<<

. 8
( 10)



>>



C:I) ,da C:I)]
A= P(lxl) [a
= -p(lxl)a C:I) ·
'P


/3 : R [0, 1] is a smooth cut-off function which is identically 0 if
In this formula -+
I˜ 1/2 and identically 1 if t ˜ 3/4, and [, ] is the Lie bracket on su(2).
The map I : 0 2 5 2 A is a homotopy equivalence.
-+
PROPOSITION.

This is proved in [17]. It immediately shows that there is a definite topological
a.spect to the study of monopoles.
There is a group of gauge transformations

9 = Mapo(R 3, SU(2))

which acts on the'space A. Here Ma·po means those maps 9 : R 3 -+ 8U(2) which
!\a,t.isfy the base point condition

= 1.
lim g(t, 0, 0)
t-+oo
194 Cohen and Jones: Representations of braid groups

Because of this base point condition g is contractible. Gauge transformations act
onA by


This formula makes sense since 9 is a matrix valued function and A is a matrix of
l-forms. The action of g on A is free and has local slices, see [17], and so we form
the quotient spaces
B = A/g.
The quotient map is a principal fibre bundle with fibre g and since g is contractible
we deduce that the projection A --+ B is a homotopy equivalence. The natural maps
fit into a diagram


1
B


and since each of these maps is a homotopy equivalence we deduce the following
proposition.
There is a natural homotopy equivalence
PROPOSITION.




In view of these results A and 8 break up into components labelled by the integers
Z and we use the subscript k to denote the k-th component. This integer k is the
charge of the pair (A, <p). The above propositions show that



where nis 2 means the space of all base point preserving maps 8 2 --+ 8 2 of degree k..
The Yang-MilIs-Higgs functional is invariant under the gauge group g and so
defines a function U on 8. The space Mk ˜ Bk of based SU(2)-monopoles of
charge k is defined to be the space of absolute minima of U. If k 2:: 0 then M k can
be identified with the space of pairs (A, c.p) which satisfy the Bogomolnyi equation



where * is the Hodge star operator on R 3, modulo gauge equivalence. If k < 0
then M k can be identified with the space of pairs (A, <p) which satisfy the equation
DAcp = - *FA .
We concentrate on the case where k ;::: 0 and so study the space of solutions of
the equation D A <p = *FA. This space M k is a smooth manifold of dimension 4k
195
Cohen and Jones: Representations of braid groups

and its geometrical properties are extensively studied in [4]. For example, when
k = 1 a monopole is uniquely determined by its centre, which is a point in R 3 , and
a "phase" parameter in 8 1 ; thus




In fact monopoles can be regarded as "time invariant instantons" in the following
sense. Given the pair (A,<p) we can form an su(2) connection




= R3
on R4 R. This connection is independent of t and it is easy to check that
X
(A, <p) satisfies the Bogomolnyi equation if and only if a is self-dual,



where * is the star operator on R4 •
The following theorem, due to Cliff Taubes [17], shows that these spaces Mk
have some very interesting topological features.
The inclusion i : Mk ˜ Bk is an "asymptotic homotopy equivalence";
THEOREM.
that is there is a function q( k) with q( k) ˜ 00 as k --+ 00 such that the map i
induces an isomorphism of homotopy groups 'Tr q provided q ˜ q(k).
Note that the homotopy type of 8k ˜ ni8 2 is independent of k so the spaces Mk
provide finite dimensional approximations to a fixed homotopy type which become
better and better approximations as k --+ 00.

§2 BRAIDS
In the previous section we saw that the space of monopoles is a finite dimen-
sional homotopical approximation to the space 0 2 8 2 of all base-point preserving
rllaps 8 2 --+ 8 2. In fact there is a more classical finite dimensional homological
approximation to this space and this is where braid groups come into the picture.
Define Ck = Ck(R 2) to be the space of unordered k-tuples of distinct points in
the plane R2. Then recall that the braid group {Jk is the fundamental group of Ck




13k,
ludeed it is well-known that Ck is the classifying space of the braid group that
IS

˜
if i 2.
It. is usual to draw braids as follows:
196 Cohen and Jones: Representations of braid groups




Let Ck = Ck(R 2) be the space of ordered k-tuples of distinct points in R2. The
symmetric group Ek acts freely on Ck by permuting points and the quotient is Ck.
The covering 6k -+ Ck corresponds to a homomorphism



In terms of the diagram representing a braid this is given by mapping a braid to
the permutation of its end points.
There is a natural map
j: C k ˜ ni5 2
defined as follows. First replace the space Ck by the homotopy equivalent space
of configurations of k disjoint disks in the plane. Now given a configuration D =
{D 1 , •• • ,Die} of k disjoint disks define a map In : 52 -+ 8 2 as follows. Regard the
domain of In as R2 U 00 and then In maps the complement of D 1 U · . · U Dk to the
base point of 8 2 and on the disk Di it is the usual identification of DilaDi with 8 2 •
Then D ˜ In gives the required map j : Ck ˜ Q%S2. The main theorem relating
nis 2
is the following result due to May, Milgram, and Segal [14], [15], [16].
Ck and
nis2 is an asymptotic homology equivalence.
The map j : Ck
THEOREM. -+

Now we see that there must indeed be a relation between the monopole space
Mk and the spa.ce BPk = Ck. There are maps



and the map i is an asymptotic homotopy equivalence whereas the map j is an
asymptotic homology equivalence. So the space Mk is a finite dimensional homo-
topical approximation to nZ82 and the space Ck is a finite dimensional homological
approximation to the same space. However the precise relation between braids and
monopoles is rather more subtle tha.n the above remarks might lead one to expect
and it is explained in deta.il in §3.
Cohen and Jones: Representations of braid groups 197


