Internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 209-221
209
TWISTORS AND GAUGE FIELDS
A.G. SERGEEV
Steklov Mathematical Institute
Moscow
(Received August 23, 1985)
We describe briefly the basic ideas and results of the twistor theory.
ABSTRACT.
The main points:
twistor representation of Minkowsky space, Penrose correspondence
and its geometrical properties, twistor interpretation of linear massless fields,
Yang-Mills fields (including instantons and monopoles) and Einstein-Hilbert equations.
KEY WORDS AND PHHASES. istons, Minowsy space, line.an massTess fiaZds, Yang-Mill.
eguations, Einstein-Hilbent egTations.
1980 AMS SUBJECT CLASSTF[CATION CODES. 8C
I.
INTRODUCTION.
The theory of twistors, first initiated by R. Penrose at the end of the sixties,
now enjoys a period of rapid development.
The increased interest in this theory is
explained, mainly, by its applications to solving the fundamental nonlinear equations
of theoretical physics, namely Yang-Mills and Einstein-Hilbert equations (YM- and EH-
equations, for short).
The twistor method together with the closely related inverse
scattering method have now become the main methods available to construct classical
solutions of these equations.
In this review we describe briefly the basic ideas and
results of the twistor theory.
see
For a more detailed exposition and further references
[I-3].
Twistor model of
Minkowsky space.
The Minkowsky space-time
M
which is the
basic space of relativistic field theory appears in the twistor theory in a rather
unusual form.
The twistor model of
M
is constructed in two steps,
which is the spinor model of Minkowsky space.
coordinates of a point
x
Let
x
(x
0
x
x
2
the first of
x
3
denote the
Mwith a fixed orgin.
The mapping
2
x
X
2
x e M
assigns to every point
matrices is called the
The vector space
S
+
ix
3
x
0
3
(1.1)
xl
an Hermitian
2
2
-matrix
spinormodelofMinkowskispace and
C
X.
The space of all such
is denoted by the same letter
2
M.
where these matrices operate, is called the spinor space.
0
The coordinates of spinors, i.e. vectors z
S are denoted by z A
(z
z );
coordinates of complex conjugate spinors, i.e. vectors
w e S, are denoted by
w
A
210
(w
A.G. SERGEEV
0’
).
w
S’,
u
Dual spinors, i.e. vectors
S’, are denoted:
v
u
(u 0,
A
Ul),
(v0,, v I,)-
VA,
In spinor notation the mapping (i.i)has the form:
A
where
o
X
o
0
0
2
(x
2
Using the standard basis in the space of Hermitian
0,I
I
0
(x a)
x
AA’
2 -matrices:
03
i
(I.I) can be written in the form
(x a)
x
o
where
AA’
(x
X
3
AA’)
(a=Z0
a AA’
x o
a
o
are entries of the matrix
a
(1.2)
a
a
0,I,2,3
in spinor notation.
"Z" sign in formulas and sum over repeated indices.
Physicists usually omit the
The mapping (1.1) has the interesting property of transforming the Lorentz norm
(x0) 2
Ixl 2
(xl) 2
(x2) 2
(x3) 2
The eton of the Lorentz group
of a vector
M
x
SL(2,C)
Lorentz metric) corresponds to the inner action of the group
i)
with determinant
SL(2,C)
det X.
into
(the group of orthogonal transformations of M
L
on Hermitian matrices:
AxA *
X
A
(2x2 -matrices
SL(2,C).
The group
L
is thus seen to be the two-fold covering of the identity component
the Lorentz group (note that elements
A
and
-A
SL(2,C)
of
in
of
O
induce the same trans-
In other words, the spinor space is a space of the two-dimensional
the two-fold covering of the
(fundamental) representation of the group SL(2,C)
formation of M).
Lorentz group.
space
The mapping (I.i) can be extended to the complex Minkowski
space of points
(z a)
z
(z a)
z
Z
C
4
by the following formula:
(zao a AA
(z
).
The image of this mapping coincides with a space of all complex
is called the spinor model of
Izl 2
(gO)
Let’s
2
(zl)
(Z2)
2
22 -matrices which
CM
(and is denoted by the same letter), the norm
(Z3)
2
det Z.
transforms into
turn now to a construction of the twistor model of Minkowski space.
