Dunajski M. Solitons, Instantons, and Twistors

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286 10 : Anti-self-dual conformal structures

4. Consider a deformation of the twistor space given by a Hamiltonian f =

f (ω

,π



) homogeneous of degree two and independent of ω

, and show

that it leads to a family of ASD pp-waves generalizing the metric (10.5.116)

and given by

g = dwdx + dzdy + F (w, y)dw

where F is an arbitrary function of two variables.

5. Find the ASD Ricci-ﬂat metric corresponding to a deformed twistor space

with a deformation Hamiltonian

f =

(π



)

[Hint: Show that the deformation equations integrate to

˜ω

= exp[t(π



)

−2

]ω

and ˜ω

= exp [−t(π



)

−2

]ω

, (10.5.129)

where Q = ω

restricts to α



on each twistor curve for some α



and β



. Obtain the splitting Q

−2

(o ·π)

h − h where, setting ∂

:= o



∂

∂α



and ∂

:= o



∂

∂β



h =2(π · o)∂

∂



α · o

(α · β)(π · α)



and h =2(π · o)∂

∂



β · o

(α · β)(π · β)



The twistor curves are now given by

=(γ · π )e

and ω

=(δ · π )e

−ht

for some spinors γ



and δ



APPENDIX A

Manifolds and Topology

The ﬁrst six chapters of this book are intended to give an elementary introduction

to the subject and the reader is expected only to be familiar with basic real and

complex analysis, algebra, and dynamics as covered in the undergraduate syllabus.

In particular no knowledge of differential geometry is assumed. One obvious

advantage of this approach is that the book is suitable for advanced undergraduate

students.

The disadvantage is that the discussion of Hamiltonian formalism and con-

tinuous groups of transformations in earlier chapters used phrases like ‘spaces

coordinatized by ( p, q)’, ‘open sets in

’, or ‘groups whose elements smoothly

depend on parameters’ instead calling these object by their real name – manifolds.

The ﬁrst part of this appendix is intended to ﬁll this gap. The second part of

the appendix contains the discussion of homotopy groups and topological degree

needed in Chapters 5–7.

Deﬁnition A.0.1 An n-dimensional smooth manifold is a set M together with a

collection of open sets U

called the coordinate charts such that

The open sets U

labelled by a countable index α cover M.

There exist one-to-one maps φ

: U

→ V

onto open sets in R

such that for any

pair of overlapping coordinate charts the maps

◦ φ

−1

: φ

∩ U

) −→ φ

∩ U

)

are smooth (i.e. inﬁnitely differentiable) functions from

to R

. (Figure A.1)

Thus a manifold is a topological space together with additional structure which

makes local differential calculus possible. The space

itself is of course a manifold

which can be covered by one coordinate chart.

Example. A less trivial example is the unit sphere

= {r ∈ R

n+1

, |r| =1}.

To verify that this is indeed a manifold, cover S

by two open sets U

= U and



U = S

/{0,...,0, 1} and



U = S

/{0,...,0, −1},

288 A: Manifolds and topology

‚

–1

Figure A.1 Manifold

and deﬁne the local coordinates by stereographic projections

φ(r

, r

,...,r

n+1



1 − r

n+1

,...,

1 − r

n+1



=(x

,...,x

) ∈ R

and

φ(r

, r

,...,r

n+1



1+r

n+1

,...,

1+r

n+1



=(x

,...,x

) ∈ R

N = (0, 0, . . . , 0, 1)

n+1

ê (P)

Using

1+r

n+1



1 − r

n+1

1+r

n+1



1 − r

n+1

, k =1,...,n,

289

where r

n+1

= ±1 shows that on the overlap U ∩



U the transition functions

φ ◦

−1

,...,x



+ ···+ x

,...,

+ ···+ x



are smooth.

The Cartesian product of manifolds is also a manifold. For example, the n-torus

arising in the Arnold–Liouville theorem (Theorem 1.2.2) is the Cartesian product

of n one-dimensional spheres.

