DETERMINANTS OF LAPLACIANS
by
Marek Kierlanczyk
M.S. Warsaw University
(1979)
Submitted to the Department of Mathematics
in Partial Fulfillment of the
Requirements for the Degree of
DOCTOR OF PHILOSOPHY IN MATHEMATICS
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Marek Kierlanczyk
1986
The author hereby grants to M.I.T. permission to reproduce and to
distribute copies of this thesis document in whole or in part.
SIGNATURE OF AUTHOR:
i
Department of Mathematics
June 10, 1986
CERTIFIED BY
Isadore M. Singer
Thesis Supervisor
ACCEPTED BY:
A
U
Nesmith C. Ankeny
Chairman, Departmental Graduate Committee
MA SSACHUSE TT S NSTITUTc
OF TECHNOLOGY
.FP 96
1986
IRARI 3
ARCHIVES
-2-
DETERMINANTS OF LAPLACIANS
by
Marek Kierlanczyk
Submitted to the Department of Mathematics on June 10, 1986,
in partial fulfillment of the Requirements for the
Degree of Doctor of Philosophy in Mathematics,
ABSTRACT
This thesis deals with determinants of Laplacian and the square of
Dirac operators.
For tori they are computed in terms of theta
functions, and for compact Riemann surfaces of genus at least two in
terms of Selberg zeta functions.
are presented.
Applications to conformal geometry
Finally, we discuss some obstructions to the existence
of parallel spinors and symplectic structures.
Thesis Supervisor: Isadore M. Singer
Title: John D. MacArthur Professor of Mathematics
-3-
TABLE OF CONTENTS
Page
INTRODUCTION ...................
5
CHAPTER 1
DETERMINANTS OF LAPLACIAN AND DIRAC
ON THE TORUS......................................... 7
CHAPTER 2
DETERMINANTS OF LAPLACIAN AND DIRAC
ON THE RIEMANN SURFACE............................
.23
CHAPTER 3
SOME ASPECTS OF CONFORMAL GEOMETRY ...............
31
CHAPTER 4
PARALLEL SPINORS AND SYMPLECTIC STRUCTURES........
40
-4-
ACKNOWLEDGEMENTS
I wish to warmly thank my thesis advisor, Professor Isadore M.
Singer, for his teaching, guidance,
and encouragement.
Thanks also go to the Mathematics Department of the Massachusetts
Institute of Technology and the University of California at Berkeley.
Finally, I would like to thank my wife, Ki, for her moral support
through the turbulent times of my studies.
-5-
INTRODUCTION
Determinants of elliptic operators, especially of
a
and Dirac
'
operators on compact Riemann surfaces, have attracted wide attention
among mathematicians and physicists.
The increasing evidence that string theories can provide models
that unify gravitation and matter into a finite quantum theory
generated a lot of interesting mathematical questions.
In this
thesis, we will give the answer to a few of them.
In Chapter 1 we derive formulas for determinants of Laplacian and
2 on two-dimensional tori in terms of theta functions.
General-
izations to the n-dimensional case are discussed.
In Chapter 2 we derive the formulas for the determinant of the
Laplacian on compact Riemann surfaces in terms of the Selberg zeta
function and for det( 2 + a) as well.
In Chapter 3 we analyze a new conformal invariant discovered
recently by Parker-Rosenberg.
is not a conformal invariant.
We also show that the determinant of
2
-6-
In Chapter 4 we prove that there are no parallel spinors on
Riemann surfaces of genus greater than one.
We also analyze some
consequences of Thurston's conjecture about the existence of a
symplectic structure on almost complex manifolds with a nonnilpotent
second cohomology class.