One may think of the map j : C" --. nls 2 as superimposing, or gluing in, a
standard map, the identity of S2 or, to put it another way, the identification of
D 2 /8D2 with the sphere 52, at each of k distinct points in the plane. There is a
similar superposition process in the theory of instantons on R4; in this particular
case this is the 't Hooft construction of instantons on R4 and in the case of a general
4-manifold this is the patching process due to Taubes. If we follow the analogy with
instantons, compare [5], we might think that there is a gluing process for monopoles
which superimposes a standard k = 1 monopole at each of k distinct points in the
plane and that this process leads to a map ,\ : C,,(R 3 ) ˜ Mk with the property
that the composite

nis2
C,,(R 2 ) ˜ C,,(R 3 ) ˜ M" 8" ˜
-+

is homotopic to the map j used above. Suppose indeed that such a map A exists,
then computing the induced homomorphisms on fundamental groups quickly gives
a contradiction;


and the induced homomorphismj. is surjective. But 7rl(Ck(R3)) is the symmetric
group Ek and the homomorphism



cannot factor through Ek.
This argument shows that the gluing proceedure for monopoles is considerably
Inore delicate than that for instantons. Indeed it is possible to superimpose "well-
separated monopoles" but great care must be taken with this construction and
t.his is one of the points where the theory of monopoles is very different from the
(˜orrespondingtheory of instantons.


§3 RATIONAL FUNCTIONS

To understand the full relation between braids and monopoles we need to use
J)onaldson's theorem relating Mk and Rat", the space of rational functions on C
which map infinity to 0 and have k poles.
There is a diffeomorphism Mk ˜ Ratk-
'I'IIEOREM.

The proof of this theorem is given in [10], [11], [12]. Now let Rat˜ be the subspace
of Ratk consisting of those rational functions with k distinct simple poles; this is
til(' subspace of generic rational functions. If f E Rat˜ then fez) can be written in
t,l1(\ form
n
f(z)=˜˜
L-t z - b·
i=l I
198 Cohen and Jones: Representations of braid groups

where the hi are distinct complex numbers and the ai are non-zero complex numbers,
therefore
Rat" = Ck XEk (C * )k .
o -

Here Ok is the space of ordered k-tuples of distinct points in C and C* is the space
of non-zero complex numbers. The symmetric group Ek acts on both Ok and (C*)k
by permutations. From this we deduce that Rat˜ is the classifying space B/32,k of
the semi-direct product
/32,k = 13k ˜ (Z)k
where {11: acts on (Z)k by permuting factors. This group f32,k can be thought of
as the group of framed braids and it is naturally a subgroup of the group f32k.
The inclusion /32,k ˜ f32k is given by the cabling process which can be described as
follows. Start with k pairs of pieces of string and twist the i-th pair ni times where
ni E Z. Now braid the k pairs according to the braid b E f3k. This gives a braid on
2k strings and the map which sends (13; nI, ..., nk) to this braid gives the inclusion
{32,k -+ {32k.
We now have the following diagram.
<p
Rat˜ = B{32,k ˜ B/32k

t/Jl
Ratk = Mk
and the following theorem follows directly from one of the main results of [8].
Let E be any cohomology theory, then
THEOREM.

cp* : E*(Bf32k) E*(Bf32,k)
-+

1/J* : E*(Mk) ˜ E*(Bf32,k)
are both split injective and the splittings induce a natural isomorphism


We will use this theorem in the case where E is K -theory but before doing so we
discuss some of its implications. It is a reasonably straightforward piece of group
theory to check the following lemma concerning the braid group, for example see [7].
(1) The abelianisation of the braid group f3k is the integers Z.
LEMMA.
(2) If k ˜ 5 then the commutator subgroup [Pk, ,Bk] ˜ {3k is perfect.
Given this lemma we can form the Quillen plus construction to kill [f32k, {32k] to
get the space B /3tk' and this space then has fundamental group Z and the same
homology, with any twisted coefficients, as B{32k. We know that ?rIMk = Z and, by
the above theorem, H*(Mk; Z) ˜ H*(B/32k; Z) and so the following question seems
very natural
199
Cohen and Jones: Representations of braid groups


QUESTION. Is there a homotopy equivalence




The essential difficulty is whether there is a map Bf32k ---+ Mk which abelianises
the fundamental group and induces an isomorphism in homology with any twisted
coefficient system. Both spaces are as nice as possible, they are 4k-dimensional
manifolds and both have natural complex structures. A theorem due to Kan and
Thurston [13] asserts that any space is homotopy equivalent to a space obtained
by applying Quillen's plus construction to a space of the form B1r. It would be
rather remarkable if it were possible to obtain the monopole space by applying
the plus constrution to B/32k. If there is such an equivalence this would give the
neatest possible way of expressing the relation between braid groups and monopoles,
however for many purposes the above theorem is completely satisfactory since it
provides a definite method of producing the isomorphism E*(Mk) --+ E*(Bf32k) as
we shall see in a specific example.

§4 THE COUPLED DIRAC OPERATOR
Now we turn to the construction of vector bundles on the space Mk using diff-
erential operators and the purpose of this section is to describe the basic operator,
the coupled Dirac operator, and its index bundle. Let 53 be the space of spinors
on R 3 and write E for the usual 2-dimensional complex representation of 5U(2).
Then the Dirac operator coupled to a pair c = (A, ep) E A is the operator




defined by the formula

3
+ (1 ® cp)f·
1)· (DA,i(f))
ac(f) = 2:(ei @
i=l


Here ei is the i-th generator of the Clifford algebra C3 of R 3 and D A,i is covariant
differentiation in the i-th direction in R 3 defined using the spinor connection on 53
Hud the given connection A on E. The ei act on 53 via the usual spin representation
of C3 and su(2) acts on E via the standard representation of the Lie algebra su(2).
In fact this operator is the time invariant Dirac operator on R 4 coupled to a time
illvariant instanton i˜ the following sense. Given the pair (A, cp) form the connection
t \' == A + 'P dt on R 4, as in §1. Now we can form the Dirac operator on R 4 coupled

tC) the connection a
200 Cohen and Jones: Representations of braid groups

st are the positive and negative spinors on R4. We can restrict this operator
where
to the subspace of functions on R4 = R3 X R which are independent of the fourth
coordinate t and since a is independent of t we get an operator