Another way to obtain interesting manifolds is to deﬁne them as surfaces in

by a vanishing condition for a set of functions. If f

,..., f

: R

→ R then the set

:= (x ∈ R

, f

(x)=0, i =1,...,k) (A1)

is a manifold if the rank of the k × n matrix of gradients ∇ f

is constant in a

neighborhood of M

in R

. If this rank is maximal and equal to k then dim M

n − k. The manifold axioms can be veriﬁed using the implicit function theorem. For

example, the sphere S

n−1

arises this way with k = 1 and f

=1−|x|

.Thereisa

theorem which says that every manifold arises as some surface in

for sufﬁciently

large n. If the manifold is m-dimensional then n is at most 2m + 1. This useful

theorem is now nearly forgotten – differential geometers like to think of manifolds

as abstract objects deﬁned by a collections of charts as in Deﬁnition A.0.1.

A map between smooth manifolds f : M →



M, where dim M = n and dim



M =

n, is called smooth if it is smooth in local coordinates. This means that the maps

◦ f ◦ φ

−1

: R

−→ R

are smooth maps in the ordinary sense. Here (U

,φ

) and (



) are coordinate

charts for M and



M, respectively.

Let γ :

R → M be a smooth curve such that

γ (0) = p ∈ M and

dγ (ε)

dε

ε=0

= V ∈ T

where V is a vector tangent to γ at p and the tangent space T

M consists of all

tangent vectors to all possible curves through p.IfdimM = n, the tangent space is

an n-dimensional vector space. The collection of all tangent spaces as p varies in

M is called the tangent bundle TM = ∪

p∈M

M. The tangent bundle is a manifold

of dimension 2n.

A smooth map f between two manifolds induces, for each p ∈ M, a smooth

map between tangent spaces

∗

: T

M −→ T

f ( p)



such that

∗

(V)=

df(γ (ε))

dε

ε=0

. (A2)

This map is called the tangent map. It depends smoothly on a point p ∈ M and

thus it extends to the tangent bundle TM.IfV

are components of the vector ﬁeld

290 A: Manifolds and topology

V with respect to the natural basis {∂/∂x

} then

( f

∗

= V

∂γ

∂x

The Lie derivative of a vector W along a vector V is deﬁned as

Lie

W = lim

→0

W( p) − γ ()

∗

W( p)



, (A3)

where γ () is the one-parameter group of transformations generated by V. Thus,

using the Leibniz rule

Lie

( f )=V( f ), Lie

(W)=[V, W], and Lie

(ω)=d(V ω)+V

(dω),

where f, W, and ω are a function, a vector ﬁeld and a one-form, respectivelly, and

is a contraction of a differential form with a vector ﬁeld.

A.1 Lie groups

We can now give a proper deﬁnition of a Lie group:

Deﬁnition A.1.1 A Lie group G is a group and, at the same time, a smooth

manifold such that the group operations

G × G → G, (g

, g

) → g

, and G → G, g → g

−1

are smooth maps between manifolds.

Example. The general linear group G = GL(n, R) is an open set in R

deﬁned

by the condition det g =0, g ∈ G. It is therefore a Lie group of dimension n

The special orthogonal group SO(n) is deﬁned by (A1), where the n(n +1)/2

conditions in

are

− 1 =0, det g =1.

The determinant condition just selects a connected component in the set of

orthogonal matrices, so it does not count as a separate condition. It can be shown

that the corresponding matrix of gradients has constant rank and thus SO(n)is

an [n(n − 1)/2]-dimensional Lie group.

In Chapter 4 a Lie algebra

g was deﬁned as a vector space with an antisymmetric

bilinear operation which satisﬁes the Jacobi identity (4.2.6).

A Lie algebra of a Lie group G is the tangent space to G at the identity element,

g = T

G with the Lie bracket deﬁned by a commutator of vector ﬁelds at e. For any

g ∈ G deﬁne left translation L

using the group multiplication

: G −→ G and L

(h)=gh.

The tangent mapping (A2) maps T

G = g to T

G and each element V ∈ g corre-

sponds to a vector ﬁeld (L

)

∗

V on the group manifold. Theses vector ﬁelds are

A.1 Lie groups 291

called left-invariant. The Lie bracket of two left-invartiant vector ﬁelds is again

left-invariant as

[(L

)

∗

(V), (L

)

∗

(W)] = (L

)

∗

[V, W]

from the properties of the tangent map (A2). (The bracket on the LHS is the Lie

bracket of two vector ﬁelds. The symbol [ , ]

on the RHS is the bracket in the Lie

algebra

g.) Therefore the elements of g can be represented by global vector ﬁelds

on G, and Lie groups are paralizable as they globally admit dim(G) non-vanishing

vector ﬁelds which are left translations of vectors in

Let L

,α =1,...,dimg be a basis of left-invariant vector ﬁelds such that

, L

]= f

αβ

and let σ

be the dual basis of one-forms such that L

= δ

. The identity

dω(V, W)=V[ω(W)] − W[ω(V)] − ω([V, W])

with V = L

, W = L

, and ω = σ

gives

dσ

βγ

∧ σ

=0.

