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Yang-Mills equations in higher dimensions(joint work with C. Shahbazi)
Vicente Muñoz
(Universidad Complutense de Madrid)
BCAM, 22 September 2016
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 1 / 22
Geometry
Universe: locally (around a point) is like Rn. It can be coordinatized.
Geometry: concept of smooth manifold.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 2 / 22
Geometry
Universe: locally (around a point) is like Rn. It can be coordinatized.
Geometry: concept of smooth manifold.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 2 / 22
Geometry
Universe: locally (around a point) is like Rn. It can be coordinatized.
Geometry: concept of smooth manifold.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 2 / 22
Geometry
Global geometry of manifolds:
Classification of smooth manifolds of dimension n.
Relevant (geometrically and physically): compact case.
Universe (space): n = 3.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 3 / 22
Geometry
Global geometry of manifolds:
Classification of smooth manifolds of dimension n.
Relevant (geometrically and physically): compact case.
Universe (space): n = 3.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 3 / 22
Differential equations on manifolds
Let M be a smooth manifold.Let (x1, . . . , xn) be (local) coordinates.
Consider a differential equation, for instance the heat equation:
∂T∂t
= 4T =∑ ∂2T
∂x2i
where T is the temperature.
If we consider other coordinates (y1, . . . , yn), then the above equationbecomes an equation on yj which has a different aspect.
need of intrinsic formulations.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 4 / 22
Differential equations on manifolds
Let M be a smooth manifold.Let (x1, . . . , xn) be (local) coordinates.Consider a differential equation, for instance the heat equation:
∂T∂t
= 4T =∑ ∂2T
∂x2i
where T is the temperature.
If we consider other coordinates (y1, . . . , yn), then the above equationbecomes an equation on yj which has a different aspect.
need of intrinsic formulations.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 4 / 22
Differential equations on manifolds
Let M be a smooth manifold.Let (x1, . . . , xn) be (local) coordinates.Consider a differential equation, for instance the heat equation:
∂T∂t
= 4T =∑ ∂2T
∂x2i
where T is the temperature.
If we consider other coordinates (y1, . . . , yn), then the above equationbecomes an equation on yj which has a different aspect.
need of intrinsic formulations.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 4 / 22
Connections
For each p ∈ M, we take a vector space Ep where the magnitudes takevalues.
Notion of fiber bundle E → M.Let X (p) ∈ Ep, for each p ∈ M (denoted X ∈ Γ(E))∂X∂xi
(p) = limh→0X(p+hei )−X(p)
h does not have sense.
To define derivatives, we need for p,q ∈ M, isomorphisms
Pγ : Ep −→ Eq
where γ is a path that joins p to q. This Pγ is called parallel transportand it gives rise to the notion of connection ∇:∇X∂xi
(p) = limh→0Pp+hej ,p
X(p+hei )−X(p)h
To parallel transport an object is to move it with zero derivative (rigidly).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 5 / 22
Connections
For each p ∈ M, we take a vector space Ep where the magnitudes takevalues. Notion of fiber bundle E → M.
Let X (p) ∈ Ep, for each p ∈ M (denoted X ∈ Γ(E))∂X∂xi
(p) = limh→0X(p+hei )−X(p)
h does not have sense.
To define derivatives, we need for p,q ∈ M, isomorphisms
Pγ : Ep −→ Eq
where γ is a path that joins p to q. This Pγ is called parallel transportand it gives rise to the notion of connection ∇:∇X∂xi
(p) = limh→0Pp+hej ,p
X(p+hei )−X(p)h
To parallel transport an object is to move it with zero derivative (rigidly).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 5 / 22
Connections
For each p ∈ M, we take a vector space Ep where the magnitudes takevalues. Notion of fiber bundle E → M.Let X (p) ∈ Ep, for each p ∈ M (denoted X ∈ Γ(E))∂X∂xi
(p) = limh→0X(p+hei )−X(p)
h does not have sense.
To define derivatives, we need for p,q ∈ M, isomorphisms
Pγ : Ep −→ Eq
where γ is a path that joins p to q. This Pγ is called parallel transportand it gives rise to the notion of connection ∇:∇X∂xi
(p) = limh→0Pp+hej ,p
X(p+hei )−X(p)h
To parallel transport an object is to move it with zero derivative (rigidly).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 5 / 22
Connections
For each p ∈ M, we take a vector space Ep where the magnitudes takevalues. Notion of fiber bundle E → M.Let X (p) ∈ Ep, for each p ∈ M (denoted X ∈ Γ(E))∂X∂xi
(p) = limh→0X(p+hei )−X(p)
h does not have sense.
To define derivatives, we need for p,q ∈ M, isomorphisms
Pγ : Ep −→ Eq
where γ is a path that joins p to q. This Pγ is called parallel transport
and it gives rise to the notion of connection ∇:∇X∂xi
(p) = limh→0Pp+hej ,p
X(p+hei )−X(p)h
To parallel transport an object is to move it with zero derivative (rigidly).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 5 / 22
Connections
For each p ∈ M, we take a vector space Ep where the magnitudes takevalues. Notion of fiber bundle E → M.Let X (p) ∈ Ep, for each p ∈ M (denoted X ∈ Γ(E))∂X∂xi
(p) = limh→0X(p+hei )−X(p)
h does not have sense.
To define derivatives, we need for p,q ∈ M, isomorphisms
Pγ : Ep −→ Eq
where γ is a path that joins p to q. This Pγ is called parallel transportand it gives rise to the notion of connection ∇:∇X∂xi
(p) = limh→0Pp+hej ,p
X(p+hei )−X(p)h
To parallel transport an object is to move it with zero derivative (rigidly).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 5 / 22
Connections
For each p ∈ M, we take a vector space Ep where the magnitudes takevalues. Notion of fiber bundle E → M.Let X (p) ∈ Ep, for each p ∈ M (denoted X ∈ Γ(E))∂X∂xi
(p) = limh→0X(p+hei )−X(p)
h does not have sense.
To define derivatives, we need for p,q ∈ M, isomorphisms
Pγ : Ep −→ Eq
where γ is a path that joins p to q. This Pγ is called parallel transportand it gives rise to the notion of connection ∇:∇X∂xi
(p) = limh→0Pp+hej ,p
X(p+hei )−X(p)h
To parallel transport an object is to move it with zero derivative (rigidly).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 5 / 22
Metrics
TpM tangent space at p ∈ M.TM → M tangent bundle, X ∈ Γ(TM) is a vector field.
A Riemannian metric is to give a scalar product g(p) on TpM for eachp ∈ M.This is given as a positive definite symmetric matrix in coordinates:g(x) = (gij(x)).A Riemannian manifold (M,g) is a physical space where lengths,angles, volumes have intrinsic meaning.Levi-Civita connection ∇: unique connection with parallel transportpreserving length.
Heat equation:
4T =∑
i
∇∂xi
∇T∂xi
on o.n. basis
=∑i,j
g ij(x)∇∂xi
∇T∂xj
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 6 / 22
Metrics
TpM tangent space at p ∈ M.TM → M tangent bundle, X ∈ Γ(TM) is a vector field.A Riemannian metric is to give a scalar product g(p) on TpM for eachp ∈ M.This is given as a positive definite symmetric matrix in coordinates:g(x) = (gij(x)).
A Riemannian manifold (M,g) is a physical space where lengths,angles, volumes have intrinsic meaning.Levi-Civita connection ∇: unique connection with parallel transportpreserving length.