st and S4 we can form the operator
Now using to identify
e4




st
Using the usual identification of the C3 module 53 with where ei E C3 acts as
oc
e4 ei this operator e4 • 00/ can be identified with the operator defined above.
Now note that Spin(3) is isomorphic to SU(2) and under this isomorphism the
spin representation 8 3 becomes the usual2-dimensional representation E of SU(2).
Therefore 8 3 ®E is the 4-dimensionaJ representation E0E of SU(2) x SU(2). This
representation has a real structure; indeed using the fact that Spine4) is isomor-
phic to SU(2) x SU(2) the underlying real representation is the double covering
Spine4) ˜ SOC4). The outcome is that there is there is a 4-dimensional real rep-
resentation R of SU(2) x SU(2) whose complexification is E ® E. Therefore the
operator Oc has a real structure and there is an operator



oc.
This operator Dc is the one we use and we refer to it
whose complexification is
as the real Dirac operator coupled to (A, cp).
In [17] Taubes shows that this operator extends to a Fredholm operator on the
appropriate Soholev spaces. Furthermore if Cl and C2 are gauge equivalent the oper-
ators OCl and OC2 are isomorphic and so we get a family of operators 6c parametrised
by the points c E 8. Taubes also shows that this is a continuous family and therefore
has an index bundle
ind(6) E KOO(B).
Recall briefly how this index bundle is defined. Let

Cc = coker8c

so both K c and Cc are finite dimensional real vector spaces. Suppose in fact that
the spaces K c form a vector bundle K over A and the spaces Cc also form a vector
bundle Cover A. Then
ind(o) = K - C.
In general the spaces K c do not form a vector bundle, nor do the spaces Cc , since
the kernel and cokernel of bc may jump in dimension, but we can reduce to this
case by a deformation, compare [2].
201
Cohen and Jones: Representations of braid groups

Now we must identify this index bundle




using the equivalence of B with U2 8 2 • There is the usual isomorphism

52 = 5p(1)jU(1)

and we can compose this equivalence with the stabilisation

=
Sp(l)/U(l) ˜ SpjU lim 5p(n)jU(n)
n˜oo




SpjU. Now apply 0 2 to this map and use the equivalence
to get a natural map 8 2 --+




occuring in real Bott periodicity, a.nd we get a map




This map gives us an element 1 E KOO(U 2 S 2 ) with corresponding components
"fA: E KOO(Ui5 2 ).
Using the natural isomorphism of1(00(8k) with KOO(QiS2 ) the index
THEOREM.
bundle ind(6) can be identified with 1k:

1(00(8A:) ˜ KOO(n˜S2)
,k
ind(6) +-+


This theorem is proved by adapting Atiyah's proof of Bott preriodicity [1] using
elliptic operators and it is given in [9]. In some sense the proof is straightforward
hut in detail it is quite tricky since it must necessarily involve the intricacies of
8-fold Batt periodicity in real K-theory. The complexification of the bundle, is
t.rivial so there is no way to avoid the extra complications of real K -theory and get
non-trivial results.
The appearance of Sp(l)jU(l) in the above description of I is very natural. In
the theory of monopoles for general Lie groups the asymptotic conditions imposed
011 the pair (A, <p) are that <p approaches a fixed orbit of the adjoint action of the
,˜roup on its Lie algebra. In our case we see that using the identification of Sp(l)
wit,h SU(2) the unit 2-sphere in the Lie algebra su(2) can be identified with the
orhit of Sp(l) acting on a fixed unit vector, that is Sp(l)jU(l).
202 Cohen and Jones: Representations of braid groups


§5 THE DIRAC OPERATOR AND REPRESENTATIONS OF THE BRAID GROUP
Now we look at the operator



where c E Mk. The first simplification is that since c now satisfies the Bogomolnyi
equation we can use the vanishing theorem of [17].
THEOREM.
=0
Cc = coker(be)
Therefore the spaces K c = ker(cc) form a k-dimensional real vector bundle over
the monopole space M k and this is the index bundle ind(6) of the family of operators
Cc where c E Mk. Now we use the theorem of [8] described in §3 to identify
the corresponding bundle over B(32k, more precisely the corresponding element in
KOO(Bf3k). To do this we must first identify ˜*(ind(6) where



is the map described in §3.
There is a natural representation

˜ O(k)
'Irk : f32,k

defined as follows. Let ai be the generator of the i-th copy of Z in (32,k = (3k t>< (Z)k,
then in the representation 'lrk, ai changes the sign of the i-th basis vector in R k
and a braid b E 13k S; f32,k permutes the basis of R k. Another way to describe this
representation is as the composition



where the first homomorphism is the obvious quotient and Ek ˜ (Z/2)k is identified
with the subgroup of O(k) generated by the permutation matrices and the diagonal
matrices with ±1's along the diagonal. This representation defines a bundle over
B(32˜k which will still be denoted by 1rk. The corresponding bundle can be described
quite explicitly as follows. Let H be the real Hopf line bundle over C* = R X 8 1 ;
then 1rk is the bundle



THEOREM.
= 'Irk E KOO(B132,k)
¢*(ind(6))
The proof of this theorem is given in [9] so here we try to describe the intuition
behind it. First we deal with the case k = 1, then M 1 = R 3 X 8 1 and ind(b) is the
Cohen and Jones: Representations of braid groups 203

non-trivial I-dimensional line bundle over M 1 , that is the (real) Hopf line bundle
over the circle 51. This of course immediately verifies the theorem in the case k = 1
since !J2 l = l, M 1 = BP2,1 and the Hopf line bundle is just ?rl. These facts have
t

