A Riemann Surfaces and Theta Functions

Home , Abelian integral, Fundamental polygon, Theta characteristic, Theta function

In this appendix we collect some basic facts on theta functions on Riemann surfaces. The idea is to give a comprehensive presentation of the mathemat- ics on theta functions and the notation used in this book without providing lengthy proofs. For more detailed accounts of the subject, the reader is referred to [128], [138], [139], [187], and [256] to [270], for topics related to the Ernst equation see also [271].

A.1 Riemann Surfaces and Algebraic Curves

A Riemann surface Σ is a connected complex one-dimensional manifold. Riemann surfaces can be associated to multi-valued functions. Let f be an analytic function on C2 with arguments w and z. The equation

f(w, z) = 0 (A.1) deﬁnes a one-dimensional complex submanifold of C2. Let the complex gra- dient gradCf be given by ∂f ∂f grad f := , (A.2) C ∂w ∂z and call (w0,z0) with f(w0,z0) = 0 regular iﬀ

gradCf| =0 . (A.3) (w0,z0) It can be shown that the corresponding Riemann surface admits a com- pactiﬁcation if equation (A.1) describes a plane algebraic curve, i.e. if f is of the form k i f(w, z)= w ai(z) , (A.4) i=0 with ai (i =0,...,k) being polynomials in z. Then w = w(z) is a multiple- valued algebraic function. It can be shown that any compact Riemann surface can be represented as an algebraic curve. A complex structure is introduced on the algebraic curve in the following way: in the neighborhoods of the

C. Klein, O. Richter: Ernst Equation and Riemann Surfaces, Lect. Notes Phys. 685, 191–208 (2005) www.springerlink.com c Springer-Verlag Berlin Heidelberg 2005 192 A Riemann Surfaces and Theta Functions points where ∂f/∂w = 0, the variable z is taken as a local parameter, in the neighborhoods of the points ∂f/∂z = 0, the parameter w is taken as a local parameter. In this volume we shall be mainly concerned with functions of the form

2 f(w, z)=w − Pn(z) , (A.5) with Pn being a polynomial of degree n in z. The corresponding multiple- valued function is denoted by w = Pn(z) and the Riemann surface L is called hyperelliptic (for n =3, 4 the surface is called elliptic). The hyperelliptic surface is everywhere regular if and only if Pn(z) has no multiple roots. ∼ For w = 0 one chooses√ z as a local parameter, for z zi, where Pn(zi)=0, − the variable τzi = z zi can be used as a local parameter. For Σ being hyperelliptic the hyperelliptic involution σ is deﬁned by

σ : Σ P =(z,w) → σ(P ) ≡ P σ =(z,−w) ∈ Σ, (A.6) i. e., σ interchanges the two sheets of the Riemann surface. Any compact Riemann surface Σg of genus g is topologically equivalent to a sphere with g handles. Any compact Riemann surface has ﬁnite genus. The ﬁrst homology group of Σg is denoted by H1(Σg, Z). A standard basis of generators of H1(Σg, Z) consists of g pairs of cycles (a1,b1),...,(ag,bg) where a pair (ai,bi) encircles the i-th handle (or surrounds the ith hole) so that ai intersects bi, see Fig. A.1

a1 a2

b1 b2

Fig. A.1. A homology basis for a Riemann surface of genus two

The way cycles intersect is described by intersection numbers. In Fig. A.2 we show for two cycles γ1 and γ2 when the intersection number γ1 · γ2 is +1 or −1. We choose oriented closed curves a1,...,ag,b1,...,bg such that their intersection numbers are

ai · aj = bi · bj =0,

ai · bj = −bi · aj = δij . (A.7) A basis with the above intersection numbers is called canonical basis.The ˜ choice of such a basis is not unique: Any other basis (˜a, b)ofH1(Σg, Z) A.1 Riemann Surfaces and Algebraic Curves 193

γ2 γ1

γ1 γ2

γ1 · γ2 =+1 γ1 · γ2 = −1

Fig. A.2. The orientation dependence of the intersection numbers

T (where a and b denote the g-dimensional vector a =(a1,...,ag) and b = T (b1,...,bg) ) is given by the transformation a˜ a = A ,A∈ SL(2g,Z) . (A.8) ˜b b

From the requirement that the new basis is also canonical we ﬁnd that the matrix A is symplectic, A ∈ Sp(g,Z), 0 −I J = AJAT ,J= . (A.9) I 0

A canonical basis is also referred to as a cut-system. If one cuts the Rie- mann surface starting from one point along the canonical cycles, the resulting surface is simply connected, a 4g-gon called the fundamental polygon. For the surface of Fig. A.1 one gets the fundamental polygon shown in Fig. A.3.

−1 a1 b1

−1 b1 a1

− a2 1 b2

− b2 1 a2

Fig. A.3. The fundamental polygon Σ˜g of a Riemann surface Σg of g =2 194 A Riemann Surfaces and Theta Functions A.2 Diﬀerentiation and Integration on Riemann Surfaces

A diﬀerential (or one form) dΩ = adx +bdy = αdz + βd¯z, where z is a local complex coordinate in the neighborhood of some point P on the Riemann surface Σg of genus g is called an Abelian diﬀerential if we have

dΩ = f(z)dz, (A.10) with f being a meromorphic function in the vicinity of P . Since ∂f ∂f d(dΩ)= dz ∧ dz + d¯z ∧ dz =0, (A.11) ∂z ∂z¯ Abelian differentials are always closed. We may classify Abelian differentials as follows: – Abelian differentials of the first kind (or holomorphic differentials): f is a holomorphic function in any local chart. – Abelian differentials of the second kind: have a single pole of order higher than 1 with vanishing residue. – differentials of the third kind: have two first-order poles with residues ±1. Furthermore, it turns out that each Abelian differential can be decomposed into Abelian differentials of the above mentioned kinds. We denote a (n) differential of the second kind with pole p of order n +1 bydΩP and a differential of the third kind with poles P (residue +1) and Q (residue −1) by dΩPQ. We have for the differential dΩPQ 1 dΩPQ = + O(1) dτP , near P, τP 1 dΩPQ = − + O(1) dτQ , near Q, (A.12) τQ where τP and τQ are local parameters at P and Q with τP (P )=0and τQ(Q) = 0. For the differential of the second kind we have (n) 1 dΩP = n+1 + O(1) dτP , near P, (A.13) τP where τP is a local parameter at P with τP (P ) = 0. A differential of the second kind can be obtained from a differential of the third kind by differen- tiating 1 dΩ(n) = ∂n dΩ . (A.14) P n! P PQ An Abelian integral on a Riemann surface is an integral of an Abelian differential. A.2 Differentiation and Integration on Riemann Surfaces 195

1 It can be shown that the vector space H (Σg) of holomorphic differentials on Σg is g-dimensional. For example if Σg is a hyperelliptic Riemann surface 2g+2 2 1 of the form µ = (λ − λi), we may define a basis in H (Σg)by i=1 λk−1dλ dν = , (A.15) k µ for i =1,...,g. The hyperelliptic differentials of the third kind have the form µ + µ µ + µ dλ dω (R)= P − Q , if µ =0 ,µ =0 , PQ λ − τ λ − τ 2µ P Q P Q µ + µ 1 dλ dω (R)= P − , if µ =0 ,µ =0, PQ λ − τ λ − τ 2µ P Q P Q 1 1 dλ dωPQ(R)= − , if µP =0,µQ =0, (A.16) λ − τP λ − τQ 2µ where the argument of dωPQ is the point R =(λ, µ) ∈ Σg.

Definition A.1. The periods along the cycles a1,...,bg of a closed differential dΩ are defined by # . Ai = dΩ, a#i . Bi = dΩ, (A.17)

bi for i =1,...,g. The periods are independent of the representatives of the cycles with the given homology classes since the differentials are closed. Let dΩ respectively dΩ be closed differentials and denote the corresponding periods by Ai and ∈ Bi respectively Ai and Bi (i =1,...,g). Let P0 Σg be fixed and define a function f on Σ˜g (the fundamental polygon) by

P f(P )= dΩ, (A.18)

∀P ∈ Σg. With these settings one obtains Riemann’s bilinear identities, Theorem A.2. The following relation holds: # g ∧ − dΩ dΩ = fdΩ = (AiBi AiBi) , (A.19) i=1 Σg ∂Σ˜g where ∂Σ˜g is the boundary of the 4g-gon Σ˜g, oriented in positive direction. 196 A Riemann Surfaces and Theta Functions

The first identity in (A.19) follows from Stokes’ theorem, the last identity follows from an evaluation of the integrand at the boundary of the fundamental domain (for details see [186]). The Riemann bilinear relations of Theorem A.2 imply useful relations for the A-andB-periods of Abelian differentials. Applying them to two holomorphic differentials dΩ and dΩ one finds g − AjBj BjAj =0. (A.20) j=1

and   g   AjBj ≤ 0 . (A.21) j=1 Thus one has Corollary A.3. (i) An Abelian diﬀerential of the ﬁrst kind where all A-

(ii) or all B-periods vanish is identically zero. (iii) An Abelian diﬀerential of the ﬁrst kind with only real periods is identically zero.

Remark A.4. With the above relations one may show that for an integral of the third kind the poles and integration limits can be interchanged

Q R

dΩRS = dΩQP , (A.22) P S

P, Q, R, S ∈ Σg.

We normalize the Abelian diﬀerentials in the following way:

– The holomorphic diﬀerentials are normalized by the condition #

dωj =2πiδij . (A.23) ai The so normalized basis of the holomorphic differentials is called canonical. – Abelian differentials of the second and third kind are only determined up to a linear combination of holomorphic differentials. This ambiguity will be fixed by demanding that all a-periods of the normalized differentials of the second and third kind vanish. A.2 Differentiation and Integration on Riemann Surfaces 197

The Riemann bilinear relations (A.19) imply that the matrix Bij, # . Bij = dωj , (A.24) bi of b-periods of the normalized holomorphic differentials is a Riemann matrix, i.e. it is symmetric Bij = Bji and has a negative definite real part. We note that the Riemann matrix transforms under a change of the homology basis from (a, b)to(ã,˜b)accordingto a˜ AB a AB = , ∈ Sp(g,Z) (A.25) ˜b CD b CD as B =2πi(DB +2πiC)(BB +2πiA)−1 . (A.26) For the b-periods of the normalized differentials of the third kind, the bilinear relations imply # P dΩPQ = dωk . (A.27) Q bk The corresponding relations for the b-periods of the normalized differentials of the second kind follow from (A.14). In physical applications so-called real surfaces (surfaces with an anti- holomorphic involution) play an important role. In the hyperelliptic case the surface Σg of genus g is given by the relation

g+1 2 µ = (K − Ei)(K − Fi) , (A.28) i=1 where Ei,Fi ∈ R or Ei = F¯i for i =1,...,g+ 1. On this surface the anti- holomorphic involution has the form

