<<

. 7
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numbers of the moduli spaces of SU(2)-instantons on 54 are given [10]. In Section 3
an analogue of Floer's instanton homology group is defined for lens spaces with odd
order fundamental groups [12]. Using the classification in Section 2 and the above
correspondence between invariant instantons on S4 and instantons on (8 3 /r) x R,
these groups are described explicitly. In Section 4 an application to cobordisms
among lens spaces is explained; this is an extension of an argument in [11].
Acknowledgement. The author is grateful to M. Crabb for his reading through the
manuscript and all his comments.

2. DESCRIPTION OF ONE-DIMENSIONAL MODULI SPACE
2.1 8 1 x 5 1 -invariant instantons.
Let T = 8 1 X 8 1 be a maximal torus of SO(4). Then topological isomorphism
classes of T-equivariant SO(3)-bundles over 8 4 with negative first Pontrjagin class
are parametrized by {(k 1 ,k2 ) : k1 ,k2 EN}. We write P(k l ,k2 ) for the T-bundle
corresponding to (k 1 , k2 ). Then P( k 1 , k 2 ) is characterized by
(i) The T-action on P(k 1 ,k2 )oo (resp. P(k 1 ,k2 )o) is given by the conjugacy
class of a homomorphism f : T ˜ SO(3) defined by

- sin 0 0 )
COS 0
f(t l ,t2) = cos (} 0 ,
sinO
(
001

where eie = t˜lt˜2 (resp. t˜lt2k2). We call f the isotropy representation at
00 (resp. 0).
(ii) PI (P(k 1 , k 2 »[S4] = -4k 1 k 2 •
In this subsection we consider T-invariant instantons on P( kl , k2 ). Taking a
double covering T of T, if necessary, we may consider T-invariant instantons on
a T-equivariant SU(2)-bundle P( k1 , k2 ) instead of P( k1 , k2 ). (The choice of the
double covering depends on k1 and k2 .) Note that c2(P(k 1 ,k2 »[84 ] = k1 k2 .
The ADHM-construction reduces classification of SU(2)-instantons to that of cer-
tain holomorphic SL 2 (C)-bundles over p3. Moreover S. K. Donaldson reduced the
classification to that of holomorphic SL 2 (C)-bundles over p2 = C 2 U 1 which are
00

trivial over Zoo. A pair of an SU(2)-instanton and a base point of the SU(2)-bundle
at 00 corresponds to a pair of a holomorphic SL2 (C)-bundle and a holomorphic
trivialization of the bundle on 100 •
By considering this procedure equivariantly, we can reduce the classification of '1'-
invariant SU(2)-instantons to that of T-equivariant holomorphic SL 2 -bundles over
p2 which are trivial on 100 • Here we regard T as a subgroup of U(2) = SO(4) n
GL2 (C) which acts on p2 naturally.
Since the T-action preserves the holomorphic structure of the bundle, it can be
extended to an action of the complexification TO. Because p2 has a dense to-orbit,
163
Furuta: Z/a-invariant SU(2) instantons over the four-sphere


it would be expected that a small nlll11ber of data should classify TO-equivariant
holomorphic bundles. In fact such equivariant SL 2 (C)-bundles were classified by
T. Kaneyama [17]. (Kaneyama assmned that the TC-acton is algebraic. This is
shown by, for instance, looking at a TC -action on monad3, which we shall mention
later. )
In our case they are parametrized by (k l , k2 ) E N and are expressed as E( k l , k2 )
in the following exact sequence.
'P
o˜ O( -k l ˜ ˜ ˜
k2 ) 0 EB O( -k2 ) EB O( -k l E(k}, k2 ) 0,
- )

where 'P(f) = (Z;t+ k2f,zflf,z;2f). (We write (ZO,Zl,Z2) for the homogeneous
coordinate of P2.) Let t E TC be a lift of (t l , t 2 ) E TO = C* x C*. Then the
action of t on E( k l , k2 ) is induced from the actions on O( - k l - k2 ), 0, O( - k2 ) and
O(-k l ) which are defined below.
= t-.;k /2{:;k /2 f(zo,t 11z 1,t;:lz2) f E O(-k} - k2)
t· f(ZO,Zl,Z2) 2
1


= t-;kl/2t:;k2/2f(zo,tllz1,t;:lz2)
t· f(ZO,Zl,Z2) f E0
kl/2t-k2/2f( Zo, t-l Zl, t- l Z2 ) f E O(-k 2)
,Zz ) = t 1
t . I( Zo, Zt 2 1 2

t· f(ZO,Zl,Z2) = t˜kl/2t˜2/2 f(zo,tllzl,t:;lz2) I E O(-k 1 )
The T-invariant instantons on the T-equivariant SU(2)-bundle P(k 1 ,k2 ) over 54
correspond to the fC-equivariant SL 2 (C)-bundle E(k 1 , k2 ). Note that the base
points of an SU(2)-bundle at 00 are parametrized by SU(2) and the trivializations of
a holomorphic bundle on 1 are, if any, parametrized by SL 2 (C). So, in general, one
00

holomorphic bundle gives rise to a family of (non-based) instantons parametrized
by SL2 (C)jSU(2). But we have now group symmetries. If we consider the base
points and the trivializations compatible with the T-action and the fC-action,
then they are parametrized by D(l) and C* respectively. Hence one fC-equivariant
holomorphic bundle gives rise to a family of T-invariant instantons parametrized by
C* jU(l) = R+. So we have
M(P(k 1 , k 2 » ˜ R+.
1 [3,13].
THEOREM

Note that the dilation r : 54 --+ 8 4 (r E R+) rex) = rx, (x E R 4 U {oo} = 54)
induces a free R+-action on M(P(k 1 ,k2 )). Theorem 1 says that M(P(k t ,k2 »
consists of exactly one orbit.
In particular the dimension of M(P(k 1 , k2 » is one. This can also be shown from
the Atiyah-Bott-Lefschetz formula. Let P k be an SU(2)-bundle over 54 such that
C2(Pk)[S4] = k and Mk be the moduli space of instantons on Pk. The SO(5)-action
on 54 cannot be lifted to P k if k -=F 0, but there is an SO(5)-action on Mk because
the action can be lifted up to gauge transformations_ Forgetting the T-action we
can think of M(P(k l , k2 )) as a submanifold of M k for k = k 1 k 2 - (Since every non-
trivial instanton on 54 is irreducible, the map M(P(k l ,k2 » -+ Mk is injective. Its
image is contained in the fixed point set Mr.) For [A] E M(P(k 1 , k2 » the tangent
space (T Mk)[A] is a T-module. From the Atiyah-Bott-Lefschetz formula we have
Furuta: Z/a-invariant SU(2) instantons over the four-sphere
164

1 [12].
LEMMA


-1+ Laijsis˜,
(„k)[A] =

1 ˜ e , IiI S k 2
= I{e E {±I} : IiI S k 1 - 1 ; e }I,
aij _


where 81 and 82 are the components of (TS 4)0 as complex representation spaces of
T: (T 5 4 )0 = 81 + 82. (We fix an identification 8 4 = C 2 U { 00 }.)
The constant term of the above two-variable Laurent polynomial is one. This gives
the dimension of M(P(k 1 , k2 )).
We already gave an expression of the holomorphic T-bundle corresponding to an ele-
ment of M(P(k 1 , k2 ». But to consider Za-invariant instantons later it is convenient
to give the monad description.
We recall the monad description [5]. Let V be a k-dimensional complex vector space
and W a 2-dimensional complex vector space. Suppose four linear maps 0.1 , 0.2, a, b
are given, where b: W ˜ V, a1,a2 : V ˜ V and a: V ˜ W. When they satisfy



= (zo, Zl , Z2) E C 3 \
one has maps Az and B z below parametrized by Z {O}:
Az Bz
o--+ V ˜ ˜
V E9 V E9 W V 0,
--+



(:::˜ ˜:˜˜:) , + z1 I k,zob).
z2 I k,zoG.1
Bz =
Az = (-zOa2 -
zoa

They satisfy BzA z = O. Moreover if A z is injective and Bz is surjective for every
Z, one has a holomorphic bundle I1[Z)EP2 Ker B z 11m Az over p2. A trivialization
on 100 corresponds to an isomorphism W ˜ C 2 •
For example P(3, 2) is given by

= (!_a -l,!a 1)'
W 2' 2'

.1,eo .1,e1 .l.,e_1_1,eo _l,e} _.l.}
V = (e_ 1'2 '2 '2 '2 , 2 '2


a, b defined below.
and aI, l}2,

a
at
at
e -I,! !!,1
eO,˜
˜ ˜
e1,t ˜




1
1 1
02 02 02

b 01
01

f-˜,-l eo,_! e1,-t
˜ ˜ ˜
e-1,-!
165
Furuta: Z/a-invariant SU(2) instantons over the four-sphere


Other matrix elements of aI, 02, a and b are defined to be zero. Here we define a
T-action on W and V by



where t E T is a lift of (t l , t z ) E T. Then these data define a T-equivariant
holomorphic SLz(C)-bundle. In general the monad for P(k 1 , k2 ) is given in a similar
way_
To recover an instanton from a holomorphic bundle, one has to solve a (finite
dimensional) variational problem [5]: find a hermitian metric on V such that
2
II al liZ + II 0z 11 + II a liZ + II b liZ attains the minimum. (The metric of W
is fixed because W is associated to the fibre of the SU(2)-bundle at 00.) Then
a solution gives rise to a set of data for the ADHM-construction from which one
can find a connection form. For P( k1 , k2 ) the variational problem is reduced to an
equation which is similar to Kirchhoff's law. Firstly let us write down solutions for
P(3,2). Let r E R+ be a parameter.

