n );
Similarly, hK manifolds of dimension 4n, with n commuting
generalised symmetries (Z → L n
i=1 O(2r i )) correspond to
solutions of a system of linear PDEs on a space of sections of
L n
i=1 O(2r i ) (generalised Legendre transform of Lindström and

Roček).
Examples include all toric hyperkähler manifolds (all r i are 1),
monopole metrics, asymptotic monopole metrics, gravitational
instantons, and various quiver varieties, in particular, hyperkähler
metrics on adjoint orbits of GL(n,C) and the natural metrics on
Calogero-Moser spaces.
In the twistor language, a generalised symmetry in Γ(E ⊗ S 2r−1 H)
corresponds to a map Z → O(2r), Hamiltonian for the twisted
symplectic form Ω. Examples: ALE spaces.
hK manifolds of dim 4n with an abelian group of symmetries of
dimension n correspond to polyharmonic functions on R n ⊗ R 3
(harmonic on every v ⊗ R 3 ) (Lindström-Roček).E.g.
A k ALE space is given by ∑ k i=1
1
|x−x i | ,
1
|x−x i |
A k ALF space is given by c + ∑ k i=1
R n ⊗ R 3 is the image of the moment map (a space of τ-invariant
sections of O(2) × C n );
Similarly, hK manifolds of dimension 4n, with n commuting
generalised symmetries (Z → L n
i=1 O(2r i )) correspond to
solutions of a system of linear PDEs on a space of sections of
L n
i=1 O(2r i ) (generalised Legendre transform of Lindström and

Roček).
Examples include all toric hyperkähler manifolds (all r i are 1),
monopole metrics, asymptotic monopole metrics, gravitational
instantons, and various quiver varieties, in particular, hyperkähler
metrics on adjoint orbits of GL(n,C) and the natural metrics on
Calogero-Moser spaces.
In the twistor language, a generalised symmetry in Γ(E ⊗ S 2r−1 H)
corresponds to a map Z → O(2r), Hamiltonian for the twisted
symplectic form Ω. Examples: ALE spaces.
hK manifolds of dim 4n with an abelian group of symmetries of
dimension n correspond to polyharmonic functions on R n ⊗ R 3
(harmonic on every v ⊗ R 3 ) (Lindström-Roček).E.g.
A k ALE space is given by ∑ k i=1
1
|x−x i | ,
1
|x−x i |
A k ALF space is given by c + ∑ k i=1
R n ⊗ R 3 is the image of the moment map (a space of τ-invariant
sections of O(2) × C n );
Similarly, hK manifolds of dimension 4n, with n commuting
generalised symmetries (Z → L n
i=1 O(2r i )) correspond to
solutions of a system of linear PDEs on a space of sections of
L n
i=1 O(2r i ) (generalised Legendre transform of Lindström and

The construction
The maps f i : Z → O(2r i ) induce maps ˆfi from the space of sections of
Z , in particular from the manifold M, to the space of sections of O(2r i ),
i.e. to the space of polynomials of degree 2r i , which we write as
α i (ζ) =
2r i
∑
a=0
The real structure τ acts on this space by
w i aζ a .
τ(w i a) = (−1) r i+a w i 2r i −a
and, consequently, we obtain a map
) nM
ˆf =
(ˆf1 ,...,ˆfn : M → R 2ri+1 .
The image of ˆf (and the hyperkähler structure of M) is, in turn,
determined by a function F : L n
i=1 R 2ri+1 → R satisfying the system of
PDE’s:
F w i =
a ,w j F
b w i c ,w j (1)
d
for all a,b,c,d such that a + b = c + d.
i=1

The construction
The maps f i : Z → O(2r i ) induce maps ˆfi from the space of sections of
Z , in particular from the manifold M, to the space of sections of O(2r i ),
i.e. to the space of polynomials of degree 2r i , which we write as
α i (ζ) =
2r i
∑
a=0
The real structure τ acts on this space by
w i aζ a .
τ(w i a) = (−1) r i+a w i 2r i −a
and, consequently, we obtain a map
) nM
ˆf =
(ˆf1 ,...,ˆfn : M → R 2ri+1 .
The image of ˆf (and the hyperkähler structure of M) is, in turn,
determined by a function F : L n
i=1 R 2ri+1 → R satisfying the system of
PDE’s:
F w i =
a ,w j F
b w i c ,w j (1)
d
for all a,b,c,d such that a + b = c + d.
i=1

The construction
The maps f i : Z → O(2r i ) induce maps ˆfi from the space of sections of
Z , in particular from the manifold M, to the space of sections of O(2r i ),
i.e. to the space of polynomials of degree 2r i , which we write as
α i (ζ) =
2r i
∑
a=0
The real structure τ acts on this space by
w i aζ a .
τ(w i a) = (−1) r i+a w i 2r i −a
and, consequently, we obtain a map
) nM
ˆf =
(ˆf1 ,...,ˆfn : M → R 2ri+1 .
The image of ˆf (and the hyperkähler structure of M) is, in turn,
determined by a function F : L n
i=1 R 2ri+1 → R satisfying the system of
PDE’s:
F w i =
a ,w j F
b w i c ,w j (1)
d
for all a,b,c,d such that a + b = c + d.
i=1