τ : Σ P =(K, µ) → τ(P ) ≡ P¯ =(K,¯ µ¯) ∈ Σ, (A.29) i.e.. it acts as a complex conjugation on each sheet of L. In the context of the Ernst equation we are interested in surfaces with Eg+1 = F¯g+1 = ξ. Since in the cut-system of Fig. A.4 the closed curves ak remain in one sheet, they are not invariant under the hyperelliptic involution σ whereas the curves bk are (as a point set). The hyperelliptic involution acts on the Abelian differentials of the first kind, as one easily finds from (A.15) and (A.23), as multiplication by −1. Similarly, the anti-involution τ acts on H1(Σg) as follows:

τ(a )=−a , i i τ(bi)=bi − ak , (A.30) k= i 198 A Riemann Surfaces and Theta Functions

ξ F1 FN

.. F E Fg EN+1 N+1 g ξ E1 EN

Fig. A.4. Homology basis for real hyperelliptic Riemann surfaces

for curves ai surrounding two complex conjugated branch points and

τ(a )=−a , i i τ(bi)=bi − ak , (A.31) k=1 for curves ai surrounding two real ones. Since the non-normalized differentials ∗ dνi (A.15) transform according to τ dνi =dνi we find with (A.23) and (A.30) for the normalized Abelian differentials of the first kind

∗ τ (dωi)=dωi . (A.32)

Therefore, the components of the Riemann matrix B transform under τ in the following way # # # # # ∗ B¯ ij = dωj = τ (dω)j = dωj = dωj − dωk b b b l=i a i i τ(bi) i l = Bij − 2πi δjl , (A.33) l= i for ovals ai surrounding two complex conjugated branch points, and # # # # g # ∗ B¯ ij = dωj = τ (dω)j = dωj = dωj − dωk l=1 bi bi τ(bi) bi al

= Bij − 2πi , (A.34) for ovals ai surrounding two real branch points. Notice that the periods of the holomorphic diﬀerentials on a hyperelliptic surface can be expressed in terms of diﬀerences of the Abel map between branch points. At the end of A.3 Divisors and the Theorems of Abel and Jacobi 199 this section we introduce the notion of the Jacobian of a Riemann surface and of the Abel map. Let Λ be the lattice

Λ = {2πiN + BM, N, M ∈ Zg} (A.35)

g generated by the periods of Σg. This defines an equivalence relation in C : two points of Cg are equivalent if they differ by an element of Λ. Definition A.5. The complex torus

g Jac(Σg)=C /Λ (A.36) is called the Jacobi variety or the Jacobian of Σg. Deﬁnition A.6. The map P ω : Σg → Jac(Σg) , ω(P )= dω , (A.37) P0 where dω is the canonical basis of the holomorphic diﬀerentials and where P0 ∈ Σg, is called the Abel map.

(i) g (i) Let {e }, i =1,...,g be a basis in C with (e )j = δij and deﬁne

(i) . (f )j = Bij . (A.38)

Then the vectors 2πie(1),...,e(g), f (1),...,f (g) are lineary independent since the real part of the Riemann matrix is negative deﬁnite. A point e ∈ Jac(Σg) is uniquely determined by the 2g real numbers pi, qj (i, j =1,...,g),

g g (k) (k) e =2πi qke + pkf . (A.39) k=1 k=1

The numbers pi, qj form the characteristic [e] of the point e, . p [e] = . (A.40) q

A.3 Divisors and the Theorems of Abel and Jacobi

A divisor on a Riemann surface Σg is a formal sum of points,

N A = niPi (A.41) i=1 with Pi ∈ Σg and ni ∈ Z. The sum 200 A Riemann Surfaces and Theta Functions

N degA = ni (A.42) i=1 is called the degree of A. The set of all divisors with the obviously deﬁned group action

N n1P + n2P =(n1 + n2)P, −A = (−ni)Pi (A.43) i=1 forms an Abelian group Div(Σg). A divisor with all ni ≥ 0 is called positive (or integral or eﬀective). This notion allows a partial ordering in Div(Σg),

A ≤ A ⇐⇒ A − A ≥ 0 . (A.44)

Let f be a meromorphic function on Σg with zeros P1,...,PM of multiplicities p1,...,pM > 0 and poles Q1,...,QN of multiplicities q1,...,qN > 0. Then the divisor

A = p1P1 + ...+ pM PM − q1Q1 − ...− qN QN =(f) (A.45) is called the divisor of f and denoted by (f). A divisor is called principal if there exists a function f with (f)=A. For two meromorphic functions f and g one has obviously (fg)=(f)+(g). Two divisors A and A are called equivalent if the divisor A − A is principal which is denoted by A ≡ A.The corresponding equivalence class is called the divisor class. Similarly one can define the divisor of an Abelian differential dΩ = f(z)dz. Since the quotient of two Abelian differentials is a meromorphic function, any two divisors of Abelian differentials are lineary equivalent. The corresponding class is called canonical. The Abel map is defined for divisors of the form (A.41) in a natural way, N Pi ω(A)= ni dω . (A.46) i=1 P0

If the divisor is of degree zero, ω(A) is obviously independent of P0. This leads to the important

Theorem A.7 (Abel’s theorem). The divisor D ∈ Div(Σg) is principal if and only if: 1. degD =0, 2. ω(D)=0on Jac(Σg). Since the divisor D is principal, it deﬁnes a class of meromorphic functions. Let F be such a function. Then we have the useful A.3 Divisors and the Theorems of Abel and Jacobi 201

Corollary A.8. Let the conditions of Abel’s theorem hold and let D = A−B where A and B are positive divisors. Then the following identity holds A F (P ) dωPQ =ln . (A.47) B F (Q)

Let D∞ be a positive divisor on Σg. A natural question is to describe the vector space of meromorphic functions with poles at D∞ only. For a general divisor D on Σg one can deﬁne the vector space

L(D)={f meromorphic functions on Σg|(f) ≥ D or f ≡ 0} . (A.48)

If we split D in positive (D0) and negative (D∞) parts D0 = niPi , D∞ = mkQk , (A.49)

where ni,mk > 0, then the space L(D) (we call its dimension r(D)) consists of the meromorphic functions with zeros of order at least ni at Pi and with poles of order at most mk at Qk. Similarly one can deﬁne the corresponding vector space of the diﬀerentials

H(D)={Ω Abelian differential on Σg|(Ω) ≥ D or Ω ≡ 0} , (A.50) the dimension of which is denoted by i(D). This dimension is called the index of speciality. Definition A.9. A positive divisor D of degree degD = g is called special if there exists a holomorphic differential dω with

(dω) ≥ D . (A.51)

It can be shown that the condition (A.51) is equivalent to the existence of a non-constant function f with (f) ≥−D. Since the space of holomorphic diﬀerentials is g-dimensional, equations (A.51) form a linear system of g equations for g variables. This shows that most of the positive divisors are non-special. In the hyperelliptic case equations (A.51) imply that a divisor is special if and only if it contains two or more points lying on diﬀerent sheets but having the same projection into the complex plane. With these notions we can state the important

Theorem A.10 (Jacobi’s inversion theorem). Let Dg be the set of positive divisors of degree g on Σg. The Abel map on this set ω : Dg !→ Jac(Σg) is surjective, i.e. for any x ∈ Jac(Σg) there exists a positive divisor of degree g with P1 + ...+ Pg ∈ Dg (the Pi are not necessarily diﬀerent) satisfying g Pi dω = x . (A.52) i=1 P0 202 A Riemann Surfaces and Theta Functions A.4 Theta Functions of Riemann Surfaces

Theta functions are a convenient tool to work with meromorphic functions on Riemann surfaces. We define them as an infinite series. Definition A.11. Let B be a g ×g Riemann matrix. The theta function with characteristics [p, q] is defined as 1 Θ (x, B)= exp B (N + p) , N + p + x +2πiq, N + p , pq 2 N∈Zg (A.53) g g with x ∈ C and p, q ∈ C ,where ·, · denotes the Euclidean scalar product g N, x = i=1 Nixi. The properties of the Riemann matrix ensure that the series converges ab- solutely and that the theta function is an entire function on Cg. A characteristic is called singular if the corresponding theta function vanishes identically. Of special importance are half-integer characteristics with 2p, 2q ∈ Zg.A half-integer characteristic is called even if 4 p, q = 0 mod 2 and odd otherwise. Theta functions with odd (even) characteristic are odd (even) functions of the argument x. The theta function with characteristics is related to the Riemann theta function, the theta function with zero characteristics . Θ = Θ00,via 1 Θ (x, B)=Θ(x + Bp +2πiq)exp Bp, p + p, x +2πiq . (A.54) pq 2

The theta function has the periodicity properties

1 2πipj −2πiqj −zj − Bjj Θpq(x+2πiej)=e Θpq(x) ,Θpq(x+Bej)=e 2 Θpq(x) , (A.55) where ej is the g-dimensional vector consisting of zeros except for a 1 in jth position, and satisﬁes the heat equation (under the assumption that all entries of the matrix B are independent)

2∂Bαβ Θpq(z, B)=∂zα ∂zβ Θpq(z, B) . (A.56)

The above deﬁnitions are possible for Riemann matrices B which are not associated to a Riemann surface. In the following we will only consider theta functions on Riemann surfaces. Then we have Proposition A.12. The theta function Θ(ω(P ) − e) either vanishes identically on Σg or has exactly g zeros (counting multiplicities).

Proposition A.13. Let K be the Riemann vector defined by A.4 Theta Functions of Riemann Surfaces 203   # P 2πi+B 1 K = jj − dω (P ) dω  , (A.57) 2 2 l j l= j al P0 where P0 is the base point of the Abel map, and let D = P1 + ...+ Pg be a non-special divisor. Then the theta function Θ(ω(P ) − ω(D) −K) vanishes if and only if P ∈ D. For general Riemann surfaces it is difficult to find an explicit form for the Riemann vector. On hyperelliptic surfaces it is related to an odd characteristic which can be expressed in terms of the Abel map of some divisor of branch points. For example for the cut-system in Fig. A.4 one has g K = ω(Fi) . (A.58) i=1 In general there is no simple relation between the complex conjugate of a theta function Θ(x) and the theta function of the complex conjugated argument Θ(x¯). However for real hyperelliptic surfaces such relations exist. For the cut-system of Fig. A.4 the relations (A.33) for the matrix of b-periods imply for the Riemann theta function,

Θ(x)=Θ (x¯ +iπ∆) , (A.59) where ∆i =1ifEi and Fi are real and ∆i = 0 otherwise. Abelian integrals can be expressed in terms of theta functions. Since for the applications in Chap. 3 the integrals of the third kind are particularly interesting, we want to show how a sum of g such integrals can be expressed via theta functions, see [187]. Let f be a meromorphic function on Σ˜g of the form Θ(ω(P ) − ω(D) −K) f(P )= , (A.60) Θ(ω(P ) − ω(E) −K) where D = P1 +···+Pg and E = P˜1 +···+P˜g, Pi, P˜j ∈ Σg are two non-special divisors of degree g and K is the Riemann vector. This function has g zeros respectively poles at the points of the divisor D respectively E. Furthermore, f(P ) has at the cut ai the jump 1 and at bi the jump exp{2(ωi(D) − ωi(E))}. On the other hand, the function    P  fˆ(P ) = exp dω ˜ , (A.61)  PiPi  i P0 ˆ where P0 ∈ Σg is ﬁxed, has the same properties as f:wehavef(Pi)=0and fˆ(P˜j)=∞. The jumps at the cuts aj respectively bj are the same as for the function f. Therefore, both functions coincide, up to a P -independent factor. By taking P = P0 one gets 204 A Riemann Surfaces and Theta Functions

P Pi dω = dω (A.62) P˜iPi P0P i i P0 P˜i Θ(ω(P ) − ω(D) −K)Θ(ω(P ) − ω(E) −K) =ln 0 . Θ(ω(P ) − ω(E) −K)Θ(ω(P0) − ω(D) −K) Here we have used that for an Abelian integral of the third kind the poles and limits of integration may be interchanged.