¥r ¥r
r
------+ ˜ ˜
eo,! ft,1
e-1,i e1,!

˜rl
rl rl
f-˜,-1 ------+
˜ ˜
eo,_˜
e-l,-t r el,-!
¥r ¥r

Here a metric on V is given so that {ep.v} is an orthononnal basis, and the positive
number written at each arrow describes the square norm of a matrix element of aI,
az, a and b for this basis. In general, if we call these positive numbers flows, the
equation is
(i) At each vertex the sum of the entering flows is zero. (In the above example
we have, for instance, r + (1 + J5)r /2 - (3 + J5)r /2 = 0 at the vertex e1 , ˜.)
2
(ii) At each unit square the two products of the flows corresponding to a1 a2 and
0Za1 agree.

(Every monad describing P(k J , kz ) satisfies (ii). So (i) is the essential equation.)
The author does not know how to solve this equation for general P( k 1 , k2 ).

2.2 SI-invariant instantons.
If one takes a sufficiently complicated subgroup of T, one could expect that its fixed
point set in M k is the same as the fixed point set for the T-action. For a natural
number p let T p be the subgroup {(t, t P ) : t E 51} of T. We show that T p satisfies
t.his property if p is odd and larger than k. The following argument is outlined in
IlO). For a fixed point rA] E M[p there is a unique lift of the Tp-action to an action
166 Furuta: Zla-invariant SU(2) instantons over the four-sphere

on the adjoint bundle adPk = Pk xadsu(2) preserving A. Suppose that the isotropy
representation of the Tp-action on (ad Pk) at 00 (resp. 0) is

0)
cose -sinO
(t, t P ) sin 0 cos (} 0 ,
t---+
( o 0 1

where ei9 = t'oo (resp. ei9 = t'o ). Then one can use the Atiyah-Bott-Lefschetz
formula to calculate the Tp-action on (TMk)[A]. The result is




The right hand side must be a Laurent polynomial in t with non-negative coefficients
and its value at t = 1 must be equal to dim M k = 8k-3. From these requirements we
can easily show, replacing 1 and 1 by -100 and -10 if necessary, that 100 = k1 +pk2 ,
0
00

10 = k1 - pk2 and k = k1 k 2 for some k 1 , k 2 EN. (Looking at the value at 1 we have
2(1˜ - 15)/p - 3 = 8k - 3, i. e. l˜ - 15 = 4pk. Looking at the value at e21ri / r we
have 1 == ±loo mod p. Note that we assumed that p is odd.)
0
Then the Tp-action on ad P is isomorphic to a restriction of the T -action on ad P( k1 , k 2
Moreover if p is larger than k, then the constant term of the Laurent polynomial is
one. Hence the dimension of M'[l' is one. Since R+ acts freely on MJp, it must be
a disjoint union of copies of R+. Because T is commutative, the T-action on Mk
preserves Mil' as a set. Since an action of a compact group on R+ must be trivial,
the T-action on M'[l' must be trivial, so we have Mil' = M'[.
We know from Theorem 1 that M'[ is a disjoint union of R+ 's and the number of
components is equal to the number d( k) of positive divisors of k. As an application
we find the Euler characteristic number of Mk.
2. X(Mk ) = d(k).
THEOREM

= Ei(-l)idim Hi(Mk,R).
Here X(Mk )
Let X be M k \ M'[. Then we have shown XTp = 0. The Tp-action on
PROOF:
X may not be free. So the quotient space X/Tp is not smooth in general, but is
an orbifold, or a V-manifold. The projection map X -;. X/Tp is not in general a
circle bundle, but is a circle bundle in the category of orbifolds (a circle V-bundle).
Then we still have a version of the Thom-Gysin exact sequence for the de Rham
cohomology groups:




where H*(X, R) is the de Rham cohomology of X and H*(X/Tp , R) is here defined
as the cohomology of the chain complex of smooth differential forms on X which are
167
Furuta: Z/a-invariant SU(2) instantons over the four-sphere

basic for the Tp-action. Then if H*(X, R) and H*(X/Tp , R) are finite dimensional,
we obtain that X(X) = X(X/Tp ) - X(X/Tp ) = 0 and

= X(M[) + X(X) = d(k) + 0 = d(k).
X(Mk )

It suffices to see the finiteness of dim H*(Mk , R). This follows from the fact that
the moduli space of based instantons Mk is an SO(3)-bundle over Mk and that Uk
is a quasi-affine variety [5]. (In fact Hashinl0to gave an explicit embedding of Mk
into an affine space by identifying Uk with a space of representations of a certain
algebra [14].) •
The above argument does not give which Betti number is not zero. But the following
facts are known.
(i) J. Hurtubise showed that the first Betti number of M k is zero. In fact
7f'1(Mk) = 0 if k is odd and 7f'l(Mk) = Z2 if even [15].
(ii) Y. Kamiyama showed that the second Betti number of Mk is also zero. In
fact H 2 (Mk ,Zp) = 0 for a prime number p larger than k [16].
2.3 Za-invariant instantons. Let Vi and V2 be two faithful complex I-dimensional
representation spaces of Za. We think of (VI EB V2 ) U {<X>} as S4 with a Za-action.
For simplicity we assume from now on that a is odd. (When a is even, the argument
is parallel except that we need a double covering of Zao For the details see J3,13Jo)
Let L be the kernel of the map Z x Z ˜ Hom(Za, Sl) defined by (i,j) ˜ V1®2 ˜ V2 J.
Then Hom(Za, 8 1 ) can be identified with lattice points on the torus (R x R)/ L.
A Za-equivariant SU(2)-bundle P over 8 4 is specified by the Za-actions on (P)oo
and (P)o and the second Chern class C2(P). A Za-action on (P)oo or (P)o corre-
sponds to a conjugacy class of a homomorphism from Za to SU(2), i. e. the isotropy
representation. These three satisfy a compatibility condition. Note that the set
Hom(Za,SU(2»/conj. can be identified with the unordered pairs {/l,f2} such that
11 + 12 == 0 mod L. The compatibility condition is described as follows [12]. Let
{fo, - fo} and {f00' - lex>} be the pairs corresponding to the isotropy representations
at 0 and 00. Take one of the rectangles on R x R which satisfy.
(i) The four edges are parallel to R x {O} or {O} x R, so the vertices are written
as (Xl,YI), (Xl,Y2), (X2,YI), (X2,Y2) for Xl < X2 and YI < Y2- Moreover
Xl, X2, Yl and Y2 are integers.
(ii) The projection of the two-point set {(Xl, Y2), (X2' Yl)} to (R x R)jL is equal
to {fa, - fo} and the projection of {(Xl, YI), (X2' Y2)} is {feo, - fool·
Then the area (X2 - Xl )(Y2 - YI) of the rectangle is well defined mod a and the
compatibility condition is



ell fact the T-equivariant bundle P(X2 - XI,Y2 - Yl) can be regarded as a Za-
('quivariant bundle P by restriction of the action, which satisfies C2(P) = (X2 -
Furuta: Z/a-invariant SU(2) instantons over the four-sphere
168

X1)(Y2 - Y1) and has the required isotropy representations at 0 and 00. (Here, since
a is odd, we have a unique lift Za --+ T of Za CT.)
If we identify Hom(Za, SU(2))/conj. with the set of isomorphism classes of flat
SU(2)-connections on the lens space S3/Z a , then the above formula could give the
Chern-Simons invariants of these flat connections.
The Za-equivariant bundles with one-dimensional moduli spaces are classified by
using Lemma 1.

