Proc. Indian Acad. Sci. (Math Sci.), Vol. 105, No. 1, February 1995, pp. 23-29.
9 Printed in India.
Flat connections, geometric invariants and energy of harmonic
functions on compact Riemann surfaces
K GURUPRASAD
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
MS received 1 September 1993; revised 23 March 1994
Abstract. A geometric invariant is associated to the space of fiat connections on a G-bundle
over a compact Riemann surface and is related to the energy of harmonic functions.
Keyaords. Principal G-bundle; fiat connections; Chern-Simons forms; energy of maps;
harmonic maps.
Introduction
This work grew out of an attempt to generalize the construction of Chern-Simons
invariants. In this paper, we associate a geometric invariant to the space of flat
connection on a SU(2)-bundle on a compact Riemann surface and relate it to the
energy of harmonic functions on the surface.
Our set up is as follows. Let G = SU(2) and M be a compact Riemann surface and
E ~ M be the trivial G-bundle. (Any SU(2)-bundle over M is topologically trivial).
Let ~g be the space of all connections and ~- the subspace of all flat connections on
this G-bundle. We endow on ~gthe Frechet topology and the subspace topology on o~-.
Given a loop ~r:S1 ~ - , we can extend tr to the closed unit disc t~:D2 ~cg since
is contractible. On the trivial G-bundle E x D 2 ~ M x D 2 we define a tautological
connection form 8 ~ as follows
8~[~e,t) -- 6(t)V(e, t)~E x 0 2.
Clearly restriction of 8 ~ to the bundle E x {t} ~ M x {t} is ~(t)VteD 2. Let K(8") be
the curvature form of 8 ~. Evaluation of the second Chern polynomial on this curvature
form K(8 ~) gives a closed 4-form on M x D 2, which when integrated along D 2 yields
a 2-form on M. This 2-form is closed since dim M = 2 and thus defines an element
in H2(M,R),~ R. It is seen that this class is independent of the extension of a. We
thus have a m a p
Z:f~(~)--. H2(M, R) ,,~ R
where f l ( ~ ) is the loop-space of ~-.
We assume that the genus of M t> 2. The energy E(f) of any smooth function
f : M - ~ G is defined using the Poincare metric on M and the bi-invariant metric on
G = SU(2) given by the Killing form.
23
24
K Guruprasad
Any smooth function f : M ~ G defines a flat connection to: = f*(/~) on the trivial
bundle M x G ~ M, where/~ is the M a u r e r - C a r t a n form on G. By a result of Hitchin
([HI), the loop in ~ is given by
a:(t) = ~(to: + (cos t)to: + (sin t)(.to:)) for te[0, 2~],
where .:A~(M, fg)~A~(M,f~) is the Hodge star operator, is actually a loop in ~- if
and only if f is harmonic. ((g is the Lie-Algebra of G).
The main result of this paper is
Theorem l f f :M ~ G is a harmonic map, then
Z(ay)=-lE(f).
1. Construction of the basic geometric invariant
In this paper we suppose M is a compact Riemann surface of genus g t> 2, G = SU(2)
with Lie algebra f~ = su(2) and re: E ~ M is the trivial G-bundle on M. c~ is the space
of connections and ~- is the subspace of all fiat connections on E ~ M. D 2 is the
closed unit disc in R 2 and dD 2 = S 1 is the unit circle, f ~ ( ~ ) = M a p ( S l , ~ ) is the
loop-space of ~ . Given a loop tr:S ~~ ~ we extend a to ti:D2 ~ ~ (c~ is contractible).
On the trivial bundle E x D 2 ~ M x D 2, let ~ be the tautological connection defined
in the introduction. Let K(0 ~) be the curvature 2-form of the connection 0". Let C2
be the second Chern polynomial on f~. For the Lie algebra f9 = su(2), C2 is essentially
the determinant. More particularly C2(A)= -(1/4rc2)det(A) for A~su(2) (cf. [KN],
Chap. XII). Now an easy computation shows that
C2(A ) = 1 . t r a c e ( A 2 ) for A~f~.
87zz
Evaluation of C 2 on K(8 ~) gives a closed 4-form C2(K(8")) on E x D 2 which projects
to a closed 4-form C2(K(~r on M x D 2. Integrating C2(K(9~ along D z yields a
closed 2-form on M(dim M = 2) and thus defines a cohomology class in H2(M, R) i.e.
We outline the proof of the following lemma (cf. [G], w1 and [GS], w2, 3).
Lemma 1.1. SD2C2(K(~)~)) is independent of the extension of a:S 1 __,cg to 0:D2:-*c~.
Proof Let ~ , a2 be two extensions of tr with corresponding connection forms ~ , 8~
and curvature forms K(8~),K(8~2) on the bundle E x D 2 ~ M x D 2. On E x D 2 we
have
dTC2(~)~) = C2(K(~) )
d T C 2 ( ~ ) - - Cz(K(~)~) )
Flat connections, #eometric invariants...