A.4.1 Elliptic Surfaces

A special case are surfaces of genus 1 which are called elliptic, see e.g. [199]. Elliptic curves can always be brought into the standard form

µ2 =(1− λ2)(1 − k2λ2) , (A.63) byusingaM¨obius transformation. The periods of the holomorphic diﬀeren- tials can be expressed in terms of complete elliptic integrals 1 dλ K(k)= , K˜ (k)=K( 1 − k2). (A.64) −1 µ

The Jacobi elliptic theta functions ϑi where i =1,...,4 have the character- 1 1 0 0 istics 1 , 1 , 1 and 1 respectively. 2 1 2 0 2 0 2 1

A.5 The Trisecant Identity for Theta Functions on Riemann Surfaces

Theta functions are subject to a number of addition theorems. A typical example is the ternary addition theorem which can be cast into the form, see [1, 270]:

1 2 Theorem A.14. Let [mi]=[mi , mi ](i =1,...,4) be arbitrary real 2g- dimensional vectors. Then

Θ[m1](u + v)Θ[m2](u − v)Θ[m3](0)Θ[m4](0) (A.65) 1 = exp(−4πi m1, a2 ) 2g 1 2g 2a∈(Z2)

× Θ[n1 + a](u)Θ[n2 + a](u)Θ[n3 + a](v)Θ[n4 + a](v) ,

1 2 where a =(a , a ),and(m1,...,m4)=(n1,...,n4)T with A.5 The Trisecant Identity for Theta Functions on Riemann Surfaces 205   1111   1  1 −1 −1 −1  T =   . (A.66) 2 1 −11−1 1 −1 −11

Each 1 in T denotes the g × g identity matrix. The above addition theorem holds for general theta functions. A very useful identity due to Fay [128] (see also [139, 272]), however, holds only for theta functions on Riemann surfaces. To generalize the cross ratio function (3.1) to the Riemann surface Σg one needs as a building block an object on Σg which has exactly one zero 1 as the difference of two points z − z0 in CP . There is no function with this property on a Riemann surface, but the so called prime form which is the − 1 − 1 × ( 2 , 2 )-differential on Σg Σg defined by P Θ( ) E(P, Q)= Q , h∆(P )h∆(Q) 2 g ∂Θ where h (P )= (0)dωα(P ), and where ≡ [p q ] is an odd non- ∆ α=1 ∂zα singular half-integer characteristic (note that the prime form is independent P of the choice of the characteristic ). Q denotes the line integral from Q to P of the vector dω(τ). With this notation we can define the cross ratio function E(P1,P2)E(P3,P4) λ1234 = , (A.67) E(P1,P4)E(P3,P2) which is a function on Σg that vanishes for P1 = P2 and P3 = P4 and has poles for P1 = P4 and P2 = P3. The generalized identity (3.2) is Fay’s trisecant identity

Theorem A.15. Let P1,P2,P3,P4 ∈ Σg be four arbitrary points and let p, q ∈ Cg be two characteristic vectors. Then the following identity holds: P3 P4 P3 P4 λ3124Θpq z+ Θpq z+ +λ3214Θpq z+ Θpq z + P2 P1 P1 P2 P3 P4 = Θpq(z)Θpq z + + , (A.68) P2 P1 where all integration contours are chosen not to intersect the canonical basic cycles; this requirement completely ﬁxes all terms of the identity (A.68). In the sequel we will use degenerate versions of Fay’s identity which lead to identities for derivatives of theta functions. Let τ be a local coordinate X ∼ near P . Then we can write the Abel map P for X P as a series in τ, 206 A Riemann Surfaces and Theta Functions X 1 1 = Uτ + V τ 2 + W τ 3 + ... . (A.69) P 2 6

Let us denote by DP the operator for the directional derivative along the basis of holomorphic diﬀerentials, acting on theta functions, and similarly DP and DP the directional derivatives along V and W ,

DP Θpq(z)= ∇Θpq(z), U , ∇ DP Θpq(z)= Θpq(z), V , ∇ DP Θpq(z)= Θpq(z), W . (A.70)

Since the theta function (A.53) depends only on the sum of the vectors z and q, the action of the operator DP on a theta function with characteristics can be written alternatively as

dω (P ) D Θ (z)= ∂ {Θ (z)} α . (A.71) P pq qα pq dτ α P

This form of DP can be easily extended to any object depending on a vector q. Diﬀerentiating (A.68) with respect to the argument P4 and taking the limit P4 → P2 one obtains Corollary A.16. The following identity for ﬁrst derivatives of theta functions on Riemann surfaces holds: P3 Θpq(z + ) D ln P1 P2 Θ (z) pq P2 P3 Θpq(z + )Θpq(z + ) P1 P2 = c1(P1,P2,P3)+c2(P1,P2,P3) , (A.72) P3 Θpq(z)Θpq(z + ) P1 where the functions of three variables c1 and c2 are given by

dωP1P3 (P2) c1(P1,P2,P3)= , (A.73) dτP2 and E(P1,P3) c2(P1,P2,P3)= . (A.74) E(P1,P2)E(P2,P3)dτP2

The derivative of (A.72) with respect to argument P3 gives in the limit P3 → P1 Corollary A.17. The following identity for second derivatives of theta functions on Riemann surfaces holds: A.6 Rauch’s Formulas and Root Functions 207 P1 P2 Θpq(z + )Θpq(z + ) P2 P1 DP1 DP2 ln Θpq(z)=d1(P1,P2)+d2(P1,P2) 2 , Θpq(z) (A.75) where the functions of two variables d1 and d2 are given by

W (P1,P2) d1(P1,P2)=− , (A.76) dτP1 dτP2 1 d2(P1,P2)= 2 ; (A.77) E (P1,P2)dτP1 dτP2

W (P1,P2)=dP1 dP2 ln E(P1,P2) is the canonical meromorphic bidiﬀerential.

In the limit P2 → P1 = P we obtain for an expansion of the terms in (A.75) in the diﬀerence of the local parameters near P1 and P2 Corollary A.18. The following identity holds:

4 2 2 − DP ln Θ(z)+6(DP ln Θ(z)) +3DP DP ln Θ(z) 2DP DP ln Θ(z) − 2 − 2 24e1(P )DP ln Θ(z) + 12(10e2(P ) 3e1(P )) = 0 . (A.78)

Here the functions e1(P ), e2(P ) turn up in the Taylor expansion of the nor- (1) malized diﬀerential dΩP of the second kind with a pole of second order at P (τ is the local parameter in the vicinity of P with τ(P )=0), 1 dΩ(1) = − +2e (P ) − (6e2(P ) − 12e (P ) τ 2 + ...)dτ. (A.79) P τ 2 1 1 2

A.6 Rauch’s Formulas and Root Functions

So far we have only studied functions on a given Riemann surface with ﬁxed branch points. In the context of the Ernst equation it is however necessary to study certain functions on a whole family of Riemann surfaces in dependence on the branch points. Rauch’s variational formulas [179] describe the dependence of the basic normalized holomorphic diﬀerentials dωα and the matrix of b-periods Bαβ on the moduli of the Riemann surface. The moduli space of hyperelliptic curves√ can be parameterized by the positions of the branch − points. Let τλm (P )= λ λm be a local parameter in the neighborhood of P . Then Rauch’s formulas read:

1 W (P, λm) dωα(λm) ∂λm (dωα(P )) = , (A.80) 2 dτλm dτλm

1 dωα(λm) dωβ(λm) ∂λm Bαβ = . (A.81) 2 dτλm dτλm 208 A Riemann Surfaces and Theta Functions

In the case of hyperelliptic Riemann surfaces, quotients of theta functions with the same argument but diﬀerent half integer characteristics are equivalent to so-called root functions, see Chap. 1 of [1], [267]. If we write the surface L in the form 2g+2 2 µ = (λ − λm) , (A.82) m=1 the following identity for root functions holds [1, 267] for any point P = (K, µ(K)) ∈L: E(P, λ ) dτ K − λ m λm = C m , (A.83) K − λ E(P, λn) dτλn n where C is a constant with respect to λ(P ). We will also need these functions in a form free of prime forms: Let Qi, i =1,...,2g + 2, be the branch points of a hyperelliptic Riemann surface Σg of genus g and Aj = ω(Qj) with ω(Q1) = 0. Furthermore let {i1,...,ig} and {j1,...,jg} be two sets of numbers in {1, 2,...,2g +2}. Then the following equality holds for an arbitrary point P ∈L, Θ [K + g A ](ω(P )) (K − E ) ...(K − E ) k=1 ik i1 ig K g = C1 − − , (A.84) Θ [ + k=1 Ajk ](ω(P )) (K Ej1 ) ...(K Ejg ) where C1 is a constant independent of K.LetX = X1 + ... + Xg with Xj =(Kj,µ(Kj )) be a divisor of degree g on L. Then the following identity is satisﬁed, g Θ [K + Ai](ω(X)) (Kk − Qi) = C2 , (A.85) Θ [K + Aj](ω(X)) (Kk − Qj) k=1 where C2 is a function independent of the Kk. B Ernst Equation and Twistor Theory

In this appendix we establish the relation between twistor theory and the solutions to the Ernst equation discussed in this volume. We basically follow the approach of Mason and Woodhouse [32]. In Sect. B.1 we review some basic facts on the quaternionic Hopf bundle. This will be used in Sect. B.2 to perform a symmetry reduction of the Penrose–Ward transform which leads to the linear system of the Yang equation discussed in Sect. 2.4. In Sect. B.3 we construct explicitly the holomorphic bundles over the reduced twistor space RV for the algebro-geometric solutions of the stationary and axisymmetric solutions to the Einstein equations according to Woodhouse and Mason [88]. We equip the space RV associated to a region V of the (, ζ)-plane with a standard cover consisting of four charts. It turns out that for solutions to the Yang equation which yield a symmetric and real Yang matrix and which have a regular behavior on the symmetry axis, one patching matrix (and a couple of integers) characterizes the bundle over RV completely. Then we pass to the construction of the holomorphic bundles associated to the solutions to the Ernst equation discussed in Chap. 4. If the solution has a regular axis, we can read oﬀ the corresponding patching matrix directly from the Ernst potential at the symmetry axis.

