(joint work with C. Shahbazi) Vicente Muñoz · g(x) = (gij(x)). A Riemannian manifold (M;g) is a...

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.

Geometry

Global geometry of manifolds:

Classification of smooth manifolds of dimension n.

Relevant (geometrically and physically): compact case.

Universe (space): n = 3.

Geometry

Global geometry of manifolds:

Classification of smooth manifolds of dimension n.

Relevant (geometrically and physically): compact case.

Universe (space): n = 3.

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.

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

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

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).

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

Pγ : Ep −→ Eq

X(p+hei )−X(p)h

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

Pγ : Ep −→ Eq

X(p+hei )−X(p)h

Connections

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

X(p+hei )−X(p)h

Connections

Pγ : Ep −→ Eq

X(p+hei )−X(p)h

Connections

Pγ : Ep −→ Eq

X(p+hei )−X(p)h

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 =∑

∇∂xi

∇T∂xi

on o.n. basis

=∑i,j

g ij(x)∇∂xi

∇T∂xj

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 =∑

∇∂xi

∇T∂xi

on o.n. basis

=∑i,j

g ij(x)∇∂xi

∇T∂xj

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 =∑

∇∂xi

∇T∂xi

on o.n. basis

=∑i,j

g ij(x)∇∂xi

∇T∂xj

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 =∑

∇∂xi

∇T∂xi

on o.n. basis

=∑i,j

g ij(x)∇∂xi

∇T∂xj

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 =∑

∇∂xi

∇T∂xi

on o.n. basis

=∑i,j

g ij(x)∇∂xi

∇T∂xj

Holonomy

Let (M,g) be a Riemannian manifold. Consider the collection of allparallel transport around loops.It yields the holonomy group Holg ⊂ O(n).

Possible holonomy groups

Theorem [Berger]Let M be a simply-connected Riemannian manifold not symmetric andnot reducible. Then Holg is one of the following:

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.

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.

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.

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.

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.

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.

Curvature

Let ∇ be a connection.In coordinates, ∇∂xi

= ∂∂xi

∇ = d + Γ, Γ =∑

Γidxi ∈∧1⊗EndE

Curvature: K (ei ,ej)(X ) = ∇∂xi

∇∂xj

X − ∇∂xj

∇∂xi

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∇

Curvature

= ∂∂xi

∇ = d + Γ, Γ =∑

∇∂xj

X − ∇∂xj

∇∂xi

K = dΓ + Γ ∧ Γ ∈∧2⊗EndE

∧2⊗hol∇

Curvature

= ∂∂xi

∇ = d + Γ, Γ =∑

∇∂xj

X − ∇∂xj

∇∂xi

K = dΓ + Γ ∧ Γ ∈∧2⊗EndE

For the Levi-Civita connection: K ∈ TM ⊗ TM ⊗ TM ⊗ TM

Ricci tensor: Ric = tr13K

∧2⊗hol∇

Curvature

= ∂∂xi

∇ = d + Γ, Γ =∑

∇∂xj

X − ∇∂xj

∇∂xi

K = dΓ + Γ ∧ Γ ∈∧2⊗EndE

∧2⊗hol∇

Curvature

= ∂∂xi

∇ = d + Γ, Γ =∑

∇∂xj

X − ∇∂xj

∇∂xi

K = dΓ + Γ ∧ Γ ∈∧2⊗EndE

∧2⊗hol∇

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).

General relativity

Ricij −12

General relativity

Ricij −12

General relativity

Ricij −12

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).

General relativity

Ricij −12

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)

Maxwell equations

F =∑

Eidxi ∧ dt +∑

Bidxj ∧ dxk

Maxwell equations

F =∑

Eidxi ∧ dt +∑

Bidxj ∧ dxk

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.

Maxwell equations

Solutions to Maxwell equations are equivalent to dd∗A = 0, d∗dA = 0,

i.e. ∆A = 0, which is of elliptic type.

Maxwell equations

On compact,

riemannian, 4-manifold, theory of elliptic operatorsprovide many results.

Maxwell equations

On compact, riemannian,

4-manifold, theory of elliptic operatorsprovide many results.

Maxwell equations

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.

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).

∧n−2, 〈∗α, β〉 = α ∧ β.Yang-Mills equation: dA ∗ FA = 0

For n = 4, ∗∗ = id,∧2 =

∧2+⊕

For n = 4, ∗∗ = id,∧2 =

∧2+⊕

For n = 4, ∗∗ = id,∧2 =

∧2+⊕

For n = 4, ∗∗ = id,∧2 =

∧2+⊕

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

:∧1⊗EndE −→

+⊗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

∧1⊗EndE , |τ | < ε, with F+A = 0.

A0τ + (τ ∧ τ)+ = 0.

:∧1⊗EndE −→

+⊗EndE

∧1⊗EndE

∧1⊗EndE , |τ | < ε, with F+A = 0.

A0τ + (τ ∧ τ)+ = 0.

:∧1⊗EndE −→

+⊗EndE

∧1⊗EndE

∧1⊗EndE , |τ | < ε, with F+A = 0.

A0τ + (τ ∧ τ)+ = 0.

:∧1⊗EndE −→

+⊗EndE

∧1⊗EndE

∧1⊗EndE , |τ | < ε, with F+A = 0.