-7-
CHAPTER 1
DETERMINANTS OF LAPLACIAN AND DIRAC
ON THE TORUS
Let W be a compact complex analytic manifold without boundary,
complex dimension N.
complex (p,q)
If D 'q(W,L)
of
denotes the space of C
forms on W with values in a flat
holomorphic vector
bundle L, then the exterior differential d splits:
d = d' + d",
where
Dp+1,q
d':D 'q
+
d":DP'
+- D Pq+
Since the notation 3 is often used for the operator which we write
as d",
the system of complexes which it
usually called the a-complex.
p,
Let Z L
We are using the notation of [1].
denote the kernel of d" in D '9(WL).
cohomology groups H '9(WL)
,
defines for each choice of p is
where
0
= Z
'q/d"Dp 'q
The Dolbeault
(WL) are isomorphic to
is the sheaf of germs of forms of type (p,O) with
holomorphic coefficients.
-8-
X be a finite dimensional unitary representation of the
Now let
fundamental group 7 (W),
bundle.
Let Dp'q(W,X)
and let L(X) be the associated complex vector
= D 'q(WL(X)).
If W is the simply connected
covering manifold of W, with 7 l acting as deck transformations, a form
f in D(W,X) may be identified with a vector valued form f on W which
satisfies foy
X(Y)f, yeff .
Suppose W has a Hermitian metric.
Since X is unitary, the
associated duality operator * satisfies
*:Dp'q(WX
*D-q,N-p(WX),
and determines the inner product (f,g) in D P).
Since the vector
bundle L(X) is flat, the exterior differential d has the formal
adjoint
-*d*
=
'+
V"
with
6'
= -
'
6" = -d*
Let A
+DP~lp
+
D
q
be the corresponding Laplacian
pq
A
p,q
-q
=-(6@@dl
+ d"":
-9-
Let HP)9(W,X) be the space of harmonic forms in
that is, satisfying Af = 0.
(X),
The Hodge theorem states each harmonic
form is closed (and coclosed), and that the resulting inclusion map of
H pq(WX) into the Dolbeault group HP q(WL) is an isomorphism onto.
When the metric is KMhler,
w =
is
closed,
2
h
that is, when the 2-form
dzi A dzk
jk
then Apq = -(1/2)(6d
+ d6),
and the Dolbeault groups may
be identified with subgroups of the cohomology of W with coefficients in
L(X).
In this case, one has A
-
3 2
A
where A is the Hodge Laplacian
dd* + d*d.
A
generates a semi-group of compact operators exp(tA
As t
on the Hilbert space completion of D p,(W,X).
operators approach the projections P
HPq (W,X).
of D '9(W,X)
+
+ c,
),
t > 0,
these
onto the sub-space
The zeta function associated with the Laplacian A
is
defined by
tA
p~ (S,X)
=
(S)
0 t itr(e
q
-
Pq
)dt
= Z X0(-x )-s
X <0
n
n
for Re s large, the sum running over the non-zero eigenvalues Xn of
A p~q.
(s,X) extends to a meromorphic function in the
s-plane, which is analytic at s = 0.
dtr (A-s)
ds
d
ds
-s
j
Here
-s
is
-10-
Formally,
at s
=
0,
this would be Z log X=
log det A =
log det A.
Hence, we define
(S)
s=0
Consider the 2-dimensional torus, T = C/F, where r is the
lattice generated by 1 and T, ImT > 0.
Laplacian A5.
p
= q =
Consider the corresponding
We assume that our representation X is trivial and
0.
Theorem 1.1
det A7 = (IMT)2 1 rI(t) 4
where
II
li
2 TrnT
(1 - e2Wn
=RelT
n=1
Proof
The eigenvalues of A on D*(W,l) are
in,n
42
(I
2
2
I-~ -
n)
Applying the Poisson summation formula we obtain
-lmit + nI2
4tme
tr(et
m,n
-11-
Now we have
CO
c(s) =
1
f ts-
-tA
1tr(e- )dt
0
_
InT
1
+
1f
IMT
4rF(s)
F(s) 47(s-1)
-I+n12
4t
s-2
e
dt
0
00
+
where E
=
fts-i (tr et)dt
sum over all (m,n) 0 (0,0).