the following intuitive interpretation. A I-monopole is determined by its centre in
R 3 and a phase, which determines its component in 51. The space of solutions of
the Dirac equation oc(/) = 0 is one dimensional and any non-trivial solution of this
equation has the property that if we rotate the phase of the monopole through 2?r
then this solution changes sign.
Now we look at the general case of Mk and restrict attention to the subspace of
"well separated monopoles"; this is the subspace obtained by gluing in I-monopoles
at k distinct points in R 3. Then it is natural to expect the solutions of the Dirac
equation coupled to such a monopole to be k-dimensional with one basic solution
associated to each of the k-distinct points. If the points are permuted then so is
the basis and if the phase of the i-th I-monopole is rotated through 2?r then the
sign of the i-th basis vector changes. Therefore the representation ?rk should appear
quite naturally. However as we have already pointed out the gluing process for
monopoles is quite subtle so that while this heuristic argument is quite convincing,
considerable care is needed to give a precise proof.
The next step is to look for the corresponding element of KOO(BP2k). First we
show that there is an element x E [(OO(BP2k) with the property that
= ?rk
c.p*(x)
where c.p : BP2,k ---. BP2k is induced by the inclusion j : {32,k --+ (32k. The obvious
starting point is the permutation representation P2k : f32k -+ O(2k) and so the first
step is to compute the representation j*(p2k). Let q : (32 k --+ Pk be the quotient
t

homomorphism Pk I>< (l)k -+ Pk.
LEMMA.
j*(p2k) = ?r2k EB q*(Pk)
The proof of this lemma is quite straightforward. If we write the basis of R 2k as
Cl, ••• ,e2k then in the representation j*(p2k) the generator ai of the i-th factor l
acts by interchanging e2i-l and e2i and leaving the other basis vectors fixed while
a. braid b E (3k permutes the k pairs (el, e2), . .. ,(e2k-l, e2k). Therefore in the basis

11 = el - e2,···, Ik = e2k-l - e2k, !k+l = el + e2,···, 12k = e2k-l + e2k
t.he representation j*(p2k) is precisely ?r2k EB q*(Pk).
To find the element x E KOO(Bf32k) such that <p*(x) = ind(o) E KOO(Mk) we
need to find a way to subtract off Pk from P2k. In KOO(Bf32k) there is a natural
way to do this; at the level of representations things are more delicate and will be
analysed carefully in [9] . The inclusion i : Pk -+ {32k induces a surjection
i* : KOO(B(32k) ---. KOO(B{jk)
and this map has a natural splitting. More precisely we use the following proposition
from [6].
204 Cohen and Jones: Representations of braid groups

There is a natural map
PROPOSITION.




such that
= 1, where i : f3k ˜ {32k is the inclusion
(1) i* · a
= q* where j : {J2,k -+ {32k is the inclusion.
(2) j* · a

From this proposition we see that the element x we have been looking for is
P2k - a(pk). Now by analysing the splittings which occur in the theorem of [8]
we end up with the following result, proved in [9], which gives the precise relation
between the index bundle for the family of Dirac operators coupled to monopoles
and the permutation representation of the braid group.
Using the natural isomorphism of KOO(Mk) with [(OO(B/32k) the in-
THEOREM.
dex bundle ind(6) can be identified with P2k - a(pk):

KOO(Mk) ˜ KOO(B{32k)
ind(6) ˜ P2k - a(Pk)

§6 FURTHER COMMENTS
It is natural to wonder which of the other representations of the braid groups
appear in index bundles of operators parametrised by the space of monopoles. The
above proceedure can be generalised as follows. We can couple the Dirac operator on
R 3 to the pair (A,cp) using any representation of SU(2). The Dirac operator has a
quaternionic structure and so if we couple it to a representation with a quaternionic
structure the corresponding index bundle will have a real structure. This gives a
homomorphism


where RSp is the quaternionic representation ring. Similarily we get a homomor-
phism


where RO is the real representation ring and KSp is quaternionic K-theory. These
two homomorphisms will be studied in detail in [9]. These constructions give the I




most general natural elements of K-theory which can be constructed by these meth-
ods.
The representations which are used in Vaughan Jones's work all occur in continu-
ous one parameter families which for special values of the parameter factor through
the permutation group ˜n. The corresponding bundles are homotopic, therefore
isomorphic, and so define the same element of K -theory. So [{-theory is not suffi-
ciently sensitive to detect the difference between these inequivalent representations
and every K-theory class which arises from these representations factors through
205
Cohen and Jones: Representations of braid groups

the permutation group. However the index bundles come as sub-bundles of a trivial
bundle of Hilbert spaces and so they inherit natural connections. It seems reason-
able to guess that the holonomy of these natural connections is closely related to
representations of the braid groups and if true this would give a more precise way
of associating representations of the braid groups to these index bundles. One final
point worth mentioning is that the space of monopoles on hyperbolic space is also
diffeomorphic to the space of rational functions Ratk' see [3], and it may be possible
to exploit this fact. We plan to return to these ideas in future work.

REFERENCES

1. M.F. Atiyah, Bott periodicity and the index of elliptic operators, Quart. J. Math., Oxford
(2) 19 (1968), 113-140.
2. M.F. Atiyah, "K-theory," W.A. Benjamin, New York, 1967.
3. M.F. Atiyah, Magnetic monopoles in hyperbolic space, in "Proceedings of the Bombay collo-
quium on vector bundles, 1984," Oxford University Press, Oxford, 1987, pp. 1-34.
4. M.F. Atiyah and N.J. Hitchin, "The Geometry and Dynamics of Magnetic Monopoles,"
Princeton Univ. Press, Princeton, 1988.
5. M.F. Atiyah and J.D.S. Jones, Topological aspects of Yang-Mills theory, Comm. Math. Phys.
61 (1978), 97-118.
6. E.H. Brown Jr. and F.P. Peterson, On the stable decomposition of n 2 s r +2 , Trans. of the
A.M.S. 243 (1978), 287-298.
7. Joan Birman, "Braids, links and, mapping class groups," Annals of mathematics studies, 82,
Princeton Univ. Press, Princeton, 1974.
8. F .R. Cohen, R.L. Cohen, B.M. Mann, and R.J. Milgram, The topology of rational functions
and divisors of surfaces, Acta Math. (to appear).
9. R.L. Cohen and J.D.S. Jones, Monopoles braid groups and the Dirac operator, to appear.
10. S.K. Donaldson, Nahm's equations and the classification of monopoles, Comm. Math. Phys.
96 (1984), 387-407.
11. N.J. Hitchin, On the construction of monopoles, Comm. Math. Phys. 89 (1983), 145-190.
12. J. Hurtubise, Monopoles and rational maps: a note on a theorem of Donaldson, Comm.
Math. Phys. 100 (1985), 191-196.
13. D.M. Kan and W.P. Thurston, Every connected space has the homology of a !«1r, 1), Topol-
ogy 15 (1976), 253-259.
14. J.P. May, "The Geometry of Iterated Loop Spaces," Lecture Notes in Mathematics 271,
Springer-Verlag, 1972.
15. R.J. Milgram, Iterated loop spaces, Ann. Math 84 (1966), 386-403.
16. G.B. Segal, Configuration spaces and iterated loop spaces, Invent. Math. 21 (1973), 213-221.
17. C.H. Taubes, Monopoles and maps from 8 2 to 8 2 ; the topology of the configuration space,
Comm. Math. Phys 95 (1984), 345-391.
18. E. Witten, Quantum field theory and the Jones polynomials, Comm. Math. Phys.121 (1989),
351-399.