An equivalent characterization of (1) is that F is given by a contour
integral of a holomorphic (possibly singular or multivalued) function of
2n + 1 variables G = G(ζ,α 1 ,...,α n )
I
F(wa) i =
( G ζ,α 1 (ζ),...,α n (ζ) ) /ζ 2 dζ
c
where α i (ζ) = ∑ 2r i
a=0 w i aζ a , or a sum of such contour integrals.
The function F determines the hyperkähler structure as follows. The
image of ˆf is given by
F w i a
= 0 if 2 ≤ a ≤ 2r i − 2. (2)
We have local complex coordinates z 1 ,...,z n ,u 1 ,...,u n :
{
z i = w0, i u i if r i ≥ 2
F w i
1
=
u i + ū i if r i = 1.
Then the hK metric of M is given by a Kähler potential:
K = F −∑(u i w i 1 + ū i ¯w 1).
i
i

An equivalent characterization of (1) is that F is given by a contour
integral of a holomorphic (possibly singular or multivalued) function of
2n + 1 variables G = G(ζ,α 1 ,...,α n )
I
F(wa) i =
( G ζ,α 1 (ζ),...,α n (ζ) ) /ζ 2 dζ
c
where α i (ζ) = ∑ 2r i
a=0 w i aζ a , or a sum of such contour integrals.
The function F determines the hyperkähler structure as follows. The
image of ˆf is given by
F w i a
= 0 if 2 ≤ a ≤ 2r i − 2. (2)
We have local complex coordinates z 1 ,...,z n ,u 1 ,...,u n :
{
z i = w0, i u i if r i ≥ 2
F w i
1
=
u i + ū i if r i = 1.
Then the hK metric of M is given by a Kähler potential:
K = F −∑(u i w i 1 + ū i ¯w 1).
i
i

An equivalent characterization of (1) is that F is given by a contour
integral of a holomorphic (possibly singular or multivalued) function of
2n + 1 variables G = G(ζ,α 1 ,...,α n )
I
F(wa) i =
( G ζ,α 1 (ζ),...,α n (ζ) ) /ζ 2 dζ
c
where α i (ζ) = ∑ 2r i
a=0 w i aζ a , or a sum of such contour integrals.
The function F determines the hyperkähler structure as follows. The
image of ˆf is given by
F w i a
= 0 if 2 ≤ a ≤ 2r i − 2. (2)
We have local complex coordinates z 1 ,...,z n ,u 1 ,...,u n :
{
z i = w0, i u i if r i ≥ 2
F w i
1
=
u i + ū i if r i = 1.
Then the hK metric of M is given by a Kähler potential:
K = F −∑(u i w i 1 + ū i ¯w 1).
i
i

An equivalent characterization of (1) is that F is given by a contour
integral of a holomorphic (possibly singular or multivalued) function of
2n + 1 variables G = G(ζ,α 1 ,...,α n )
I
F(wa) i =
( G ζ,α 1 (ζ),...,α n (ζ) ) /ζ 2 dζ
c
where α i (ζ) = ∑ 2r i
a=0 w i aζ a , or a sum of such contour integrals.
The function F determines the hyperkähler structure as follows. The
image of ˆf is given by
F w i a
= 0 if 2 ≤ a ≤ 2r i − 2. (2)
We have local complex coordinates z 1 ,...,z n ,u 1 ,...,u n :
{
z i = w0, i u i if r i ≥ 2
F w i
1
=
u i + ū i if r i = 1.
Then the hK metric of M is given by a Kähler potential:
K = F −∑(u i w i 1 + ū i ¯w 1).
i
i

The construction is very similar to the case of an abelian symmetry, as
explained by Dunajski and Mason. At least in the case, when all r i = r,
the function F (without any extremisation conditions) produces a
generalised hypercomplex manifold N with an actual n-dimensional
symmetry.
N has a twistor space Z(N) → CP 1 ; N is a space of real sections
with normal bundle a sum of O(r)’s.
N has dimension n(2r + 1) + n and is foliated by 4n-dimensional
hK manifolds with generalised symmetries, cf. equation (2).
The foliation corresponds to a sequence of blow-ups of Z(N) at a
pair of points interchanged by τ.
End of introduction.
New examples, properties, asymptotics:

The construction is very similar to the case of an abelian symmetry, as
explained by Dunajski and Mason. At least in the case, when all r i = r,
the function F (without any extremisation conditions) produces a
generalised hypercomplex manifold N with an actual n-dimensional
symmetry.
N has a twistor space Z(N) → CP 1 ; N is a space of real sections
with normal bundle a sum of O(r)’s.
N has dimension n(2r + 1) + n and is foliated by 4n-dimensional
hK manifolds with generalised symmetries, cf. equation (2).
The foliation corresponds to a sequence of blow-ups of Z(N) at a
pair of points interchanged by τ.
End of introduction.
New examples, properties, asymptotics:

The construction is very similar to the case of an abelian symmetry, as
explained by Dunajski and Mason. At least in the case, when all r i = r,
the function F (without any extremisation conditions) produces a
generalised hypercomplex manifold N with an actual n-dimensional
symmetry.
N has a twistor space Z(N) → CP 1 ; N is a space of real sections
with normal bundle a sum of O(r)’s.
N has dimension n(2r + 1) + n and is foliated by 4n-dimensional
hK manifolds with generalised symmetries, cf. equation (2).
The foliation corresponds to a sequence of blow-ups of Z(N) at a
pair of points interchanged by τ.
End of introduction.
New examples, properties, asymptotics:

The construction is very similar to the case of an abelian symmetry, as
explained by Dunajski and Mason. At least in the case, when all r i = r,
the function F (without any extremisation conditions) produces a
generalised hypercomplex manifold N with an actual n-dimensional
symmetry.
N has a twistor space Z(N) → CP 1 ; N is a space of real sections
with normal bundle a sum of O(r)’s.
N has dimension n(2r + 1) + n and is foliated by 4n-dimensional
hK manifolds with generalised symmetries, cf. equation (2).
The foliation corresponds to a sequence of blow-ups of Z(N) at a
pair of points interchanged by τ.
End of introduction.
New examples, properties, asymptotics:

The construction is very similar to the case of an abelian symmetry, as
explained by Dunajski and Mason. At least in the case, when all r i = r,
the function F (without any extremisation conditions) produces a
generalised hypercomplex manifold N with an actual n-dimensional
symmetry.
N has a twistor space Z(N) → CP 1 ; N is a space of real sections
with normal bundle a sum of O(r)’s.
N has dimension n(2r + 1) + n and is foliated by 4n-dimensional
hK manifolds with generalised symmetries, cf. equation (2).
The foliation corresponds to a sequence of blow-ups of Z(N) at a
pair of points interchanged by τ.
End of introduction.
New examples, properties, asymptotics:

GLT on spectral curves
Recall that F = H c G/ζ2 dζ, where G can be multivalued. Instead, we
can consider single-valued (meromorphic) functions on some
branched covering of E = ⊕ n i=1O(2r i ).
For example, when E = O(2) ⊕O(4) we can consider a 2-fold
covering:
D 1 = {(η,α 1 ,α 2 ) ∈ O(2) ⊕O(2) ⊕O(4); η 2 + α 1 η + α 2 = 0},
or a 3-fold one:
D 2 = {(η,α 1 ,α 2 ) ∈ O(2) ⊕O(2) ⊕O(4); (η + α 1 )(η 2 + α 2 ) = 0}.
The space of real sections of E should be now viewed as a space of
spectral curves - all compact curves in |O(4)| for D 1 , and a subset of
the set of reducible compact curves in |O(6)| for D 2 .

GLT on spectral curves
Recall that F = H c G/ζ2 dζ, where G can be multivalued. Instead, we
can consider single-valued (meromorphic) functions on some
branched covering of E = ⊕ n i=1O(2r i ).
For example, when E = O(2) ⊕O(4) we can consider a 2-fold
covering:
D 1 = {(η,α 1 ,α 2 ) ∈ O(2) ⊕O(2) ⊕O(4); η 2 + α 1 η + α 2 = 0},
or a 3-fold one:
D 2 = {(η,α 1 ,α 2 ) ∈ O(2) ⊕O(2) ⊕O(4); (η + α 1 )(η 2 + α 2 ) = 0}.
The space of real sections of E should be now viewed as a space of
spectral curves - all compact curves in |O(4)| for D 1 , and a subset of
the set of reducible compact curves in |O(6)| for D 2 .

GLT on spectral curves
Recall that F = H c G/ζ2 dζ, where G can be multivalued. Instead, we
can consider single-valued (meromorphic) functions on some
branched covering of E = ⊕ n i=1O(2r i ).
For example, when E = O(2) ⊕O(4) we can consider a 2-fold
covering:
D 1 = {(η,α 1 ,α 2 ) ∈ O(2) ⊕O(2) ⊕O(4); η 2 + α 1 η + α 2 = 0},
or a 3-fold one:
D 2 = {(η,α 1 ,α 2 ) ∈ O(2) ⊕O(2) ⊕O(4); (η + α 1 )(η 2 + α 2 ) = 0}.
The space of real sections of E should be now viewed as a space of
spectral curves - all compact curves in |O(4)| for D 1 , and a subset of
the set of reducible compact curves in |O(6)| for D 2 .