When Re s < 0,
we apply the expression for tr(e-t
) in the
last term on the right to obtain
-
lInt
-mt+n1 2
4t
ts-2 e
'
IT
=
4TrF(s)
dt
=
0
F(1-s)
4U
F
'
j
s)
4
ImT + ni
1-s
2*
Consider
E(T,u)
(- IMT
ImT+nI
u , Re u > 1
This function can be continued analytically into a function of u
regular for Re u > 1/2 except for a simple pole at u=1 with the
residue
T.
E(Tu)
At u=1 it has an expansion:
=
+ 2R(Y - log2 - log((ImT)
|T1(T)1 )) + a (u-1) +
It is called the first limit formula of Kronecker.
-12-
Rewriting C(s) further we have
=
C (S)
lInT
IT
(I
41
F(l-s)
r-
1-s
E (T,1-s)
(ImT)
lint Si1
= ( )
[S.-2F'(l)s
2
[s-2F1(1)s+...]
=( -)
1
[-
+..]
[-
2.
+ 2Tr(y-log(2(Imt) z
2
(t)I ))+...]
+ 2 (Y-log(2(Imt) 2
2
1
=
[1 + 2log(2(ImT)2 If(T) 2) s + ... ]
-(n)s
as F'(l)
=
-y
Therefore
1
V(0)
=-[log(
2
+ 2log(2(ImT)
I)
2
and finally
det A-
= exp(-C'(0)) = (Im2T)2 1(T)14
which ends the proof.
Observe that the obtained expression is not SL(2,Z) invariant;
that is
not surprising as the eigenvalues X
do not have this
property also.
Notice that all of the essential ingredients: the Poisson
summation formula,
the Kronecker limit formula,
and the explicit
formula for eigenvalues of the Laplacian have n-dimensional
analogues.
Although we are considering the Hodge Laplacian dd* + d*d
acting on smooth functions, our result extends to forms.
The identity
-13-
I....d x
A (f dx
)
dx
(A f)dx
0
p
1
P 11
i
1
shows that the p-forms spectrum is the same as the function spectrum, each
repeated (n) times.
p
determines det A
Therefore the det A
completely.
p
0
Now we will gather some information which will be useful in deriving
a formula for the determinant of Laplacian on an n-dimensional torus
T = Rn/L,
L = AZn,
The metric structure of Rn
A C GL(n;R).
projects to T such that vol(T) = Idet Al.
The set L = {a C Rn
n
t
V a E L} is the dual lattice of L; L = (A-i) Zn.
2
~2
are 42 Hal
a a C Z,
The eigenvalues of T
for a arbitrary in L where IIIIis the Euclidean norm.
The Poisson summation formula states
-a
trke
-tA
)L
det A
n
2
(41 t)
e
aEZ
tA)a
4t
n
so ?(s) reads
CO
detA
n
S)=
(47)2
det A
n 7
TT
4
r()
f
acZ-
0
s- n_
t
2
-attA)a
4t
e
0
F(-n - s)
2
(S)
n-s
(at (AtA)a)
-
dt
-14-
Let S be a n x n matrix of a positive definite real quadratic form
and let p be a complex variable with Re p > n/2.
Then Epstein's
zeta function is defined by
Z (S,p)=!
n
2
I
,
(ta Sa)
aeZn-O
where the sum is over all column vectors with integral coordinates,
not all
of which are zero.
Define
n
k (S)
n
n n
lim
m.
{Z (SP)
n
2
I'( )IdetSI
2
1
12 ~
-
n
2
2
This is just the constant term in the Laurent expansion of the Epstein
zeta function at p = n/2.
We will introduce some notation following A. Terras [Z]*
n = n, + n 2 , with 1 < ni < n-l.