A0τ + (τ ∧ τ)+ = 0.

:∧1⊗EndE −→

+⊗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,

∧1⊗EndE

∧1⊗EndE , |τ | < ε, with F+A = 0.

A0τ + (τ ∧ τ)+ = 0.

:∧1⊗EndE −→

+⊗EndE

∧1⊗EndE

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

A0be the homology of the complex,

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 ⊗ EndE

dA0−→∧

1 ⊗ EndEd+

A0−→∧

2+ ⊗ EndE

= 0/GE around A0

A0⊕ d∗A0

Let H0A0,H1

A0= 0.

∧0 ⊗ EndE

dA0−→∧

1 ⊗ EndEd+

A0−→∧

2+ ⊗ EndE

= 0/GE around A0

, i.e., the first homology of the ellipticcomplex.

The operator LA0 = d+A0⊕ d∗A0

Let H0A0,H1

A0= 0.

∧0 ⊗ EndE

dA0−→∧

1 ⊗ EndEd+

A0−→∧

2+ ⊗ EndE

= 0/GE around A0

A0⊕ d∗A0

Let H0A0,H1

A0= 0.

∧0 ⊗ EndE

dA0−→∧

1 ⊗ EndEd+

A0−→∧

2+ ⊗ EndE

= 0/GE around A0

A0⊕ d∗A0

Let H0A0,H1

A0= 0.

∧0 ⊗ EndE

dA0−→∧

1 ⊗ EndEd+

A0−→∧

2+ ⊗ EndE

= 0/GE around A0

A0⊕ d∗A0

Let H0A0,H1

A0= 0.

∧0 ⊗ EndE

dA0−→∧

1 ⊗ EndEd+

A0−→∧

2+ ⊗ EndE

= 0/GE around A0

A0⊕ d∗A0

Let H0A0,H1

A0= 0.

Let N be the space of metrics. Consider

1 + ∗g2 dA0 : N ×

∧1⊗EndE −→

+⊗EndE

The linearisation with respect to g ∈ N is

12µ d−,gA0

+ d+,gA0

where µ :∧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

1 + ∗g2 dA0 : N ×

∧1⊗EndE −→

+⊗EndE

12µ d−,gA0

+ d+,gA0

ψ ∈∧2

1 + ∗g2 dA0 : N ×

∧1⊗EndE −→

+⊗EndE

12µ d−,gA0

+ d+,gA0

ψ ∈∧2

1 + ∗g2 dA0 : N ×

∧1⊗EndE −→

+⊗EndE

12µ d−,gA0

+ d+,gA0

ψ ∈∧2

), 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.

1 + ∗g2 dA0 : N ×

∧1⊗EndE −→

+⊗EndE

12µ d−,gA0

+ d+,gA0

ψ ∈∧2

), 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.

1 + ∗g2 dA0 : N ×

∧1⊗EndE −→

+⊗EndE

12µ d−,gA0

+ d+,gA0

ψ ∈∧2

1 + ∗g2 dA0 : N ×

∧1⊗EndE −→

+⊗EndE

12µ d−,gA0

+ d+,gA0

ψ ∈∧2

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.

such that 〈ei ,ej ,ek ,el〉 is a quaternionic line in the octonions O = R8.∇Ω = 0

=⇒ dΩ = 0.

Higher dimensional gauge theory

The operator ∗(• ∧ Ω) :∧2 →

∧2 has eigenvalues 3 and −1 anddecomposes: ∧2

7⊕∧2

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

7⊕∧2

So ∗FA = −FA ∧ Ω ⇐⇒ F 7A = π7(FA) = 0.

A = 0.

A = 0/GE

7⊕∧2

So ∗FA = −FA ∧ Ω ⇐⇒ F 7A = π7(FA) = 0.

A = 0.

A = 0/GE

7⊕∧2

So ∗FA = −FA ∧ Ω ⇐⇒ F 7A = π7(FA) = 0.

A = 0.

A = 0/GE

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−→

7⊗EndE

The homology groups H0A0,H1

A0are finite dimensional and give the

following information:

is the Lie algebra of the automorphism group of A0. HenceH0

A0= 0 if A0 is irreducible.

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.

Deformation theory

∧0⊗EndE

d7A0−→

7⊗EndE

following information:H0

A0is the Lie algebra of the automorphism group of A0. Hence

= 0 if A0 is irreducible.

smooth.

Deformation theory

∧0⊗EndE

d7A0−→

7⊗EndE

is the tangent space to the moduli space ofSpin(7)-instantons (the first order deformations of the solutions).

is the obstruction space. If H2A0

= 0, the moduli space issmooth.

Deformation theory

∧0⊗EndE

d7A0−→

7⊗EndE

smooth.

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.

Main results

∧2 →∧2 of rank 7.

Main results

∧2 →∧2 of rank 7.

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.

Main results

∧2 →∧2 of rank 7.

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.

Main results

∧2 →∧2 of rank 7.

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.

Main results

∧2 →∧2 of rank 7.

Main results

∧2 →∧2 of rank 7.

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(joint work with C. Shahbazi) Vicente Muñoz · g(x) = (gij(x)). A Riemannian manifold (M;g) is a...

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