3 [3,13,12]. Let {fo, - fo} and {foo, - fool be any two unordered pairs
THEOREM
corresponding to two elements ofHom( Z a, SU(2) / conj. If the projection ofa rectan-
gle on RxR satisfying (i) and (ii) above to (Rx R)/L does not have self-intersection
except vertices, then P( X2 - Xl, Y2 - YI) with the restricted Za -action has a one-
dimensional moduli space of invariant instantons. Conversely all Za-equivariant
SU(2)-bundles with one-dimensional moduli spaces are given in this way_ Moreover
for each such Za-equivariant SU(2)-bundle, the one-dimensional moduli space is
diffeomorphic to R+.
Suppose that a rectangle with vertices {(Xi,Yj) : i,j = 1,2} satisfies the
PROOF:
above assumption. From Lemma 1 it is equivalent to say that the number of iden-
tity representations in a Za-module (TMk)[A] is one for [A] E M(P), where P is
P(X2 - Xl, Y2 - Yl) with the restricted Za-action. Hence M(P) is one-dimensional.
Conversely if the moduli space for a Za-equivariant SU(2)-bundle P is non-empty
and one-dimensional, then, as in the previous subsection, the Za-action can be ex-
tended to a T-action for a double cover T of T and Za-invariant instantons are
actually T-invariant. Hence we can use the classification of T-invariant instantons
to obtain the result. I
The monad description of P( X2 - Xl, Y2 - Yl) can be explained using the corre-
sponding rectangle: the space V which appears in the monad is spanned by vectors
corresponding to unit squares {(Zl, Z2), (Zl + 1, Z2), (Zl, Z2 + 1), (Zl + 1, Z2 + I)}
(Zl' Z2 E Z) sitting in the rectangle. The maps 01 and 02 could be seen as 'flows'
on the rectangle from the left to the right and from the bottom to the top respec-
tively. The space W can be understood as a vector space spanned by two vectors
corresponding to the two-point set {leo, - fool. The maps a and b are local 'flows'
around these two points.
When a rectangle on (R x R)/L has self-intersection only on an edge, Lemma 1
can be used to see that the dimension of the corresponding moduli space is three.
In fact in this case the 'flow' obtains one more (complex) dimensional freedom at
the tangential edge. Similarly suppose that two rectangles corresponding to one-
dimensional moduli spaces have intersection only on their edges and that the two
vertices of the one rectangle for the isotropy representation at 00 are equal to the
two vertices of the other for the isotropy representation at 0, then the union of the
two rectangles could be used to construct a three-dimensional moduli space. These
pictures could give the mona.d description of all three-dimensional moduli spaces
169
Furuta: Zla-invariant SU(2) instantons over the four-sphere


to show that the the possibility of the diffeomorphism type of three-dimensional
moduli space is only (52 - { n-points }) X R+, where n = 0,1,2 [3].
In general it is convenient to describe a Za-equivariant monad as a collection of
finite dimensional vector spaces assigned to each unit square of (R x R)J L and
{foo, - fool together with four 'flows'. The dimension of the vector space assigned
to 100 or - 100 should be one.
When we fix the vector spaces, we can construct a deformation of a monad by
deforming the 'flows', which gives a family of instantons. Take an oriented closed
path on the torus (R x R)jL which lies in (Z x R)JL U (R x Z)JL. Then for each
non-zero complex number c new flows are defined as follows.
(i) In the place where the flow does not go across the path, the new flow is the
same as before.
(ii) To give the new flow, each component of flow (which is a homomorphism
between vector spaces) is multiplied by ci when it goes across the path,
where i is the multiplicity of the intersection between the path and the flow.
(When the path goes through 100 or -100' we can arrange the new flow so that, for
instance, the local flow a is the same as before and b is multiplied by ci as above.)
Since a torus has two closed paths homologically independent of each other, we can
construct a family of Za-equivariant holomorphic SL 2 (C)-bundles parametrized by
C* x C* , which gives a map from C* x C* to the moduli space divided by dilations.
If P is a Za-equivariant bundle such that M(P)JR+ is compact, then the image of
t.his map should have a compact closure.
A similar idea is used in [13] to classify P such that M(P)JR+ is compact: the
eases we have described (M(P) ˜ R+ and M(P) ˜ 52 x R+) turn out to be the
only possibilities.
[t is now well known that an end of a moduli space corresponds to a splitting of the
hundle associated with 'bubbles'. Therefore by collecting our results, the following
eriterion can be shown.
4 [3,13]. A Za-equivariant SU(2)-bundle allows an invariant instanton
THEOREM
on it, if and only ifit is isomorphic to a connected sum offinitely many T-eqwvariant
SU(2)-bundles with non-negative second Chern class. The connected is con-
SU.ID
structed by gluing neighbourhoods of 0 and 00 together.
'Lo glue P( k1 , k 2 ) and P(ll' 1 ) at 0 and 00, one needs the condition that the isotropy
2
r<˜presentation of P( k} , k2 ) at 0 is isomorphic to that of P(11 , 1 ) at 00 if restricted
2
t.o Za. (In [13] the above theorem is shown under the assumption a > c2(P)[54 ].
Austin proved it in general by using a different argument in [3].)
li˜XAMPLE 1. Let Vo be the standard complex one-dimensional representation space
®12 and V2 =
of Za. Let I} and 1 be natural numbers coprime to each other, Vi = Vo
2
\˜;"˜-lt.Suppose that a is sufficiently large compared with 11 and 1 • (The precise
2
l'oudition will be given soon later.) Then P( I}, 1 ) has the following properties.
2
(i) P(l}, [2) with the restricted Za-action has a one-dimensional moduli space.
170 Furuta: Zla-invariant SU(2) instantons over the four-sphere

The isotropy representation at 00 is trivial and that at 0 is VO@1112 0 Vo
@-1112.

(ii) Let P be a Za-equivariant SU(2)-bundle over S4 such that the isotropy rep-
resentation at 0 is the same as that of P(ll, 12 ). Suppose that there exists at
least one invariant instanton on P and C2(P)[S4] ˜ c2(P(11,12 »[S4](= 1 1 ).
12
Then P ˜ P(1},12).


= Vi and 82 = V2 in the Laurent polynomial given
(i) If we substitute
PROOF: 81
in Lemma 1, then the number of identity representations of Za in it is equal to the
dimension. It turns out to be one if a is sufficiently large so that the conditions
x1 2 -y11 == 0 mod a, Ik1t :::; 11 and Ik2 1 :::; 1 for integers x and y imply x = 11, y = 12
2
or x = -11, y = -12 •
(ii) From Theorem 4 we may assume that P = P( k 1 , k 2 ) for some k 1 , k 2 and the
isotropy representations of P( k1 , k2 ) and P( I} , 1 ) at 0 agree. Therefore if a is
2
sufficiently large so that the conditions k l l 2 + k2 1} == ±211 1 mod a and k1 k2 ˜ 11 12
2
imply k1 = 11 and k 2 = 1 , then we obtain the result .•
2

We use this example in Section 4.

3. AN ANALOGUE OF FLOER'S INSTANTON HOMOLOGY FOR LENS SPACES

For an oriented homology 3-sphere ˜, A. Floer defined the instanton homology
groups H I*(˜) using instantons on Ex R [9]. For closed 4-manifold or V-manifolds,
it has been important to consider some cohomology classes on moduli spaces 6f
instantons [6,7]. For instance Donaldson's polynomial invariant is defined by using
certain cohomology classes. For 4-manifolds with boundaries, R. Fintushel and
R. Stern used some Z2-cohomology classes introduced by Donaldson [6] to compute
the polynomial invariants valued in the instanton homology groups in a special case
[8]. However it has not been made clear how these Z2-cohomology classes are related
to the instanton homology in a general context. In this section we use one of the
Z2-cohomology classes to define an analogue of the instanton homology groups for
lens spaces with odd order fundamental groups, as an attempt to understand this
cohomology class. For the details see [12].
A lens space is not a homology 3-sphere. It is only a rational homology 3-sphere and
every flat SU(2)-connection is reducible. When the instanton homology is defined,
a reducible fiat connection gives rise to a difficulty which ought to be solved in
itself: a reducible connection has a non-trivial symmetry and it causes a quotient
singularity in the space of connections. We do not deal with this problem here.
However when one uses a co˜omology class of degree one, as we shall see, it is
rather easier to evaluate the class on certain moduli spaces if a flat connection has
exactly one-dimensional symmetry.
Recall that the instanton homology groups are defined by using a chain complex
(C,8) under a certain transversality assumption.
(i) C is spanned by the classes of irreducible flat SU(2)-connections on E.
171
Furuta: Z/a-invariant SU(2) instantons over the four-sphere