B.1 The Quaternionic Hopf Bundle and the Twistor Transform

It is well known that methods of complex analysis are also suited for the description of real analysis of two variables. This is due to the fact that one may identify R2 C (i. e. one may introduce in R2 a complex structure) uniquely, if an orientation and a metric in R2 are given. If one wants to make a similar identification for R4 one would naively identify R4 C2. Unfortunately, for R4 this identification is not possible, because in this case there is no natural complex structure in C2 induced by orientation and metric in R4. It turns out that the correct four-dimensional analogue of C is the three-dimensional complex projective space CP3, the so called Penrose twistor space. Using this complex manifold it turns out that one may describe (Euclidean) (anti-)self dual Yang–Mills fields by algebraic constructions. To be more precise, we first

C. Klein, O. Richter: Ernst Equation and Riemann Surfaces, Lect. Notes Phys. 685, 209–235 (2005) www.springerlink.com c Springer-Verlag Berlin Heidelberg 2005 210 B Ernst Equation and Twistor Theory introduce the basic notions of quaternion geometry. We deﬁne three formal symbols i, j and k which fulﬁl the following requirements

i2 = j2 = k2 = −1 , ij = −ji = k, jk = −kj = i, ki = −ik = j . (B.1)

A general quaternion q ∈ H is then given as a linear combination

q = x0 + x1i + x2j + x3k , (B.2) with x0,...,x3 ∈ R.Theconjugated quaternionq ¯ is deﬁned by

q¯ = x0 − x1i − x2j − x3k . (B.3)

The conjugation is an anti-involution, i. e. (qq˜)=¯q˜q¯. With (B.1) one ﬁnds

3 2 qq¯ =¯qq = xµ . (B.4) µ=0

We denote the above expression, which is zero only for q =0,by|q|2.Each q with |q| = 0 has a unique inverse q−1, which is given by

q−1 =¯q/|q|2 . (B.5)

The unit quaternions, i. e. all quaternions with |q| = 1, form a multiplicative group which is geometrically the sphere S3

3 2 xµ =1 . (B.6) µ=0 If we identify i with the usual complex number we may regard the complex numbers C as contained in H (by taking x2 = x3 = 0). Furthermore, each q has a unique representation

q = z1 + z2j , (B.7)

2 with z1 = x0 + x1i and z2 = x2 + x3i. Therefore, we may identify H C . In analogy to real and complex projective spaces, one may define quaternionic projective spaces HPn. An element of HPn is an equivalence class n+1 n+1 of lines in H , i. e. we call q =(q0,...,qn) ∈ H equivalent to n+1 ∗ q˜ =(˜q0,...,qñ) ∈ H , q ∼ q˜,iffq = λ q˜ for some λ ∈ H = H \{0}. For the further analysis it is useful to introduce two important principal fibre bundles, the Hopf bundle and the quaternionic Hopf bundle. The Hopf bundle is the simplest example of a family of U(1)-bundles S2n+1 → CPn, which can be defined as follows. From the equivalence class of (n + 1)-vectors n+1 2n+2 z =(z0,...,zn)inC R we may always choose a representative whose tip lies on the unit sphere in S2n+1, i. e. it satisfies B.1 The Quaternionic Hopf Bundle and the Twistor Transform 211

2 2 |z0| + ···+ |zn| =1 , (B.8) −1/2 n | | by multiplying z by the scalar λ = µ=0 zµ . Of course, the resulting vector is only unique up to multiplication by scalars of the form eiφ.In other words, we find that CPn can be obtained from S2n+1 by identifying all points eiφz on S2n+1 with z. Therefore, we have a principal fibre bundle S2n+1(CPn, U(1)) with base space CP3 and structure group U(1). Now, for n = 1 (using the identifications CP1 S2 and U(1) S1) the corresponding bundle is called the Hopf bundle S3(S2,S1), i. e. S3 is a principal fibre bundle with base space S2 and structure group S1. Analogously, one may define the quaternionic Hopf bundle as follows. We first endow R4n+4 with the structure of the (n + 1)-dimensional quaternion n+1 space H with coordinates q0,...,qn. In these coordinates the unit sphere S4n+3 in R4n+4 is given by

n 2 |qµ| =1 . (B.9) µ=0

We have a natural (left) action of unit quaternions on S4n+3 as follows

4n+3 4n+3 SU(2) × S (q, (q0,...,qn)) → (qq0,...,qqn) ∈ S , (B.10) n | |2 n | |2 because with µ=0 qµ =1wehave µ=0 qqµ = 1. The resulting orbit space S4n+3/SU(2) is just the quaternion projective space HPn.Thuswe have Proposition B.1. S4n+3(HPn, SU(2)) is a principal ﬁbre bundle with standard ﬁbre SU(2).

Remark B.2. For n = 1 we have the identification HP1 S4. With SU(2) S3 we find the quaternionic Hopf bundle S7(S4,S3), i. e. S7 is a principal fibre bundle with base space S4 and structure group S3.

With these notions at hand we can now introduce the twistor bundle, a fibre bundle which is associated to the quaternionic Hopf bundle. We construct this fibre bundle with standard fibre SU(2)/U(1) S2 as follows. We remark that SU(2) acts naturally on the homogeneous space SU(2)/U(1) by adjoining to [g] ∈ SU(2)/U(1) and g ∈ SU(2) the equivalence class [gg] ∈ SU(2)/U(1). Now we construct the bundle

7 7 3 E = S ×SU(2) SU(2)/U(1) = S /U(1) = CP . (B.11)

Here the second equality reﬂects a standard proposition in the theory of associated bundles [273], and the third equality is nothing but the above construction of U(1)-bundles S2n+1 → CPn for n = 3. The above construction motivates the following 212 B Ernst Equation and Twistor Theory

Definition B.3. The twistor bundle CP3(S4,S2) is a bundle with base space S4 and standard fibre S2 associated to the quaternionic Hopf bundle S7(S4,S3). The twistor bundle is of interest for the construction of solutions to the Yang–Mills equations for the following reason. Suppose we are given a Yang– Mills field over the four-dimensional Euclidean space E4,i.e.wearegivena principal fibre bundle P (E4,G) with structure group (gauge group G)and bundle space P . If one compactifies E4 by adding the point “∞”, one gets a bundle with compact base space S4 (this is just the four-dimensional analogue to the one point compactification in complex analysis). This is due to a fundamental result of Uhlenbeck [29], which states that all Euclidean finite- action Yang–Mills fields over E4 are smoothly extendible to Yang–Mills fields over S4. Thus, we are led to the investigation of principal fibre bundles over S4. The projection π : CP3 → S4 in the twistor bundle can be applied to the principal bundle P (S4,G) and gives an induced bundle over CP3, the pull- back bundle π∗P (CP3,G). A fundamental result states then, see e. g. [274],

Proposition B.4. There is a natural one-to-one correspondence between (i) anti-self-dual U(n)-gauge potentials over S4 (up to gauge equivalence) and (ii) holomorphic vector bundles E of rank n over CP3 with a positive deﬁnite real form (up to isomorphism).

B.2 Symmetry Reductions of the Penrose–Ward Transform

B.2.1 The Reduced Twistor Space

An interesting feature of the Penrose–Ward transform (see for instance [30]) is that it allows for symmetry reductions. It turns out that the Yang equation can be obtained as such a reduction. In fact, in [32] most of the known integrable non-linear equations have been solved by a symmetry reduction of the above Penrose–Ward transform. For the stationary, axisymmetric Einstein equations this procedure has been worked out in [88], see also [163] and [164]. A detailed description of the procedure for the Kerr solution can be found in [165]. The important point for us is that in the solution process linear systems for the Yang equation are generated and get a geometric meaning. Let us recall this symmetry reduction procedure. To start with we consider the action of the Abelian isometry group

G = S1 × R , (B.12) on the four-dimensional Euclidean space E4, deﬁned by

(φ0,t0) · (t, φ, , ζ):=(t + t0,φ+ φ0,,ζ) , (B.13) B.2 Symmetry Reductions of the Penrose–Ward Transform 213 for all φ0 ∈ [0, 2π), t0 ∈ R. This action extends to a conformal action on the compactiﬁcation S4 of E4 and we may lift this conformal action to an action on CP3. In terms of local coordinates t, φ, , ζ, δ, δ¯ on CP3, adapted to the Hopf bundle structure, which are related to holomorphic coordinates ξ1,ξ2,ξ3 by

ξ1 = δ, ξ2 = ζ +it − δe−iφ ,ξ3 = eiφ + δ(ζ − it) , (B.14) we ﬁnd for the generators Φ and T of this action ∂ ∂ ∂ Φ = +iδ − iδ¯ , ∂φ ∂δ ∂δ¯ ∂ T = . (B.15) ∂t 1 2 3 3 In holomorphic coordinates (ξ ,ξ ,ξ )onCP we also ﬁnd Φ =2(YΦ)and T =2(YT ), with ∂ ∂ Y =iξ3 +iξ1 , Φ ∂ξ3 ∂ξ1 ∂ ∂ Y =i − iξ1 . (B.16) T ∂ξ2 ∂ξ3

The holomorphic vector fields YΦ and YT generate the action of the complexification GC = C∗ ×C of G on CP3. Let us consider the orbits of these Killing fields. We have Proposition B.5. The orbits of the Killing fields are parametrized by w ∈ C and given by subsets of 3 1 2 3 3 ξ 2 −1 iφ −iφ Qw = ξ ,ξ ,ξ ∈ CP + ξ = δ e +2ζ − δe =2w . ξ1 (B.17) Proof. Let Y := αYΦ + βYT (B.18) with α, β ∈ C, denote an arbitrary linear combination of YΦ and YT .Inthe holomorphic coordinates ξ1,ξ2,ξ3 we have ∂ ∂ ∂ Y =iαξ1 +iβ +i αξ3 − βξ1 . (B.19) ∂ξ1 ∂ξ2 ∂ξ3 For the integral curve (with curve parameter t)ofY we have to solve the following system of equations (a dot denotes the derivative with respect to t)

ξ˙1 = αiξ1 , ˙2 ξ = βi , (B.20) ξ˙3 =i αξ3 − βξ1 , 214 B Ernst Equation and Twistor Theory from which we immediately obtain for ξ1(t)andξ2(t)

ξ1(t)=eiαt+C1 , 2 ξ (t)=iβt + C2 , (B.21) with arbitrary complex constants C1 and C2. Thus, we obtain

ξ˙3 − iαξ3 = −iβeiαt+C1 , (B.22) with general solution 3 iαt C1 ξ (t)=e C3 − iβte , (B.23)