Assign to a matrix
tity
2
CM, which is a
Z
CM
the complex
4x2 -matrix
(-iZ)
12
2x2 -matrix, and consider the two-dimensional plane in
where
C
4
12
is the iden-
which is generated
by the basis consisting of the two four-dimensional columns of this matrix.
of all two-dimensional subspaces in
C
4
mapping constructed above assigns to every matrix
We shall write this mapping in coordinates.
(Z
Z
0
O,Z I,Z 2,Z 3)
0
and consider a vector
ZI I
Z
2
0’
Z3
Z
i’
G2(C 4),
i.e. the Grassmann manifold
called the twistor model of the complex Minkowski
C
4
spa.ce
Z
and is denoted by
CM
The space
is
CM
The
CM
a point of the space
Denote coordinates in
C
as a pair of spinors
Z
4
by
Z
(Z)
(A,A,),
i.e.
The above mapping can be rewritten then in
TWISTORS AND GAUGE FIELDS
the form
(z
Z
AA
f linear equations:
space
CP
space
PT
=-iz
AA’
A
A’’
In other words, we assign to a matrix
=>
(oA,nA,)
A
-+ {(mA,A,):
AA’
-lZ
A
(1.3)
0,
(z
AA’
the plane defined by a pair
Since these equations are homogeneous in
hA’
they also define a proective line in the 3-dimensional complex projective
3.
T
The space
CP
3
C
4
with coordinates
are called the spaces of twistors
(A,A,)
and the corresponding
twistors respective-
an._ective
ly.
The superposition mapping
{(A,A, ):
(z a)
asigns to a point
z
(z a)
plane in the twistor space
m
A
-iz
’ A’
of the complex Minkowski space
T
CM
or a projective line in the space
extends also to the conformal compactification
CM
a two-dimensional
PT.
The mapping (1.4)
"at infinity" (cf. [2]).
GEOMETRY OF TWISTORS.
We have defined the mapping (1.4) which assigns to every point of CM
tive line in
The associated correspondence between points of
PT
respondence.
CM
and
a projec-
PT
is
We now consider geometric properties of this cor-
called the Penrose correspondence.
CM
CM,
of the complex Minkowski space
a complex light cone
which is obtained by "adding" to CM
2.
(1.4)
These properties are formulated in diagrams where geometrical objects of
occupy the left side and corresponding objects of
PT
the right side.
We have:
{projective line of PT
{point of CM}
The converse assertion:
ig"
i =>{null
complex
in CM
2-planei
a-plane
bundle of projective
point of PT
lines passing through a fixed point
of PT
A null plane is by definition, a plane such that the distance
Lorentz metric) between any two of its points is zero.
a point of
A null plane corresponding to
by the Penrose correspondence is called an a-plane.
PT
in the complex
The dual asser-
tion:
complexin
[.=> {null
CM
2-plane
B-plane
projective plane of PT E point
of PT*
pencil of projective
lines lying in a fixed projective
plane
The "intersection" of the last two diagrams gives:
(0,2)-flag in PT E (point of PT
projective plane in PT including
this point) E pair of incident
points of PT and PT*
null line in CM
complex light ray
It follows from the last assertion:
complex light cone in CME
bundle of complex light
rays passing through a fixed
point of CM
These are the main facts of the twistor
Let’s
now consider the real
projective line of PT E (0,1,2)|
-flags in PT with a fixed
projective line
Beometry for the complex Minkowski space.
compactified Minkowski space M. Denote by N the
A.G. SERGEEV
212
T
quadric in
PN
and let
M
{point of
T- <=>
T- (Z
points
z
T+ (Z
(Z)
T+
0).
+ iye CM
x
0,
of PN
(point of PN;
tangent plane to PN in this point)
complex}
{point
T
0)
(Z)
lyl
2
PN
interseccomplex tangent
point of a fixed
the
space of
nesative
twistors
into the two subspaces
and the
CM+ (resp.