(n)
S(n)
[T(nP)
0
0
S'2
12
1
1 and T
I
n2)
(n2 _
t
Let n
Then S can be represented as
t
22]
-01
2
Let
-
t.
*) I am grateful to P. Sarnak for this reference.
L_
1
-15-
=t
Define
z{b} =
1
1
bQ + i t
2
(S
-l
[b])
2
,
n
Y - log 2 -
for n even,
r=1
c
n
=
<
1
-(n-3)
,
(2r+l)
y -E
for n odd
r=0
Then
k(S)
= Zn-_1
2p
n
2
+
Idet S 2
(-)
2
2
1
X{c n -
log(t2
II
(1 - exp{27iz{b}})I 2
bEZn-1-0,b(mod+l)
This is an n-dimensional generalization of the Kronecker first
limit formula.
forts
In the 2-dimensional case
s =
s
s12
s2
-q
]
2
2
1
q
t
0
0
1
0
s2
1 2
12
>
1
q
,andqs
/s 2
12
'
-16-
Then
1
lim {Z2 (S,p) -
r(det 5)
p-i
p+1
TT
=
1 {X
-
log 2 - log(t
2
TI(z)I
2
)}
(det S)2
1
for z = q + i t
11
22
s2
=
-1
s2
(S12 + i(det S?
To obtain the expression for the determinant of the Laplacian on
the n-dimensional torus, one uses Terras'
functional equation [2] for
the Epstein zeta function:
1
-.
Pp)Z
p
2
- I(P)Z (S P) = (detS) 2i
1
~-2n
F(
1
-p)
Z (S
_-11
)
-P)
together with the generalizations of Kronecker's limit formula
included on the previous page.
-17-
We assume the reader knows about the Dirac operator so we discuss
it only briefly.
Let X be a compact,
orthonormal frames E is
oriented riemannian manifold.
a principal SO-bundle.
The bundle of
Suppose E lifts
to
give a principal Spin-bundle E; then X is a spin manifold and we can
define via the spin representation a vector bundle V = E x S, the
E lifts
bundle of spinors.
differ by a Z 2 1-cocycle,
to E iff w2 (x)
= 0 and any two liftings
so the number of inequivalent liftings
(the number of spin structures) is # H (X,Z2).
V splits into V+
: C (V)
-
V
C (V).
In even dimensions,
so one can define the desired Dirac operator
In particular we get a decomposition of harmonic
and negative H_ harmonic
spinors H into the space of positive H
spinors.
Let X be a complex manifold.
It turns out that X is a spin manifold
when the canonical line bundle K has a holomorphic square root (W
2
c1 ()mod
2).
Furthermore,
the set of spin structures on X is
correspondence with the [L:L
2
=
K,L a holomorphic line bundle]. Choosing
a spin structure means choosing an L,
same as
:C (AOeven d L)
+
in 1-1
so
jf:V
C (AO,odd 3 L).
manifold we have H+ ~ Heven(X,(L)),
H~
+ V
turns out to be the
For a compact KAhler
Hodd(X,0(L)).
-18-
Consider now a 1-dimensional torus T and the square of the Dirac
operator on T.
There are four of them,
spin structures.
The eigenvalues of these operators are
-41T2
m,n
1
m +
4I2 2
where (e1 ,E2 ),
corresponding to the underlying
1
2
2 - t(n + f cE) 2
i = 0,1 classifies the spin structures on
T (which can be thought as Z2 characters of the fundamental group).
See Friedrich [3] where the case of Tn is also to be found.
A straightforward analysis,
using Epstein's functional equation,
leads to a formula for the determinant which amounts to the
Kronecker's second limit formula.
Theorem 1.4
1
det 2
det
where (EE2)
1
1 2
e4 1
2
ak
2
-
1
=Ie2
(0,0) and 01 is the theta function
2ni(IkIT - akw)
Tri(w + 1) C
(w-n(T)
1
6
e
1
sign (k + )
k
2
-
(l-e
2
Ik + 11
2
-19-
Proof
We will follow closely Ray-Singer [1].