Heat equation:
4T =∑
i
∇∂xi
∇T∂xi
on o.n. basis
=∑i,j
g ij(x)∇∂xi
∇T∂xj
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 6 / 22
Metrics
TpM tangent space at p ∈ M.TM → M tangent bundle, X ∈ Γ(TM) is a vector field.A Riemannian metric is to give a scalar product g(p) on TpM for eachp ∈ M.This is given as a positive definite symmetric matrix in coordinates:g(x) = (gij(x)).A Riemannian manifold (M,g) is a physical space where lengths,angles, volumes have intrinsic meaning.
Levi-Civita connection ∇: unique connection with parallel transportpreserving length.
Heat equation:
4T =∑
i
∇∂xi
∇T∂xi
on o.n. basis
=∑i,j
g ij(x)∇∂xi
∇T∂xj
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 6 / 22
Metrics
TpM tangent space at p ∈ M.TM → M tangent bundle, X ∈ Γ(TM) is a vector field.A Riemannian metric is to give a scalar product g(p) on TpM for eachp ∈ M.This is given as a positive definite symmetric matrix in coordinates:g(x) = (gij(x)).A Riemannian manifold (M,g) is a physical space where lengths,angles, volumes have intrinsic meaning.Levi-Civita connection ∇: unique connection with parallel transportpreserving length.
Heat equation:
4T =∑
i
∇∂xi
∇T∂xi
on o.n. basis
=∑i,j
g ij(x)∇∂xi
∇T∂xj
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 6 / 22
Metrics
TpM tangent space at p ∈ M.TM → M tangent bundle, X ∈ Γ(TM) is a vector field.A Riemannian metric is to give a scalar product g(p) on TpM for eachp ∈ M.This is given as a positive definite symmetric matrix in coordinates:g(x) = (gij(x)).A Riemannian manifold (M,g) is a physical space where lengths,angles, volumes have intrinsic meaning.Levi-Civita connection ∇: unique connection with parallel transportpreserving length.
Heat equation:
4T =∑
i
∇∂xi
∇T∂xi
on o.n. basis
=∑i,j
g ij(x)∇∂xi
∇T∂xj
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 6 / 22
Holonomy
Let (M,g) be a Riemannian manifold. Consider the collection of allparallel transport around loops.It yields the holonomy group Holg ⊂ O(n).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 7 / 22
Possible holonomy groups
Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:
SO(n)
U(n), Kähler manifold, dim M = 2n.SU(n), Kähler manifold Ricci flat (Calabi-Yau).Sp(n), hyperKähler manifold, dim M = 4n.Sp(n) · Sp(1), quaternionic Kähler manifold, dim M = 4n.G2 < SO(7), dim M = 7.Spin(7) < SO(8), dim M = 8.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 8 / 22
Possible holonomy groups
Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:
SO(n)
U(n), Kähler manifold, dim M = 2n.SU(n), Kähler manifold Ricci flat (Calabi-Yau).Sp(n), hyperKähler manifold, dim M = 4n.Sp(n) · Sp(1), quaternionic Kähler manifold, dim M = 4n.G2 < SO(7), dim M = 7.Spin(7) < SO(8), dim M = 8.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 8 / 22
Possible holonomy groups
Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:
SO(n)
U(n), Kähler manifold, dim M = 2n.
SU(n), Kähler manifold Ricci flat (Calabi-Yau).Sp(n), hyperKähler manifold, dim M = 4n.Sp(n) · Sp(1), quaternionic Kähler manifold, dim M = 4n.G2 < SO(7), dim M = 7.Spin(7) < SO(8), dim M = 8.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 8 / 22
Possible holonomy groups
Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:
SO(n)
U(n), Kähler manifold, dim M = 2n.SU(n), Kähler manifold Ricci flat (Calabi-Yau).
Sp(n), hyperKähler manifold, dim M = 4n.Sp(n) · Sp(1), quaternionic Kähler manifold, dim M = 4n.G2 < SO(7), dim M = 7.Spin(7) < SO(8), dim M = 8.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 8 / 22
Possible holonomy groups
Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:
SO(n)
U(n), Kähler manifold, dim M = 2n.SU(n), Kähler manifold Ricci flat (Calabi-Yau).Sp(n), hyperKähler manifold, dim M = 4n.
Sp(n) · Sp(1), quaternionic Kähler manifold, dim M = 4n.G2 < SO(7), dim M = 7.Spin(7) < SO(8), dim M = 8.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 8 / 22
Possible holonomy groups
Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:
SO(n)
U(n), Kähler manifold, dim M = 2n.SU(n), Kähler manifold Ricci flat (Calabi-Yau).Sp(n), hyperKähler manifold, dim M = 4n.Sp(n) · Sp(1), quaternionic Kähler manifold, dim M = 4n.
G2 < SO(7), dim M = 7.Spin(7) < SO(8), dim M = 8.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 8 / 22
Possible holonomy groups
Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:
SO(n)
U(n), Kähler manifold, dim M = 2n.SU(n), Kähler manifold Ricci flat (Calabi-Yau).Sp(n), hyperKähler manifold, dim M = 4n.Sp(n) · Sp(1), quaternionic Kähler manifold, dim M = 4n.G2 < SO(7), dim M = 7.
Spin(7) < SO(8), dim M = 8.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 8 / 22
Possible holonomy groups
Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:
SO(n)
U(n), Kähler manifold, dim M = 2n.SU(n), Kähler manifold Ricci flat (Calabi-Yau).Sp(n), hyperKähler manifold, dim M = 4n.Sp(n) · Sp(1), quaternionic Kähler manifold, dim M = 4n.G2 < SO(7), dim M = 7.Spin(7) < SO(8), dim M = 8.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 8 / 22
Curvature
Let ∇ be a connection.In coordinates, ∇∂xi
= ∂∂xi
+ Γi
∇ = d + Γ, Γ =∑
Γidxi ∈∧1⊗EndE
Curvature: K (ei ,ej)(X ) = ∇∂xi
∇∂xj
X − ∇∂xj
∇∂xi
X
K = dΓ + Γ ∧ Γ ∈∧2⊗EndE
For the Levi-Civita connection: K ∈ TM ⊗ TM ⊗ TM ⊗ TMRicci tensor: Ric = tr13K
Relation to holonomy:Holonomy Hol∇ ⊂ EndE , with Lie algebra hol∇ ⊂ EndEK ∈
∧2⊗hol∇
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 9 / 22
Curvature
Let ∇ be a connection.In coordinates, ∇∂xi
= ∂∂xi
+ Γi
∇ = d + Γ, Γ =∑
Γidxi ∈∧1⊗EndE
Curvature: K (ei ,ej)(X ) = ∇∂xi
∇∂xj
X − ∇∂xj
∇∂xi
X
K = dΓ + Γ ∧ Γ ∈∧2⊗EndE
For the Levi-Civita connection: K ∈ TM ⊗ TM ⊗ TM ⊗ TMRicci tensor: Ric = tr13K
Relation to holonomy:Holonomy Hol∇ ⊂ EndE , with Lie algebra hol∇ ⊂ EndEK ∈
∧2⊗hol∇
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 9 / 22
Curvature
Let ∇ be a connection.In coordinates, ∇∂xi
= ∂∂xi
+ Γi
∇ = d + Γ, Γ =∑
Γidxi ∈∧1⊗EndE
Curvature: K (ei ,ej)(X ) = ∇∂xi
∇∂xj
X − ∇∂xj
∇∂xi
X
K = dΓ + Γ ∧ Γ ∈∧2⊗EndE
For the Levi-Civita connection: K ∈ TM ⊗ TM ⊗ TM ⊗ TM
Ricci tensor: Ric = tr13K
Relation to holonomy:Holonomy Hol∇ ⊂ EndE , with Lie algebra hol∇ ⊂ EndEK ∈
∧2⊗hol∇
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 9 / 22
Curvature
Let ∇ be a connection.In coordinates, ∇∂xi
= ∂∂xi
+ Γi
∇ = d + Γ, Γ =∑
Γidxi ∈∧1⊗EndE
Curvature: K (ei ,ej)(X ) = ∇∂xi
∇∂xj
X − ∇∂xj
∇∂xi
X
K = dΓ + Γ ∧ Γ ∈∧2⊗EndE
For the Levi-Civita connection: K ∈ TM ⊗ TM ⊗ TM ⊗ TMRicci tensor: Ric = tr13K
Relation to holonomy:Holonomy Hol∇ ⊂ EndE , with Lie algebra hol∇ ⊂ EndEK ∈
∧2⊗hol∇
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 9 / 22
Curvature
Let ∇ be a connection.In coordinates, ∇∂xi
= ∂∂xi
+ Γi
∇ = d + Γ, Γ =∑
Γidxi ∈∧1⊗EndE
Curvature: K (ei ,ej)(X ) = ∇∂xi
∇∂xj
X − ∇∂xj
∇∂xi
X
K = dΓ + Γ ∧ Γ ∈∧2⊗EndE
For the Levi-Civita connection: K ∈ TM ⊗ TM ⊗ TM ⊗ TMRicci tensor: Ric = tr13K
Relation to holonomy:Holonomy Hol∇ ⊂ EndE , with Lie algebra hol∇ ⊂ EndEK ∈
∧2⊗hol∇
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 9 / 22
General relativity
Universe (space-time): n = 4.Special relativity: R4, (t , x , y , z),g = −dt2 + dx2 + dy2 + dz3, of signature (3,1).General relativity: (M4,g) a 4-manifold with a Lorentz metric.