Identify elements of L m l
i=1 R2i+1 with τ-invariant polynomials
P l (ζ,η) = η m l
+ ∑ m l
i=1 α i(ζ)η m l−i
= 0;
Identify elements of S = L k
L ml
l=1 i=1 R2i+1 with τ-invariant
singular curves S = S 1 ∪ ··· ∪ S k given by the equation
k
∏
l=1
P l (ζ,η) = 0.
Consider GLT given by a meromorphic function G(ζ,η) on S,
and a the cycle c defined on these curves.
Example
Let k = 1 and let
F = − 1 I
η 2
2πi ˜0 ζ dζ + η
Ic
3 ζ dζ, 2
where ˜0 is the sum of simple contours around points in the fibre ζ = 0
of S. We obtain the natural hyperkähler metric on the moduli space of
SU(2)-monopoles of charge m (Ivanov-Roček and Houghton).

Identify elements of L m l
i=1 R2i+1 with τ-invariant polynomials
P l (ζ,η) = η m l
+ ∑ m l
i=1 α i(ζ)η m l−i
= 0;
Identify elements of S = L k
L ml
l=1 i=1 R2i+1 with τ-invariant
singular curves S = S 1 ∪ ··· ∪ S k given by the equation
k
∏
l=1
P l (ζ,η) = 0.
Consider GLT given by a meromorphic function G(ζ,η) on S,
and a the cycle c defined on these curves.
Example
Let k = 1 and let
F = − 1 I
η 2
2πi ˜0 ζ dζ + η
Ic
3 ζ dζ, 2
where ˜0 is the sum of simple contours around points in the fibre ζ = 0
of S. We obtain the natural hyperkähler metric on the moduli space of
SU(2)-monopoles of charge m (Ivanov-Roček and Houghton).

Identify elements of L m l
i=1 R2i+1 with τ-invariant polynomials
P l (ζ,η) = η m l
+ ∑ m l
i=1 α i(ζ)η m l−i
= 0;
Identify elements of S = L k
L ml
l=1 i=1 R2i+1 with τ-invariant
singular curves S = S 1 ∪ ··· ∪ S k given by the equation
k
∏
l=1
P l (ζ,η) = 0.
Consider GLT given by a meromorphic function G(ζ,η) on S,
and a the cycle c defined on these curves.
Example
Let k = 1 and let
F = − 1 I
η 2
2πi ˜0 ζ dζ + η
Ic
3 ζ dζ, 2
where ˜0 is the sum of simple contours around points in the fibre ζ = 0
of S. We obtain the natural hyperkähler metric on the moduli space of
SU(2)-monopoles of charge m (Ivanov-Roček and Houghton).

Identify elements of L m l
i=1 R2i+1 with τ-invariant polynomials
P l (ζ,η) = η m l
+ ∑ m l
i=1 α i(ζ)η m l−i
= 0;
Identify elements of S = L k
L ml
l=1 i=1 R2i+1 with τ-invariant
singular curves S = S 1 ∪ ··· ∪ S k given by the equation
k
∏
l=1
P l (ζ,η) = 0.
Consider GLT given by a meromorphic function G(ζ,η) on S,
and a the cycle c defined on these curves.
Example
Let k = 1 and let
F = − 1 I
η 2
2πi ˜0 ζ dζ + η
Ic
3 ζ dζ, 2
where ˜0 is the sum of simple contours around points in the fibre ζ = 0
of S. We obtain the natural hyperkähler metric on the moduli space of
SU(2)-monopoles of charge m (Ivanov-Roček and Houghton).

Triviality of line bundles
In this example, the extremisation condition F w i a
= 0, 2 ≤ a ≤ 2i − 2,
becomes equivalent to L 2 |S ≃ O, where L2 is the line bundle on TP 1
with transition function exp(2η/ζ) (from ζ ≠ 0 to ζ ≠ ∞). In fact, this
turns out to be always true:
If F = ∑ q 1
p=1
Hc p
G
ζ 2 p (ζ,η)dζ on S = S(m 1 ,...,m k ), then the
extremisation conditions for the GLT imply are equivalent to
triviality of certain line bundles on each S l , l = 1,...,k.
Example
Let F be as for SU(2)-monopoles, but defined on S(m 1 ,...,m k ). The
cycle c defines on each S l two divisors ∆ − l
,∆ + l
(entrance and exit
points of c) of the same degree. The extremisation conditions are
equivalent to
L 2 |S l
⊗ [∆ − l
− ∆ + l
] ≃ O
on each S l .

Triviality of line bundles
In this example, the extremisation condition F w i a
= 0, 2 ≤ a ≤ 2i − 2,
becomes equivalent to L 2 |S ≃ O, where L2 is the line bundle on TP 1
with transition function exp(2η/ζ) (from ζ ≠ 0 to ζ ≠ ∞). In fact, this
turns out to be always true:
If F = ∑ q 1
p=1
Hc p
G
ζ 2 p (ζ,η)dζ on S = S(m 1 ,...,m k ), then the
extremisation conditions for the GLT imply are equivalent to
triviality of certain line bundles on each S l , l = 1,...,k.
Example
Let F be as for SU(2)-monopoles, but defined on S(m 1 ,...,m k ). The
cycle c defines on each S l two divisors ∆ − l
,∆ + l
(entrance and exit
points of c) of the same degree. The extremisation conditions are
equivalent to
L 2 |S l
⊗ [∆ − l
− ∆ + l
] ≃ O
on each S l .