Denote by
such that
line of
projective
with a
tion ofto PN
PN in any
plane
projective line
divides the twistor space
positive twistors
IZ312
PT lying in PN}
projective line of
M
light cone of M
bundle of light rays
passing through a fixed
point of M
N
IZ212
A restriction of the Pen-
has the following properties:
null line of M
light ray of M
The quadric
IZII 2
+
PT.
be the associated projective quadric in
rose correspondence to
-ig"5]
IZ012
(Z)
given by the equation:
0
space
CM-)
coincides with a space of
y0
and
of the
lyl 2
(resp.
0
y0
0
0).
<
A restriction of the Penrose correspondence to these spaces gives:
point of CM+
CM-
projective line of
{(resp.
So,
{(resp.
in the real case we have the following duality:
PN; points of
proSective lines of
PN
correspond
M
rays which can intersect each other in
points of
lisht rays
to
PT
lying in
PT
correspond
to
PT-)
M
+}
Note that light
M.
split into separate points of
PN
This
fact is of the great importance in the twistor theory.
The transformation group
with determinant
SU(2,2) (the group of unitary transformations of
(Z)) preserves the quadric
preserving the form
duces transformations of the Minkowski space
light cones.
of
M
M
T
hence it in-
which carry light cones again into
SU(2,2)
In other words, the group
N
induces conformal transformations
Moreover, this group is the four-fold covering of the identity component of
M
the conformal group of
transformation of
M).
(note that elements
+/-A, +/-iA
of
SU(2,2)
induce the same
Hence, we can define the twistor space (by analogy with the
spinors) as a space of the 4-dimensional (fundamental) representation of the sroup
SU(2,2)
the four-fold covering of the
conformal
sroup.
It is also interesting to consider the Penrose correspondence for the compactifled Euclidean
space
(we recall the Euclidean space
E
the complex Lorentz metric
Izl
2
E
is a subspace of
coincides with Euclidean metric).
PT
projective line in
{point of
E
under mapping
j:
CM
where
We have
invariant
(Z 0, Z I, Z 2, Z 3)
(_i, 0, _3, 2)
It appears also that a restriction of the Penrose correspondence to
the natural bundle:
the
CP
3
HP
j-invariant projective lines in
S
4
PT
E
E
coincides with
fibers of this bundle are exactly
i.e.
or images of points of
E
under the Penrose
correspondence.
It is useful to introduce the Klein model of
The Klein model can be constructed as follows.
CM
along with the twistor model.
We have defined the twistor model of
CM as the Grassmann manifold
of two-dimensional subspaces in T. Every such
subspace can be defined (up to a non-zero complex multiplier) by a byvector p in
G2(T
TWISTORS AND GAUGE FIELDS
A2T
of
e. A e.
given by byvectors
T. Assign to a byvector
i}
is a basis
{P12’ P13’ P14’ P23’ P24’ P34
six complex numbers
p
{e
1,2,3,4, where
i,j
i
213
(defined up to proportionality) which are called the PiHcker coordinates of the subp A p
The evident condition
space.
is rewritten in the PiHcker coordinates as
0
P13P24
P12P34
Nence, we have assigned to every plane of
Q
given by (2.1) in
CP
5
p+p
+
G2(T)
(2.1)
0
P14P23
a point of the projective quadric
Q
This is a i-I correspondence and we call the quadric
CM.
In suitable coordinates it can be written in
p q+q+q
The basic objects of the twistor geometry
spac
the Klein model of Minkowski
the form:
+
have the following interpretation in terms of the Klein model:
CM}
{point of
{point of
Q
straight generators
of Q
{ipplanes and
of CM
{(2-planes)
lanes
intersection of the tangent space
Q in the corresponding point
Q with Q
{complex
light cone
point of CM
of
of
point of the quadric:
+
+
M}
{point of
1
+
Q
lying in
point of the real quadric:
x++x+x+x-x--0
E}
{point of
3.
Q
lying in
Fig. 6
TWISTOR INTERPRETATION OF
LM(LINEAR MASSLESS)-EQUATIONS.
2 we have given a twistor interpretation of geometric objects of
In
How
CM.
do relativistic fields or solutions of conformally-invariant equations on CM transform
According to the "twistor programme" of Penrose
([4]) relativistic fields are to be interpreted in terms of complex geometry of PT,
under the Penrose correspondence?
i.e. in terms of holomorphic bundles, cohomologies with coefficients in such bundles
and so on.