Einstein’s field equations:
Ricij −12
gij + Λgij =8πGc4 Tij
relevance of 4-manifolds.
String theory:Particles are strings vibrating on different sates, of the length of thePlanck size.They vibrate on an extra direction→ universe is M3,1 × N,N is a riemannian manifold of dimension 6,7,8 with holonomySU(3),G2,Spin(7).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 10 / 22
General relativity
Universe (space-time): n = 4.Special relativity: R4, (t , x , y , z),g = −dt2 + dx2 + dy2 + dz3, of signature (3,1).General relativity: (M4,g) a 4-manifold with a Lorentz metric.
Einstein’s field equations:
Ricij −12
gij + Λgij =8πGc4 Tij
relevance of 4-manifolds.
String theory:Particles are strings vibrating on different sates, of the length of thePlanck size.They vibrate on an extra direction→ universe is M3,1 × N,N is a riemannian manifold of dimension 6,7,8 with holonomySU(3),G2,Spin(7).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 10 / 22
General relativity
Universe (space-time): n = 4.Special relativity: R4, (t , x , y , z),g = −dt2 + dx2 + dy2 + dz3, of signature (3,1).General relativity: (M4,g) a 4-manifold with a Lorentz metric.
Einstein’s field equations:
Ricij −12
gij + Λgij =8πGc4 Tij
relevance of 4-manifolds.
String theory:Particles are strings vibrating on different sates, of the length of thePlanck size.They vibrate on an extra direction→ universe is M3,1 × N,N is a riemannian manifold of dimension 6,7,8 with holonomySU(3),G2,Spin(7).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 10 / 22
General relativity
Universe (space-time): n = 4.Special relativity: R4, (t , x , y , z),g = −dt2 + dx2 + dy2 + dz3, of signature (3,1).General relativity: (M4,g) a 4-manifold with a Lorentz metric.
Einstein’s field equations:
Ricij −12
gij + Λgij =8πGc4 Tij
relevance of 4-manifolds.
String theory:Particles are strings vibrating on different sates, of the length of thePlanck size.They vibrate on an extra direction→ universe is M3,1 × N,
N is a riemannian manifold of dimension 6,7,8 with holonomySU(3),G2,Spin(7).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 10 / 22
General relativity
Universe (space-time): n = 4.Special relativity: R4, (t , x , y , z),g = −dt2 + dx2 + dy2 + dz3, of signature (3,1).General relativity: (M4,g) a 4-manifold with a Lorentz metric.
Einstein’s field equations:
Ricij −12
gij + Λgij =8πGc4 Tij
relevance of 4-manifolds.
String theory:Particles are strings vibrating on different sates, of the length of thePlanck size.They vibrate on an extra direction→ universe is M3,1 × N,N is a riemannian manifold of dimension 6,7,8 with holonomySU(3),G2,Spin(7).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 10 / 22
Maxwell equations
Describe the electromagnetic field (photon).E = (Ex ,Ey ,Ez) electric field,B = (Bx ,By ,Bz) magnetic field.
F =∑
Eidxi ∧ dt +∑
Bidxj ∧ dxk
Then F = dA, A is a potential, A ∈∧1⊗EndE
E → M complex fiber bundle of rank 1Maxwell equation: d∗F = 0
Interpretation: ∇A = d + A connectionF = dA curvature (the group U(1) is abelian, so A ∧ A = 0)
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 11 / 22
Maxwell equations
Describe the electromagnetic field (photon).E = (Ex ,Ey ,Ez) electric field,B = (Bx ,By ,Bz) magnetic field.
F =∑
Eidxi ∧ dt +∑
Bidxj ∧ dxk
Then F = dA, A is a potential, A ∈∧1⊗EndE
E → M complex fiber bundle of rank 1Maxwell equation: d∗F = 0
Interpretation: ∇A = d + A connectionF = dA curvature (the group U(1) is abelian, so A ∧ A = 0)
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 11 / 22
Maxwell equations
Describe the electromagnetic field (photon).E = (Ex ,Ey ,Ez) electric field,B = (Bx ,By ,Bz) magnetic field.
F =∑
Eidxi ∧ dt +∑
Bidxj ∧ dxk
Then F = dA, A is a potential, A ∈∧1⊗EndE
E → M complex fiber bundle of rank 1Maxwell equation: d∗F = 0
Interpretation: ∇A = d + A connectionF = dA curvature (the group U(1) is abelian, so A ∧ A = 0)
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 11 / 22
Maxwell equations
Gauge group: (super)symmetryGE = Aut E acts onAE = space of connections on Eeiϕ : E → E sends A 7→ A + dϕ
So the connections modulo gauge (i.e. the space AE/GE ) isparametrized by the equation d∗A = 0.
Solutions to Maxwell equations are equivalent to dd∗A = 0, d∗dA = 0,i.e. ∆A = 0, which is of elliptic type.
On compact, riemannian, 4-manifold, theory of elliptic operatorsprovide many results.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 12 / 22
Maxwell equations
Gauge group: (super)symmetryGE = Aut E acts onAE = space of connections on Eeiϕ : E → E sends A 7→ A + dϕ
So the connections modulo gauge (i.e. the space AE/GE ) isparametrized by the equation d∗A = 0.
Solutions to Maxwell equations are equivalent to dd∗A = 0, d∗dA = 0,i.e. ∆A = 0, which is of elliptic type.
On compact, riemannian, 4-manifold, theory of elliptic operatorsprovide many results.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 12 / 22
Maxwell equations
Gauge group: (super)symmetryGE = Aut E acts onAE = space of connections on Eeiϕ : E → E sends A 7→ A + dϕ
So the connections modulo gauge (i.e. the space AE/GE ) isparametrized by the equation d∗A = 0.
Solutions to Maxwell equations are equivalent to dd∗A = 0, d∗dA = 0,
i.e. ∆A = 0, which is of elliptic type.