Triviality of line bundles
In this example, the extremisation condition F w i a
= 0, 2 ≤ a ≤ 2i − 2,
becomes equivalent to L 2 |S ≃ O, where L2 is the line bundle on TP 1
with transition function exp(2η/ζ) (from ζ ≠ 0 to ζ ≠ ∞). In fact, this
turns out to be always true:
If F = ∑ q 1
p=1
Hc p
G
ζ 2 p (ζ,η)dζ on S = S(m 1 ,...,m k ), then the
extremisation conditions for the GLT imply are equivalent to
triviality of certain line bundles on each S l , l = 1,...,k.
Example
Let F be as for SU(2)-monopoles, but defined on S(m 1 ,...,m k ). The
cycle c defines on each S l two divisors ∆ − l
,∆ + l
(entrance and exit
points of c) of the same degree. The extremisation conditions are
equivalent to
L 2 |S l
⊗ [∆ − l
− ∆ + l
] ≃ O
on each S l .

Depending on the cycle chosen, this last example gives asymptotic
monopole metrics (when an SU(2)-monopole of charge m becomes
asymptotically a superposition of monopoles of charges m 1 ,...,m k ) or
the natural metric on the moduli space of SU(k + 1)-monopoles of
charge (m 1 ,...,m k ). Or (conjecturally) asymptotic metrics on moduli
spaces of SU(k + 1)-monopoles.
Replace L 2 with other line bundles on T P 1 .
Most generally, define F on S(m 1 ,...,m k ) by
I
η
F =
ζ dζ − 1 k I
1
2 2πi ˜0l ζ 2 H l(ζ,η)dζ,
c
∑
l=1
where each H l is a linear combination of monomials η i /ζ j ,
i,j > 0,
get a (pseudo)-hyperkähler metric which lives on an open subset
of reducible spectral curves S 1 ∪ ··· ∪ S k such that on each S l
E l |Sl ≃ [ ∆ + l
− ∆ − ]
l ,
where E l is the line bundle on T P 1 with transition function
exp ∂H l
∂η .

Depending on the cycle chosen, this last example gives asymptotic
monopole metrics (when an SU(2)-monopole of charge m becomes
asymptotically a superposition of monopoles of charges m 1 ,...,m k ) or
the natural metric on the moduli space of SU(k + 1)-monopoles of
charge (m 1 ,...,m k ). Or (conjecturally) asymptotic metrics on moduli
spaces of SU(k + 1)-monopoles.
Replace L 2 with other line bundles on T P 1 .
Most generally, define F on S(m 1 ,...,m k ) by
I
η
F =
ζ dζ − 1 k I
1
2 2πi ˜0l ζ 2 H l(ζ,η)dζ,
c
∑
l=1
where each H l is a linear combination of monomials η i /ζ j ,
i,j > 0,
get a (pseudo)-hyperkähler metric which lives on an open subset
of reducible spectral curves S 1 ∪ ··· ∪ S k such that on each S l
E l |Sl ≃ [ ∆ + l
− ∆ − ]
l ,
where E l is the line bundle on T P 1 with transition function
exp ∂H l
∂η .

Depending on the cycle chosen, this last example gives asymptotic
monopole metrics (when an SU(2)-monopole of charge m becomes
asymptotically a superposition of monopoles of charges m 1 ,...,m k ) or
the natural metric on the moduli space of SU(k + 1)-monopoles of
charge (m 1 ,...,m k ). Or (conjecturally) asymptotic metrics on moduli
spaces of SU(k + 1)-monopoles.
Replace L 2 with other line bundles on T P 1 .
Most generally, define F on S(m 1 ,...,m k ) by
I
η
F =
ζ dζ − 1 k I
1
2 2πi ˜0l ζ 2 H l(ζ,η)dζ,
c
∑
l=1
where each H l is a linear combination of monomials η i /ζ j ,
i,j > 0,
get a (pseudo)-hyperkähler metric which lives on an open subset
of reducible spectral curves S 1 ∪ ··· ∪ S k such that on each S l
E l |Sl ≃ [ ∆ + l
− ∆ − ]
l ,
where E l is the line bundle on T P 1 with transition function
exp ∂H l
∂η .

Depending on the cycle chosen, this last example gives asymptotic
monopole metrics (when an SU(2)-monopole of charge m becomes
asymptotically a superposition of monopoles of charges m 1 ,...,m k ) or
the natural metric on the moduli space of SU(k + 1)-monopoles of
charge (m 1 ,...,m k ). Or (conjecturally) asymptotic metrics on moduli
spaces of SU(k + 1)-monopoles.
Replace L 2 with other line bundles on T P 1 .
Most generally, define F on S(m 1 ,...,m k ) by
I
η
F =
ζ dζ − 1 k I
1
2 2πi ˜0l ζ 2 H l(ζ,η)dζ,
c
∑
l=1
where each H l is a linear combination of monomials η i /ζ j ,
i,j > 0,
get a (pseudo)-hyperkähler metric which lives on an open subset
of reducible spectral curves S 1 ∪ ··· ∪ S k such that on each S l
E l |Sl ≃ [ ∆ + l
− ∆ − ]
l ,
where E l is the line bundle on T P 1 with transition function
exp ∂H l
∂η .