(C3 ∈ C). Inserting (B.21) yields

3 ξ (t) − + ξ2(t)=C e C1 + C =: 2w. (B.24) ξ1(t) 3 2 It is well known, that if G acts freely and properly on a manifold M then the space of orbits is a Hausdorﬀ manifold. Since YΦ and YT are the lifts of the G-action on E4 which is obviously not free, the situation is a bit more involved. First, we have for the zeros of the Killing vector ﬁelds Proposition B.6. For the Killing vectors of the action of GC on CP3 we have (i) L0 = { = δ =0} (YΦ(p)=0)⇐⇒ p ∈ , (B.25) L1 = { =0,δ = ∞} (ii) −1 (YT (p)=0)⇐⇒ p ∈ I = π (∞) . (B.26)

1 3 Proof. ad (i) From (B.16) we ﬁnd YΦ =0forξ = ξ =0,i.e.,forr = δ =0. In order to prove that YΦ vanishes on L1 we consider its local expression in the chart (CP3 \{Z1 =0}) with coordinates

Z0 1 ξ˜1 = = , Z1 δ Z2 1 ξ˜2 = = (ζ +it) − e−iφ , Z1 δ Z3 1 ξ˜3 = =(ζ − it)+ eiφ . (B.27) Z1 δ

Thus, we ﬁnd for YΦ in this local chart

1 ∂ 2 ∂ YΦ = −iξ˜ − iξ˜ , (B.28) ∂ξ˜1 ∂ξ˜2 B.2 Symmetry Reductions of the Penrose–Ward Transform 215 which vanishes for ξ˜1 = ξ˜2 =0,i.e.,for =0andδ = ∞. The proof of ii) follows similarly. With this proposition we find that the different orbits of GC on CP3 are characterized by the intersections of Qw with the singularities L0, L1,andI of YΦ and YT . In the following we will often consider regions D ⊂ CP3 which are related to subsets of the (, ζ)-plane. Let D ⊂ CP3 be open. Then a holomorphic function f on D, invariant under the action of G has also to be invariant under the action of its complexification GC,i.e.,wehave

YΦ(f)=YT (f)=0 . (B.29)

Of course, the orbits of YΦ and YT may intersect D in disconnected sets. Then we find with (B.29) that f is locally constant on each component of the intersection of D with the orbits of GC. But, on different intersections it can take different values, i. e., f is not the pull-back of a function on the space C 3 of orbits of G on CP . But, since YΦ and YT generate an Abelian isometry group, they commute [YΦ,YT ]=0 , and span, by Frobenius’ theorem, an integrable distribution of TD which is non-singular at all points p ∈ D \{L0 ∪ L1 ∪ I}. In other words, we have a codimension 1 foliation F of D, except at the intersections of D with L0,L1, and I, generated by YΦ and YT . Let H ⊂ C be the upper half of the (, ζ)-plane, i. e., H = {w = ζ +i ∈ C|>0} (B.30) and V ⊂ H be connected and simply connected and define V¯ by V¯ = {w¯| w ∈ V }. We may consider w = ζ +i as a stereographic coordinate on a Riemann sphere CP1. We define pr : E4 \{ =0}−→H (B.31) by pr((t, φ, , ζ)) := ζ +i. (B.32) 4 We denote EV an open subset of E with

pr(EV )=V. (B.33)

Remark B.7. Often we shall be interested in a simply connected set EV ,i.e., −1 in general we have EV =pr (V ). 4 Let V ⊂ H be ﬁxed and choose EV ⊂ E to be an open subset of the Euclidean four-space (see above) and, correspondingly, of S4. We deﬁne 3 PV := ξ ∈ CP |π(ξ) ∈ EV , (B.34) where π : CP3 → S4 is the projection in the twistor bundle CP3 S4,S2 . 216 B Ernst Equation and Twistor Theory

3 Deﬁnition B.8. Let PV ⊂ CP be given as above. The set of connected components of Qw ∩ PV (w ∈ C) is the reduced twistor space RV associated with V .

Let us now show that the reduced twistor space RV can be represented as a compact but in general non-Hausdorff Riemann surface. We consider the surjection 3 1 Γ : CP \ (I ∪ L0 ∪ L1) −→ CP , (B.35) defined by 1 ξ3 1 1 Γ ξ1,ξ2,ξ3 := + ξ2 = δ−1eiφ + ζ − δe−iφ , (B.36) 2 ξ1 2 2 or, with (B.17), Γ ξ1,ξ2,ξ3 = w. (B.37) Let w be fixed and x ∈ E4 have the coordinates x =(t, φ, , ζ). Let π−1(x) CP1 be the fibre over x with stereographic coordinate δ. This fibre intersects the quadric Qw labelled by w in points δ± with

1 2 −iφ 1 iφ δ e +(w − ζ)δ± − re =0 , (B.38) 2 ± 2 see Fig. B.1.

L0 3 PTV CP

=0 E EV 4

Fig. B.1. The twistor space above EV B.2 Symmetry Reductions of the Penrose–Ward Transform 217

We have

Lemma B.9. Let w be ﬁxed. The intersection Qw ∩ PV is connected if there exists a path γ :[0, 1] → Qw with γ(0) = δ+ and γ(1) = δ− such that π(γ(t)) ∈ EV for t ∈ [0, 1] and has two connected components otherwise. Proof. The proof follows directly from the deﬁnition of a connected set. This lemma yields another possibility to characterize points of the reduced 2 2 twistor space RV .Forw = ζ ±i the discriminant ∆ = (w−ζ) + of (B.38) vanishes such that δ+ = δ−.Thuswehave

Lemma B.10. If w ∈ V ∪ V¯ then Qw ∩ PV has one connected component and two two connected components otherwise.

A direct consequence of this Lemma is that RV does not possess the Hausdorﬀ property.

Lemma B.11. For V being open the reduced twistor space RV associated to V is not Hausdorﬀ.

Proof. Let x ∈ ∂V and Ux a neighborhood of x. Then x/∈ V and according to Lemma B.10 there are two points x1 and x2 in RV associated to x. Schematically we have

Fig. B.2. The non-Hausdorﬀ property of RV

Obviously, each neighborhood Ux1 of x1 contains elements of the neighbor- ∩ ∅ hood Ux2 of x2,i.e.,Ux1 Ux2 = , and the Hausdorﬀ property fails.

NowwewanttoshowthatRV is a compact Riemann surface. More precisely, we have

Proposition B.12. Let V ⊂ H be simply connected. Then RV is a compact 1 Riemann surface, i. e., there is a covering map Γ : RV → CP .

− Proof. We have just shown that Γ 1(w) has either one or two points depending on w. Thus, it remains to show that Γ −1 CP1 \ V ∪ V¯ is a connected double cover of the complex projective plane CP1.Letx ∈ E4 be ﬁxed. Let 218 B Ernst Equation and Twistor Theory w(t) be a closed path which encircles V such that w(0) = w(1) = w0.From (B.38) it follows that ∆ changes its sign, i. e., δ± changes continuously to δ∓. Thus, a closed curve in CP1 lifts to a path in Γ −1 CP1 \ V ∪ V¯ which −1 joins the two points in the preimage Γ (w0).

Thus, following [88], we can construct RV associated to V in the following way: (i) Choose points c ∈ V andc ¯ ∈ V¯ and make a cut C from c toc ¯. (ii) Take two copies S0 and S1 of the extended w-plane with cuts C0 and C1. (iii) Identify points in V0 with the corresponding points in V1 and points in V¯0 with corresponding points in V¯1. (iv) Identify points on the +-branch of the cut C0 in S0 with points on the −-branch of the cut C1 in S1.

B.2.2 Holomorphic Vector Bundles over the Reduced Twistor Space This subsection deals with the symmetry reduced Penrose–Ward transform. We will ﬁnd that holomorphic vector bundles over RV for some region V of spacetime will correspond to a solution of the Yang equation on V .Let us ﬁrst recall the Penrose–Ward transform for our model. Let V ⊂ H and J = J(, ζ) a solution to the Yang equation (2.85) over V . Suppose, we have

H, Hˆ : EV −→ Gl(2; C) . (B.39) such that J = H · H.ˆ (B.40) Using the complex coordinates p ≡ w˜ = eiφ and q ≡ z¯ = ζ +it we deﬁne by −1 −1 Υ := H (∂p¯Hd¯p + ∂q¯Hd¯q)+Hˆ ∂pHˆ dp + ∂qHˆ dq (B.41)

2 a gl(2; C)-valued connection D =d+Υ in the trivial vector bundle (BV ,EV , C ) with total space 2 BV = EV × C . (B.42) In Sect. 2.4 it has been shown 2 Lemma B.13. The connection D=d+Υ in (BV ,EV , C ) is anti-self-dual if and only if J is a solution to the Yang equation (2.111).

Let us now deﬁne by means of Υ a gl(2; C)-valued one-form on PV . We take Ψ := π∗Υ (0,1) , (B.43) where π is the projection in the twistor bundle π : CP3 → S4 and where ∗ (0,1) ∗ π Υ denotes the (0, 1)-component of the one form π Υ on PV in the decomposition (ξ,ξ¯). With λ := δe−iφ we have B.2 Symmetry Reductions of the Penrose–Ward Transform 219

Proposition B.14. The connection one form Ψ is given by 1 −1 − ¯ ¯ Ψ = 2 H (∂H λ∂ζ H) ∂ +(∂ζ H + λ∂H) ∂ζ 1+λ −1 2 2 +Hˆ λ ∂Hˆ + λ∂ζ Hˆ ∂¯ + λ ∂ζ Hˆ − λ∂Hˆ ∂ζ¯ . (B.44)

Proof. We will prove the proposition for H = J. Then we have eiφ ∂H 1 ∂H π∗Υ = H−1 e−iφd − ie−iφdφ + H−1 (dζ − idt) , (B.45) 2 ∂ 2 ∂ζ such that we ﬁnd eiφ ∂H 1 ∂H Ψ = H−1 e−iφ∂¯ + ∂¯ e−iφ + H−1 ∂ζ¯ − i∂t¯ . (B.46) 2 ∂ 2 ∂ζ From (B.14) we know ξ2 = ζ +it − δe−iφ ,ξ3 = eiφ + δ(ζ − it) , (B.47) and using the fact that ξ1 and ξ2 are holomorphic coordinates, i. e., ∂ξ¯ 1 = ∂ξ¯ 2 = 0, we obtain 0=∂ζ¯ +i∂t¯ − δ ∂¯ e−iφ − δe−2iφ∂¯ eiφ , 0=eiφ∂¯ + ∂¯ eiφ + δ∂ζ¯ − iδ∂t¯ . Therefore, we ﬁnd λ2 − 1 ∂¯ − 2λ∂ζ¯ ¯ iφ ∂ e = −iφ 2 , e (λ +1) λ2 − 1 ∂ζ¯ +2λ∂¯ i∂t¯ = , (λ2 +1) and inserting into (B.46) yields (B.44).