afor irreducible flat connections Al and A 2 is given
(ii) The matrix element of
by counting the number (with sign) of the components of a one-dimensional
moduli space of instantons on E x R which connect Al to A 2 •
For a lens space S3/Z a with a odd, firstly we define (C', a') similarly.
(i)' C' as a Z2-vector space spanned by the classes of non-trivial (reducible) flat
SU(2)-connections on S3/Z a •
(ii)' The matrix element of a' for non-trivial flat connections Al and A 2 is given by
counting the number (up to mod 2) of the components of a one-dimensional
moduli space of instantons on (S3/Z a ) X R which connect Al to A 2 .
a at
While Floer showed 2 = 0, there is no reason for the square of to be zero as we
°depends
2=
shall see. Recall that the proof of 8 on the two facts below.
(iii) The matrix element of 8 2 for Al and A 3 is given by counting the number
(with sign) of the ends of a two-dimensional moduli space of instantons on
E X R which connect Al to A 3 •
(iv) The two-dimensional moduli space has a free R action and the quotient is
bne-dimensional. Hence the nwnber of ends (with sign) is zero.
The reason why the two dimensional moduli space comes in is explained as follows
[9]. Suppose we are given a non-empty one-dimensional moduli space of instantons
which connect Al to A 2 , and similarly A 2 to A 3 . Then one can construct an end
of a moduli space M(A l ,A3 ) of instantons which connect At to A 3 • When A 2
is irreducible, the dimension of the moduli space is two, which is the sum of the
dimensions of two moduli spaces. However if A 2 is a non-trivial reducible connec-
tion, then M(A 1 ,A3 ) becomes three-dimensional, where the extra one dimension
comes from the dimension of the symmetry of A 2 • In this case the number of ends
M(A l ,A3 )/R is not necessary zero (mod 2) since its dimension is two.
The idea to define an analogue of instanton homology groups is as follows.
(iv)' In the above situation suppose A 2 is a non-trivial reducible fiat connection
and suppose we have a Z2-cohomology class u of degree one. An end of
M(A l ,Ag )/R is diffeomorphic to 51 X (0,1). (Here 8 1 is the symmetry of
A 2 which gives an extra parameter in gluing two connections.) Then the
number of ends such that u[Sl] = 1 should be zero mod 2, since it is the
evaluation of u by the boundary of truncated M(A l , A 3 )/R.
We define u by using the Dirac operator D( AI, A 3 ) twisted by the bundle on which
Al and A 3 are connected. If the numerical index of D(A I , A 3 ) is even, then the
(leterminant line bundle for the family of the Dirac operators descends to a real line
hundle on M(A t ,A3 ) [6]. Then u is defined as its first Stiefel-Whitney class. Let
fJ(A t ,A2 ) and D(A 2 ,A3 ) be similar twisted Dirac operators. Then we have
= indD(A t , A 2 ) + indD(A 2 , A 3 )
indD(A 1 , A 3 )
(when ˜ is a lens space). Hence if ind D( AI, A 3 ) is even, then the parities of
illdD(A t ,A2) and indD(A 2 ,A3 ) agree and moreover we can show that it is also
equal to U[SI] [12 Proposition 3.2].
172 Furuta: Z/a-invariant SU(2) instantons over the four-sphere

a" :C' ---+ C' as follows.
Now we define a map
a" for Al and A 2 is given by the number (up to mod
(ii)" The matrix element of
2) of components of a one-dimensional moduli space of instantons which
connect Al to A 2 such that the associated twisted Dirac operator has odd
index.
Then the argument in (iv)' implies 8"2 = o. An analogue of the instanton homology
is defined as the homology group of (C', a").
We can introduce a Zs-grading for (C', a") [12].
REMARK.

Recall that instantons on (S3 /Za) x R with L 2 -bounded curvatures can be regarded
as Za-invariant instantons on 54. We can use the classification of Za-equivariant
SU(2)-bunc:lles which have one-dimensional moduli spaces of invariant instantons to
1
describe the boundary map a" explicitly. In addition to Theorem 3 we only have

I
to calculate the index of the twisted Dirac operator up to mod 2. It can be shown
that the index is equal to the second Chern nwnber mod 2. Hence we obtain the
following description.
J
1. [12 Theorem 4.2]
PROPOSITION .˜

(i) C' is a Z2-vector space spanned by the pairs {f, - f} (f i= 0) of lattice points
j
of(R X R)jL.
˜':
(ii) The matrix elements of a" correspond to rectangles on (R x R)/ L with-
out self-intersection such that the centre of the rectangles are of the form
(a/2, a/2) mod L
4. COBORDISMS AMONG LENS SPACES
a:moduli space M of instantons on a 4-manifold X as a cobordism
One could regard
between its ends and its singularities. When both can be described by some topo-
logical data of X, each cohomology class of M of degree dim M - 1 gives rise to a
certain equation for the data. Such an idea was first developed by Donaldson [4] and
subsequently used by Fintushel and Stern [7] and T. Lawson [18] for V-manifolds.
Suppose given a sequence of instantons. Then an end of the moduli space of instan-
tons corresponds to divergence of their curvatures at some points on the 4-manifold.
On a smooth 4-manifold the divergence could be captured by instantons on S4 [4].
On the other hand the divergence on a V-manifold could be understood by using
instanton on 54 jf, where r is a finite subgroup of SOC4).
Lawson used a certain non-existence result of invariant instantons on 8 4 to see
compactness of some moduli spaces for V-manifolds and used it to obtain some
results in topology [18].
In this section we use Example 1 in Section 2.3 to give an application to topology.
Let I} and 1 be natural numbers coprime to each other and a be an odd num-
2
ber sufficiently large compared with 11 and 1 - Let us identify 8 3 with the unit
2
tZl
tZll2
sphere of the Za-module Vo EB VO - 11 , where Vo is the standard one-dimensional
representation. We write L(a; 1 ,12 ) for the quotient S3/Z a •
1
173
Furuta: Z/a-invariant SU(2) instantons over the four-sphere

5. Suppose X be an oriented closed 4-dimensional V-manifold which
THEOREM
satisfies
(i) 7r"l(X) = 1. (The fundamental group of the underlying space of X, not the
orbifold fundamental group of X.)
(ii) H 2 (X, Q) = O.
(iii) There is a singular point p whose neighbourhood is of the form cL(a; 11 ,12 )
(the cone on L(a; 1 ,12 )).
1

Then there is a singular point (# p) whose neighbourhood is of the form S3 jr with
Irl 2:: a/(1112).
Let us identify S4 with (Vo E9 Vo®-h) U {oo}. Then we
SKETCH OF A PROOF: @12

can think of X as a 'connected sum' of X and S4 jZa at p and 00. Let P be an
SU(2)-V-bundle defined by a 'connected sum' of X x SU(2) and P(ll, 12)jZa. (We
can take the connected sum because the Za-action on (P(ll' 1 ))00 is trivial.) ˜
2
Let M(P) be the moduli space of instantons on P. Then, after deforming slightly if
necessary, M(P) has a structure of a smooth one-dimensional manifold. Using the
one-dimensional moduli space of invariant instanton on P(ll' 1 ), we can construct
2
an end of M(P) diffeomorphic to an interval [4,6,11]. The property of P(ll, 1 ) 2
showed in Example 1 (ii) says that this is the only end where curvature of instantons
diverges at p. Since the number of ends of a one-dimensional manifold is even, there
must be an end where curvature of instantons diverges at some other point q. The
amount of L 2 -norm of the curvature concentrated on q is equal to or less than the
total amount of the L2-norm. When we write the neighbourhood of q as c(S3 jr),
then this inequality implies Irl ˜ aj(I}12). I

1. Let ˜ be the connected sum L(a;11,12)#(#'i=lS3jZai)' where
COROLLARY
S3 jZai is a lens space with fundamental group Zai' Suppose that a is an odd num-
ber sufficiently large compared with 11 and 12 , and that ai < aj( /1 1 ) for i == 1, ... ,n.
2
Then ˜ cannot be smoothly embedded in 54.

Suppose there is an embedding ˜ C S4. Then S4 is divided into two pieces.
PROOF:
l˜rom one of them a counterexample of the previous theorem can be constructed. •
()ther applications of P(11,12) are given in [11] when I} = 12 = 1.
We remark that, using an argument similar to the above, one could conversely show
t.he existence of some invariant instantons on 8 4 topologically without appealing to
auy classification. The simplest example would be to show that M I is non-empty:
I(˜t P be an SU(2)-bundle over p2 with C2(P)[P2] = 1, then the moduli space M(P)
()f instantons on P cannot be compact because a singular point of M{P) requires an


˜'lld of M(P), which implies M I =/:. 0. In order to consider invariant instantons on S4
one could use, instead of p2, the quotient of a weighted 5 1 -action on 55 = 5(C 3 ),
which is a rational p2 with (at most) three singular points of the form c(S3 jZa).
A similar argument was used by Fintushel and Stern for another direction [7].
174 Furuta: Z/a-invariant SU(2) instantons over the four-sphere