Mathematics Department, Stanford University, Stanford California 94305
Mathematics Institute, University of Warwick, Coventry CV4 7AL England
Extremal Immersions and the Extended Frame
Bundle
D H Hartley, R W Tucker
Department of Physics, University of Lancaster, UK
1989




Abstract


We present a computationally powerful formulation of variational problems that
depend on the extrinsic and intrinsic geometry of immersions into a manifold. The
approach is based on a lift of the action integral to a larger space and proceeds by
systematically constraining the variations to preserve the foliation of a Pfaflian sys-
tem on an extended frame bundle. Explicit Euler-Lagrange equations are computed
for a very general class of Lagrangians and the method illustrated with examples
('clevant to recent developments in theoretical physics. The method provides a
Ineans of determining spatial boundary conditions for immersions with boundary
and enables a construction to be made of constants of the motion in terms of Euler-
Lagrange solutions and admissible symmetry vectors.

INTRODUCTION

(˜urrent trends in theoretical physics have focussed attention on the properties of
sl>acetime immersions that extremalise various aspects of their geometrical struc-
r nrc. Thus string theories are based on models that extremalise the. induced area
of two dimensional time-like world sheets. Their generalisations to p-dimensional
iUUllersions provide a dynamical prescription for (p - I)-dimensional membranes.
I'˜xt,remalising the integral of the natural induced measure has provided a very rich
plu'llomenological interpretation in the context of particle physics and has led to
" 11lunber of speculations connecting gravitation to the other forces of Nature. In
111˜\S(' developments the properties of the ambient embedding space for the various
Hartley & Tucker: Extremal immersions and the extended frame bundle
208

immersions playa minor role at the classical level. At the quantum level consistency
conditions constrain their dimensionality when the ambient space is flat.

A number of recent papers [1] have begun to investigate the properties of spacetime
immersions that extremalise integrals of certain of their extrinsic properties. Such
an approach incorporates the ambient space into the fundamentals of the theory
in a non-trivial way. At the classical level such models promise a rich and varied
phenomenological interpretation and offer important challenges for the quantisation
program of non-linear systems.

In order to appreciate the properties of individual models constructed from their
extrinsic geometrical properties it is important to be able to study a class of models.
f
In this manner we may put individual properties into their proper perspective.. I
J
However the classical methods of the variational calculus applied to such higher- }
I
order Lagrangian systems become increasingly unwieldy for the models that we have
.\
in mind. The traditional approach is to formulate the calculus of variations on a
bundle of k-order jets. In plinciple this formulation is available to us but we have .
found a more economical approach based on the use of exterior differential systems
on an extended frame bundle. This idea has been inspired by the work of Griffiths
[2] who has considered 1-dimensional variational problems in this context. For
variational problems that are concerned with the extremal properties of immersions
this approach seems both natural and powerful.

We consider below a class of immersions determined from properties of their shape
tensor. (The immersions with extremal volume are included as a special case.)
Immersions in this class lie at the basis of a number of recent membrane mod-
els in theoretical physics and have been studied by Willmore and others in the
mathematical literature [3],[4]. We believe that the approach to be described will
provide a unifying route to the Euler-Lagrange equations of these and more general
geometrical actions.


1. The Darboux Frame and Second Fundamental Forms


Let C be a p-dimensional submanifold of an m-dimensional {pseudo-)Riemannian
=
manifold (M,g) with Levi-Civita connection V. For any point pEe write TpM
Hartley & Tucker: Extremal immersions and the extended frame bundle 209

V the Levi-Civita connection of the induced metric 9
TpC E9 (TpC).L and denote by
on C. Then
= VyZ +h(Y,Z) W,ZETC 1.1
V'yZ

defines the shape tensor h E Tl(Tlvl). For the ranges a,b,c = 1, ... ,m; It, {3, 'Y =
1, ... ,p; i,j,k = p + 1, ... ,me Let {e a } = {ei,e Q } be a local g-orthonormal
coframe for M such that {eQ}p E T;C and {ei}p E (T;C)l.. We introduce the
= g(e a , eb) with inverse TJ4b- If C is the image of the embedding map
matrix TJab


f:D˜M 1.2

then {e a } constitutes a local Darboux coframe adapted to C if

'Vi 1.3

Denoting the dual orthonormal frame as {Xa } = {Ni,X a } we may expand the
shape tensor h in terms of the normal basis to define the set of m - p second
fundamental forms Hi on M:

VX,Y ETC 1.4

Since V'x a X{1 = -w˜C(XQ)Xc in terms of the connection l-forms wah in a Darboux
frame:


or
1.5

Since
1.6

and each e i restricts to zero on C we see that:

1.7

here and below ˜ denotes restriction by pullback. The restricted structure
\v here

Hi is a symmetric second degree tensor.
"(Illations imply that each
210 Hartley & Tucker: Extremal immersions and the extended frame bundle

2. Differential Ideals of Exterior Forms (5]


A set of differential forms {a A } (1 ˜ PA ˜ n) with a A E APA (T* B) on an
n-dimensional smooth manifold B generates a differential ideal I( B) in the exterior
algebra A(T*B) consisting of all forms of the type E A(T*B).
where
1rA 1\ etA 'irA
B for some pA B E A(T*B).
The differential ideal is closed if for all A, do: A = pA B l\a