Remark B.15. (i) Since ∂¯ = ∂ζ¯ =0forx ∈ EV ﬁxed, we have | Ψ X = 0 (B.48) along all ﬁbres X = π−1(x). (ii) We have LT Ψ = LΦΨ =0 . (B.49) In fact, since H and λ are independent of t,wehave 1 ∂H ∂H T H−1 − λ =0 . (B.50) 1+λ2 ∂ ∂ζ

Furthermore, LT ∂¯ = LT ∂ζ¯ = 0. The proof of the LΦΨ = 0 follows similarly. 220 B Ernst Equation and Twistor Theory

(iii) Let us consider the quadric lim Qw in PV . Since w→∞ 1 w = ζ + λ−1 − λ , (B.51) 2 this means λ, δ → 0orλ, δ →∞. We denote

Λ0 := PV ∩{λ =0} ,

Λ1 := PV ∩{λ = ∞} , (B.52) 1 i. e., Λ0 and Λ1 are the points in RV above ∞ in CP . Then we ﬁnd from (B.44) ∂H ∂H Ψ(Λ )=H−1 ∂¯ + ∂ζ¯ = H−1∂H¯ , 0 ∂ ∂ζ 1 2 ˆ −1 ˆ ¯ 2 ˆ −1 ˆ ¯ Ψ(Λ1) = lim 2 A + λB + λ H ∂H ∂ + λ H ∂ζ H ∂ζ λ→∞ 1+λ −1 ¯ −1 ¯ = Hˆ ∂ζ Hˆ ∂ζ + Hˆ ∂Hˆ ∂ = Hˆ −1∂¯H.ˆ (B.53) The curvature form of Υ is anti-self-dual and we want to investigate the consequences for Ψ. To do this, we use the following theorem, due to Atiyah, see [274]. Proposition B.16. A two-form ω on U ⊂ S4 is anti-self-dual if and only if π∗ω is a (1, 1)-form over U˜ ⊂ CP3. Remark B.17. This proposition is closely related to a lemma which states that a two-form ω over U ⊂ R4 is anti-self-dual if and only if ω is of type (1, 1) for all compatible complex structures on R4, see [274]. The above proposition implies the following result Lemma B.18. A necessary condition for the anti-self-duality of the curvature two-form F =dΥ + Υ ∧ Υ is ∂Ψ¯ + Ψ ∧ Ψ =0 . (B.54) Proof. Let us decompose π∗Υ = Ψ + Ψˆ =: Ψ,˜ (B.55) with Ψˆ being a (1, 0)-form. Then we ﬁnd for the pull-back π∗F of the curvature form F π∗F = π∗ (dΥ )+π∗ (Υ ∧ Υ )=d(π∗Υ )+π∗Υ ∧ π∗Υ = ∂ + ∂¯ Ψ + Ψˆ + Ψ + Ψˆ ∧ Ψ + Ψˆ = ∂Ψ + ∂Ψ¯ + ∂Ψˆ + ∂¯Ψˆ + Ψ ∧ Ψ + Ψ ∧ Ψˆ + Ψˆ ∧ Ψ + Ψˆ ∧ Ψ.ˆ B.2 Symmetry Reductions of the Penrose–Ward Transform 221

The (2, 0)-part of the pull-back of F does not depend on Ψ and, according to Proposition B.16,the(0, 2)-part of the pull-back of F has to vanish, i. e.,

∂Ψ¯ + Ψ ∧ Ψ =0 . (B.56) In the following we will make use of the decomposition of the connection ∞ 2 ∇ = d + Ψ˜ in the trivial C -bundle B = PV × C into a (1, 0)-part and a (0, 1)-part ∇ = ∇(1,0) + ∇(0,1) . (B.57) Then we have the following proposition, for the proof of which we refer to [275], see also [276]. ∗ Proposition B.19. The (0, 1)-form Ψ ∈ T (PV )⊗gl(2; C) with ∂Ψ¯ +Ψ ∧Ψ = 0 deﬁnes a holomorphic structure on B,if

∇(0,1)b = ∂b¯ + Ψb =0 , (B.58) for b ∈ Γ (PV ,B). Remark B.20. The proof makes essential use of the following proposition Proposition B.21. Let (B,M,π) be a vector bundle over a complex manifold M and let ∇ be a connection in B with curvature being of type (1, 1).Then there exists a holomorphic structure on (B,M,π). | Corollary B.22. According to (B.48), the restriction Ψ X of Ψ to the ﬁbres X = π−1(x) vanishes for x ∈ E4.Thus(B.58) reduces along the ﬁbres X to

∂b¯ =0 . (B.59) | From this it follows easily that the restriction B X is holomorphically trivial. Thus, Proposition B.19 gives the condition for locally holomorphic sections | × C2 in the vector bundle B, such that B X = X is trivial not only as a smooth bundle but also as holomorphic bundle. However, for a bundle over RV , it is also necessary that the sections are invariant under the G-action, i. e., the sections should fulﬁl

∂b¯ + Ψb =0 ,Φ(b)=0=T (b) . (B.60)

The existence of such sections is implied by (B.49) and (B.54). We will prove

Lemma B.23. The sections in (B.60) depend only on ζ, and λ.

Proof. That b is independent of t is a direct consequence of the deﬁnition of T as ∂ T = . (B.61) ∂t 222 B Ernst Equation and Twistor Theory

Let us now show that b depends on φ and δ only via λ.Wehave

∂ eiφ ∂ ieiφ ∂ = + , (B.62) ∂λ 2 ∂δ 2δ ∂φ and ∂b ∂b ∂b +iδ − iδ¯ =0 , (B.63) ∂φ ∂δ ∂δ¯ because Φ(b) = 0. The derivative with respect to δ¯ vanishes due to the holo- morphicity condition of Proposition B.19 and we have ∂b ∂b ∂b ∂b = −iδe−iφ , =e−iφ . (B.64) ∂φ ∂λ ∂δ ∂λ

Since the leaves of the foliation of PV are simply connected, the value of b on a leaf Λ is completely determined by its value in one point of Λ via Yφ(b) − Y¯φΨ b =0=Yt(b) − Y¯tΨ b, (B.65) and the complex conjugate equations which are a direct consequence of (B.60) and the deﬁnitions of Yφ and YT . Let now EΛ be the two-dimensional complex space of solutions to these equations.

Definition B.24. The vector bundle E → RV is generated in each point Λ ∈ RV by the two-dimensional fibre EΛ. By taking the locally holomorphic invariant sections of B, i. e., solutions to (B.60), as locally holomorphic sections of E, the holomorphic structure of E is induced by that of B. The matrices H and Hˆ distinguish bases in the fibres over Λ0 and Λ1 in the following way. We define 0 −1 ◦ f i = H π Λ ei , 0 1 −1 f i = Hˆ ◦ π ei , (B.66) Λ1

(i =1, 2), with 1 e = , 1 0 0 e = . (B.67) 2 1

Now we have 0 0 1 1 Proposition B.25. The sets f 1,f 2 and f 1,f 2 form ﬁbre bases over Λ0 and Λ1. B.2 Symmetry Reductions of the Penrose–Ward Transform 223

0 0 Proof. Since H and Hˆ have non-vanishing determinants, f 1 and f 2 respec- 1 1 tively f 1 and f 2 are linearly independent. Then we have for the i-th column of H−1 ∂¯ H−1 + Ψ H−1 = ∂¯ H−1 + H−1∂H¯ H−1 (B.68) i i i i − −1 ¯ −1 −1 ¯ −1 = H ∂(H)H i + H ∂H H i =0 . 1 1 The proof for f 1,f 2 follows similarly.

Remark B.26. A diﬀerent choice of H and Hˆ generates a bundle isomorphic to E.

To summarize, we have shown, how to associate to a solution to the Yang equation on some region V of spacetime a holomorphic vector bundle over the reduced twistor space RV associated to V . In fact, the converse is also true. We have Theorem B.27. There is a 1-1-correspondence between: (i) Gl(2; C)-valued solutions J to the Yang equation on V ⊂ H, uniquely determined up to transformations J !→ AJB−1 with A, B ∈ Gl(2; C) and C ∗ | (ii) holomorphic vector bundles (E,RV , Gl(2; )) such that Π E X is holo- −1 morphically trivial for any ﬁbre X = π (x) ⊂ PV (x ∈ EV ). 0 1 The ﬁxing of frames f respectively f over Λ0 respectively Λ1 in E determines J completely.

Remark B.28. The transformations J !→ AJB−1 correspond to a left multiplication of H by A and of Hˆ by B. Since A and B are constant matrices the one-form Υ remains constant.

Proof. It remains to be shown how a solution to the Yang equation with the required properties can be constructed by means of a holomorphic vector bundle over RV . Let (E,RV , Gl(2; C)) be a vector bundle fulfilling the above requirements. Due to the triviality in any fibre we may construct for B a | globally smooth frame (b1,b2) such that bi is a holomorphic section of B X for all X. Taking the derivative of bi with respect to the coordinates of PV we define globally a gl(2; C)-valued (0, 1)-form Ψ by

∂b¯ i + Ψbi =0 , (B.69) which in turn yields a holomorphic structure on B. Writing

Ψ =(π∗Υ )(0,1) , (B.70) we obtain, according to Proposition B.16, a one-form Υ with anti-self-dual curvature. Obviously, the choice of bi is not unique: with bi we may also 224 B Ernst Equation and Twistor Theory

j j choose ai bj with (ai ) ∈ Gl(2; C) depending smoothly on , ζ, φ and t. This transformation induces a gauge transformation of Υ . In order to determine J completely, we deﬁne frames over Λ0 respectively Λ1 by the following formulae

0 ∗ 0 ∗ s = Π f0 ,s= Π f0 . (B.71)

Here Π : PV / F−→RV (B.72) is the projection of PV onto RV which associates to each p ∈ PV the corresponding leaf. We define b| := s0 H, (B.73) Λ0 and obtain b| := s1 H,ˆ (B.74) Λ1 ¯ where Υ is determined by (B.41). Then we have ∂b + Ψb =0onΛ0 and Λ1 and obtain J by J := H · Hˆ −1 . (B.75) Therefore, the chosen gauge fixes the splitting of J into H and H¯ and the fixing of f 0 and f 1 determines the entries of J. In particular, for b| = s1 Λ1 we get b| = s0 · J,˜ (B.76) Λ0 0 1 with J˜ : Λ0 → Gl(2; C) given by E, f and f . Then J˜ is just the solution of the Yang equation one is looking for. Furthermore, we have Ψ| = J˜−1∂¯J˜ Λ0 and it follows with (B.75) that H = J˜ = J. A change of the frames s0 respectively s1 on Λ0 respectively Λ1 can, according to the definition of these frames, only lead to a transformation J !→ AJB−1 with constant matrices A and B. For the same reason we have J = J(, ζ). Of course, a solution to the Yang equation has to be real and symmetric in order to describe a stationary and axisymmetric solution to the Einstein equations. From Sect. 2.4 we know that with J being a solution to the Yang equation also J −1, J¯ and J T are solutions (J = J¯). Therefore, we may construct holomorphic vector bundles over RV corresponding to these solutions and relate them to E. To do this we have to define some natural involutions on RV :

(i) Let i : RV → RV be the holomorphic mapping, which interchanges the two spheres S0 and S1 in RV ,i.e.,

i(w±):=w∓ . (B.77)

(ii) Let j : RV → RV denote the anti-holomorphic mapping

j(w±):=¯w∓ . (B.78) B.3 Transition Matrices for the Holomorphic Vector Bundles 225

Then we have the following Proposition B.29. Let J be a solution to the Yang equation corresponding 0 1 to the triple (E,f ,f ) over the reduced twistor space RV by Theorem B.27. Then (i) the inverse solution J −1 is generated by i∗(E,f0,f1):= i∗(E),i∗(f 0),i∗(f 1) , (B.79)

(ii) the complex conjugate solution J¯ is generated by the triple ¯j∗(E,f0,f1):= j∗(E¯),j∗(f¯0),j∗(f¯1) , (B.80)

(iii) and the solution (J −1)T is generated by the triple of dual objects (E,f0,f1)∗ := E∗, (f 0)∗, (f 1)∗ . (B.81)

Proof. We will prove (i), the proof of (ii) and (iii) follows similarly. Let J = H · Hˆ −1. Then J −1 = Hˆ · H−1. Therefore, J −1 generates by (B.44)a(0, 1)- form Ψ in which H and Hˆ have changed their role. But, from the form of Ψ is obvious that Ψ(λ)=Ψ (−λ−1) and, therefore, for the ﬁbre it follows −1 −1 E (λ)=E(−λ ). With i(w±)=w∓ it follows i(λ)=−λ andweﬁnd E = i∗(E).