On compact, riemannian, 4-manifold, theory of elliptic operatorsprovide many results.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 12 / 22
Maxwell equations
Gauge group: (super)symmetryGE = Aut E acts onAE = space of connections on Eeiϕ : E → E sends A 7→ A + dϕ
So the connections modulo gauge (i.e. the space AE/GE ) isparametrized by the equation d∗A = 0.
Solutions to Maxwell equations are equivalent to dd∗A = 0, d∗dA = 0,i.e. ∆A = 0, which is of elliptic type.
On compact, riemannian, 4-manifold, theory of elliptic operatorsprovide many results.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 12 / 22
Maxwell equations
Gauge group: (super)symmetryGE = Aut E acts onAE = space of connections on Eeiϕ : E → E sends A 7→ A + dϕ
So the connections modulo gauge (i.e. the space AE/GE ) isparametrized by the equation d∗A = 0.
Solutions to Maxwell equations are equivalent to dd∗A = 0, d∗dA = 0,i.e. ∆A = 0, which is of elliptic type.
On compact,
riemannian, 4-manifold, theory of elliptic operatorsprovide many results.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 12 / 22
Maxwell equations
Gauge group: (super)symmetryGE = Aut E acts onAE = space of connections on Eeiϕ : E → E sends A 7→ A + dϕ
So the connections modulo gauge (i.e. the space AE/GE ) isparametrized by the equation d∗A = 0.
Solutions to Maxwell equations are equivalent to dd∗A = 0, d∗dA = 0,i.e. ∆A = 0, which is of elliptic type.
On compact, riemannian,
4-manifold, theory of elliptic operatorsprovide many results.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 12 / 22
Maxwell equations
Gauge group: (super)symmetryGE = Aut E acts onAE = space of connections on Eeiϕ : E → E sends A 7→ A + dϕ
So the connections modulo gauge (i.e. the space AE/GE ) isparametrized by the equation d∗A = 0.
Solutions to Maxwell equations are equivalent to dd∗A = 0, d∗dA = 0,i.e. ∆A = 0, which is of elliptic type.
On compact, riemannian, 4-manifold, theory of elliptic operatorsprovide many results.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 12 / 22
Yang-Mills equations
Other forces require higher dimensional complex vector bundles, andinvolve non-abelian groups U(r) (and others).
Let E → M be a (unitary) complex vector bundle of rank r ,M compact, riemannian n-manifold.AE = space of connections on E .
Locally a connection is d + A, A ∈∧1⊗u(r)
FA = dA + A ∧ A curvature of A.Bianchi identity: dAFA = 0.
Yang-Mills Functional:YM(A) =
∫M |FA|2 (energy).
Critical points: d∗AFA = 0.(Second order) elliptic equation on M.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 13 / 22
Yang-Mills equations
Other forces require higher dimensional complex vector bundles, andinvolve non-abelian groups U(r) (and others).
Let E → M be a (unitary) complex vector bundle of rank r ,M compact, riemannian n-manifold.AE = space of connections on E .
Locally a connection is d + A, A ∈∧1⊗u(r)
FA = dA + A ∧ A curvature of A.Bianchi identity: dAFA = 0.
Yang-Mills Functional:YM(A) =
∫M |FA|2 (energy).
Critical points: d∗AFA = 0.(Second order) elliptic equation on M.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 13 / 22
Yang-Mills equations
Other forces require higher dimensional complex vector bundles, andinvolve non-abelian groups U(r) (and others).
Let E → M be a (unitary) complex vector bundle of rank r ,M compact, riemannian n-manifold.AE = space of connections on E .
Locally a connection is d + A, A ∈∧1⊗u(r)
FA = dA + A ∧ A curvature of A.Bianchi identity: dAFA = 0.
Yang-Mills Functional:YM(A) =
∫M |FA|2 (energy).
Critical points: d∗AFA = 0.(Second order) elliptic equation on M.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 13 / 22
Yang-Mills equations
Other forces require higher dimensional complex vector bundles, andinvolve non-abelian groups U(r) (and others).
Let E → M be a (unitary) complex vector bundle of rank r ,M compact, riemannian n-manifold.AE = space of connections on E .
Locally a connection is d + A, A ∈∧1⊗u(r)
FA = dA + A ∧ A curvature of A.Bianchi identity: dAFA = 0.
Yang-Mills Functional:YM(A) =
∫M |FA|2 (energy).
Critical points: d∗AFA = 0.(Second order) elliptic equation on M.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 13 / 22
Yang-Mills equations
Other forces require higher dimensional complex vector bundles, andinvolve non-abelian groups U(r) (and others).
Let E → M be a (unitary) complex vector bundle of rank r ,M compact, riemannian n-manifold.AE = space of connections on E .
Locally a connection is d + A, A ∈∧1⊗u(r)
FA = dA + A ∧ A curvature of A.Bianchi identity: dAFA = 0.
Yang-Mills Functional:YM(A) =
∫M |FA|2 (energy).
Critical points: d∗AFA = 0.(Second order) elliptic equation on M.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 13 / 22
Yang-Mills equations
Other forces require higher dimensional complex vector bundles, andinvolve non-abelian groups U(r) (and others).
Let E → M be a (unitary) complex vector bundle of rank r ,M compact, riemannian n-manifold.AE = space of connections on E .
Locally a connection is d + A, A ∈∧1⊗u(r)
FA = dA + A ∧ A curvature of A.Bianchi identity: dAFA = 0.
Yang-Mills Functional:YM(A) =
∫M |FA|2 (energy).
Critical points: d∗AFA = 0.(Second order) elliptic equation on M.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 13 / 22
Anti-self-dual connections (instantons)
Hodge operator: ∗ :∧2 →
∧n−2, 〈∗α, β〉 = α ∧ β.
Yang-Mills equation: dA ∗ FA = 0
For n = 4, ∗∗ = id,∧2 =
∧2+⊕
∧2−, both of rank 3.
Minima of YM(A) =∫
M |FA|2 (instantons) given by:∗FA = −FA (i.e. F+
A = 0).These are called anti-self-dual connections.
Moduli space: ME = A ∈ AE |F+A = 0/GE
Theorem [Donaldson, Atiyah-Hitchin-Singer]Let r = 2. For a generic metric g,ME is a smooth manifold of finitedimension (at irreducible instantons).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 14 / 22
Anti-self-dual connections (instantons)
Hodge operator: ∗ :∧2 →
∧n−2, 〈∗α, β〉 = α ∧ β.Yang-Mills equation: dA ∗ FA = 0
For n = 4, ∗∗ = id,∧2 =
∧2+⊕
∧2−, both of rank 3.
Minima of YM(A) =∫
M |FA|2 (instantons) given by:∗FA = −FA (i.e. F+
A = 0).These are called anti-self-dual connections.
Moduli space: ME = A ∈ AE |F+A = 0/GE
Theorem [Donaldson, Atiyah-Hitchin-Singer]Let r = 2. For a generic metric g,ME is a smooth manifold of finitedimension (at irreducible instantons).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 14 / 22
Anti-self-dual connections (instantons)
Hodge operator: ∗ :∧2 →
∧n−2, 〈∗α, β〉 = α ∧ β.Yang-Mills equation: dA ∗ FA = 0
For n = 4, ∗∗ = id,∧2 =
∧2+⊕
∧2−, both of rank 3.
Minima of YM(A) =∫
M |FA|2 (instantons) given by:∗FA = −FA (i.e. F+
A = 0).These are called anti-self-dual connections.