In other words, relativistic equations are "coded" in the complex struc"disappear" when we pass to
ture of the twistor space and in this sense equations
twistors.
We explain these heuristic considerations first in the case of Maxwell
equations and, more generally, LM -equations.
The Maxwell equations on M can be written in the form:
F
where
Fabdxa A dx
b
is a 2-form defined by the tensor
electromagnetic field, and
If
F
F
dA
for some
where
F
d(*F)
O.
into a sum of its self-dual and anti-self-dual components:
F+
(iF
+/-
*F)
*F+
+/-
self-dual, i.e.
F
F+ (or F
of the
M
dF
0 is
We can split the
F
F+
+ F_
The Maxwell equations in terms of these com-
iF+
ponents can be written in the form:
0
(a,b =0,1,2,3)
l-form A, the the equation
automatically satisfied, soMaxwell’s equation reduce to
form
d(*F)
is the Hodge operator defined by the metric of
*
has a potential, i.e.
Fab
O,
dF
dF+
O.
(If
F
has a potential and is (anti)-
then Maxwell’s equations are automatically satisfied).
A. G. SERGEEV
214
In order to obtain a twistor interpretation of the Maxwell equations we must
transforms under (1.2)
The tensor
apply the mappings (1.1) and (1.3) to F
Fab
b
a
FAA, BB
into the spinor
o
where
FaboAA,OBB,
a
are Pauli matrices
Splitting this
spinor into a sum of its self-dual and anti-self-dual components we obtain the spinor
version of Maxwell’s equations:
-
0,
dF_
where
M
with
0
0
s
O, where
AB’ A’B’
the second- to the self-dual equation
More generally, we call the following system
XAA AB...L
XAA A’B’
0
The first equation corresponds to the anti-
M.
are symmetric spinor functions on
self-dual Maxwell equation
AB
XAA
s:
LM-equations of spin
XAA A ’B’...L’
-i
0
s
0.
dF+
0
AB...L
A’B’...L’
are symmetric spinor functions on the spinor model of
21s
spinor indices.
Again the first equation is called anti-self-dual,
the second- self-dual.
The case
-linearized EH-equations,
s
+/-I
corresponds to Maxwell equations,
+/-2
s
0 -wave equation.
s
It is more convenient to begin with a twistor interpretation of holom0r_p_hoi.
Since every distribution solution of LM-equations
solutions of the above equations.
can be represented as a jump of boundary values of holomorphic solutions in the
future and past tube (proved for the wave equation in
vAA AB...L(z AA’
vAA’A,B,...L,(zAA’)
vAA’ /ZAA’
where
21sl
0
22
it is sufficient to con-
O,
s
0,-,-I
(3.1)
s
0,,I
(3.2)
AB...L’ A’B’...L’
are symmetric spinor functions (with
Note that the spinor model of
indices), holomorphic in the future tube.
a space of
[]),
LM-equations in the future tube:
sider the following system of
CM+
is
-matrices with positive imaginary part.
A twistor interpretation of the anti-self-dual Maxwell equations (equations (3.1)
for
i) is given by the theorem of Penrose ([6]) which asserts that there is a
s
I-i correspondence
anti-self-dual holomorphic
solutions of Maxwell
equations in CM+
}
H
(PT+,O)
(3.3)
In Dolbeault’s representation the cohomology group on the rimht side coincides with a
-closed (0,1) -forms on PT + modulo
-exact forms. The self-
space of smooth
dual Maxwell equations have the following interpretation:
self-dual holomorphic
solutions of Maxwell
equations in CM+
where
0(-4)
1
is a sheaf of holomorphic functions of
given by homogeneous functions of degree
-4
H
(PT+, 0(-4))
PT
whose local sections are
in homogeneous coordinates of
PT
The
apparent asymmetry between self-dual and anti-self-dual cases can be removed if we re-
place
T
ir self-dual case by the dual space
become analogous.
(PT+) *.