Two sets of differential forms {a A} and {f3B}, not necessarily with the same number
of elements, are said to be algebraically equivalent if they generate the same ideal:
in which case any subspace of T(B) that annuls all forms in one set will also annuls
all forms in the equivalent set. In this case we write {aA} == {,BB}. The associated
space of an ideal I( B) generated by {etA} is the smallest subspace Q* C T* B such
that the exterior algebra generated by the l-forms in Q* contains a subset that
genera.tes I(B). The associated Pfaff system of {etA} is this set {6 1 ,S2, •.• sr} of
I-forms that span Q* and the rank of the ideal I(B) is defined as the dimension
of Q*. In the following we are mainly concerned with ideals generated by a set
of I-forms, known as a Pfaffian system. Then any Pfaffian system of rank r is
algebraically equivalent to a set of r linearly independent I-forms. An integral
manifold of an ideal I(B) is a mapping

F:D--+B 2.1

of maximal rank such that all forms in the ideal vanish when restricted to D, i.e.

Yo. E I(B)
Po. = 0 2.2

The closure of a Pfaffian system {SA} is the ideal containing {SA, dS A }. Since d
commutes with pull backs˜ integral manifolds of the Pfaffian system {(JA} are also
integral manifolds of its closure and conversely. Clearly the closure of a Pfaffian
system is closed. The system is said to be completely integrable if it is algebraically
equivalent to a set of r exact I-forms where r is the rank of the system. This occurs
iff the system is (locally) closed. In this case one has a (local) foliation on B whose
e {deA} More generally a
==
leaves A = constant are the integral manifolds of {IJA}
system S of forms {o.A}, not containing O-forms, has as its characteristic system
Ch(S) the associated Pfaff system of its closure S = {o:A, da A } [6]. If r is the rank
of this associated Pfaff system then Ch( S) always possesses local integral manifolds
Hartley & Tucker: Extremal immersions and the extended frame bundle 211

of dimension dim B - r. These are the characteristic manifolds for S, and can be
used to determine integral manifolds of S itself. In the following we have been able
to bypass the explicit construction of Ch( S) by examining directly the structure of S
in order to determine the appropriate integral manifold for our particular problem.

3. Variations of Exterior Systems


A regular exterior differential system on a smooth manifold B consists of a differen-
tial ideal I(B) C A(T*B) and a preferred set {nO'} of linearly independent I-forms
on B. An integral manifold of such a system consists of a smooth mapping


3.1
j:D--+B

1is an integral manifold of I(B) together with the condition that {j*OO:}
such that
is non-zero. The problem discussed below entails finding an appropriate space B
for an ideal I(B) in formulating our variational principle.

Ie f3 of a p-form f3 on a manifold M.
In the first instance we deal with an integral
In our applications we have in mind that M is a spacetime. Thus we wish to find
the extremum of the integral
Lf3
= 3.2
A(C,j)

= f(D) f3 some prescribed
where f : D --+ M is a p-dimensional immersion, C and
form constructed from properties of the immersion and the ambient space M. In
terms of the Hodge p-form, *1, defined with respect to the induced metric on C
\ve define the Lagrangian O-form £, by

3.3

A variation of f is a map
F : D x [-e, e] M
-+


= F(p,O) It :
that f(p) for all p E D. This defines a family of immersions
h'ICh

=
/) x {t} --+ M with ft(D) t E [-e,e], and we seek a critical immersion
Ct,
l ˜ 10 such that
1 I3lt=o °
dA t d 3.4
't=o
dt = dt =
Ct
Hartley & Tucker: Extremal immersions and the extended frame bundle
212

= F. :t
aco- Denoting V
subject to conditions on to be a variational vector field
on M we have in terms of the Lie derivative

1
dA t 3.5
8A[V] == -d It=o = £vf3
t Co

f3 depends on properties of the immersion.
The problem is to compute LV f3 when
A relatively simple case occurs when t3 = *1 and p = m - 1. In this case suppose
V = N, the normal vector field to the varied hypersurfaces and choose f3 = iN * 1
in terms of the Hodge m-form on M, since this pulls back to the volume form on
= i.˜rd/3 = iNd *N, where Nis the metric dual of N, so using
D. Clearly £N/3

3.6

we have
r(TrH)*l
e5A[V] = - 3.7
10 0

Hence closed immersions of zero mean curvature {"minimal immersions"} will ex-
tremalise our integral. To accommodate the induced variations in (3 in the more
general case we reformulate our problem by lifting f to a map j : D -+ B for some
suitable space B such that 6 = j(D) becomes a p-dimensional integral manifold
P
of a regular Pfaffian differential system {(JA, d()A} ,{na } on B. We now regard
f* f3 = i* i3 j that
as a p-form on B such that and look for integral manifolds
extremalise the integral
= fc˜ 3.8
A(C,j)

x [-e, e] --+ B , generated by the vector field V =
F :D
for admissible variations
:t
F* on B, and appropriate conditions on 86_By an admissible variation we shall
it :D x {t} -+ B are also integral manifolds of our
mean that the various maps
system {(JA, dfJ A }. This implies that admissible variations are generated by fields
V that satisfy:
'1.4 3.9
i.e.
3.10

Since
3.11
Hartley & Tucker: Extremal immersions and the extended frame bundle 213


for any forms AA it follows that admissible variations satisfy

3.12

Thus given some p-form f3 on M we have elevated the variational problem to the
f
determination of an integral manifold of a differential exterior system such that
for all variations satisfying 3.12 we have

3.13

modulo boundary data.