Moduli space: ME = A ∈ AE |F+A = 0/GE
Theorem [Donaldson, Atiyah-Hitchin-Singer]Let r = 2. For a generic metric g,ME is a smooth manifold of finitedimension (at irreducible instantons).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 14 / 22
Anti-self-dual connections (instantons)
Hodge operator: ∗ :∧2 →
∧n−2, 〈∗α, β〉 = α ∧ β.Yang-Mills equation: dA ∗ FA = 0
For n = 4, ∗∗ = id,∧2 =
∧2+⊕
∧2−, both of rank 3.
Minima of YM(A) =∫
M |FA|2 (instantons) given by:∗FA = −FA (i.e. F+
A = 0).These are called anti-self-dual connections.
Moduli space: ME = A ∈ AE |F+A = 0/GE
Theorem [Donaldson, Atiyah-Hitchin-Singer]Let r = 2. For a generic metric g,ME is a smooth manifold of finitedimension (at irreducible instantons).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 14 / 22
Anti-self-dual connections (instantons)
Hodge operator: ∗ :∧2 →
∧n−2, 〈∗α, β〉 = α ∧ β.Yang-Mills equation: dA ∗ FA = 0
For n = 4, ∗∗ = id,∧2 =
∧2+⊕
∧2−, both of rank 3.
Minima of YM(A) =∫
M |FA|2 (instantons) given by:∗FA = −FA (i.e. F+
A = 0).These are called anti-self-dual connections.
Moduli space: ME = A ∈ AE |F+A = 0/GE
Theorem [Donaldson, Atiyah-Hitchin-Singer]Let r = 2. For a generic metric g,ME is a smooth manifold of finitedimension (at irreducible instantons).
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 14 / 22
ProofLet A0 ∈ME . Locally, solutions to the ASD equation are given byA = A0 + τ , τ ∈
∧1⊗EndE , |τ | < ε, with F+A = 0.
We have FA = FA0 + dA0τ + τ ∧ τ , so the equation isd+
A0τ + (τ ∧ τ)+ = 0.
The linearization of the equation is
d+A0
:∧1⊗EndE −→
∧2
+⊗EndE
The gauge group GE acts as h ∈ Γ(Aut E), ∇ 7→ h−1 ∇ h,i.e., h 7→ h−1A0h + h−1dh.The linearization of the action is given by ϕ 7→ dϕ+ [A0, ϕ] = dA0ϕ,
dA0 :∧0⊗EndE −→
∧1⊗EndE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 15 / 22
ProofLet A0 ∈ME . Locally, solutions to the ASD equation are given byA = A0 + τ , τ ∈
∧1⊗EndE , |τ | < ε, with F+A = 0.
We have FA = FA0 + dA0τ + τ ∧ τ , so the equation isd+
A0τ + (τ ∧ τ)+ = 0.
The linearization of the equation is
d+A0
:∧1⊗EndE −→
∧2
+⊗EndE
The gauge group GE acts as h ∈ Γ(Aut E), ∇ 7→ h−1 ∇ h,i.e., h 7→ h−1A0h + h−1dh.The linearization of the action is given by ϕ 7→ dϕ+ [A0, ϕ] = dA0ϕ,
dA0 :∧0⊗EndE −→
∧1⊗EndE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 15 / 22
ProofLet A0 ∈ME . Locally, solutions to the ASD equation are given byA = A0 + τ , τ ∈
∧1⊗EndE , |τ | < ε, with F+A = 0.
We have FA = FA0 + dA0τ + τ ∧ τ , so the equation isd+
A0τ + (τ ∧ τ)+ = 0.
The linearization of the equation is
d+A0
:∧1⊗EndE −→
∧2
+⊗EndE
The gauge group GE acts as h ∈ Γ(Aut E), ∇ 7→ h−1 ∇ h,i.e., h 7→ h−1A0h + h−1dh.The linearization of the action is given by ϕ 7→ dϕ+ [A0, ϕ] = dA0ϕ,
dA0 :∧0⊗EndE −→
∧1⊗EndE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 15 / 22
ProofLet A0 ∈ME . Locally, solutions to the ASD equation are given byA = A0 + τ , τ ∈
∧1⊗EndE , |τ | < ε, with F+A = 0.
We have FA = FA0 + dA0τ + τ ∧ τ , so the equation isd+
A0τ + (τ ∧ τ)+ = 0.
The linearization of the equation is
d+A0
:∧1⊗EndE −→
∧2
+⊗EndE
The gauge group GE acts as h ∈ Γ(Aut E), ∇ 7→ h−1 ∇ h,i.e., h 7→ h−1A0h + h−1dh.The linearization of the action is given by ϕ 7→ dϕ+ [A0, ϕ] = dA0ϕ,
dA0 :∧0⊗EndE −→
∧1⊗EndE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 15 / 22
ProofLet A0 ∈ME . Locally, solutions to the ASD equation are given byA = A0 + τ , τ ∈
∧1⊗EndE , |τ | < ε, with F+A = 0.
We have FA = FA0 + dA0τ + τ ∧ τ , so the equation isd+
A0τ + (τ ∧ τ)+ = 0.
The linearization of the equation is
d+A0
:∧1⊗EndE −→
∧2
+⊗EndE
The gauge group GE acts as h ∈ Γ(Aut E), ∇ 7→ h−1 ∇ h,i.e., h 7→ h−1A0h + h−1dh.
The linearization of the action is given by ϕ 7→ dϕ+ [A0, ϕ] = dA0ϕ,
dA0 :∧0⊗EndE −→
∧1⊗EndE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 15 / 22
ProofLet A0 ∈ME . Locally, solutions to the ASD equation are given byA = A0 + τ , τ ∈
∧1⊗EndE , |τ | < ε, with F+A = 0.
We have FA = FA0 + dA0τ + τ ∧ τ , so the equation isd+
A0τ + (τ ∧ τ)+ = 0.
The linearization of the equation is
d+A0
:∧1⊗EndE −→
∧2
+⊗EndE
The gauge group GE acts as h ∈ Γ(Aut E), ∇ 7→ h−1 ∇ h,i.e., h 7→ h−1A0h + h−1dh.The linearization of the action is given by ϕ 7→ dϕ+ [A0, ϕ] = dA0ϕ,
dA0 :∧0⊗EndE −→
∧1⊗EndE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 15 / 22
The deformation complex is elliptic:
∧0 ⊗ EndE
dA0−→∧
1 ⊗ EndEd+
A0−→∧
2+ ⊗ EndE
The local model forME = A0 + τ |F+A0+τ
= 0/GE around A0
is given by ker d+A0∩ ker d∗A0
, i.e., the first homology of the ellipticcomplex.The operator LA0 = d+
A0⊕ d∗A0
is Fredholm, so this is finite-dimensional.Surjectivity of LA0 implies that the space of solutions is smooth.
Let H0A0,H1
A0,H2
A0be the homology of the complex,
H0A0
is the Lie algebra of the h ∈ GE such that h · A0 = A0.So A0 is irreducible (not a direct sum of lower rank connections) if andonly if H0
A0= 0.
Smoothness is equivalent to H2A0
= 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 16 / 22
The deformation complex is elliptic:
∧0 ⊗ EndE
dA0−→∧
1 ⊗ EndEd+
A0−→∧
2+ ⊗ EndE
The local model forME = A0 + τ |F+A0+τ
= 0/GE around A0
is given by ker d+A0∩ ker d∗A0
, i.e., the first homology of the ellipticcomplex.The operator LA0 = d+
A0⊕ d∗A0
is Fredholm, so this is finite-dimensional.Surjectivity of LA0 implies that the space of solutions is smooth.
Let H0A0,H1
A0,H2
A0be the homology of the complex,
H0A0
is the Lie algebra of the h ∈ GE such that h · A0 = A0.So A0 is irreducible (not a direct sum of lower rank connections) if andonly if H0
A0= 0.