Then both formulations will
215
TWISTORS AND GAUGE FIELDS
Let
We now explain the idea of the proof of above results in the self-dual case.
HI(pT+,
be an element of
0(-4)).
Since a point
(1.3) is calculated by averaging
f
of the form
e CM+
f
of pre-images of
m
A
under
-izAA A’
This
along the indicated line with a standard
f
[0’’I’]
in homogeneous coordinates
A’’B’
corresponds by (1.3)to
along the projective line:
averaging is given by a fiber-integral of
kernel
AA’
A,B,(Z AA’)
PT +, the value
a whole projective line in
z
on the
line.
For the same construction in the anti-self-dual case we need to consider the "normal
derivatives" of an element
HI(pT+,0)
g
In other words, we have to go out into an
infinitesimal neighbourhood of the line.
The value
AB(Z AA’)
of pre-images of
/mA. /B
under (1.3)isgivenby a fiber-integral of the "normal derivative"
g
(g)
The results formulated above for Maxwell equations can be generalized naturally to
IM-equations.
Namely, the following correspondence
solutions of
LM-equations with spin s
in CM+
is
I-I
{HI(pT+,
0(-2s-2))
(3.4)
The mapping (3.4) is constructed in the same way as in the Maxwell case.
There is also a direct twistor interpretation of solutions of
LM-equations on
not using their representation as a jump of holomorphic solutions in
PT+
this interpretation one has to replace the cohomologies of
PN (cf. [7]).
cohomologies of the quadric
equations in terms of (0,2) -flags in
PT
M
To obtain
CM/
(3.4) by tangent
in
LM-
A characterization of solutions of
or incident pairs of points of
PT*
PT
can be given without splitting these solutions into their anti- and self-dual components.
In the case
s=0
of a complex wave equation (d’Alembertian) we have a representa-
tion of solutions as fiber-integrals of elements of
HI(pT +,
0(-2))
along projective
It is interesting to note that an analogue of this representation for the ultra-
lines.
hyperbolic equation was proved in 1938 by F. John.
(Recall that the ultrahyperbolic
operator coincides with the restriction of the d’Alembertian to the real subspace of
where the complex Lorentz metric has signature (2,2).)
CM
At the same time this repre-
sentation for solutions of the real wave equations and Laplacian, obtained by restric-
of D’Alembertian to
tion
4.
YM(YANG-MILLS) EQUATIONS.
YM-equations are matrix analogues of Maxwell’s equations.
consider a principal
A dx
a
on
a
M
G
More precisely,
If a Lie
with connection over Minkovski space.
A
A
A
F
Fabdxa A dx
b
a
Fab
is the commutator of matrices
A
a
F
bAa
=VAA
+
dA
[Aa’Ab]
A connection
+ 1/2[A,A]
where
A
In
N -matrices
N
The curva-
is called otherwise a matrix potential.
is given by the covariant derivative
valued 2-form:
We shall consider
G
of
of the connection are anti-Hermitian
a
The connection
with zero trace.
g
related to the most important physical applications.
SU(N)
this case the coefficients
[Aa ,]
M
with values in the Lie algebra
further the case
ture of
P
G -bundle
is a matrix group, then the connection is given by a matrix valued l-form
G
group
respectively, was apparently unknown.
E
and
TWISTOR INTERPRETATION OF
The
A
M
and is a matrixa
/x
is called a
a
YM(YangX
A. G. SERGEEV
216
[A,*FI
A field
0
A
VA(*F)
YM -equation:
Mills) -field if its curvature satisfies the
*F
is (anti)-self-dual if
(-)iF
+
<=> d(*F)
0
(Anti)-self-dual
YM-equations (by the Bianchi identity).
fields automatically satisfy the
YM
interpretation of holomorphic anti-self-dual
By the theorem of Ward ([8]) there is a I-I correspondence:
I,et us describe first a twistor
CM+
-fields on
{anti-self-dual
YM -fields in
holomorphic vector bundles
on PT+ holomorphically
trivial on projective
lines in PT+
images of
points of CM+
holomorphic
CM+
(One can check that in the scalar case this theorem coincides with (3.3)).