REFERENCES

1. M. F. Atiyah, Magnetic monopoles in hyperbolic spaces, Vector bundles on algebraic varieties,
Tata Institute of Fundamental Research, Bombay, 1984.
2. M. F. Atiyah. V. G. Drinfeld, N. J. Hitchin and Yu. I. Manin, Constructions of instantons,
Phys. Lett. 65 A (1978), 185-187.
3. D. M. Austin, SO(3)-Instantons on L(p, q) X R, preprint.
4. S. K. Donaldson, An application of gauge theory of -4-dimensional topology, J. Differential
Geometry 18 (1983), 279-315.
5. S. K. Donaldson, Instantons and geometric invariant theory, Commun. Math. Phys. 93
(1984), 453-46l.
6. S. K. Donaldson, Connections cohomology and the intersection forms of 4-manifolds, J.
Differential Geometry 24 (1986), 295-341.
7. R. Fintushel and R. Stern, Pseudofree orbifolds, Ann. of Math. 122 (1985), 335-364.
8. R. Fintushel and R. Stern, Homotopy K3 surfaces containing E(2, 3, 7), preprint.
9. A. Floer, An instanton invariant for 3-manifolds, Commun. Math. Phys. 118 (1988),215-240.
10. M. Furuta, Euler number of moduli spaces of instantons, Proc. Japan Acad. 63, Ser. A
(1987), 266-267.
11. M. Furuta, On self-dual pseudo-connections on some orbifolds, preprint.
12. M. Furuta, An analogue of Floer homology for lens spaces, preprint.
13. M. Furuta and Y. Hashimoto, Invariant instantons on 8 4 , preprint.
14. Y. Hashimoto, Instantons and representations of an associative algebra, preprint, University
of Tokyo.
15. J. Hurtubise, Instantons and jumping lines, Commun. Math. Phys. 105 (1986), 107-122.
16. Y. Kamiyama, The 2 dimensional cohomology group of moduli space of instantons, preprint,
University of Tokyo.
17. T. Kaneyama, On equivariant vector bundles on an almost homogeneous variety, Nagoya
Math. J. 57 (1975),65-86.
18. T. Lawson, Compactness results for orbifold instantons, Math. Z. 200 (1988), 123-140.
19. K. K. Uhlenbeck, Removable singularities in Yang-Mills fields, Commun, Math. Phys. 83
(1982), 11-30.
PART 3
DIFFERENTIAL GEOMETRY AND
MATHEMATICAL PHYSICS
177

Differential geometry has proved to be a natural setting for large parts of mathe-
matical physics and conversely mathematical physics has provided a supply of new
ideas and problems for differential geometers. Perhaps the best known example of
this two-way interaction is given by the instanton solutions of the Yang-Mills equa-
tions - first noted by physicists but now playing an important role in several areas
of mathematics.. There are, however, many other very interesting special equations
and theories; some of them, like the "monopole" equations, close relatives of the
Yang-Mills instantons and some rather different . In this section we have a number
of papers which exemplify this rich interaction.
The paper of Manton describes the Skyrme model, which is of practical interest
in physics and is also very attractive mathematically. The theory appears to offer
a number of challenging problems in the calculus of variations; being a variant of
the well-known harmonic map theory in which the energy integrand is modified by
a quartic term. The intriguing scheme described by Manton, relating the Skyrme
Inodel to instantons, also displays very well the beautiful classical geometry involved
in the explicit description of the instanton solutions.
An important line of research on instantons, going back to the seminal paper of
Atiyah and Jones [AJ), bears on the limiting behaviour of the homotopy and ho-
lIlology groups of the instanton moduli spaces over S4, for large Chern numbers.
The "Atiyah-Jones conjecture" suggests that these agree with the homotopy and
homology groups of the third loop space of the structure group. From quite differ-
('fit directions, Taubes and Kirwan have made important advances on this problem
recently. Analogous problems and results apply to the moduli spaces of monopoles.
rrhe paper of Cohen and Jones below describes more refined results in this direction,
/!;iving a complete description of the homology of all the monopole moduli spaces in
t.erms of the braid groups (which also enter into the Jones theory of link invariants,
as described in Atiyah's lecture in Durham).
Next we have two papers on the oldest branch of differential geometry-the geome-
t.ry of submanifolds. The article of Hartley and Thcker develops a general framework
for dealing with variational problems for submanifolds, more complicated than the
sirnple minimal surface problem. A well-known instance of this kind of theory in
t.he mathematical literature is the work initiated by Willmore, for surfaces in R 3 •
,I'lle paper of Burstall gives a fine illustration of the application of the holomorphic
J!.()ometry of the Penrose twistor space to minimal sunaces..
'I'lte papers of Tod and Wood are quite closely related. Both consider special differ-
«'Iltial geometric structures in 3-dimensions, with particular reference to Thurston's
!loInogeneous geometries, which have to do with the space of geodesics in a 3-
Iliallifoid (the mini-twistor space, in Tod's terminology). These structures seem to
Ilave a good deal of potential, posing many natural questions (for example the ex-
1˜;f.()llce of Einstein-Weyl structures on general 3-manifolds) and offering scope for
:.i,t;llificant interactions with 4-dimensional geometry. Note that there are similari-
t i.>s between the diS(˜llH:..;iOll ill the last section of Wood's paper, on the passage from
178

a Seifert 3-manifold to an elliptic surface, and the technique described in the paper
of Okonek for studying the representations of the fundamental group of a Seifert
manifold.


[AJ] Atiyah, M.F. and Jones, J.D.S. Topological aspects of Yang-Mills theory
Commun. Math. Phys. 61 (1978) 97-118
Skyrme Fields and Instantons
N.S. MANTON
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
Silver Street, Cambridge CB3 9EW
England


ABSTRACT The first part of this paper is a brief review of the Skyrme model,
and some of the mathematical problems it raises. The second part is a summary
of the proposal by M.F. Atiyah and the author to derive families of Skyrme fields
from Yang-Mills instantons.


1 THE SKYRME MODEL
IIadronic physics at modest energies (a few GeV) is concerned with the interactions
of nucleons (protons and neutrons) and of pions. About 30 years ago, Skyrme
:-luggested a model for these particles which is still useful (Skyrme, 1962), despite
t.he fact that the particles are now believed to be bound states of quarks. In the
Skyrme model only the pion field appears, and the nucleons are quantum states of
classical soliton solution of the pion field equations, known as the Skyrmion.
a.


Nucleons have baryon number 1, their antiparticles have baryon number -1, and
,)ions have baryon number o. In any physical process the total baryon number
is unchanged. In the Skyrme model, a field configuration has a conserved integral
t.opological charge which Skyrme identified with the baryon number. The Skyrmion
has charge 1, and there is a similar solution with charge -1.

Sl<yrme's pion field is a scalar field U taking values in SU(2). I shall mainly consider
fi(˜lds at a given time, and not discuss dynamics much. In this case, U is a map
from physical space R 3 to SU(2). The uniform field U = 1 represents the vacuum,
a.nd all field configurations are assumed to be asymptotically like the vacuum so
"(x) ˜ 1 as Ixt -+ 00. Space may therefore be compactified to a 3-sphere of
ill finite radius. Let S3(R) denote a 3-sphere with its standard metric and ra.dius R.
,'l'/ f (2) with its standard metric is S3 (1) . U is effectively a map


(1)

degree, deg U, is a topological invariant and an integer. Skyrme identified deg U
11.:-;
\vith the baryon number.
180 Manton: Skynne fields and instantons

U maps an infinitesimal sphere of radius E, centred at x, to a neighbourhood of
U(x). To lowest order in E, this neighbourhood is an ellipsoid, with principal axes
J.LIE,J.t2f. and P,3f., say. The energy density at x, proposed by Skyrme, is (Manton,
1987)
e(x) = J.t˜ + JL˜ + p,˜ + J.t˜J.t˜ + JL˜p,˜ + #L˜P˜ (2)
and the total field energy is

(3)

Note the following about this energy expression:
1) E is the potential energy of the Skyrme field at a given time. Using Lorentz
invariance one can obtain the kinetic energy and hence the Lagrangian for dynamical
fields. The kinetic energy expression defines a metric on the function space of static
fields.
2) The vacuum field U = 1 has zero energy, and degU = O.
Presumably, finite energy implies that U (x) --+ const as Ixl ˜ 00, but this
3)
may not have been rigorously proved.
4) E is in dimensionless form. The energy unit and length unit are determined
from experimental properties of hadrons.
5) The symmetries of E are the Euclidean group of RS , and the 0(4) group of
S3(1). The latter is the "chiral symmetry" group. The choice of a vacuum U = 1
breaks this down to 0(3) which is "isospin symmetry".

It follows immediately from (2) that

e(x) ˜ 6PIJ.L2Pa, (4)
and since J.ttJ.t2JLa is the modulus of the Jacobian of the map U, and the volume of
S3 (1) is 211"2, the energy satisfies the inequality (Fadeev, 1976)

E ˜ 1211" 2 1deg UI . (5)


Let En denote the infimum of the energy for fields of degree n. The symmetry
U ˜ U-l (the inverse in SU(2» changes the sign of deg U, so En = E_ n . It has
been shown (Castillejo and Kugler, 1987) that En < E,+En-l for any integer I not
equal to 0 or n, and Esteban has shown, assuming this inequality, that the infimum
is attained for each integer n by a smooth field whose energy is concentrated in a
single region of space (Esteban, 1986). The physical meaning of the strict inequality
is that there are attractive forces in the Skyrme model. It is easy to prove that
En ˜ E, + En-l by considering fields of degrees 1and n -I glued together at a large
separation, but the strict inequality is less obvious.
181
Manton: Skynne fields and instantons


The vacuum is the lowest energy field of degree o. The lowest energy field of degree
1 is not known for certain, but physicists have assumed, and numerical evidence
makes it likely, that it is spherically symmetric. The lowest energy spherically
symmetric field is known as the Skyrmion, and its standard form is

= + (6)
U(x) i sin f(r)x · 0"
cos f(r)1 •



Here r = Ixl,i = xfr and 0'1,0'2 and uS are the Pauli matrices. The profile
function fer) has been determined numerically, by solving the variational equation
for f obtained from the energy functional. The boundary conditions are /(0) = 1r
and f(r) -+- 0 as r -+- 00, so U is continuous at the origin and deg U = 1. The energy
of the Skyrmion is 1.231 ... X 121r 2 (Adkins, Nappi and Witten, 1983; Jackson and
Rho, 1983). A six-parameter family of Skyrmions is generated from (6) by the action
of symmetries. The centre can be moved to an arbitrary point, and the orientation
changed by conjugating U with some fixed element of SU(2). Replacing / by - f
gives the anti-Skyrmion with the same energy but degree -1. The physical nucleons
and anti-nucleons are obtained by promoting the position and orientation collective
coordinates to dynamical variables and quantizing these. The quantum states are
characterized by their momentum, their spin and their isospin. The nucleons have
! !'
˜pin and isospin with the isospin "up" for the proton and "down" for the
rleutron.