Depending on the cycle chosen, this last example gives asymptotic
monopole metrics (when an SU(2)-monopole of charge m becomes
asymptotically a superposition of monopoles of charges m 1 ,...,m k ) or
the natural metric on the moduli space of SU(k + 1)-monopoles of
charge (m 1 ,...,m k ). Or (conjecturally) asymptotic metrics on moduli
spaces of SU(k + 1)-monopoles.
Replace L 2 with other line bundles on T P 1 .
Most generally, define F on S(m 1 ,...,m k ) by
I
η
F =
ζ dζ − 1 k I
1
2 2πi ˜0l ζ 2 H l(ζ,η)dζ,
c
∑
l=1
where each H l is a linear combination of monomials η i /ζ j ,
i,j > 0,
get a (pseudo)-hyperkähler metric which lives on an open subset
of reducible spectral curves S 1 ∪ ··· ∪ S k such that on each S l
E l |Sl ≃ [ ∆ + l
− ∆ − ]
l ,
where E l is the line bundle on T P 1 with transition function
exp ∂H l
∂η .

Example
A huge class of hyperkähler metrics in dimension
4n = 4(m 1 + ··· + m k )
Many complete examples
Further investigation necessary
Let S be a spectral curve of degree k and let W S ⊂ S(1,...,k) (i.e.
curves S 1 ∪ ··· ∪ S k , degS i = i) be defined by setting S k = S. Then
the GLT w.r.t.
I
F = −
c
η
dζ (3)
ζ2 on W S defines a hyperkähler metric on (an open subset of) a coadjoint
orbit of GL(k,C). When S is fully reducible, i.e. union of lines, then
these are the metrics of Kronheimer-Biquard-Kovalev, otherwise we
get their generalisation due to D’Amorim Santa-Cruz.
(-) “Line bundles on spectral curves and the generalised Legendre
transform”, arXiv:0806.0510

Example
A huge class of hyperkähler metrics in dimension
4n = 4(m 1 + ··· + m k )
Many complete examples
Further investigation necessary
Let S be a spectral curve of degree k and let W S ⊂ S(1,...,k) (i.e.
curves S 1 ∪ ··· ∪ S k , degS i = i) be defined by setting S k = S. Then
the GLT w.r.t.
I
F = −
c
η
dζ (3)
ζ2 on W S defines a hyperkähler metric on (an open subset of) a coadjoint
orbit of GL(k,C). When S is fully reducible, i.e. union of lines, then
these are the metrics of Kronheimer-Biquard-Kovalev, otherwise we
get their generalisation due to D’Amorim Santa-Cruz.
(-) “Line bundles on spectral curves and the generalised Legendre
transform”, arXiv:0806.0510

Example
A huge class of hyperkähler metrics in dimension
4n = 4(m 1 + ··· + m k )
Many complete examples
Further investigation necessary
Let S be a spectral curve of degree k and let W S ⊂ S(1,...,k) (i.e.
curves S 1 ∪ ··· ∪ S k , degS i = i) be defined by setting S k = S. Then
the GLT w.r.t.
I
F = −
c
η
dζ (3)
ζ2 on W S defines a hyperkähler metric on (an open subset of) a coadjoint
orbit of GL(k,C). When S is fully reducible, i.e. union of lines, then
these are the metrics of Kronheimer-Biquard-Kovalev, otherwise we
get their generalisation due to D’Amorim Santa-Cruz.
(-) “Line bundles on spectral curves and the generalised Legendre
transform”, arXiv:0806.0510

Example
A huge class of hyperkähler metrics in dimension
4n = 4(m 1 + ··· + m k )
Many complete examples
Further investigation necessary
Let S be a spectral curve of degree k and let W S ⊂ S(1,...,k) (i.e.
curves S 1 ∪ ··· ∪ S k , degS i = i) be defined by setting S k = S. Then
the GLT w.r.t.
I
F = −
c
η
dζ (3)
ζ2 on W S defines a hyperkähler metric on (an open subset of) a coadjoint
orbit of GL(k,C). When S is fully reducible, i.e. union of lines, then
these are the metrics of Kronheimer-Biquard-Kovalev, otherwise we
get their generalisation due to D’Amorim Santa-Cruz.
(-) “Line bundles on spectral curves and the generalised Legendre
transform”, arXiv:0806.0510

Asymptotics
I
F =
c
η
ζ 2 dζ − 1
2πi
k
∑
l=1
I
˜0l
1
ζ 2 H l(ζ,η)dζ =
I
c
k
η
ζ dζ − ∑ 2 h l (S l ),
l=1
where h l is a polynomial depending only on the coefficients of P l
defining S l .
We consider the asymptotics of the manifold M, when the curves
S tend to infinity (as polynomials).
If all h l ≡ 0, the function F and the metric are homogeneous: M is
a cone (in particular, Euclidean volume growth)
If some h l ≢ 0, let m n be a sequence of points M such that the
coefficients of the corresponding P l tend to ∞.
Rescale P l (ζ,η) ↦→ P l /R m l
(ζ,R˜η) so that the limit exists and
defines a compact curve S ∞ l
.
If ∂h l
(2 ≤ a ≤ 2i − 2) has a nonzero limit after rescaling, then S ∞ wa
i l
cannot have components of degree ≥ 1.