Corollary B.30. We have: J is symmetric iﬀ

(E,f0,f1)∗ = i∗(E,f0,f1) , (B.82)

and J is real iﬀ (E,f0,f1)∗ = ¯j∗(E,f0,f1) . (B.83)

T Proof. The ﬁrst assertion follows from the fact that J = J is equivalent to T J −1 = J −1 . The second is just the deﬁnition.

B.3 Transition Matrices for the Holomorphic Vector Bundles

B.3.1 The Covering of the Reduced Twistor Space

It is well known, that the two-sphere S2 can be covered completely by two local charts. As it has been shown in Sect. B.2.1, the reduced twistor space RV , is a two-sheeted covering of the Riemann sphere. This suggests that RV may be covered by four charts R0,...,R3. These can be viewed as covers of S2 induced by the compactiﬁcation of E2 with the complex coordinate w.Let 226 B Ernst Equation and Twistor Theory

us now, following [88], but see also [166] for a detailed description, construct a standard cover of RV . To begin with, we cover the Riemann sphere by two charts U and U in the following way. Let U denote an open subset of the complex w-plane such that (i) u ∈ =⇒ u¯ ∈ U, (ii) V ∪ V¯ ⊂ U, (iii) U \ V , U \ V¯ ,andU itself are connected and simply connected. Furthermore, let U be a neighborhood of w = ∞ such that (i) u ∈ U =⇒ u¯ ∈ U and (ii) the intersection A = U ∪ U is an annular region, see Fig. B.3.

w w

V V

A+

w w A−

V¯ V¯

Fig. B.3. The covering of the w plane

Then A is the union of two simply connected sets A+ and A−, A = A+ ∪A−, −1 and we have A+ ∩ A− =(A ∩ V ) ∪ A ∩ V¯ . The set Γ (U ) ⊂ RV contains −1 two copies R0 and R1 of U ,i.e.,Γ (U )=R0 ∪ R1. The subset R0 is a neighborhood of w = ∞ in the Riemann sphere S0 and, similarly, R1 is a neighborhood of w = ∞ in S1. The intersection R0 ∪ R1 on RV can be identified with the intersection A+ ∩ A− in the complex w-plane. − Analogously, Γ 1(U) consists of two copies, which both contain the cut −1 C and intersect in Γ V ∪ V¯ . We denote these copies by R2 and R3 where the labelling is fixed by −1 −1 R0 ∩ R2 ⊂ Γ (A−) ,R1 ∩ R3 ⊂ Γ (A+) . (B.84) −1 −1 In Γ (A+ ∩ A−) ⊂ Γ (V ∪ Vˆ ) the four covering sets are identified. But, over each point of the Riemann sphere they are in one sphere and, therefore, they are Hausdorff and may be used as coordinate patches for the basis RV of the holomorphic vector bundles. B.3 Transition Matrices for the Holomorphic Vector Bundles 227

B.3.2 Patching Matrices for Real, Symmetric Framed Bundles

2 Let us now consider the holomorphic vector bundle (E,RV , C ) with typical 2 ﬁbre C and projection pr : E → RV . This bundle is trivial if restricted to the contractible Hausdorﬀ manifolds Rα. Since the Rα are Stein manifolds, we may choose globally a holomorphic frame eα in the form of a 2 × 2-matrix

α α α (e )=(e1 ,e2 ) . (B.85)

(Note that α enumerates the different covering sets and not the different vectors of one particular frame.) Then, the trivialization of E takes the form i α ∈ Rα,y ei for pr(y) Rα. In particular, we may have e0 = f 0 ,e1 = f 1 , (B.86) Λ0 Λ1 which is well defined because of Λ0 ∈ R0 and Λ1 ∈ R1. In an intersection α Rα ∩ Rβ the patching matrix Pαβ transforms holomorphic frames e into holomorphic frames eβ according to

β α e = e Pαβ . (B.87)

In general, there are six patching matrices which we denote as follows:

P01 : A+ ∩ A− −→ Gl(2; C) , P02 : A− −→ Gl(2; C) , P : A− −→ Gl(2; C) , 13 (B.88) P03 : A+ −→ Gl(2; C) , P12 : A+ −→ Gl(2; C) , P23 : V ∪ Vˆ −→ Gl(2; C) .

1 Since the covering of RV by Rα suspends locally the twofold covering of CP , the patching matrices depend only on w. The cocycle condition

P01P12P20 = I (B.89)

− fixes P01 on its whole domain A+ ∩ A in terms of P12 and P02,i.e.,we are left with five transition matrices. This number may be reduced further by requiring particular properties of the holomorphic bundle E. Here the starting point is that the Yang matrix J which corresponds to a vacuum solution of the stationary and axisymmetric Einstein equations has to be real and symmetric, see (2.84). Recall that we have seen in Sect. B.2.2 what these requirements mean for the corresponding holomorphic vector bundles. In the sequel we shall assume that the conditions (B.82) and (B.83) for the holomorphic bundles are fulfilled. (How det J = −2 can be achieved will be discussed in Corollary B.37.) In order to transfer conditions like (B.82) and (B.83) to the patching matrices we first observe for J being symmetric 228 B Ernst Equation and Twistor Theory the isomorphism between E∗ and i∗(E) can be understood as a family of invertible linear mappings depending holomorphically on Λ ∈ RV −→ ∗ SΛ : EΛ Ei(Λ) . (B.90)

Since i2 =1wehave −→ ∗ Si(Λ) : Ei(Λ) EΛ , (B.91) ∗ and we denote by Si(Λ) the map dual to Si(Λ) ∗ −→ ∗ Si(Λ) : EΛ Ei(Λ) . (B.92) Then we may deﬁne by −1 ∗ ◦ −→ σΛ := Si(Λ) SΛ : EΛ EΛ (B.93) a holomorphic section in the automorphism bundle Aut(E)=E ⊗ E∗ of E and have

Lemma B.31. σ ≡ I on RV . ∗ ∗ 0 1 1 0 Proof. We have SΛ0 (f )= f and SΛ1 (f )= f and, therefore, σ = I over Λ0 and Λ1. Now we pull back the bundle E to the ﬁbre X ⊂ PV | ∗ B X = Π (E) (B.94) | | ∗ and use the triviality of B X and Aut(B X ). Thus, Π (σ) is a holomorphic section in a trivial bundle which is the identity map over λ =0andλ = ∞. By Liouville’s theorem σ is then the identity on the whole of RV .

Thus, we have for the standard cover of RV with R0 = i(R1)andR2 = i(R3) and the corresponding frames ∗ ∗ ∗ i Ei(Λ) =(i E)Λ = EΛ , (B.95) i. e., ∗ ∗ i∗(e0)= e1 i∗(e1)= e0 , ∗ ∗ i∗(e2)= e3 i∗(e3)= e2 . (B.96)

Analogously, for a vector bundle corresponding to a real solution of the Yang equation there exists an isomorphism ¯ TΛ : EΛ −→ Ej(Λ) , (B.97) and one deﬁnes ¯ τΛ := Tj(Λ) ◦ TΛ . (B.98) B.3 Transition Matrices for the Holomorphic Vector Bundles 229

Since τ is also a holomorphic section in the automorphism bundle Aut(E)of E, which is the identity mapping over Λ0 and Λ1, we find again, that τ is the identity map everywhere on RV . Then we have ∗ ∗ ¯ j Ej(Λ) =(j E)Λ = EΛ , (B.99) i. e., j∗ (eα)=¯eα , (B.100) for all α =0,...,3. Using (B.96) and (B.100) we find for the patching matrices of a real, symmetric framed bundle the following. Lemma B.32. Let J be a symmetric real solution to the Yang equation (2.85) and (E,RV , Gl(2; C)) the corresponding holomorphic vector bundle over RV with patching matrices as defined above. Then we have

Pαβ(w)=P¯αβ(w) , (B.101) and

T T T P02(w)=(P31(w)) ,P03(w)=(P21(w)) ,P23(w)=(P23(w)) . (B.102)

β α β α Proof. Let us show ﬁrst (B.101). Since e = e Pαβ, we also havee ¯ =¯e P¯αβ and, therefore, ∗ β ∗ α α j e = j (e Pαβ)=¯e Pαβ , (B.103) and ∗ β β α j e =¯e =¯e P¯αβ . (B.104) In order to prove (B.102), we ﬁrst note that ∗ 2 3 ∗ T 1 ∗ i e = e =(P31) e , (B.105) and ∗ 2 ∗ 0 1 ∗ i e = i e P02 =(P02) e (B.106) is an identity, because the domains of P02 and P31 coincide. Furthermore, we have ∗ 3 2 ∗ T 3 ∗ i e = e =(P23) e , (B.107) and ∗ 3 ∗ 2 3 ∗ i e = i e P23 =(P23) e . (B.108) According to the Lemma B.32 the framed bundle (E,f0,f1) corresponding to a real and symmetric solution of the Yang equation (2.85), is completely characterized by the following three patching matrices 230 B Ernst Equation and Twistor Theory

F := P02 : A− −→ Gl(2; C) ,

G := P12 : A+ −→ Gl(2; C) , | −→ C P := P23 V : V Gl(2; ) . (B.109) Furthermore, we have | ¯ | ¯ P23 V¯ = P23 V = P, (B.110) and we way obtain from the identities given above further requirements for the patching matrices. From the above Lemma we immediately ﬁnd P = P T . In addition, we have G(w)=G¯(w)andF (w)=F¯(w), i. e., on the real axis these matrices become real. Finally, on A ∩ V = A− ∩ V = A+ ∩ V we ﬁnd

T P23P30P02 = I = PG F. (B.111)

We have Proposition B.33. The patching matrices are only determined up to gauge transformations of the form −1 (P, F, G) −→ LP LT ,KFL−1, KT GL−1 , (B.112) with the holomorphic and (in the above sense) real matrices L : U → Gl(2; C) and K : U → Gl(2; C), fulﬁlling L = L¯, K = K¯ and K(∞)=I.HereU and U are the open subsets of the w-plane introduced at the beginning of Sect. B.3.1.