side of (4.1) can be redefined as a space of
u
+ [u,u]
0
and is
-exact iff
v-lv
u
The right
-closed matrix-valued (0,I) -forms
(A matrix-valued (0,I) -form
exact forms.
modulo
I
(4.1)
u
-closed iff
is
for some non-degenerate matrix-valued
v).
function
The idea of the
A
-planes. Let
on CM+ and E
a
We have a geometric criterion of anti-
is the following.
proof
A
a connection
self-duality:
<=> its curvature
is anti-self-dual
be an anti-self-dual connection in a principal
F
vanishes on
SU(N)
bundle
P
It is convenient to use the following dia-
-associated vector bundle.
gram:
F+
where
(0,I) -flags in
is a space of
PT+
and a projective line in
tion
E’
Denote by
tions.
A
-
and
A’
including this point,
over a point
Z
s’
by sections
component of
PT+
VA,
is given by
E’
of
over
,
and
-i
A’
(Z)
along the fibers of
E’’
PT+
on
as follows.
-horizontal sections of
VA,S
such that
v.
0
PT+
are natural projec-
E
the trivial liftings of the bundle
F+, and define the bundle
to
PT+, i.e. pairs consisting of a point of
and the connec-
E’’
The fiber of
E’
where
-I
over
VA’
(Z),
i.e.
is the fiber
(This definition is correct due to anti-
self-duality of A). The bundle E’’ is an image of the field A under (4.1). To
prove that this bundle is holomorphic, we define an almost complex structure (CauchyRiemann operator)
given by a section
on
s’
E’’
by
Sde f VA(?,l)s,,
of the bundle
where
VA(,0’I)
E’, and
s’’
is the section of
is the
(0, l)-component of
To show this definition to be correct it is sufficicent to check that
a section of
E’’
i e. that
VA(? ’l)(V,s’) + [VA,, VA(,0’I)] s’
VA’
V
V (0’I)
V,, A’
0 because
But
0
s
V, s’ =Oby
(0’I)]
E’’
s’’
VA,.
is again
s
VA’"V(0,1)
A’
the given condition, and
2
0 due to anti-self-duality of A. One can show also that
0
A’
(almost complex analogue of the Frobenius condition) hence by the Newlander-Nirenberg
theorem ([9]) this almost complex structure defines, in fact, a complex structure
on
E’’.
lines in
By construction, the bundle
PT
+
E’’
which are images of points of
is holomorphically trivial on projective
+.
CM
TWISTORS AND GAUGE FIELDS
E’’
Conversely, let
E’
and
be a holomorphic bundle on
F+
be its trivial lifting to
is given by holomorplic sections of
CM+
217
PT+
E’
-1(z).
over
E
+.
CM
E’
CpI.)
E
We shall con-
along a complex
(corresponding to the ray).
The infinitesmial version of this definition defines a connection in
self-dual by construction (its curvature is zero on
E
which is anti-
-planes corresponding to the
PT+).
points of
The self-dual
PT*.
YM-fields have an analogous interpretation in terms of the dual
An arbitrary
YM-field cannot be decomposed into the sum of anti- and self-
dual fields owing to nonlinearity of the
PT*
PT
YM-fields
However, general
YM-equations.
also have a natural twistor interpretation in terms of (0,2)-flags in
pairs in
on
over projective lines (corresponding
to the different points of the ray with their common point
space
E
It is sufficient to define
The parallel transport in
light ray is given by identifying fibers of
,
CM+
z
over a point
-l(z)
by means of parallel transport.
it along complex light rays in
E
This defines the bundle
(This definition is correct due to compactness of
struct a connection in
trivial on projective lines
Since the fiber of
PT
or incident
(cf. [10,11]).
The most interesting physical applications of the above results are related to the
Euclidean case.
The instantons are by definition (anti)-self-dual
which realize minima of the action functional:
f(F,*F) d4x
S(F)
E
YM-fields on
By the Ward’s
theorem there is a I-I correspondence between instantons and holomorphic vector bundles
on
PT
which are holomorphically trivial on j-invariant projective lines (cf. 2) and
With the help of this re-
have an additional quaternionic structure generated by
suit there was given in [12] a description of the space of instantons in terms of
Considering physical appli-
quaternion matrices, satisfying some quadratic constraints.
cations of this result we note that it would be desirable to have (if possible) a more
explicit description of the space of instantons because one has to integrate over this
space for quantisation.