Asymptotics
I
F =
c
η
ζ 2 dζ − 1
2πi
k
∑
l=1
I
˜0l
1
ζ 2 H l(ζ,η)dζ =
I
c
k
η
ζ dζ − ∑ 2 h l (S l ),
l=1
where h l is a polynomial depending only on the coefficients of P l
defining S l .
We consider the asymptotics of the manifold M, when the curves
S tend to infinity (as polynomials).
If all h l ≡ 0, the function F and the metric are homogeneous: M is
a cone (in particular, Euclidean volume growth)
If some h l ≢ 0, let m n be a sequence of points M such that the
coefficients of the corresponding P l tend to ∞.
Rescale P l (ζ,η) ↦→ P l /R m l
(ζ,R˜η) so that the limit exists and
defines a compact curve S ∞ l
.
If ∂h l
(2 ≤ a ≤ 2i − 2) has a nonzero limit after rescaling, then S ∞ wa
i l
cannot have components of degree ≥ 1.

Asymptotics
I
F =
c
η
ζ 2 dζ − 1
2πi
k
∑
l=1
I
˜0l
1
ζ 2 H l(ζ,η)dζ =
I
c
k
η
ζ dζ − ∑ 2 h l (S l ),
l=1
where h l is a polynomial depending only on the coefficients of P l
defining S l .
We consider the asymptotics of the manifold M, when the curves
S tend to infinity (as polynomials).
If all h l ≡ 0, the function F and the metric are homogeneous: M is
a cone (in particular, Euclidean volume growth)
If some h l ≢ 0, let m n be a sequence of points M such that the
coefficients of the corresponding P l tend to ∞.
Rescale P l (ζ,η) ↦→ P l /R m l
(ζ,R˜η) so that the limit exists and
defines a compact curve S ∞ l
.
If ∂h l
(2 ≤ a ≤ 2i − 2) has a nonzero limit after rescaling, then S ∞ wa
i l
cannot have components of degree ≥ 1.

Asymptotics
I
F =
c
η
ζ 2 dζ − 1
2πi
k
∑
l=1
I
˜0l
1
ζ 2 H l(ζ,η)dζ =
I
c
k
η
ζ dζ − ∑ 2 h l (S l ),
l=1
where h l is a polynomial depending only on the coefficients of P l
defining S l .
We consider the asymptotics of the manifold M, when the curves
S tend to infinity (as polynomials).
If all h l ≡ 0, the function F and the metric are homogeneous: M is
a cone (in particular, Euclidean volume growth)
If some h l ≢ 0, let m n be a sequence of points M such that the
coefficients of the corresponding P l tend to ∞.
Rescale P l (ζ,η) ↦→ P l /R m l
(ζ,R˜η) so that the limit exists and
defines a compact curve S ∞ l
.
If ∂h l
(2 ≤ a ≤ 2i − 2) has a nonzero limit after rescaling, then S ∞ wa
i l
cannot have components of degree ≥ 1.

Asymptotics
I
F =
c
η
ζ 2 dζ − 1
2πi
k
∑
l=1
I
˜0l
1
ζ 2 H l(ζ,η)dζ =
I
c
k
η
ζ dζ − ∑ 2 h l (S l ),
l=1
where h l is a polynomial depending only on the coefficients of P l
defining S l .
We consider the asymptotics of the manifold M, when the curves
S tend to infinity (as polynomials).
If all h l ≡ 0, the function F and the metric are homogeneous: M is
a cone (in particular, Euclidean volume growth)
If some h l ≢ 0, let m n be a sequence of points M such that the
coefficients of the corresponding P l tend to ∞.
Rescale P l (ζ,η) ↦→ P l /R m l
(ζ,R˜η) so that the limit exists and
defines a compact curve S ∞ l
.
If ∂h l
(2 ≤ a ≤ 2i − 2) has a nonzero limit after rescaling, then S ∞ wa
i l
cannot have components of degree ≥ 1.