Proof. The proof follows by direct calculations. Since P = P T we have

T LP LT = LP LT . (B.113)

Furthermore, the cocycle condition (B.111) leads to T −1 T LP LT L−1 GT KT KFL−1 = LP GT FL−1 = LL−1 = I. (B.114) Thus, we have a one-to-one correspondence between real symmetric solutions to the Yang equation (2.85) and equivalence classes of triples of patching matrices [P, F, G] with properties discussed above.

B.3.3 The Axis-simple Case

As we have shown in Sect. 4.2, the solutions (4.19) to the Ernst equation are regular outside a contour which can correspond to the surface of a rotating body. It has been shown in [88] that under these circumstances there is a further reduction of the set of patching matrices [P, G, F]. To ﬁnd this reduction B.3 Transition Matrices for the Holomorphic Vector Bundles 231

one investigates the behavior of the holomorphic bundles by approaching the symmetry axis =0. Let U ⊂ E2 be deﬁned as in the beginning of Sect. B.3.1. We denote by RU the analogue of RV , constructed for U. Let the projection

η : RV −→ RU (B.115)

be deﬁned by w; w ∈ U \ V ∪ V¯ η(w±):= , (B.116) w±; w ∈ V ∪ V¯

for w± ∈ RV . Deﬁnition B.34. The triple (E,f0,f1) is called axis simple, iﬀ there exists a triple (E,f0 ,f1 ) such that (E,f0,f1)=η∗ E,f0 ,f1 . (B.117)

Proposition B.35. The following statements are equivalent: (i) A bundle E exists. | (ii) E Γ −1(U) is trivial. (iii) If E is, furthermore, the holomorphic vector bundle of a real, symmetric solution to the Yang equation, then the patching matrix P can be ana- lytically continued to a function on U, which is real on the symmetry axis.

Proof. (i) =⇒ (ii): Let E exist, such that η∗(E)=E. Then we have for the −1 ﬁbre over w± ∈ Γ (U \ (V ∪ V¯ ))

∗ Ew+ = Ew− = η (E )w . (B.118) | | Since U is contractible, E U is trivial and, therefore, also E Γ −1(U). ⇒ | (ii) = (i): Let E Γ −1(U) be trivial. Then, there exists a biholomorphic mapping φ :pr−1(Γ −1(U)) → Γ −1(U) × C2. We have to construct E which is, being a bundle over a contractible manifold, trivial. Due to its triviality, E can globally be written as E = (p, e)| p ∈ Γ −1(U),e∈ C2 . (B.119) | We construct E by choosing the fibre E p over η(p). Then, the condition η(p)=pr(e) is fulfilled, and E is trivial. (iii) =⇒ (i): For vector bundles related to real symmetric solutions to the Yang equation, the patching matrix P is well defined and its analytic continuation P˜ yields a globally defined holomorphic section s in the vector bundle over Γ −1(U) putting: 232 B Ernst Equation and Twistor Theory 3 e (w±);w± ∈ R s = (1) 3 . (B.120) 2 ˜ ∈ e1P(i1)(w±);w± R2

3 2 2 Here we sum over the index i of P˜ and make use of e = e P˜23 ≡ e P˜ in V ∪V¯ . | Then, E Γ −1(U) is trivial. Conversely, a global section as constructed above yields an analytic continuation of P . Using (B.96) respectively P (w)=P¯(w) we eventually find the reality on the real axis. Since R = S0∪S1, E is constructed out of two bundles E| and E| over U S0 S1 the two spheres which form RU . Being holomorphic bundles over the Riemann sphere these two bundles can, according to a theorem by Grothendieck (see e. g. [277] for a proof), be decomposed into direct sums of line bundles E| = Lp ⊕ Lq , S0 E| = L−p ⊕ L−q , (B.121) S1 with p ⊗···⊗ L = L L , (B.122) p times and L−1 = L∗. Here L is the tautological bundle over CP1. The explicit form of E| for a given E| is a direct consequence of the symmetry of S1 S0 J (or i∗(E)=E ∗). Thus, the patching matrices of an axis-simple bundle corresponding to a real and symmetric solution to the Yang equation, have the following form P : U −→ Gl(2; C) , (B.123) with P (w)=P¯(w), P (w)=P (w)T , can now be defined on U. Furthermore, we have p −1 ∩ −1 −→ C (2w) 0 F : Γ (U) Γ (U ) Gl(2; )= q (B.124) S0 0(2w) and, finally, G : Γ −1(U) ∩ Γ −1(U ) −→ Gl(2; C) , S1 −p (2w) 0 − G = P = · P (w) 1 , (B.125) 12 0(2w)−q because of the cocycle condition (B.111). Thus, in the axis-simple case, the holomorphic vector bundle E can be characterized by the triple (P, p, q) consisting of one patching matrix and two integers. It turns out that these parameters can be comparatively easily fixed and that we have a close relation between the value of P (w) and the asymptotic behavior of J(, ζ) in the limit → 0. The following proposition, for the proof of which we refer to [88] (for p = q = 0) and also [166], shows that the axis behavior of J can be read off from the patching data. B.3 Transition Matrices for the Holomorphic Vector Bundles 233

Proposition B.36. Let p>qand let g,gˆ and Ω be functions of w, such that the ﬁrst two do not vanish for → 0,andletP denote the transition matrix in an axis simple vector bundle of the form g −gΩ P = . (B.126) −gΩ gΩ2 +ˆg−1 Then, the corresponding Yang matrix has for → 0 the following behavior: p h(ζ) −h(ζ)L 0 − J(, ζ)= −1 + O rp q+1 0 q −h(ζ)Lh(ζ)L2 + hˆ(ζ) p 0 × . (B.127) 0 q

Here we have set h(ζ)=(−1)pg(ζ) and hˆ(ζ)=(−1)qgˆ(ζ) and (−1)pp−q dp−qΩ L = . (B.128) p−q − p−q 2 (p q)! dw w=ζ Then, for the determinants of the Yang matrix J and the patching matrix P it follows immediately Corollary B.37. Under the assumptions of the above proposition we have: p+q If det P =1then det J = −2 . For our purposes more important is the converse, see [163, 164]. Proposition B.38. Let E =e2U +ib be a solution to the Ernst equation with regular symmetry axis. Then the patching matrix P is given by 1 −b(0,ζ) P (ζ)=e−2U(0,ζ) . (B.129) −b(0,ζ)e4U(0,ζ) + b(0,ζ)2 Before we turn to the construction of the holomorphic vector bundles for the solutions of Chap. 4 we discuss the construction for several examples. Examples:

(i) The Weyl class. The solutions of this class have real Ernst potentials E =e2U , and we ﬁnd for the patching matrix e−2U(0,ζ) 0 P (ζ)= . (B.130) 0e2U(0,ζ) Let us consider the particular case of the Schwarzschild solution. From(1.8) with ϕ = 0 we get at the symmetry axis for ζ>m ζ − m e2U(0,ζ) = . (B.131) ζ + m 234 B Ernst Equation and Twistor Theory

(ii)The Kerr metric. Recall from (1.8) that the Ernst potential of this solution has the form −iϕ iϕ − E e r+ +e r− 2m cos ϕ = −iϕ iϕ , (B.132) e r+ +e r− +2m cos ϕ with r± = (ζ ± m cos ϕ)2 + 2. Then we ﬁnd by a simple calculation for ζ>mcos ϕ ζ2 − m2 cos2 ϕ 2m2 sin2 ϕ E(0,ζ)= − i , (B.133) (ζ + m)2 + m2 sin2 ϕ (ζ + m)2 + m2 sin2 ϕ and we have for the patching matrix 1 (ζ + m)2 + m2 sin2 ϕ 2m2 sin2 ϕ P (ζ)= . ζ2 − m2 cos2 ϕ 2m2 sin2 ϕ (ζ − m)2 + m2 sin2 ϕ (B.134)

B.4 Patching Matrices for the Class of Hyperelliptic Solutions

In Sect. 4.2 we have distinguished a class of solutions to the Ernst equation with physically interesting properties. The corresponding Ernst potentials are of the form   +   Θpq(ω(∞ )+u) 1 E ∞+∞− (, ζ)= + exp ln G(τ)dω (τ) , (B.135) Θpq(ω(∞ ) − u) 2πi  Γ 1 00 where the theta characteristic consists of blocks of the form and 2 00 1 00 . An essential property of these solutions is for Θ (ω(∞+) − u) =0 2 11 pq their regularity outside Γξ, which can possibly be interpreted as the surface of the rotating body one is interested in. Thus, the solutions are regular on V being an open subset of H \ Γξ, V ⊂ H \ Γξ. In particular, the axis behavior is regular as long as Θpq (u ) =0 , (B.136) which is the condition for an ergosphere not to hit the symmetry axis, see Sect. 7.3. Thus, we have Proposition B.39. Let E be a solution of the form (B.135). Furthermore, let RV be the reduced twistor space for V ⊂ H \ Γz. Then the holomorphic vector bundle E → RV corresponding to E is characterized by one patching matrix iﬀ Θpq (u ) =0 . B.4 Patching Matrices for the Class of Hyperelliptic Solutions 235

Corollary B.40. From Proposition B.38 it is obvious that the patching matrix is determined by the axis potential which is for the class of solutions (4.19) given by ∞+ − ∞+ ∞+ | − ε (ωg ( )+ug ) | Θpq ω ζ+ + u +( 1) e Θpq ω ζ− + u E(0,ζ)= ∞+ − ∞+ − ∞+ | − − ε (ωg ( ) ug ) | − Θ ω + u +( 1) e Θ ω − u p q ζ  p q ζ   1 exp ln G(τ)dω + − (τ)+u . (B.137) 2πi ∞ ∞ g Γ Remark B.41. (i) For the more general case of an algebro-geometric solution, obtained in Theorem 3.7, we have proven in Proposition 4.3, that the Ernst potential is regular on the axis except at the points where ξ coincides with the singularities of the Abelian integral of the second kind Ω, points of Γ and branch points, provided (B.136) holds (with u replaced by u + b). If, e. g, a pole of Ω is on the axis then we have an essential singularity, see Proposition 4.1, and the solution is no longer axis-simple. (ii) In proving Theorem 3.7 we have constructed explicitly a matrix Φ on L0, see (3.48), with column vectors given by ψ(Pˆ) ± X(Pˆ)= , Pˆ ∼ ξ , (B.138) ±χ(Pˆ)

with ψ and χ being of the form (3.38) and (3.46). These vectors just form a basis in the ﬁbre of E over any point of RV .