Let us first consider the case where V is chosen to vanish on 8C. (One might
equivalently postulate 86 = 0 although in the classical spacetime context this is
a somewhat unphysical condition.) Then since £"(r = iyd + di y equation 3.13 is
equivalent to
kivd˜ = 3.14
0

Note this statement is invariant under p-+ p+AA 1\ (}A since d(}A pulls back to zero
˜ +dp. Furthermore 3.14 is satisfied if
on D, and under ˜ --+


3.15

evanishes on 8e.
V
provided is admissible and

V, CVI BC =
Now suppose we seek a set of (p -I)-forms AA so that for all 0)

3.16

'Then 3.15 becomes with the aid of 3.12

3.17

Such an equation respects the invariances above and will be adopted as the set
of Euler-Lagrange equations for the integral manifolds that extremalise 3.8 for all
ac.
variations that vanish on

VIBe =
6
\Vhen has a boundary on which we relax the condition 0 the above
4'ollclusions need refining. In general we do not expect to be able to impose either
I
Hartley & Tucker: Extremal immersions and the extended frame bundle
214

arbitrary or unique boundary conditions that determine (: from the Euler-Lagrange
1
equations. On the other hand we. do not expect that 3.17 will need modification in
the presence of boundaries. Thus we adopt 3.17 as before but re-examine

3.18



Zi"Q"djj + Jac_i"Q"jj
f
= 3.19
Jc
where we have used Stoke's theorem in the second integral. At this point we can no
longer assert that A be stationary under arbitrary variations. However from 3.17
we may write
3.20

or using 3.10
= f _ + AA 1\ OA)
6A[Vj i"Q"(jj 3.21
Jac
At this point it is useful to introduce the notion of an
for admissible variations.
V is an admissible symmetry vector, denoted f', if in
admissible symmetry vector.
addition to 3.12
3.22

If such a vector field exists




or
f jt =0 3.23
JaG
where
3.24

When M has a Lorentzian structure we may classify subspaces of TpD according to
the signature of the restriction of f* 9 to TpD. (Recall f : D M.) We shall suppose
-+

that for physical applications f(D) is an orientable time-like (or null) p-chain rep-
resenting the history in Al of some spacelike (p - 1)-chain. (For timelike signature
f* 9 contains one timelike eigenvalue.) Then suppose we write

3.25
Hartley & Tucker: Extremal immersions and the extended frame bundle 215

where D± are spacelike (p - 1)-chains (corresponding to parameter time sli(˜es tha.t.
"cap" D) and Dr is a timelike (p - I)-chain (corresponding to the history of a
spacelike (p - 2)-chain). Conditions on D± constitute initial and final data. Thus
we adopt spatial boundary conditions that ensure:

3.26


V, so that
for all admissible

3.27


V.
for admissible

r is an admissible sym-
It follows from the global boundary condition 3.26 that if
metry vector
3.28

is a constant of the motion for any spacelike (p -I)-chain DO' C D with boundary
in Dr'


4. The Extended Frame Bundle and its Darboux Leaves


Having set up a general variational formalism we wish to apply it to the problem of
f :D M is an immersion and f3 depends on properties
extremalising 3.2 where --+
of its extrinsic geometry.

Consider the orthonormal frame bundle (OM'I1r) of M, elements of which are
{p, {X a }} for p E M. Extend this to the bundle B = (OM X R k , 1r) for some k, A
local coordinate system for UB C B is given by {x a , nab, Hi Otp} where {x a } are chart
= _a ba
functions for UM C M, parameterise the m-dimensional (pseudo)-
aab

= Hi {30/ are coordinates for R k. With the index ranges
orthogonal group and Hi a{3
d(˜clared in section 1, we have k = tp(p+ I)(m-p) and dim B = m+ !m(m-l)+k.
IJ(˜t us denote a local coframe by a set of I-forms

4.1
Hartley & Tucker: Extremal immersions and the extended frame bundle
216

Indices attached to forms on B will be raised and lowered using the matrix 'flab

introduced in section 1. A variation of C gives rise to a local foliation of M by
leaves Ct. Our first aim is to construct a local foliation V of B by subbundles
(1)(Ct ),1[') over C t such that local sections (1M : UM ˜ B give rise to a choice of
a local Darboux frame field adapted to each leaf C t - Having established a set of
"Darboux leaves", V(C t ), we shall lift our variational problem into B in order to
determine a critical leaf V(Co ) by the Euler-Lagrange equations on B. From such
a leaf we may obtain the solution Co on M of our original problem by projection.
˜
that restricts to (1eo : Co 1'(Co) fixes a Darboux frame adapted
Any section (1M
to Co-

Thus we seek an exterior differential system on B whose integral manifolds include
= {e i , 6i O'}
the leaves 1)(C t ). Consider then the system S where in terms of the
cofrarne 4.1
4.2
Hi O'P eP
wi _
ili - 4.3
0'
(1 Q -



together with the preferred forms nO' = eO.

We claim that S possesses as a local integral manifold the leaf 1)(C) where C is a
p-dimensional submanifold of M and that locally

'D(C) ˜ C x SO(C) 4.4
(SO(C))J..
X


where SO(C) is the structure group for the orthonormal frame bundle of C and
(80(C)).1. the structure group for the bundle of orthonormal frames normal to C.
To justify this assertion we must obtain from S its closure S and examine its integral
manifolds. From the first structure equations:

4.5

+ eO /\ Bi o
= _Wij 1\ Bi 4.6
using the symmetry of Hi o{3. Furthermore

4.7

4.8
Hartley & Tucker: Extremal immersions and the extended frame bundle 217


using both structure equations. Splitting

_lRi
R i 0-2 obc e6/\ e c

into different index ranges and using 4.2,4.. 3 gives:

.
= -R" atJI etJ A e˜ + R" °tJJ /\ 8J + -R" OJtfi . /\ 0k
1· ·1·
Q • Q
'J˜8J
dOl a Q'el-'
2 2
Q




+ Hi /3,e'Y) /\ w/3 a + Hi o/3e/3)
(oj 0
-(Oi f3 - wi j /\


+ Hi o/3W P'Y /\ e'Y - Hi ap(8 j P + HjP"'Y /\ ej') /\ f)i
-dH i 0/3 A e/3 4.9

Under pull back to integral manifolds we may work modulo products of the forms
4.2,4.3 hence:




0., symmetric except for the curvature
In the first term the expression in brackets is
term. However using the symmetry properties

R i 1 0 /3 - - R i j'Po 4.11
-

4.12

4.13

we may write




in terms of the I-forms:

e. . 1· .
= e'Ya = - 3'(R" a/3'Y + R" 'Y/3o)e +dH10'Y -
P · {J' {J' ••
'Y Hz 0/3
a a
+ WI j HJ 0'Y
aHa /3˜ -
W W
OJ'

4.15
The closure of S is the system S = {8i,f)io,d8i,dBia}. From 4.6 and 4.14 we see
that integral manifolds of the system .