If G is a matrix group the one-form

−1

is called the Maurer–Cartan one-from on G. The Maurer–Cartan one-form is

invariant under left multiplication of g by a constant group element. This one-

form takes its values in

g, as for any smooth curve g(s)inG we have

−1

(s)g(s + ε)=1+εg

−1

ε=0

+ O(ε

so g

−1

(dg/ds) is a tangent vector to G at g, and so it is an element of g. Thus we

can write

−1

dg = σ

where T

are matrices spanning g and σ

are left-invariant one-forms on G. The

metric

h = −Tr(g

−1

dg g

−1

dg)=−Tr(T

)σ

is the left-invariant metric on the Lie group.

The right-invariant one-forms ˜σ

are deﬁned by

−1

dg = T

˜σ

and the right-invariant vector ﬁelds R

are deﬁned by the duality R

˜σ

= δ

They satisfy

, L

]=0 and [R

, R

]=− f

αβ

292 A: Manifolds and topology

Example. The left-invariant one-forms σ

on the Lie group SO(3) can be

explicitly given in terms of Euler angles:

= cos ψ dθ + sin ψ sin θ dφ, σ

= −sin ψ dθ + cos ψ sin θ dφ, and

= dψ + cos θ dφ.

Example. Consider the three-dimensional Heisenberg group (4.1.1) with the

corresponding Lie algebra (4.1.3). We have

−1

⎛

⎜

⎝

1 −m

−m

+ m

01 −m

00 1

⎞

⎟

⎠

−1

dg = T

= T

+ T

(dm

− m

This gives

dσ

=0, dσ

=0, and dσ

= −σ

∧ σ

The left-invariant metric

h =(σ

)

+(σ

)

+(σ

)

= dm

+ dm

+(dm

− m

)

has a Kaluza–Klein interpretation: Its geodesics, when projected to the (m

, m

plane, are trajectories of a particle moving in a uniform magnetic ﬁeld F = dm

∧

with a potential A = m

. The ﬁrst integral

− m

= const of the

geodesic motion corresponds to charge conservation.

If G is a transformation group of some manifold X, then the elements of

g can

also be represented by vector ﬁelds on X.Ifρ : G × X −→ X then for any V ∈

we deﬁne ρ(V) to be a vector in X by demanding that its ﬂow coincides with a

one-parameter subgroup e

εV

of G in X. This induces a Lie algebra homomorphism

form

g to the Lie algebra of vector ﬁelds on X, that is,

[ρ(V),ρ(W)] = ρ([V, W]

), V, W ∈ g.

We will usually omit the reference to the map ρ, and denote the vectors in

g and

the corresponding vector ﬁelds in TXby the same symbol.

Example. The group SO(3, R) acts on R

as the group of rotations. The action

x −→ Ax is inﬁnitesimaly generated by three vector ﬁelds

abc

∂

∂x

The group preserves the Euclidean distance, and so its action descends to the

two-sphere S

⊂ R

given by |x| = 1. The S

-volume form ω = d(cos θ ) ∧ dψ is

A.1 Lie groups 293

preserved by this action, and the corresponding vector ﬁelds are Hamiltonian,

that is, X

ω = −dh

with Hamiltonians h

: S

−→ R:

= sin θ sin ψ, h

= −sin θ cos ψ, and h

= cos θ

such that

, h

} =

abc

. (A4)