YM-
There is also one more important class of (anti)-self-dual solutions of
equations called
A
(A)
a
on
E
monopoles.
They are by definition (anti)-self-dual
x)
not depending on "time" (i.e. coordinate
boundary condition:
]I Ao II
+ k/r + 0 (l/r) kZ)
YM-fields
and satisfying the
.
(xl) 2 + (x2) 2 + (x3) 2
f(F,*F)dxldx2dx3.
E(F)
r2
as
This condition realizes minima of the energy functional:
It is natural to consider the monopoles as fields, so called Yang-Mills-Higgs fields,
on the 3-dimensional Euclidean space
E
The space
T(CP I)
(the tangent bundle
space of the Riemann sphere) appears to be an analogue of the twistor space
this case.
PT
in
We have again an analogue of Ward’s theorem ([13]) which states that there
is a I-1 correspondence between Yang-Mills-Higgs fields on
vector bundles on
E
and holomorphic vector
T(CP I) which is holomorphically trivial on projective lines in
T(CP I) and has an addio-al quaternionic structure. Hitchin [13] used this theorem
to construct for every monopole some algebraic curve (called a spectral curve) thus
reducing the problem of description of monopoles to the description of such curves.
is interesting to note that a similar construction appeared in the paper by K.
Weirstrass in 1866 dedicated to the minimal surfaces in
E
The twistor approach
It
A. G. SERGEEV
218
(with suitable modifications) can be applied also to the other important classes of solutions of yMoequations (e.g. to the vortices, that is, (anti)-self-dual YM-fields not
depending on two variables).
5.
EH(EINSTEIN-HILBERT)-EQUATIONS.
TWISTOR INTERPRETATION OF
CM.
So far our considerations were related to the flat space-time
of gravitation?
What can be
Do our constructions continue to be valid in the presence
said if this space is curved?
It appears that under some essential restrictions the answer is affir-
mative.
M
Let
metric
be a 4-dimensional complex manifold with a complex non-degenerate Riemann
gabdzada b
g
R
of
M
(holomorphic) 2-forms on
M.
Denote by
the Riemann curvature
A+
A_
-operator with the eigenvalues +/-I).
R
represented as a block matrix
trA
of the Ricci tensor,
W
W+(W_
C
trC
is the eigenspace of the
A+A_
A
so the operator
is the scalar curvature, and
If the metric
g
A manifold
fies the complex
M
A -I/3trA
W+
and
coincides with the direct sum
(resp.
will be called self-dual
EH-equations and
W+
0 (resp.
W_
anti-self-dual iff it satis-
0).
(Anti)-self-dualmanJfolds
Let us call a conformal structure
have the following natural twistor interpretation.
on
*
can be
EH-equations then the Ricci
satisfies to the complex
tensor and scalar curvature vanish and the operator
W+W_
R
is defined by the traceless part
B
where
of
consisting re-
are respectively self-dual and anti-self-dual components of Weyl tensor
C -/3trC
W_
Then
A
the subsapces of
(A+
A2(M)
A
as a linear operator on the space
spectively of self-dual and anti-self-dual forms
We can consider
is non-Hermitian.)
g
(Note that the metric
A manifold
the class of conformally equivalent complex Riemann metrics on
with such structure will be called conformal.
The theorem of Penrose ([14]) asserts
that there is a 1-I correspondence:
deformations of complex
structures of domains in
PT ruled by projective
lines
conformal}
{anti-self-dual
manifolds
Self-dual conformal manifolds have a similar interpretation in terms of
The idea of the
proof of (5.1)
is the following.
plex manifold which is a deformation of the domain of
Then by the theorem of Kodaira ([15])
T
T
Let
L
P
corresponding to some point
This defines a conformal structure on
anti-self-dual.
fixed point of
point of
a space
M "in
The manifold
is anti-self-dual
every null directon".
the criterion of anti-self-duality of
surfaces).
is called a "light
CP
).