25
where TC2(~), TC2(~)~) are the Chem-Simons secondary forms with respect to
8 I, 8~ respectively (cf. [CS, 3]). We can easily check that C2(K(8~)) - C2(K(8~)) is an
exact form on E (cf. [G,I]). Since n*:H2(M,R)~H2(E,R) is an isomorphism it
follows that {C2 (K (9 I) )} = {C 2(K (~))} e H 2(M, R) and this proves the lemma.
We thus have a map
l)(o~-) ~ H2(M, R) ~ R
where f~(~) is the loop-space of ~-. It is easy to check that z(troa')= X(a)+ X(a')
where aotr' is the composite of two loops in o~'. We call this map X the geometric
invariant.
2. Energy of functions and a class of special loops
We recall the definition of energy of a function. Let X and Y be Riemannian manifolds.
Given a smooth map f : X ~ Y, the energy density of f is a function e(f):X--*R
defined by
e(f)(x) = Ildf(x)II ~
where Ildf(x)II denotes the Hilbert-Schmidt norm of the differential df(x)e T*(x)|
TI(,~ (Y). If X is compact and oriented, the energy off, denoted by E(f) is given by
E(f)=(fue(f)(x)dx) 1/2
where dx is the volume form of X with respect to its Riemannian metric, f is harmonic
if it is a critical point of the energy functional.
Using the Poincare metric on the compact Riemann surface of genus i> 2 and the
bi-invariant metric on G = SU(2) given by the Killing form, we can define the energy
E(f) of a smooth functionf:M--,G by the above formula.
Any smooth function f:M-~ G defines a fiat connection o91 = f*(g) on the trivial
bundle E ~ M where
/t=(
iPl
#2 q- i#3 ~
--/~2 + i/~3
-- iPl /
is the Maurer-Cartan form on G. In the case of the trivial bundle E--, M, clearly the
space of all connections c~ can be.identified with the space AI(M, c~) of all q-valued
1-forms on M. For any smooth function f:M--*G, consider the loop in ~ given
by as(t ) = 89 s + (cos t)~os + (sin t)(,o9 s)) for t e [0, 2~], where ,:A i (M, ~) ~ A 1(M, (~)
is the Hodge star operator. By a result of Hitchin (I-HI), we know that as([0, 2n]) c ~ilffis harmonic, i.e. a s is a loop in ~ ifffis harmonic.
26
K Guruprasad
3. Relation between the geometric invariant and the energy of harmonic maps
We prove the following result
Theorem 3.1.
If f : M ~ G is a harmonic map, then Z(af)= - •
1
4n
Proof. At the outset we show that the closed 2-form which represents Z(aI)~H2(M, R)
1
is -1r2
( * c ~ 1Am 1 + *02 A 0) 2 -Jr-*0) 3 A 0)3) where
ter = f ' P = (
i0)l
-- 0)2 + i0)3
te2 + i0)3).
-- i0)1
We extend the loop af in o~ to a m a p ~f:D2---~(~ in an obvious way. We drop the
suffix f and simply use a and 8 in the computations that follow.
Let (s, t) be the polar coordinates on D E = {(s, t), 0 ~ s ~< 1, 0 ~< t ~< 27r}.
Set t~(s, t) = sa(t). We now compute the curvature K ( 8 ~) of the connection form 11"
on the bundle E x D2-~ M x D 2.
K(O ~) = dO~ + g1 [ 8 ,a8 " ] ,
= dl) ~ + 8 ~ A 8 ~,
= deS" + do~8 ~ + 8 ~ A ~,
= daD;+ K(e(s, t)),
where K(cT(s, t) is the curvature of cT(s,t)) and dE and dD2 are respectively the exterior
differentials on E and D 2.
If we set
a(t)
=
_
is(t)
fl(t) + iT(t)~
fl(t) iT(l)
-- ict(t) ,1
+
as a form on M for each teS ~, then after a straightforward calculation (see [G],
L e m m a 4.1), it follows that ~D~C2(K(8~)) is cohomologous to the form
41zr2f s, ( dct(t) A ~t(t) + d fl(t) A fl(t) + dy(t) A y(t) )dt
Now
co=f'p=(
ico,
te2 + ite3)
-- ~2 + i023
-- itel
so that
,7(t)
=
i(0)1 + c o s t t e l + s i n t , 0 ) x )
(O92 q- COS tO)2 d- sin t.0)2) q/(te a + cos to) a + sin t , t e a ) ]
- (022 + cos ttez + sin t . 0)2) +
i(02a + cos t0)3 + sin t,0)3)
-- i(te 1 + cos t0)1 + sin t.0)2) I
)
Flat connections, geometric invariants...
27
i.e.