Skyrme's motivation for choosing the energy density (2) was to have a simple gen-
eralization of the harmonic map energy density p,i + J.t˜ + JLi whose variational
equations had non-trivial solutions in R 3 • Skyrme's energy density is geometrically
natural in three dimensions, and one may use it to define the energy of maps from
a.ny 3-dimensional Riemannian manifold M to another Riemannian manifold N.
In this general context one may ask: (i) Is the infimum of the energy for maps in
(\etch homotopy class attained by some smooth map that satisfies the variational
pCJuations? (ii) Are there saddle-point solutions, i.e. non-minimal solutions of the
variational equations? (iii) Is the Skyrme energy functional in some sense a Morse
fu nction? These questions are open.


Some explicit (numerical) solutions to the equations have been found for special
,˜('ometries. For example, solutions of all degrees are known for maps U : S3(R) -+-
,t,<t(l), where R is finite (Jackson, Manton and Wirzba, 1989). Most of these are
:laddle points, and most do not have good limiting behaviour as R -+- 00. Other
!l()]utions, representing Skyrme crystals, are known for maps from a flat 3-torus to
8='(1) (Kugler and Shtrikman, 1988; Castillejo et aI, 1988). On the other hand,
uo solutions other than the vacuum are known for maps of degree zero from R 3 to
.˜.:l ( l), despite an attempt to find a saddle-point solution (Bagger, Goldstein and
Soldate, 1985).
182 Manton: Skynne fields and instantons

A more detailed problem is the following. It is fairly easy to understand that the
energy bound 1211"2 is exceeded by the Skyrmion because R3 and' 8 3 (1) are not
isometric (Manton, 1987). More generally, there is a topological lower bound on
the energy for maps U : M ˜ N which cannot be attained unless U is an isometry.
The problem is to find a stronger lower bound in the case that M and N are
geometrically distinct and there are no isometric maps between them.

For maps of degree 2 from R 3 to 8 3 (1), two solutions of the Skyrme field equations
are known. The first has the spherically symmetric form (6), but with /(0) = 211".
The energy is 1.83 ... X 2411"2, which is greater than that of two well-separated
Skyrmions (Jackson and Rho, 1983). It is a saddle-point of the energy functional
and has six unstable modes as well as six zero modes (Wirzba and Bang, 1989). The
other solution is axisymmetric and has energy 1.18 ... X 2411"2, which is less than
that of two well-separated Skyrmions (Kopeliovich and Shtern, 1987; Verbaarschot,
1987; Schramm, Dothan and Biedenharn, 1988). The energy density is concentrated
in a toroidal region. It is likely that this solution is the lowest energy Skyrme field
of degree 2, and it has eight zero modes.

One of the central problems in hadronic physics is to understand the interaction
of two nucleons at low energy. It is known experimentally that there is one bound
state of a proton and neutron - the deuteron - and there is a wealth of scattering
data. Much can be described with semi-phenomenological nucleon-nucleon poten-
tial models, but there is no deep understanding of these. It is not yet possible to
calculate low energy phenomena using QeD, the theory of quarks and their inter-
actions. It is therefore a challenge to see if the Skyrme model can describe them.
In principle, one should treat the Skyrme model as a quantum field theory and re-
strict attention to the sector where the fields have degree 2. In practice, this leads
to all sorts of conceptual and computational difficulties. Instead, one may try to
select a finite dimensional submanifold of Skyrme fields, whose coordinates are the
physically relevant degrees of freedom at low energy, i.e. collective coordinates, and
one should quantize.

The simplest version of this idea is to quantize the eight collective coordinates of
the orbit of the lowest energy degree 2 solution (Braaten and Carson, 1988). One
of the quantum states is qualitatively like the deuteron. However, to describe the
deuteron quantitatively and to describe nucleon-nucleon scattering one needs at
least 12 collective coordinates since two well-separated Skyrmions have 6 collective
coordinates each, namely their positions and orientations. A candidate for a 12-
dimensional set of Skyrme fields of degree 2 is the unstable manifold of the orbit
of the spherically symmetric solution (Manton, 1988). This manifold has not been
investigated in detail, but it probably includes well-separated Skyrmions in all po-
sitions and orientations, as well as the orbit of lowest energy fields. It is an open
183
Manton: Skynne fields and instantons

problem to determine numerically which fields lie on this manifold, and to ascertain
whether the manifold is smooth at the lowest energy fields or has cusps there. It
needs to be smooth to give a physically sensible model.

2 SKYRME FIELDS FROM INSTANTONS
One natural way to obtain static Skyrme fields is as the holonomy of instantons
(Atiyah and Manton, 1989).

Suppose AJ.& is any SU(2) Yang-Mills gauge potential in (Euclidean) R 4 with finite
action and 2nd Chern class k. In a suitable gauge, AI£(x) decays faster than Ixl- 1 as
Ixl ---+ 00. Let the time-lines in R4 denote the lines parallel to the time axis. They
are labelled by points of R3. Let U(x) be the holonomy of AI£ along the time-line
labelled by x. Formally


f-:
= (7)
U(x) A,.(x, T) dT
p exp -

where 'f is the (Euclidean) time-cordinate and P denotes path ordering. U takes
values in SU(2), and hence may be regarded as a Skyrme field in R 3 •

U is unchanged under a large class of gauge transformations, but to ensure a com-
pletely gauge invariant definition of U one should regard AI£ as defined on R 4
conformally compactified to 8 4 , and U as the holonomy along a circle on 8 4 which
starts and ends at the point at infinity. In most cases, this is equivalent to closing
the contour in (7) with a semi-circle at infinity. U is then well-defined up to conju-
gation by a fixed (x-independent) element of 8U(2). Also, in this way, one ensures
that U(x) --+ 1 as Ixl --+ 00. It is a basic topological fact that the field U, regarded
a.s a Skyrme field, has degree k.

l'he moduli space of k-instantons (anti-self-dual Yang-Mills fields of 2nd Chern
class k), Mk, is an 8k-dimensional connected manifold (not 8k-3 because one allows
conjugation by fixed elements of 8U(2» (Atiyah, 1979). The holonomies have one
dimension less, as a time-translation doesn't affect them. These instantons therefore
˜enerate a connected (8k - I)-dimensional manifold of Skyrme fields of degree k,
Mk =Mlc/R .

'I'here is no precise relationship between the anti-self-duality equations for Yang-
M i lIs fields in R 4 and the Skyrme equations in R 3 , so it is not surprising that none
()f the instanton-generated Skyrme fields are solutions of the Skyrme equations.


lIowever, some are good approximations to solutions discussed in Sect. 1, and Uk
:ilnoothly interpolates between these approximate solutions. In fact, the coordinates
of'Mk seem to correspond well to the collective coordinates of Skyrme fields relevant
184 Manton: Skynne fields and instantons

to k-nucleon physics at low energy. The symmetry group acting on Mle is the prod-
uct of 80(3) (the adjoint action of 8U(2) on U) and the 15-dimensional conformal
group of R4. The 80(3) survives the holonomy construction as the isospin sym-
metry of Skyrme fields, but the conformal group is broken down to the Euclidean
group of R 3 and dilations.

The l-instantons generate a 7-dimensional set of spherically symmetric Skyrme
fields. The seven coordinates define the centre, the orientation and the scale size.
In the standard position, the Skyrme field is of the form (6), with


(8)

,\ is the scale parameter. For this simple profile, the minimal value of the Skyrme en-
ergy is 1.24 ... X 1211"2 when A2 = 2.11 ..., which exceeds the energy of the Skyrmion
by less than 1%.

The 2-instantons generate a 15-dimensional manifold of Skyrme fields. Some of
these 2-instantons may be identified with two well-separated l-instantons, and as
the time-separation tends to infinity, the resulting Skyrme field tends to a product
of two Skyrme fields of degree 1. 14 of the 15 dimensions are accounted for by the
positions, orientations and scales of these two degree 1 fields, and the last is the
time separation of the instantons which has little effect in the limit. H the spatial
separation is also large, then the time-separation may be continuously increased
from -00 to 00. The effect is to reverse the order of the product of the Skyrme
fields. A particularly symmetric configuration can occur when the time-separation
is zero.