Smoothness is equivalent to H2A0
= 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 16 / 22
The deformation complex is elliptic:
∧0 ⊗ EndE
dA0−→∧
1 ⊗ EndEd+
A0−→∧
2+ ⊗ EndE
The local model forME = A0 + τ |F+A0+τ
= 0/GE around A0
is given by ker d+A0∩ ker d∗A0
, i.e., the first homology of the ellipticcomplex.
The operator LA0 = d+A0⊕ d∗A0
is Fredholm, so this is finite-dimensional.Surjectivity of LA0 implies that the space of solutions is smooth.
Let H0A0,H1
A0,H2
A0be the homology of the complex,
H0A0
is the Lie algebra of the h ∈ GE such that h · A0 = A0.So A0 is irreducible (not a direct sum of lower rank connections) if andonly if H0
A0= 0.
Smoothness is equivalent to H2A0
= 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 16 / 22
The deformation complex is elliptic:
∧0 ⊗ EndE
dA0−→∧
1 ⊗ EndEd+
A0−→∧
2+ ⊗ EndE
The local model forME = A0 + τ |F+A0+τ
= 0/GE around A0
is given by ker d+A0∩ ker d∗A0
, i.e., the first homology of the ellipticcomplex.The operator LA0 = d+
A0⊕ d∗A0
is Fredholm, so this is finite-dimensional.Surjectivity of LA0 implies that the space of solutions is smooth.
Let H0A0,H1
A0,H2
A0be the homology of the complex,
H0A0
is the Lie algebra of the h ∈ GE such that h · A0 = A0.So A0 is irreducible (not a direct sum of lower rank connections) if andonly if H0
A0= 0.
Smoothness is equivalent to H2A0
= 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 16 / 22
The deformation complex is elliptic:
∧0 ⊗ EndE
dA0−→∧
1 ⊗ EndEd+
A0−→∧
2+ ⊗ EndE
The local model forME = A0 + τ |F+A0+τ
= 0/GE around A0
is given by ker d+A0∩ ker d∗A0
, i.e., the first homology of the ellipticcomplex.The operator LA0 = d+
A0⊕ d∗A0
is Fredholm, so this is finite-dimensional.Surjectivity of LA0 implies that the space of solutions is smooth.
Let H0A0,H1
A0,H2
A0be the homology of the complex,
H0A0
is the Lie algebra of the h ∈ GE such that h · A0 = A0.So A0 is irreducible (not a direct sum of lower rank connections) if andonly if H0
A0= 0.
Smoothness is equivalent to H2A0
= 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 16 / 22
The deformation complex is elliptic:
∧0 ⊗ EndE
dA0−→∧
1 ⊗ EndEd+
A0−→∧
2+ ⊗ EndE
The local model forME = A0 + τ |F+A0+τ
= 0/GE around A0
is given by ker d+A0∩ ker d∗A0
, i.e., the first homology of the ellipticcomplex.The operator LA0 = d+
A0⊕ d∗A0
is Fredholm, so this is finite-dimensional.Surjectivity of LA0 implies that the space of solutions is smooth.
Let H0A0,H1
A0,H2
A0be the homology of the complex,
H0A0
is the Lie algebra of the h ∈ GE such that h · A0 = A0.So A0 is irreducible (not a direct sum of lower rank connections) if andonly if H0
A0= 0.
Smoothness is equivalent to H2A0
= 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 16 / 22
The deformation complex is elliptic:
∧0 ⊗ EndE
dA0−→∧
1 ⊗ EndEd+
A0−→∧
2+ ⊗ EndE
The local model forME = A0 + τ |F+A0+τ
= 0/GE around A0
is given by ker d+A0∩ ker d∗A0
, i.e., the first homology of the ellipticcomplex.The operator LA0 = d+
A0⊕ d∗A0
is Fredholm, so this is finite-dimensional.Surjectivity of LA0 implies that the space of solutions is smooth.
Let H0A0,H1
A0,H2
A0be the homology of the complex,
H0A0
is the Lie algebra of the h ∈ GE such that h · A0 = A0.So A0 is irreducible (not a direct sum of lower rank connections) if andonly if H0
A0= 0.
Smoothness is equivalent to H2A0
= 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 16 / 22
Let N be the space of metrics. Consider
1 + ∗g2 dA0 : N ×
∧1⊗EndE −→
∧2
+⊗EndE
The linearisation with respect to g ∈ N is
12µ d−,gA0
+ d+,gA0
where µ :∧2− →
∧2+
ψ ∈∧2
+ is orthogonal to the image means that d∗Aψ = 0, and〈µ(F−A0
), ψ〉 = 0, for all µ.Using generic µ, we get that tr(FA0 ⊗ ψ) = 0,where ψ,FA0 are harmonic.This implies that ψ = 0 or A0 reducible, for r = 2.
Therefore for generic metric, we have surjectivity of d+,gA0
.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 17 / 22
Let N be the space of metrics. Consider
1 + ∗g2 dA0 : N ×
∧1⊗EndE −→
∧2
+⊗EndE
The linearisation with respect to g ∈ N is
12µ d−,gA0
+ d+,gA0
where µ :∧2− →
∧2+
ψ ∈∧2
+ is orthogonal to the image means that d∗Aψ = 0, and〈µ(F−A0
), ψ〉 = 0, for all µ.Using generic µ, we get that tr(FA0 ⊗ ψ) = 0,where ψ,FA0 are harmonic.This implies that ψ = 0 or A0 reducible, for r = 2.
Therefore for generic metric, we have surjectivity of d+,gA0
.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 17 / 22
Let N be the space of metrics. Consider
1 + ∗g2 dA0 : N ×
∧1⊗EndE −→
∧2
+⊗EndE
The linearisation with respect to g ∈ N is
12µ d−,gA0
+ d+,gA0
where µ :∧2− →
∧2+
ψ ∈∧2
+ is orthogonal to the image means that d∗Aψ = 0, and〈µ(F−A0
), ψ〉 = 0, for all µ.Using generic µ, we get that tr(FA0 ⊗ ψ) = 0,where ψ,FA0 are harmonic.This implies that ψ = 0 or A0 reducible, for r = 2.
Therefore for generic metric, we have surjectivity of d+,gA0
.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 17 / 22
Let N be the space of metrics. Consider
1 + ∗g2 dA0 : N ×
∧1⊗EndE −→
∧2
+⊗EndE
The linearisation with respect to g ∈ N is
12µ d−,gA0
+ d+,gA0
where µ :∧2− →
∧2+
ψ ∈∧2
+ is orthogonal to the image means that d∗Aψ = 0, and〈µ(F−A0
), ψ〉 = 0, for all µ.
Using generic µ, we get that tr(FA0 ⊗ ψ) = 0,where ψ,FA0 are harmonic.This implies that ψ = 0 or A0 reducible, for r = 2.
Therefore for generic metric, we have surjectivity of d+,gA0
.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 17 / 22
Let N be the space of metrics. Consider
1 + ∗g2 dA0 : N ×
∧1⊗EndE −→
∧2
+⊗EndE
The linearisation with respect to g ∈ N is
12µ d−,gA0
+ d+,gA0
where µ :∧2− →
∧2+
ψ ∈∧2
+ is orthogonal to the image means that d∗Aψ = 0, and〈µ(F−A0
), ψ〉 = 0, for all µ.Using generic µ, we get that tr(FA0 ⊗ ψ) = 0,where ψ,FA0 are harmonic.
This implies that ψ = 0 or A0 reducible, for r = 2.