~t(t) = (to 1 + cos t09 1 + sin t , 091 )
fl(t) = (092 + COS t09 2 + sin t.092)
~(t) = (093 + cos t093 + sin t,093)
Now
d
~ at(t)A~t(t) = ( ( - sin 0091 + cos t*091)A(091 + cos t,091 + sin t,091)
= - sin2t091A,091 + cos 2 t,09~A091
= ,091 A091.
Similarly
d f l ( t ) A fl(t)=,092 A 092
d
__~(t) A ~(t) = *093 A 093"
(it
It follows that
1
47t 2
fD2C2(K(O*)) is c o h o m o l o g o u s
fs t (.O91 A 091 +
to the form
*092 A 092 + *093 A 093)dt
I
----- - ('091 A 091 "31-*092 A 092 "[" *093 A 093).
2n
Thus the closed 2-form on M representing
X(ay)eH2(M,R)
1
is ~ ( ' 0 9 1 A091 +
*09")2 A 092 -]- *09-)3 A (,03).
T o prove that X(a:) = - ~ E(f), we check using local coordinates that the forms
47r
l ( d~t(t) A ~(t) + d fl(t) A fl(t) + d~(t) A ~(t))
!
and
- --~e(f)(m)dm (dm is the volume form on M) are equal
at any arbitrary point.
Since any left translation in G is an isometry, for any m~M, Ildf(m)ll =
IId(L:~,, r , of)(m)II where L:o,)_l :G--, G is left translation by f(m)-1. We can therefore
assume that f m a p s some point m~M to the identity element in G, i.e. f(m)= 1.
Since we intend to use local coordinates to p r o v e the equality of forms, we can go
to the universal cover D 2 of M with Poincare metric and assume f:D 2---,G and
f(m) = 1 for some fixed meD 2. Since there exist an isometry of D 2 which m a p s the
origin to m, we can assume f(0) = 1 and check equality of forms at the origin.
At the origin we have
axax/=
l=(a,
28
K Guruprasad
and
~/=0
O O
where O-x'Oyyare the usual coordinate vector fields. Let dx and dy be the dual 1-forms.
Clearly at the origin ,dx = dy and ,dy = - dx. Since d m = dx A dy we have
e(f)(m)dm ( f---~, ~--~) = e(f)(m).
We prove that
1(*031A COl + *(.02 A 072 + *r
A o)3)(~x,~y ) -
-le(f)(m).
If cos = asdx + bsdy (1 ~<j~< 3, aj, bj are functions on D 2) then ,w~ =
for 1 ~<j <~ 3 so that ,cos A 0~ = - (a2 + b2)dx A dy for 1 ~<j ~<3
ajdy - bjdx
x 2 A dy
=> ~ , < ~ , ^~1 +,<,,~ ^ <02+,<% ^ <,,,~= - ~(a~, +b~, +<,~ +b:, + a32+ b3)d
F o r f : D 2 -->SU(2) with f(O)= 1
lidf(O)il2= df(O)(o_~)2+
=
df(O)(~y) 2
Of(0) 2 I~_._y
O) 2
~
+
By definition of Maurer-Cartan form
if(o)="~--L-~
( ~i (o)'~
] =
"'t,--L-~)
\ o,,/+~"~t,-77-~)|
The pairing (A, B)v-+trace (AB) for A, B ~su(2) gives the Killing form on su (2) so that
Of(O)
-~x
2=trace(Of(O)Of(O)~\
Ox Ox ]
Similarly
Of(0)
~ Ildf(O)ll= = 2
J
.,
,,.<o,,,=}..__.
+/~lA-r--//
.
Flat connections, geometric invariants...
29
Noting that f * p j = col(1 ~<j ~<3) we have
).
Now
coJ(~x)=(ajdx+bidy)(f---~)=aJ
a
9
Therefore
J\--~--x] = aj (l~ j
~ 3).
Similarly
= bj
\ ay ,/
(1
3).
Thus
Ildf(0) [l2 = 2{a~z + b~ + azz + bzz + a~ + b~} =
e(f)(m).
Therefore we have
( - - (*(D 1 A O)1 "[- *(.02 A (,02 .-~ ,033 A (.,03)) = ~ e ( f ) ( m ) d x A dy.
In other words
(~-~"~(*~1
A(O1 + *O72 A ~ 2 "k *~3 A ~ 3 ) ) = - l e ( f ) ( m ) d m
ConsequentlyX(o'f)=-1E(f)
and the theorem follows.
Acknowledgement
it is a pleasure to thank A R Aithal, I Biswas and N Hitchin for helpful discussions.
References
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(1974)
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[GSI Guruprasad K and Shrawan Kumar, A new geometric invariant associated to the space of flat
connections. Compos. Math. 73 199-222 (1990)
[HI Hitchin N J, Harmonic maps from 2-torus to the 3-sphere. 2. Differ. Geom. 31, 627-710 (1990)
[KN] Kobayashi S and Nomizu K, Foundations of Differential Geometry, VoL II Interscience Publications,
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