Sujeción del pie al pedal

Now consider a principalG-bundle,π :P →_R3_{, with a gauge potential A=A} 1dx1+

A2dx2 +A3dx3 ∈ Ω1(R3, ad(P)) and a Higgs field Φ ∈ Ω0(R3, ad(P)), satisfying the

Bogomolny equation.

Now consider the projection map introduced in Chapter 1

f :R4→R3

(x0, x1, x2, x3)7→(x1, x2, x3).

Then on the pullback bundleπ0 :f∗(P)→_R4_{, we claim the following defines a gauge} potential which is invariant under translation in the x0 direction:

¯

A= Φdx0+f∗(A) = ¯Φdx0+ ¯A1dx1+ ¯A2dx2+ ¯A3dx3∈Ω1(R4,g),

where ¯X =f∗(X).

It is clear that the gauge potential is invariant under translation in thex0 direction as

∂iA¯i = 0. Furthermore under a gauge transformation, the gauge potential ¯A transforms as required.

Thus this gauge potential ¯A is R invariant in the x0 direction, and substituting it’s components into equations (3.2.2), we see that it satisfies the self-duality equations, given (A,Φ) satisfy the Bogomolny equations.

3.3

The ADHM construction

Before we get to the Atiyah-Drinfeld-Hitchin-Manin-Nahm (ADHMN) construction, which defines an equivalence between certain BPS monopoles and a set of Nahm data, we will first introduce the Atiyah-Drinfeld-Hitchin-Manin (ADHM) construction, from which the ADHMN construction was derived. The ADHM construction defines an equivalence be- tween all instanton solutions on R4 and a set linear algebraic data.

Definition 3.3.1(ADHM data). Let M be a 4-dimensional space with a complex structure, and define the complex coordinates on M to be (z1, z2) ∼= (x1, x2, x3, x4) ∈R4. A set of

ADHM data consists of:

1. Complex vector spaces V and W of dimension k and nrespectively.

2. Complexk×k matrices which we will denote B1 and B2, a complex k×nmatrixI

and a complex n×k matrixJ,

• B1,B2, I, J satisfy the ADHM equations given by

[B1, B2] +IJ = 0, [B1, B₁†] + [B2, B₂†] +II†−J J†= 0.

• For all (x, y)6= (0,0)∈M2, where x= (z1, z2), y = (w1, w2), the map α(x,y):V →

W ⊕V ⊗M α(x,y)=    w2J−w1I† −w2B1−w1B2†−z1 w2B2−w1B † 1+z2   

is injective, while β(x,y):W ⊕V ⊗M →V given by

β(x,y)=

h

w2I+w1J† w2B2−w1B₁†+z2 w2B1+w1B₂†+z1

i

is surjective.

Lemma 3.3.2. Let g∈U(k),h∈SU(n) act on (B1, B2, I, J) by (B1, B2, I, J)7→(gB1g−1, gB2g−1, gIh−1, hJ g−1).

If(B1, B2, I, J)satisfy the ADHM equations, then so do their transformed matrices defined

above.

Theorem 3.3.3(ADHM construction). There exists a one-to-one correspondence between the equivalence classes of ADHM data under the action of the groupsU(k)andSU(n), and gauge equivalent classes of anti-self-dualSU(n)gauge potentialsAoverR4, with instanton

number −k

We will now briefly outline the process from which the instanton data can be con- structed from the ADHM data. The first step involves constructing an operator Ơ which is used to construct the instanton data.

For the case at hand ∆(x,y) and in particular, its adjoint operator ∆

†

(x,y), are given by ∆(x,y)= " β(x,y) α†_(x,y) # :W ⊕V ⊗U →V ×V, ∆†_(x,y)= h β₍†_x,y) α(x,y) i :V ×V →W ⊕V ⊗U.

The next step is determining the kernel of the operator ∆†, and its dimension. This will require determining the kernel of ∆ and the index of ∆† and then applying the index formula. We then consider a set of orthonormal solutions {ψi}∀i∈I of the equation

3.3. THE ADHM CONSTRUCTION 43 In addition to the fact that ker∆ = 0, it can be shown that there are exactly nsolutions to this equation. That is,i∈N≤n.

Now we want toconstruct the instanton dataand thenprove that the data sat- isfies the required equations. The instanton data of charge −kis given, in component form, by the gauge potential

Ai,ab =ψa

∂ ∂xi

ψb, which can be shown to satisfy the self duality equation.

Lastly we need to show thatthis solution is unique up to gauge transformations. We will see that this process is replicated in the ADHMN construction for both the Nahm transform and its inverse. This is due to the ADHMN construction being derived from the ADHM construction under the equivalence proven in Section 3.2, and thus the construction for monopoles is obtained by using similar methods to that for the instanton case.

The ADHM transform as a Fourier transform

Before we go on to the ADHMN construction, we will briefly emphasize an important per- spective on the ADHM transforms. In particular, that they behave as Fourier transforms. Consider the standard Fourier transform between functions of a single variable. So consider f :R→C some integrable function of variablex. Then the Fourier transform of

f which we denote ˆf(η), over the variableη∈_Ris given by ˆ f(η) = √1 2 Z ∞ −∞ f(x)e−ixηdx,

and what we see is that under this transformation,

x7→ d

dη, d

dx 7→η.

Therefore, the Fourier transform maps algebraic operators to differential operators and vice-versa. This allows for a differential problem to be Fourier transformed to an algebraic problem which will generally be much simpler to solve.

The transforms arising in the ADHM construction can equivalently be viewed as a type of Fourier transform between two sets of data with four variables. The transform from the vector space data to the instanton data focuses around the operator Ơ in its

construction. In particular, this operator is an algebraic operator, with four algebraic vari- ables (x1, x2, x3, x4). When considering the ADHM construction in the reverse direction, that is the transform from the instanton data to the vector space data, the focus is on a differential operator, which we denoteD/†. This operator is a differential operator in four variables (x1, x2, x3, x4), with no algebraic operators.

Thus what we see is that the transforms involved in this theory are mapping the algebraic operators to differential ones, which is a generalization of the above Fourier transform to four variables. This allows for a simpler method of solving for instanton solutions rather than just solving directly the self-duality equations, which correspond to non-linear PDE’s, to instead solving algebraic equations and then applying this Fourier- like transform to obtain the instanton solutions.

What we will now see is that this Fourier transform generalizes to a Fourier-like trans- form between a set of data consisting of one differential variable and three algebraic vari- ables (the Nahm data), and a set of data consisting of one three differential variables and one algebraic variable (the monopole data). This is the case of the ADHMN construction.