Proof of the ﬁrst part of Arnold–Liouville’s Theorem 1.2.2. The gradients ∇ f

are

independent, thus the set

:= {(p, q) ∈ M; f

(p, q)=c

where c

, c

,...,c

are constant deﬁnes a manifold of dimension n.Letξ

=(p, q)

be local coordinates on M such that the Poisson bracket is

{ f, g} = ω

∂ f

∂ξ

∂g

∂ξ

, a, b =1, 2,...,2n,

where ω is the constant antisymmetric matrix



−1



The vanishing of the Poisson brackets { f

, f

} = 0 implies that each Hamiltonian

vector ﬁeld

= ω

∂ f

∂ξ

∂

∂ξ

is orthogonal (in the Euclidean sense) to any of the gradients ∂

, a =

1,...,2n, j, k =1,...,n. The gradients are perpendicular to M

, thus the Hamil-

tonian vector ﬁelds are tangent to M

. They are also commuting as

, X

]=−X

{ f

, f

}

=0,

so the vectors generate an action of the abelian group

on M. This action restricts

to an

action on M

. Let p

∈ M

, and let  be the lattice consisting of all vectors

which ﬁx p

under the group action. Then  is a discrete subgroup of R

and

(by an intuitively clear modiﬁcation of the orbit-stabiliser theorem) we have

= R

/.

Assuming that M

is compact, this quotient space is diffeomorphic to a torus T



In fact this argument shows that we get a torus for any choice of the constants

. Thus, varying the constants, we ﬁnd that the phase-space M is foliated by

n-dimensional tori.

294 A: Manifolds and topology

A.2 Degree of a map and homotopy

Deﬁnition A.2.1 Let M

and M

be oriented, compact D-dimensional manifolds

without boundary, and let ω be a volume-form on M

. A degree deg( f ) of a smooth

map f : M

→ M

is given by



∗

ω =[deg( f )]



ω. (A5)

Rescaling ω by a constant does not change the degree, and neither does choosing a

different volume-form because



ω =





implies that ω − ω



= dα for some (D − 1)-form α (the D-dimensional cohomology

of M

is one-dimensional) and

∗



= f

∗

ω − f

∗

dα = f

∗

ω − d( f

∗

α)

gives



( f

∗



− f

∗

ω)=0

by application of Stokes’ theorem. We conclude that the deg( f ) depends only

on f .

There is another useful way of calculating degree by counting a number of pre-

images. Let y ∈ M

be a generic point, that is, the set f

−1

(y)={x; f (x)=y} is

ﬁnite, and the Jacobian J ( f ) = 0 (recall that if x ∈ U has local coordinates x

, and

y ∈ f (U) has local coordinates y

, then J = det (∂ y

/∂x

)ify

= y

,...,x

)).

Proposition A.2.2 deg( f ) is the integer given by

deg( f )=



x∈f

−1

(y)

sign[J (x)]. (A6)

Proof Let f

−1

(y)={x

}, where α is a discrete index with possibly inﬁnite range.

The set of critical values of f has measure zero (by the Sard theorem [43]), and

such points do not contribute to the integral. Clearly (A6) is an integer, but it may

depend on the choice of y. Choose a neighbourhood V of y and U

of x

for some

ﬁxed value of α such that f : U

→ V is one-to-one and onto. Let ω have support

f (U

) ⊂ V (i.e. ω = 0 outside V). Therefore



∗

(ω)=





∗

ω.

Now change coordinates from x

to y

near x

. Then



∗

ω =







sign[J (x

)],

A.2 Degree of a map and homotopy 295

where we have used

ω = ρ(y)dy

∧···∧dy

and



∗

ω =



ρ[ f (x)]Jdx

∧···∧dx





ρ(y)dy

∧···∧dy



sign(J )

as the last change of variables from x to y introduces the term |J |

−1

. We obtain

similar relations on all other open sets U

, which proves (A6). 

Example. f : S

→ S

. In this case the topological degree is called the winding

number. Formula (A6) gives

N =



θ: f (θ )= f

sign(df/dθ ).

If the graph of f looks like

2

f(Ë)

2π

+ ++−−

then applying (A6) we ﬁnd

N =1− 1+1+1− 1=1.

Alternatively the degree can be calculated from the deﬁnition (A5) which yields

N = (vol

)

−1



df =

2π



2π

dθ

dθ.

If we think of S

as the unit circle |z| = 1 in the complex plane then every map of

degree k is homotopic to f (z)=z

Example. For any k ∈ Z there exist smooth maps from S

to S

of degree k. Let

f : S

n−1

−→ S

n−1

be a smooth map which takes a unit vector n in R

to another

such vector f (n). We can construct the suspension  f : S

−→ S

of f by the