2-instantons with rotational symmetry about the time-line x = 0 generate Skyrme
fields of the form (6). The profile function f(r) is quite complicated in general.
However, the 2-instantons with time reversal symmetry which correspond to two
well-separated single instantons of the same scale size ,\ (they have the same orien-
tation because of the symmetry) give, in the limit of infinite separation, the simple
profile

(9)

The Skyrme energy is minimized when A2 = 2.62 ... and then E = 1.86 ... x 2411"2.
This is probably the lowest energy Skyrme field of degree 2 with 80(3) symmetry
that is generated from instantons. Its energy again exceeds the energy of the 80(3)
symmetric solution of the Skyrme equations by about 1%.
185
Manton: Skyrme fields and instantons


A general formula for 2-instantons is known (Jackiw, Noh} and Rebbi, 1977).
Hartshorne has given a geometric characterisation of 2-instantons, and shown that
all can be expressed in this form (Hartshorne, 1978). These instantons are obtained
from an SU(2) matrix Uo and a potential

+
+
Al A2 As (10)
=
p(x) (x - X 2 )2 (x - Xs)2
(:I: - X 1)2

where Xl, X 2 and Xs are distinct points in R4 (poles), and AI,A2 and As are
positive constants (weights). (x- Xi)2 denotes the square of the Euclidean distance
from x to Xi. In terms of Uo and p the time-component of the gauge potential is

i Vp
·U ) -1
Uo ( 2 p
= (11)
A.. Uo ·

An arbitrary Uo is necessary to obtain the full 16-dimensional moduli space of
instantons. The recipe for obtaining the associated Skyrme fields is to take the
formula (7) and multiply by -1. The factor -1 comes from closing the contour.

rrhe formula (11) depends on 17 parameters (the pole positions, ratios of the weights,
and Uo), but, as shown by Jackiw et aI., there is a I-parameter family of changes
(,0 the poles and weights whose effect is simply a gauge transformation. This may
be described geometrically as follows, according to Hartshorne. Suppose for the
rnoment that X I ,X2 and Xg are not collinear. Then associated with the poles and
weights are two coplanar conics in R 4 • (See Figure). Let Al,A2 and As be the
interior points on the sides of the triangle Xl X 2 X g , defined by

(12)

l'he first conic is the unique ellipse which is tangent to the sides of the triangle at
11 1 , A 2 and Ag. The existence of the ellipse follows, by the converse of Brianchon's
theorem, from the concurrency of the lines Al Xl, A2 X 2 and Ag Xg, and this in
t.1lrn follows, by Ceva's theorem, because

Xl As ·X2 A 1 ·Xs A 2
= (13)
1.
As X 2 • Al Xs · A 2 Xl
'rite second conic is the circumcircle of the triangle Xl X 2 X g •

Xl X2 Xg circumscribing one and
Now, we have a pair of conics with a triangle
Illscribed in the other. By Poncelet's theorem, there is a porism (a one-parameter
Ll.lnily) of such triangles. A second triangle Xi X˜X˜ is shown in the Figure, tangent
t.o the ellipse at A˜, A˜ and A˜. Each triangle of the porism has associated poles
186 Manton: Skyrme fields and instantons

(the vertices) and weights (defined up to an irrelevant multiplicative constant by
the analogue of formulae (12», but they all give the same instanton, up to gauge
tranformations. The pair of conics is the gauge invariant data which defines the
instanton.




This geometrical characterization of instantons is easy to visualise, but not very
convenient for computations. An equivalent algebraic characterization is very useful.
Here, the porism of triangles is described by a one-parameter linear family of cubic
19,
equations. Let t denote the (real) rational coordinate along the circle t = tan
where (J is an angular coordinate. Suppose that the vertices of one triangle of the
porism have coordinates tl, t2 and ta- Associated with the triangle is the cubic
equation

(14)
pet) - (t - tl) (t - t2) (t - ta) = 0 .
187
Manton: Skynne fields and instantons


Associated with a second triangle of the porism, with vertices t˜, t˜ and t˜ is the
cubic equation
p'(t) == (t - t˜) (t - t˜) (t - t˜) = o. (15)
It is a remarkable fact that any triangle of the porism is associated with a cubic
equation of the form
pet) + J,t'p'(t) = 0 . (16)
Il

The porism is therefore given by a (projective) line of cubic equations, with inho-
mogeneous parameter Il' / p,. The same characterization of the porism as a line of
cubics also applies when the circle degenerates to a line. In this case, t is simply a
linear coordinate.

2-instantons are rather well understood, but the evaluation of the associated Skyrme
fields involves computing the holonomy. In practice, this means integrating the
ordinary differential equation
(17)

along a time-line. C is a 2 x 2 matrix of rational functions of r which depend on the
instanton parameters as well as on x, and \It is a 2-component vector. Eq. (17) is
of Fuchsian type, and since the integral is from -00 to +00 one may complete the
path of integration with a large semi-circle in the complex r-plane. The holonomy
is therefore a monodromy of the operator C. Since the problem is non-abelian, the
Inonodromy cannot be calculated by simply adding residues. It would be very inter-
(\sting if these monodromies could be determined without numerically integrating
( 1.7).

So far, it has only been possible to calculate the Skyrme fields for special instan-
t.ons where the holonomy is abelian along each,time-line. Expressions (8) and (9)
are examples. For the general 80(3) symmetric 2-instanton one can also give an
('xpression for the profile fer). The potential p depends only on r and r = lxi,




and it may be written as the ratio of two polynomials in r and r. The numerator is
quartic in r (and in r), and since p is positive for real r its roots are two complex
« onjugate pairs a ± ib and c ± id, with b and d positive. Then the Skyrme profile is



1I"(b + d) •
= (19)
fer)

'I'ltis is not a simple expression, because b and d depend in a complicated (but
dll-'tebraic) way on r. An explicit formula could be found, since the quartic equation
188 Manton: Skynne fields and instantons

is solvable by radicals. Things simplify when there is time-reversal symmetry and
the quartic reduces to a quadratic in ,,2.

Another special 2-instanton generates a good approximation to the minimal energy
Skyrme field of degree 2. The conics associated with this instanton are a pair of
concentric circles in a spatial plane (perpendicular to the time-lines) with the ratio
of the radii equal to 2. The triangles tangent to the inner circle, with vertices on
the outer, are all equilateral. The Skyrme field generated from this 2-instanton is
axisymmetric about the spatial line perpendicular to the plane of the circles and
passing through their centres. It has not been possible to compute the Skyrme field
at a general point because the holonomy is non-abelian, but on the axis of symme-
try and in the plane of the circles the holonomy is abelian and can be computed
straightforwardly. Suppose the field is in its standard position and orientation,
with the xg-axis as symmetry axis, and suppose the outer circle has radius R. Let
Xl = r cos 4> , X2 = r sint/> and Xs = z. Then, on the axis



(20)

and in the plane

U(r cos t/>, r sin 4>, 0) = exp i f(r) (0"1 cos 24> - (21)
sin 24» ,
(12

where

IR 1_ -IR 1].
r+
f(r) = r (22)
1(" [

(r 2 +rR+R2)2 (r 2 -rR+R2)2
There is qualitative agreement with the numerical solution of the Skyrme equations,
if one chooses the scale R appropriately. It would be interesting to quantitatively
compare the instanton-generated Skyrme field with the numerical solution, and to
compute its energy.

M2 , the set of Skyrme fields of degree 2 generated from 2-instantons, appears to
provide a sensible subset of Skyrme fields with which to model low-energy two .
nucleon interactions. However, one needs to understand the topology of M better. :
2
The Skyrme energy functional can be regarded as a Morse function on M2 and one
should verify that all the critical points correspond closely to true critical points of .
the Skyrme model.

M2 is a set of static fields, but if the coordinates (moduli) of M2 vary with time,
then the fields become dynamical. The Skyrme Lagrangian, restricted to M , de- 2 I



fines a Lagrangian on M2˜ but the computations which are necessary to find and
189
Manton: Skynne fields and instantons


solve the equations of motion are heavy. Since the Skyrme model is itself only an
approximation, it would be natural to seek a Lagrangian on M2 , defined directly
in terms of instanton moduli, and of the same qualitative form as the Skyrme La-
grangian. This requires that one find a metric 9 and potential energy V directly
in terms of instanton moduli. Given such data, the natural Hamiltonian to use
for the quantized dynamics is H = - V 2 + V, where V 2 is the Laplacian on M2
constructed using the metric g. The wave functions are (complex) scalar functions
on M2 • The novel feature of such a model in a nuclear physics context is that the
curvature and the non-trivial topology of M2 are important. Curvature alone can
lead to non-trivial scattering and quantum bound states.