4.16
Hartley & Tucker: Extremal immersions and the extended frame bundle
218

are integral manifolds of S. The rank of PI(S) is the number of linearly independent
I-forms in it. Hence:

1 1
= (m- p )(1+P+2 P(p+I»
rank PI(S) = (m- p )+p(m- p )+2(m- p )p(p+1) 4.17

It follows that the integral manifolds of Pf(S) give rise to a local foliation V in
B whose leaves have codimension equal to the rank of P I( S). Since dim B =
!m(m + 1) + t(m - p)p(p + 1) we have:

= p+ ˜p(p -1) + ˜(m -
= dim B -
dim (leaf V) rank Pf(S) p)(m - p -1) 4.18

Thus the dimension of lea! V coincides with that of a bundle locally isomor-
phic to C x SO(C) X (SO(C»l.. We may reorganise the local coframe for B
e
as {eO', w O'P ,w ij , oj, 8i 0" i 0',8} and we observe that a suitable coframe for leaf 1)
would be {eO/,w O'f3 ,w ii }. Thus the maximal integral manifolds of P/(S) are just
the Darboux bundles (V(C),1r) which are those subbundles of B whose cotangent
spaces are locally spanned by such coframes. Since integral manifolds of Pf(S) are
integral manifolds of S and hence S we identify the bundles (1'( C), 1r) as leaves
of the foliation associated ,vith system 4,2,4.3. We have already seen in section 1
c
that the equations Oi ˜ 0, (Ji O' ˜ 0 (pulled back to C = 1r(V(C» M enable us
to identify the pull backs of {e i } and {H i O'p } as the normal coframe and second
fundamental forms of C respectively.

eOt{3 ˜ o.
i
It is of interest to examine the pull back equation This equation may be
written in terms of a pull back of the exterior covariant derivative jj on sections of
A(T*C) as
ecry = '3(R OtI3"Y + R "YpOt)ef3 + DH O'-Y +w jH Ot"'(
-
iIi i i i j
4.19

Thus from 4.11,4.12,4.13




4.20

In terms of the curvature operator R.xy on M:

4.21
Hartley & Tucker: Extremal immersions and the extended frame bundle 219

Furthermore:
4.22


-l .
= eJ(Vx 4.23
V X X·I X·)X·
J
t
01
OI




in terms of the connection V induced on TC and the normal connection V -l on
(TC)-l. Thus
= e i(r7 XfJ X) - e i( v xI' X j )
j=
T7 -l
tf3W i j
· 4.24
v

and 4.20 becomes:



'.L .
. -l .
+el(V˜YJJXj)HJ a-r - 4.25
eZ(Vx-yXj)HJ af3
which is just the Codazzi equation for C C M.


5. The Variational Problem on the Extended Frame Bundle


In section 4 we established a set 'D of integral manifolds for the differential system
(8, {n Ol } ) on B. Leaves 1)(Ct) of its foliation contain the orthonormal frame bundle
for a p-dimensional submanifold C t C M together with the bundle of normal
frames. For any C t let
5.1

be any section of (V(Ct ),1r). Extend the map arbitrarily to give a local section

5.2

of B such that O'M restricts to u' on Ct - The section UM defines an orthonormal
coframe {O'A/ea} and connection forms {O'Mwa b} on M together with a set of func-
tions {O'MH i ol,8} which we have seen in 4 are the components of the shape tensor
of Ct C M. The map 0'1\/ also defines a lift Ct of C t to B:

5.3

it :D --+ B of the immersion It = Fit: D X {t} ˜ M;
= It(D),
and since C t a lift

it = (I'M It 5.4
0
Hartley & Tucker: Extremal immersions and the extended frame bundle
220

so that
5.5

Pon B
We may now lift our original variational problem onto B. For any p-form
such that
5.6

have
lfJ 1u"Mii
we
fcii == A(C,j) 5.7
A(C,f) = = =

We now seek to extremalise A(C, i) under variations that preserve the foliation V
and are subject to conditions on ac. Thus for variations that vanish on 80 we seek
an immersion
10 :D 5.8
--. B


1..
such that
-d 5.9
,8lt=o = 0
c
dt t


=
where Ct It(D) and the variations preserve the exterior differential system
(S,nO):
[-e, e]
\:It E 5.10

Then the solution (fo, Co = fo(D») to the original problem is obtained by simple
projection 1r : 'D(Co) --+ At.

We illustrate this procedure with

P= £(g(Trh,Trh»n 5.11

where on B
g(Tr h, Tr h) = r/irr,c:ra' 1JPf31 Hi no,Hi f3f3'

n=
corresponds to the square of the length of the trace of the shape tensor and
IIo:(eO'A) so O'uO = *1. According to 3.17 we must compute;

5.12

where
5.13
Hartley & Tucker: Extremal immersions and the extended frame bundle 221

= W Q - H OtI3C P
iii
5.14
()
Q




and the (p - I)-forms Ai and AiQ are to be determined. We have




5.16

= (-I)Q.,p for any q-form.,p.
where 1/'t/; Using equations 4.6 and 4.14 we have




"Normal"variations of B are maps between leaves of the foliation V generated by the
e
vector fields {Xi, XiO!, XiQ,s} on B, dual to the system {Oi, 8i Q, i 0,8}. Contracting
5.17 successively with these vectors yields:




i*{77iXj {1d˜ + efJ 1\ Ai + (6{wfJQ - 6pWii) /\ Aia} = 0 5.19

j*{77iXi(J({1dP+e,8/\AiQ}=0 5.20

These are the Euler-Lagrangian equations for any Lagrangian f3 on C. From the
structure equations it follows that

5.21

P
so for our choice 5.11 of
5.22

˜0
iXi {1dP 5.23

˜ 2£'Hi˜˜1JQf3n
iXi (J({1 d/3 5.24

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