This follows from: the derivative of the first term with respect to a
coefficient w i a (2 ≤ a ≤ 2i − 2) of P l is R γ l
Ω ai , where Ω ai is a
holomorphic differential on S l .
Consider the region U of M, where there exists a rescaling with
the limit being the union of rational curves, without multiple
components.
The metric on U approaches the metric defined by the ordinary
Legendre transform (i.e. with n commuting tri-holomorphic Killing
vector fields) exponentially fast.
The generalised symmetries become asympotically genuine
symmetries.
U is asymptotically an R p × T n−p -bundle over ∏ k l=1 C m l
(R 3 )
(C ml (R 3 ) - the configuration space of m l distinct points in R 3 ).
At least in the case, when all E l correspond to L s , the infinity of M
should be the quotient of an R p × T n−p -bundle over the unit
sphere in R 3n by the product of symmetric groups ∏ k l=1 Σ ml .

This follows from: the derivative of the first term with respect to a
coefficient w i a (2 ≤ a ≤ 2i − 2) of P l is R γ l
Ω ai , where Ω ai is a
holomorphic differential on S l .
Consider the region U of M, where there exists a rescaling with
the limit being the union of rational curves, without multiple
components.
The metric on U approaches the metric defined by the ordinary
Legendre transform (i.e. with n commuting tri-holomorphic Killing
vector fields) exponentially fast.
The generalised symmetries become asympotically genuine
symmetries.
U is asymptotically an R p × T n−p -bundle over ∏ k l=1 C m l
(R 3 )
(C ml (R 3 ) - the configuration space of m l distinct points in R 3 ).
At least in the case, when all E l correspond to L s , the infinity of M
should be the quotient of an R p × T n−p -bundle over the unit
sphere in R 3n by the product of symmetric groups ∏ k l=1 Σ ml .

This follows from: the derivative of the first term with respect to a
coefficient w i a (2 ≤ a ≤ 2i − 2) of P l is R γ l
Ω ai , where Ω ai is a
holomorphic differential on S l .
Consider the region U of M, where there exists a rescaling with
the limit being the union of rational curves, without multiple
components.
The metric on U approaches the metric defined by the ordinary
Legendre transform (i.e. with n commuting tri-holomorphic Killing
vector fields) exponentially fast.
The generalised symmetries become asympotically genuine
symmetries.
U is asymptotically an R p × T n−p -bundle over ∏ k l=1 C m l
(R 3 )
(C ml (R 3 ) - the configuration space of m l distinct points in R 3 ).
At least in the case, when all E l correspond to L s , the infinity of M
should be the quotient of an R p × T n−p -bundle over the unit
sphere in R 3n by the product of symmetric groups ∏ k l=1 Σ ml .

This follows from: the derivative of the first term with respect to a
coefficient w i a (2 ≤ a ≤ 2i − 2) of P l is R γ l
Ω ai , where Ω ai is a
holomorphic differential on S l .
Consider the region U of M, where there exists a rescaling with
the limit being the union of rational curves, without multiple
components.
The metric on U approaches the metric defined by the ordinary
Legendre transform (i.e. with n commuting tri-holomorphic Killing
vector fields) exponentially fast.
The generalised symmetries become asympotically genuine
symmetries.
U is asymptotically an R p × T n−p -bundle over ∏ k l=1 C m l
(R 3 )
(C ml (R 3 ) - the configuration space of m l distinct points in R 3 ).
At least in the case, when all E l correspond to L s , the infinity of M
should be the quotient of an R p × T n−p -bundle over the unit
sphere in R 3n by the product of symmetric groups ∏ k l=1 Σ ml .

This follows from: the derivative of the first term with respect to a
coefficient w i a (2 ≤ a ≤ 2i − 2) of P l is R γ l
Ω ai , where Ω ai is a
holomorphic differential on S l .
Consider the region U of M, where there exists a rescaling with
the limit being the union of rational curves, without multiple
components.
The metric on U approaches the metric defined by the ordinary
Legendre transform (i.e. with n commuting tri-holomorphic Killing
vector fields) exponentially fast.
The generalised symmetries become asympotically genuine
symmetries.
U is asymptotically an R p × T n−p -bundle over ∏ k l=1 C m l
(R 3 )
(C ml (R 3 ) - the configuration space of m l distinct points in R 3 ).
At least in the case, when all E l correspond to L s , the infinity of M
should be the quotient of an R p × T n−p -bundle over the unit
sphere in R 3n by the product of symmetric groups ∏ k l=1 Σ ml .

This follows from: the derivative of the first term with respect to a
coefficient w i a (2 ≤ a ≤ 2i − 2) of P l is R γ l
Ω ai , where Ω ai is a
holomorphic differential on S l .
Consider the region U of M, where there exists a rescaling with
the limit being the union of rational curves, without multiple
components.
The metric on U approaches the metric defined by the ordinary
Legendre transform (i.e. with n commuting tri-holomorphic Killing
vector fields) exponentially fast.
The generalised symmetries become asympotically genuine
symmetries.
U is asymptotically an R p × T n−p -bundle over ∏ k l=1 C m l
(R 3 )
(C ml (R 3 ) - the configuration space of m l distinct points in R 3 ).
At least in the case, when all E l correspond to L s , the infinity of M
should be the quotient of an R p × T n−p -bundle over the unit
sphere in R 3n by the product of symmetric groups ∏ k l=1 Σ ml .