ACKNOWLEDGEMENTS
Section 2 is the result of joint work with Professor Sir Michael Atiyah. A fuller
account of the Skyrme fields generated from 2-instantons will be forthcoming. The
author is grateful for the hospitality of the Theoretical Physics Institute, University
of Helsinki, where this paper was written.

REFERENCES
Adkins, G.S., Nappi, C.R. and Witten, E. (1983). 'Static Properties of Nucleons in
t,he Skyrme Mode!'. Nucl. Phys. B228, 552.
Atiyah, M.F.(1979). 'Geometry of Yang-Mills Fields.' Lezioni Fermiane, Scuola
Normale Superiore, Pisa.
Atiyah, M.F. and Manton, N.S. (1989). 'Skyrmions from Instantons.' Phys. Lett.
222B, 438.

IJagger, J. Goldstein, W. and Soldate, M. (1985). 'Static Solutions in the Vacuum
Sector of the Skyrme ModeL' Phys. Rev. D31, 2600.
Braaten, E. and Carson L. (1988). 'Deuteron as a Toroidal Skyrmion.' Phys. Rev.
1)38,3525.
(˜astillejo,
L., Jones, P.S.J., Jackson, A.D., Verbaarschot, J.J.M. and Jackson, A.
(1988). 'Dense Skyrmion Systems.' To be published.
(˜astillejo, L. and Kugler, M. (1987). 'The Interaction of Skyrmions.' Unpublished.
Il;˜teban, M.J.(1986). 'A Direct Variational Approach to Skyrme's Model for Meson
Ir'ields.' Comm. Math. Phys. 105, 571.
1'adeev, L.D. (1976). 'Some Comments on the Many-Dimensional Solitons.'
4


Id˜lt. Math. Phys. I, 289.
lIa.rtshorne, R. (1978). 'Stable Vector Bundles and Instantons'. Comm. Math.
/'hys. 59, 1.
190 Manton: Skynne fields and instantons


Jackiw, R., Nohl, C. and Rebbi, C. (1977). 'Conformal Properties of Pseudoparticle
Configurations.' Phys. Rev. D15, 1642.
Jackson, A.D. and Rho, M. (1983). 'Baryons as Chiral Solitons.' Phys. Rev. Lett.
51, 751.
Jackson, A.D., Manton, N.S. and Wirzba, A. (1989). 'New Skyrmion Solutions on
a 3-Sphere.' Nucl. Phys. A495, 499.
Kopeliovich, V.B. and Shtern, B.E. (1987). 'Exotic Skyrmions.' JETP Lett. 45,
203.
Kugler, M. and Shtrikman, S. (1988). 'A New Skyrmion Crystal.' Phys. Lett.
208B, 491.
Manton, N. S. (1987). 'Geometry of Skyrmions.' Comm. Math. Phys. 111, 469.
Manton, N.S. (1988). 'Unstable Manifolds and Soliton Dynamics.' Phys. Rev. Lett.
60,1916.
Schramm, A.J., Dothan, Y. and Biedenharn, L.O. (1988). 'A Calculation of the
Deuteron as a Biskyrmion.' Phys. Lett. 205B, 151.
Skyrme, T.R.H. (1962). 'A Unified Theory of Mesons and Baryons.' Nucl. Phys.
31,556.
Verbaarschot, J.J.M. (1987). 'Axial Symmetry of Bound Baryon Number-Two So-
lution of the Skyrme Model.' Phys. Lett. 19SB, 235.
Wirzba, A. and Bang, H. (1989). 'The Mode Spectrum and the Stability Analysis
of Skyrmions on a 3-Sphere.' To be published.
REPRESENTATIONS OF BRAID GROUPS AND
OPERATORS COUPLED TO MONOPOLES
L. D.S.
RALPH COHEN AND JOHN JONES


Let Mk be the moduli space of based SU(2)-monopoles in R3 of charge k, see
for example [4]. Associated to each monopole c there is a natural real differential
operator 6c , the coupled Dirac operator. The space of solutions of the Dirac equation
6c f = 0 is k-dimensional and as c varies these k-dimensional spaces form a real
vector bundle over the space of monopoles. One of the main purposes of this paper
is to explain how this bundle is related to representations of the braid group.
Braid groups appear in the theory of monopoles in the following way. Let Ratk
be the space of rational functions on C which map infinity to zero and have k poles,
counted with multiplicity. In [10], Donaldson showed that there is a diffeomorphism



The topological properties of the space Ratk are extensively studied in [8] and in
particular it is shown that, for large enough N, there is a homotopy equivalence



Here ˜N means N-fold suspension, (32k is the braid group on 2k strings, and B(32k
is its classifying space, an Eilenberg-MacLane space of type !{(/32k, 1). Combining
t.hese two results shows that the space of monopoles Mk and the space B/32k have
tIle same homology and cohomology; indeed if E is any (generalised) cohomology
theory, then E*(Mk) and E*(Bf32k) are isomorphic. Our aim is to investigate this
isomorphism between the K-theory of Mk and the K-theory of B/32k.
Vector bundles over a classifying space B'Ir arise most naturally from representa-
t.ions of the group 'Ir. Indeed a well-known theorem, due to Atiyah [2], shows that
if the group 'Ir is finite then the K-theory of B'Ir can be computed from the repre-
:˜(\ntation ring of 'Ir by completion. The braid group /32k is not finite and Atiyah's
theorem does not hold; but nonetheless many interesting bundles arise from repre-
˜\(·lltations. On the other hand the moduli space Mk consists of .analytic objects,
n.llnections. One very natural way of constructing vector bundles over Mk is to
4'()llstruct differential operators using the connections and to form the index bundles
!"c H' the corresponding families of operators parametrised by M k. The isomorphism
IH·t.ween the K-theory of Mk and the K-theory of B(32k suggests that there may
1)(\ a. natural correspondence between representations of /32k and operators coupled

I'hp first author was partially supported by grants from the NSF and PYI. Both authors were
by NSF grant. 1)MS-R505550
'.Ilpported
192 Cohen and Jones: Representations of braid groups

to monopoles. Here we will study the coupled Dirac operator and the permutation
representation of the braid group.
It is natural to look at other representations of the braid group, in particular
those used in Vaughan Jones's construction of polynomial invariants of knots and
links, and to see if these representations have some interpretation in terms of the
space of monopoles. We will make some comments on this project in §6. The Jones
polynomials have been given a gauge theoretic interpretation by Witten [18] and it
is natural to wonder if there is any direct connection between these various points
of view..
In this article we will summarise some of the ideas involved in establishing the
relationship between the permutation representation of the braid group and the
index bundle for the family of Dirac operators; full details will be given in [9].
The authors are indebted to Michael Atiyah, Simon Donaldson, Nigel Hitchin, and
Cliff Taubes, for helpful conversations and correspondence concerning this project.
The first author would like to thank the mathematics departments at Oxford Uni-
versity and University of Paris VII for their hospitality while some of this work was
being carried out.
§1 MONOPOLES

The purpose of this section is to summarise the basic facts concerning monopoles
and Yang-Mills-Higgs theory and to show that there are many interesting topological '
features in the theory. Fix the structure group to be SU(2) with Lie algebra su(2).
We use the standard invariant inner product on su(2) and so su(2) becomes a three
dimensional Euclidean space. We study SU(2)-monopoles on R 3 equipped with its
standard (Euclidean) metric and orientation.
We consider pairs (A, cp) where:
(1) A is a connection on the trivial bundle SU(2) on R 3 ,
(2) t.p is an su(2) valued function on R 3 •
So A is a I-form on R 3 with values in su(2);
3
L AIL(x)dX#t
A=
".=1

where AI' : R3 --+ su(2) and the x ll are the coordinates on R3. The connection A
is called the gauge potential and ep the Higgs field. The pair (A, cp) is required
to satisfy several conditions. First we require the Yang-MilIs-Higgs action of (A, <p)
to be finite, that is

(11FA1I 2 + II DAepIl2) dvol <
=f
(I) U(A,ep) 00.
iR3
\
Here FA = dA + A A A is the curvature of the connection A, D A is the covariant
derivative operator defined by A, a.nd dvol is the usual volume form on R 3. We also
193
Cohen and Jones: Representations of braid groups

-
require a condition on the behaviour of <p at infinity. There are several different
conditions which may be imposed. Here we use the weakest condition which seems
to be sufficient [1 7]

(II)

Notice that we do not assume any asymptotic conditions on the gauge potential A.
In addition we impose a base point condition

= (1,0,0).
lim <pet, 0, 0)
(III)
t-+oo

Let A be the space of pairs (A, <p) E A which satsify these three conditions.
There is a natural map
I: (228 2 -+ A,
where (2282 is the space of all smooth maps 52 -+ 52, which preserve the base
point (1,0,0) in 52. This map I is defined explicitly as follows. We identify the
unit sphere 52 with the sphere in su(2). Now given a map



define the pair I( 0') = (A, <p) by the formula

<<

. 7
( 10)



>>