Therefore for generic metric, we have surjectivity of d+,gA0
.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 17 / 22
Let N be the space of metrics. Consider
1 + ∗g2 dA0 : N ×
∧1⊗EndE −→
∧2
+⊗EndE
The linearisation with respect to g ∈ N is
12µ d−,gA0
+ d+,gA0
where µ :∧2− →
∧2+
ψ ∈∧2
+ is orthogonal to the image means that d∗Aψ = 0, and〈µ(F−A0
), ψ〉 = 0, for all µ.Using generic µ, we get that tr(FA0 ⊗ ψ) = 0,where ψ,FA0 are harmonic.This implies that ψ = 0 or A0 reducible, for r = 2.
Therefore for generic metric, we have surjectivity of d+,gA0
.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 17 / 22
Let N be the space of metrics. Consider
1 + ∗g2 dA0 : N ×
∧1⊗EndE −→
∧2
+⊗EndE
The linearisation with respect to g ∈ N is
12µ d−,gA0
+ d+,gA0
where µ :∧2− →
∧2+
ψ ∈∧2
+ is orthogonal to the image means that d∗Aψ = 0, and〈µ(F−A0
), ψ〉 = 0, for all µ.Using generic µ, we get that tr(FA0 ⊗ ψ) = 0,where ψ,FA0 are harmonic.This implies that ψ = 0 or A0 reducible, for r = 2.
Therefore for generic metric, we have surjectivity of d+,gA0
.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 17 / 22
Higher dimensional gauge theory[Donaldson-Thomas]
Let M be a compact riemannian 8-manifold with Holg = Spin(7).
Then there is a 4-form Ω preserved by the parallel transport.At a point Ω =
∑dxi ∧ dxj ∧ dxk ∧ dxl , the sum over those (i , j , k , l)
such that 〈ei ,ej ,ek ,el〉 is a quaternionic line in the octonions O = R8.∇Ω = 0 =⇒ dΩ = 0.
Let E → M be a complex vector bundle of rank r .Let A be a Yang-Mills connection. ThendA ∗ FA = 0.
If ∗FA = c FA ∧ Ω, c ∈ R, thendA ∗ FA = c d(FA ∧ Ω) = 0, as dAFA = 0 and dΩ = 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 18 / 22
Higher dimensional gauge theory[Donaldson-Thomas]
Let M be a compact riemannian 8-manifold with Holg = Spin(7).
Then there is a 4-form Ω preserved by the parallel transport.At a point Ω =
∑dxi ∧ dxj ∧ dxk ∧ dxl , the sum over those (i , j , k , l)
such that 〈ei ,ej ,ek ,el〉 is a quaternionic line in the octonions O = R8.∇Ω = 0
=⇒ dΩ = 0.
Let E → M be a complex vector bundle of rank r .Let A be a Yang-Mills connection. ThendA ∗ FA = 0.
If ∗FA = c FA ∧ Ω, c ∈ R, thendA ∗ FA = c d(FA ∧ Ω) = 0, as dAFA = 0 and dΩ = 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 18 / 22
Higher dimensional gauge theory[Donaldson-Thomas]
Let M be a compact riemannian 8-manifold with Holg = Spin(7).
Then there is a 4-form Ω preserved by the parallel transport.At a point Ω =
∑dxi ∧ dxj ∧ dxk ∧ dxl , the sum over those (i , j , k , l)
such that 〈ei ,ej ,ek ,el〉 is a quaternionic line in the octonions O = R8.∇Ω = 0 =⇒ dΩ = 0.
Let E → M be a complex vector bundle of rank r .Let A be a Yang-Mills connection. ThendA ∗ FA = 0.
If ∗FA = c FA ∧ Ω, c ∈ R, thendA ∗ FA = c d(FA ∧ Ω) = 0, as dAFA = 0 and dΩ = 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 18 / 22
Higher dimensional gauge theory[Donaldson-Thomas]
Let M be a compact riemannian 8-manifold with Holg = Spin(7).
Then there is a 4-form Ω preserved by the parallel transport.At a point Ω =
∑dxi ∧ dxj ∧ dxk ∧ dxl , the sum over those (i , j , k , l)
such that 〈ei ,ej ,ek ,el〉 is a quaternionic line in the octonions O = R8.∇Ω = 0 =⇒ dΩ = 0.
Let E → M be a complex vector bundle of rank r .Let A be a Yang-Mills connection. ThendA ∗ FA = 0.
If ∗FA = c FA ∧ Ω, c ∈ R, thendA ∗ FA = c d(FA ∧ Ω) = 0, as dAFA = 0 and dΩ = 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 18 / 22
Higher dimensional gauge theory[Donaldson-Thomas]
Let M be a compact riemannian 8-manifold with Holg = Spin(7).
Then there is a 4-form Ω preserved by the parallel transport.At a point Ω =
∑dxi ∧ dxj ∧ dxk ∧ dxl , the sum over those (i , j , k , l)
such that 〈ei ,ej ,ek ,el〉 is a quaternionic line in the octonions O = R8.∇Ω = 0 =⇒ dΩ = 0.
Let E → M be a complex vector bundle of rank r .Let A be a Yang-Mills connection. ThendA ∗ FA = 0.
If ∗FA = c FA ∧ Ω, c ∈ R, thendA ∗ FA = c d(FA ∧ Ω) = 0, as dAFA = 0 and dΩ = 0.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 18 / 22
Higher dimensional gauge theory
The operator ∗(• ∧ Ω) :∧2 →
∧2 has eigenvalues 3 and −1 anddecomposes: ∧2
=∧2
7⊕∧2
21
So ∗FA = −FA ∧ Ω ⇐⇒ F 7A = π7(FA) = 0.
YM(A) =∫|FA|2 has a minimmum when ∗FA = −FA ∧ Ω.
DefinitionA ∈ AE is a Spin(7)-instanton if F 7
A = 0.
Moduli space of Spin(7)-instantons:ME = A ∈ AE |F 7
A = 0/GE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 19 / 22
Higher dimensional gauge theory
The operator ∗(• ∧ Ω) :∧2 →
∧2 has eigenvalues 3 and −1 anddecomposes: ∧2
=∧2
7⊕∧2
21
So ∗FA = −FA ∧ Ω ⇐⇒ F 7A = π7(FA) = 0.
YM(A) =∫|FA|2 has a minimmum when ∗FA = −FA ∧ Ω.
DefinitionA ∈ AE is a Spin(7)-instanton if F 7
A = 0.
Moduli space of Spin(7)-instantons:ME = A ∈ AE |F 7
A = 0/GE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 19 / 22
Higher dimensional gauge theory
The operator ∗(• ∧ Ω) :∧2 →
∧2 has eigenvalues 3 and −1 anddecomposes: ∧2
=∧2
7⊕∧2
21
So ∗FA = −FA ∧ Ω ⇐⇒ F 7A = π7(FA) = 0.
YM(A) =∫|FA|2 has a minimmum when ∗FA = −FA ∧ Ω.
DefinitionA ∈ AE is a Spin(7)-instanton if F 7
A = 0.
Moduli space of Spin(7)-instantons:ME = A ∈ AE |F 7
A = 0/GE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 19 / 22
Higher dimensional gauge theory
The operator ∗(• ∧ Ω) :∧2 →
∧2 has eigenvalues 3 and −1 anddecomposes: ∧2
=∧2
7⊕∧2
21
So ∗FA = −FA ∧ Ω ⇐⇒ F 7A = π7(FA) = 0.
YM(A) =∫|FA|2 has a minimmum when ∗FA = −FA ∧ Ω.
DefinitionA ∈ AE is a Spin(7)-instanton if F 7
A = 0.