Consider first ?(s)
' (s) - the zeta function for the Laplacian on the space
D0 (W,X) of C
sections of L(X).
It is easy to see that the eigenvalues of W on D*(W,X) are
m,
=
m,n
-
42
(ImT)2
1 u+m - T(v+n)2
and the eigenfunctions corresponding to
,n are
2Vi
Pmn(z)
=
exp
Im(z(u + m - T(v + n))}.
We have, of course,
x
tr(e
=
)
l
t
e m,n
We can also write the heat kernel explicitly as
Pt(z,z') =
MnX(mT + n)pt (z,z' + mT + n),
where
p (zz')
1
t
t4
-Iz-z'1 2/4t
e
is the euclidean heat kernel on C.
tr(e t)
=
IMT
Z
M,n 4nt
From this we obtain
-|mt+n 2/4te 2Ti(imu+nv)
i.e., the Poisson summation formula again.
=
-20-
The above yields the analytic continuation of the zeta function.
For Re s large we can write
c(s)
=
Zm,n (-X m,n )-s
00 s-1 tr(e tA)dt
1ft
1
(s)
IMT
+IMT
4n(s-1)
4ur s
+
s
2
2
in+n >0
f
2Tri(mu+nv)
1s-2
f t0
e
0
2-m+n12/t
etdt
ts- tr(e tA)dt.
r~)1
The right side defines a meromorphic function in the s-plane, which is
the desired continuation; note that
vanishes at s
=
0.
When Re s < 0, we can insert the second expression for tr(e
) in
the last term on the right to obtain
2
_
2Tri(mu+nv) f
_4IMT
411F(s)
IMT
40
2
2
m +n >0
F(l-s)
F(s)
ts-2e-Imt+n1 /4tdt
0
27i(mu+nv)(
m2
2>0
4
1-s
Imt + n 12
Since the series does not converge absolutely when s
explain how to evaluate
'(0).
0, let us
We will assume v ' 0 (mod 1);
if v = 0 we would have u 0 0 (mod 1), since the character is not
trivial, and the interchange of u and -v is the same as replacing T
by -1/T.
Then following Siegel, one can establish the uniform
convergence by summing first
over n.
Write
-21-
r(-s)
c(s) =IT
+Z
2lTinv
n
e2Timu zo
e2
n=-M
M-YO
4 )1-s
n
4
2rinv(
1S
lMT + ni
Introduce the Dirichlet kernel
A
n
and set bn
= erinv sin TI(n+1)v
sin Tv
linT + n1
2s-2
Then
1
e 2minv
zo
n=--we
I
-
-n(A
-n
lmT+n2-2s
= z'
<
A)bl
n-inb~
.co An (bn+l - bn )
1
Z'
= sin rv
-O
1
<
= sinnv
b
n+l
-bi
n
2
(mI)2-2s*
We see that if v 0 0 (mod 1), the series for C(s) converges
uniformly for Re s < 1/2, and we may evaluate C'(0) by this method
of summation as:
=2
Im T
'Tr
n1
n
2
co2Tnv
(*)
+ ---
1
e
2TrM
2
imu
c
=-o
2ninv
2lmImTl
2
emT + n
The first series above is standard, and the first term on the
right side of (*) is given by
TrImT(2v
2
-2v
1
+),
0 < v <.
-22-
To compute the second series, define
e-2n(imlImTlx+kl+imReT(x+k))
F(x) = Z
k=-00
F is periodic and Lipschitz; hence its Fourier series converges at
each point to the value of F.