Let
be a
T intersecting the
cone" with vertex p
line
with this structure is
In fact, let’s call the set of rational curves in T passing through a
We formulate the following geometric criterion
an s-surface in M
of anti-self-duality:
throush every
be a 3-dimensional com-
has a 4-complex-parameter collection of
The set of rational curves in
p e
PT*
PT, ruled by projective lines.
rational curves (i.e. curves isomorphic to the Riemann sphere
manifold of all these curves.
(5.1)
As the constructed manifold
<=> there is an
s-surface passing
This criterion is analogous to
-
YM-fields (Riemann curvature vanishes on null
does have a sufficient number of
TWISTORS AND GAUGE FIELDS
Conversely, an anti-self-dual manifold
surfaces it is ani-self-dual by the criterion.
M
has a 3-complex-parameter collection of
faces is identified with
M
i.e. to obtain a complex solution of
have to introduce according to the "twistor
T
a-surfaces and the manifold of these sur-
T
To define a metric on
structure on
2|9
not belonging to the
programme"
of Penrose some additional
"complex geometry" of
identify
T
with the normal bundle of
T
3-forms on
L
KI
to
L
in
PT
Then we can
with holo-
and other rational curves in
K
Denote also by
the canonical line bundle of
(written in local coordinates in the form
a restriction of
For instance let
L
be a sufficiently small neighbourhood of a rational curve
morphic sections of this bundle.
EH-equations, we
f
dz
coincides with the standard bundle
Adz 2 Adz
0(-4) on
).
CP
Then
The
theorem of Penrose ([14]) asserts that these data are sufficient for the construction
of anti-self-dual solutions of
EH-equations.
i-I correspondence
More precisely, there is (locally) a
holomorphic bundles :
CP
with 4-parameter collection of
sections and isomorphism
{anti-self-dual manifolds}
*0 (-4)
KT
Unfortunately, this result cannot be applied to the construction of real Lorentz solutions of EH-equations because real anti-self-dual manifolds always have even signature.
Nevertheless, this theorem can be successfully applied to the construction of Euclidean
anti-self-dual manifolds which are considered in quantum gravity.
To this class belong
ALE (asymptotically locally Euclidean)-manifolds introduced in [16] which have topology
of
S3/F
on
S
6.
FINAL REMARKS.
R
in a neighbourhood of infinity where
F
is a finite group of isometrics
The twistor interpretation of these spaces in kind of (I0) was given in [17].
We want to underline first a connection between Penrose transformation and another
interesting mathematical result
theorem of Sato-Kawai-Kashiwara ([18]).
By this
theorem an arbitrary overdetermined system of pseudodifferential equations in general
position can be transformed microlocally (i.e. locally in the cotangent bundle) by means
of a canonical transformation of infinite order into a system of tangential CauchyRiemann equations.
This transformation (which does not reduce in general to a
coordinate transformation in the base space) preserves only analytic singularities of
solutions. The Penrose transform also carries the field theory equations into systems
of tangential Cauchy-Riemann equations on twistor manifolds and, moreover, it allows
one to obtain explicit formulas for solutions.
There is a close connection between the twistor approach and the method of RiemannHilbert boundary, problem (cf.[19]) which is also applied to solution of the self-dual
YM-equations.
The solution of these equations by the indicated method reduces to
solving the factorisation problem for a rational matrix-valued function on the Riemann
sphere CP
(which is equivalent to a trivialization of a ho!omorphic bundle on CP
defined by this matrix function) depending on three complex parameters.
lent to a construction of a holomorphic vector bundle on
projective lines which are images of points on
CM
PT
This is equiva-
holomorphically trivial on
A. G. SERGEEV
220
The Penrose transform is attached to 4-dimensional manifolds.
Its existence on the
group-theoretical language is due to the following local group isomorphlsms:
S0(3)
SU(2,2)
S0(3) (note that the group
S0(4,2)
p
S0(n) is simple for
n
3,
n
S0(4)
# 4), SL(4,C)
All these isomorphisms are "fasten" to the dimension four.
-
Figol
lp
Fig.
Fig.
Fig.5
Fig.6
0(6,C)
TWISTORS AND GAUGE FIELDS
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