Moduli space of Spin(7)-instantons:ME = A ∈ AE |F 7
A = 0/GE
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 19 / 22
Deformation theory
Let A0 ∈ AE be a Spin(7)-instanton. Then the linearization of thegauge group action, and the linearization of the Spin(7)-instantonequation give an elliptic complex:
∧0⊗EndE
dA0−→∧1⊗EndE
d7A0−→
∧2
7⊗EndE
The homology groups H0A0,H1
A0,H2
A0are finite dimensional and give the
following information:
H0A0
is the Lie algebra of the automorphism group of A0. HenceH0
A0= 0 if A0 is irreducible.
H1A0
is the tangent space to the moduli space ofSpin(7)-instantons (the first order deformations of the solutions).H2
A0is the obstruction space. If H2
A0= 0, the moduli space is
smooth.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 20 / 22
Deformation theory
Let A0 ∈ AE be a Spin(7)-instanton. Then the linearization of thegauge group action, and the linearization of the Spin(7)-instantonequation give an elliptic complex:
∧0⊗EndE
dA0−→∧1⊗EndE
d7A0−→
∧2
7⊗EndE
The homology groups H0A0,H1
A0,H2
A0are finite dimensional and give the
following information:H0
A0is the Lie algebra of the automorphism group of A0. Hence
H0A0
= 0 if A0 is irreducible.
H1A0
is the tangent space to the moduli space ofSpin(7)-instantons (the first order deformations of the solutions).H2
A0is the obstruction space. If H2
A0= 0, the moduli space is
smooth.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 20 / 22
Deformation theory
Let A0 ∈ AE be a Spin(7)-instanton. Then the linearization of thegauge group action, and the linearization of the Spin(7)-instantonequation give an elliptic complex:
∧0⊗EndE
dA0−→∧1⊗EndE
d7A0−→
∧2
7⊗EndE
The homology groups H0A0,H1
A0,H2
A0are finite dimensional and give the
following information:H0
A0is the Lie algebra of the automorphism group of A0. Hence
H0A0
= 0 if A0 is irreducible.
H1A0
is the tangent space to the moduli space ofSpin(7)-instantons (the first order deformations of the solutions).
H2A0
is the obstruction space. If H2A0
= 0, the moduli space issmooth.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 20 / 22
Deformation theory
Let A0 ∈ AE be a Spin(7)-instanton. Then the linearization of thegauge group action, and the linearization of the Spin(7)-instantonequation give an elliptic complex:
∧0⊗EndE
dA0−→∧1⊗EndE
d7A0−→
∧2
7⊗EndE
The homology groups H0A0,H1
A0,H2
A0are finite dimensional and give the
following information:H0
A0is the Lie algebra of the automorphism group of A0. Hence
H0A0
= 0 if A0 is irreducible.
H1A0
is the tangent space to the moduli space ofSpin(7)-instantons (the first order deformations of the solutions).H2
A0is the obstruction space. If H2
A0= 0, the moduli space is
smooth.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 20 / 22
Main results
Spin(7)-forms: Ω giving a reduction Spin(7) < SO(8) but dΩ 6= 0.Projectors: P :
∧2 →∧2 of rank 7.
Perturbed moduli space: MPE = A ∈ AE |P(FA) = 0/GE
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle,A ∈ AE an Spin(7)-instanton which is not reducible.For generic Spin(7)-form, we have that tr(FA ∧ ψ) = 0, for any ψorthogonal to H2.For generic projector, we have that tr(FA ⊗ ψ) = 0, for any ψorthogonal to H2.
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle ofrank 2. For generic projector,MP
E is smooth at irreducible points.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 21 / 22
Main results
Spin(7)-forms: Ω giving a reduction Spin(7) < SO(8) but dΩ 6= 0.Projectors: P :
∧2 →∧2 of rank 7.
Perturbed moduli space: MPE = A ∈ AE |P(FA) = 0/GE
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle,A ∈ AE an Spin(7)-instanton which is not reducible.For generic Spin(7)-form, we have that tr(FA ∧ ψ) = 0, for any ψorthogonal to H2.For generic projector, we have that tr(FA ⊗ ψ) = 0, for any ψorthogonal to H2.
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle ofrank 2. For generic projector,MP
E is smooth at irreducible points.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 21 / 22
Main results
Spin(7)-forms: Ω giving a reduction Spin(7) < SO(8) but dΩ 6= 0.Projectors: P :
∧2 →∧2 of rank 7.
Perturbed moduli space: MPE = A ∈ AE |P(FA) = 0/GE
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle,A ∈ AE an Spin(7)-instanton which is not reducible.
For generic Spin(7)-form, we have that tr(FA ∧ ψ) = 0, for any ψorthogonal to H2.For generic projector, we have that tr(FA ⊗ ψ) = 0, for any ψorthogonal to H2.
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle ofrank 2. For generic projector,MP
E is smooth at irreducible points.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 21 / 22
Main results
Spin(7)-forms: Ω giving a reduction Spin(7) < SO(8) but dΩ 6= 0.Projectors: P :
∧2 →∧2 of rank 7.
Perturbed moduli space: MPE = A ∈ AE |P(FA) = 0/GE
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle,A ∈ AE an Spin(7)-instanton which is not reducible.
For generic Spin(7)-form, we have that tr(FA ∧ ψ) = 0, for any ψorthogonal to H2.For generic projector, we have that tr(FA ⊗ ψ) = 0, for any ψorthogonal to H2.
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle ofrank 2. For generic projector,MP
E is smooth at irreducible points.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 21 / 22
Main results
Spin(7)-forms: Ω giving a reduction Spin(7) < SO(8) but dΩ 6= 0.Projectors: P :
∧2 →∧2 of rank 7.
Perturbed moduli space: MPE = A ∈ AE |P(FA) = 0/GE
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle,A ∈ AE an Spin(7)-instanton which is not reducible.For generic Spin(7)-form, we have that tr(FA ∧ ψ) = 0, for any ψorthogonal to H2.
For generic projector, we have that tr(FA ⊗ ψ) = 0, for any ψorthogonal to H2.
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle ofrank 2. For generic projector,MP
E is smooth at irreducible points.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 21 / 22
Main results
Spin(7)-forms: Ω giving a reduction Spin(7) < SO(8) but dΩ 6= 0.Projectors: P :
∧2 →∧2 of rank 7.
Perturbed moduli space: MPE = A ∈ AE |P(FA) = 0/GE
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle,A ∈ AE an Spin(7)-instanton which is not reducible.For generic Spin(7)-form, we have that tr(FA ∧ ψ) = 0, for any ψorthogonal to H2.For generic projector, we have that tr(FA ⊗ ψ) = 0, for any ψorthogonal to H2.
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle ofrank 2. For generic projector,MP
E is smooth at irreducible points.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 21 / 22
Main results
Spin(7)-forms: Ω giving a reduction Spin(7) < SO(8) but dΩ 6= 0.Projectors: P :
∧2 →∧2 of rank 7.
Perturbed moduli space: MPE = A ∈ AE |P(FA) = 0/GE
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle,A ∈ AE an Spin(7)-instanton which is not reducible.For generic Spin(7)-form, we have that tr(FA ∧ ψ) = 0, for any ψorthogonal to H2.For generic projector, we have that tr(FA ⊗ ψ) = 0, for any ψorthogonal to H2.
Theorem [M-S]Let M be a Spin(7)-manifold, let E → M be a complex vector bundle ofrank 2. For generic projector,MP
E is smooth at irreducible points.
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 21 / 22
THANKS!
Vicente Muñoz (UCM) Yang-Mills equations in higher dimensions 22 September 2016 22 / 22
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