The Fourier coefficients are given by
f1e-2 inxF(x)dx = f
0
e-2ff(lmImTlxl+i(mReT+n)x)dx
ImIImT
1
|ImT + n
2
So the second term on the right in (*) becomes
1
M O
= -COC
2nimu
Im,
log
o
-27(ImIImTlv+kl+imReT(v+k))
k=--W
- e2Tri(
Ik1T-Ck(u-Tv)) 1 2
The proof is completed by noting that eigenvalues for the Dirac
operator on T with underlying spin structure (e
the special twisted case.
Namely,
,
2)
are identical to
instead of considering a general
non-trivial character given by X(mT+n) = e2-i(um+vn) 0 < u,v < 1,
take u
Remark:
1
e2
1
2 el.
It gives the formula of
Theorem 1.4.
There is no analogue of the second Kronecker limit formula in
n-dimension so the extension of the Theorem 1.2 for h2 seems to be
difficult (see the footnote on page 39).
-23-
CHAPTER 2
DETERMINANTS OF LAPLACIAN FOR COMPACT RIEMANN SURFACES
We remind the reader that every compact Riemann surface M of genus
at least two can be realized as H/F where r is a discrete subgroup
of SL2(R) (more precisely in SL2 (R)/+I) defined up to a conjugacy
and acting on H without fixed points.
There exists a 6g - 6 dimensional family of nonisomorphic Riemann
surfaces of genus g.
In terms of a factor representation H/T
there
is a 6g - 6 dimensional family of nonconjugate, discrete subgroups of
SL 2 (R), each of them is isomorphic to the fundamental group of the
Riemann surface of genus g.
For compact Rieman surface H/P each element of r is of the form
z + e 2P
z
, with real fixed points zo,z
an element of F primitive if
it
and p = p
is not a power in F.
union of disjoint conjugacy classes.
PY
=
> 0.
r is the
P , if Y,Y'
are conjugate and simultaneously primitive or not primitive.
We call
-24-
Selberg introduced the zeta function:
-2p Y(6+k)
Z (6,x) = H
H
{y} k=0
)
det (1-X(y) e
which is well defined by the above product for Re 6 > 1.
It can be
extended to an entire function with zeros at the non-positive integers
and also at the points 1 + i
2
Laplacian.
-
-X-
4
for each eigenvalue X of the
Z satisfies the following functional equation:
6-1/2
Zp (1-6 ,X) = Zr(6,X) exp(-4n(g-1)
f
0
r tanh
Let F(T), F(T') be discrete subgroups of SL(2,R)
H/F(T) and H/F(T') have the same genus g > 1.
Tr
dr)
such that
Consider the
standard -1 curvature metric on those Riemann surfaces induced from
H.
Let X : Fr(T) +* U(l), X' : r(T')
nontrivial representations.
S)
f
+
U(l) be
Following [1], consider
t
s-1
e
tA(T)
dt
if X is nontrivial;
0
and
(s
F(S)
Remember that Co,0 = C0,1.
ts-1 (tr etA(T) -l)dt,
if X is trivial.
0
Notice then in both cases
00?
(S)Th c
)
r
rAeIdr
=
f
S 1 (tr
e A(T) -tr
e tA(v ))dt
0 ts
The Selberg trace formula applied to the heat kernel on D 0 (H/]F(tE),X)
states
-25-
tr(e
(
t0
)
=
(2g-2) f
+ r2)t
dr +
r tanh (Trr) e
0
2
(*)
t
1
Y
sinh(kp Y)
tr(Xk
I
+ I
{Y} k=1
k
4 t
2
P
-
where the sum is taken over conjugacy classes of primitive elements of
F.
Notice that the first term on the right-hand side of the trace
formula is the same for H/F(T) and H/F(t') because our Riemann
surfaces have the same genus (or equivalently, via the Gauss-Bonnet
theorem, they have the same volume).
Let's call the second term on the right-hand side of (*)
(t).
Fx
We have IF
I < Ce -at-(b/t) with a,b > 0 and
Is) f
=
-AV,(S)
CO (S)
(t)
ts-1(F .C(t)
dt
0()
- F
r~)0
When Re s < 1, we can write
CO
ts-1t
=
___0
(6(6-1))-se-6 (6-l)t(26-1)d6
1
Using this and interchanging integrations
CO
(S
(6)-G
(6(6-1))-s(G
f1XT
1'(s)'(l-s)
)
where
G)
Gx
( 6 ) = Xj~(~d
(26-1)
6 ( 6 l1)t F
fe
0~-
(t,d
=
dlo
log-rd
r
MX
(6))d6
-26-
So we conclude
det A(T)
det A(T)
that is, Zet (,X)
-
P(-)
lX)
is independent of the complex structure and
det AX(T) =
Constant
ZF(T)(1,X)
The same technique also applies if both representations X, X'
are trivial.
Then the Selberg zeta function has a simple root at 1 as
opposed to the twisted case.
detLA(T)
= constant *
Therefore,
d
F Z
161
|()
-
Our constants are universal for all Riemann surfaces of a given
genus g. Recently, one of them has been expressed by D'Hoker-Phong in
terms of the genus, Euler constant, fr and the derivative of the
Riemann zeta function at -1. See [4].
We remark that the above formulas look almost identical for
Laplacian acting on 1 and 2-forms.
The point is that for any
eigenvalue for Laplacian on 1-forms, one can find, by the Hodge
theorem, an eigenform of the form df,*df, where f is an eigenfunction
corresponding to X. So the spectrum for 1-forms is twice the
spectrum on functions, and the spectrum for 2-forms is just identical
to the latter.
-27-
Finally, one should remark that the formulas for the determinant
of the Laplacian (acting on functions) can be almost automatically
extended from the two-dimensional hyperbolic manifolds to the compact
space forms of symmetric spaces of rank one.
All the necessary tools,
the Selberg trace formula and the Selberg zeta function, have been
established for such spaces by Gangolli [5] and really nothing else is
needed.
Now we will discuss some recent results of D'Hoker-Phong [6] on
the determinant of the Dirac operator on Riemann surfaces.
Let Tn
denote the underlying space of a complex line bundle {f(z)dzn} for a
compact Riemann surface M with a fixed hermitian metric of constant
curvature -1.
If we fix a spin structure among the
2 2g
possible ones,
we may also consider n = (odd integer)/2, and view T1/2 as the space of
spinors and Tn as the spaces of spinor-tensor fields.
derivative V sends Tn into Tn e (T1
accordingly as
Vz
n
Tn
V
Tn 0
+
=V
n +
Vz,
n
z
with
'iT), and it
V": TX'+
z
(that is V = 3 9
T1 T n-
The covariant
can be decomposed
Tne TTf
+
on sections of Tn).
The
spaces of L2-sections of Tare Hilbert spaces, since spinor-tensor
fields can be paired at each point of M, and then integrated over M
using ds2.
With this pairing, (V1)+
= -
V
VZ is the Dirac operator,
and the natural covariant Laplacians on T7" are
ZV 'M
A+ = -V
n
n
Z
-
n
z
n
z
n
-28-
In local isothermal coordinates z, we can write
ds
2
= 2g -
dz dz,
n
zz
Vzf = gZZ
n
zz
z
f dzdz g - (g
)nf*g.
zz
zz
M
<fig>
Tn
Our M = H/F, F is a discrete subgroup of SL(2,R)/{+1},
all of whose elements are hyperbolic.
SL(2,R)
Let P be the subgroup of
containing -I which projects to P, and let Yi,. .,Y2g
fixed set of generators for P.
be a
A spin structure v on M corresponds to a
choice of multipliers v(Y) E {+l} on y E r which is multiplicative and
satisfies v(-I) = -1.
Such a choice is determined by the values of v on the generators
Y
and there are 22g of them.
D'Hoker-Phong [61 developed trace
formulas for such operators which yield
Theorem 2.1
For arbitrary half-integer n
-t(A
Tr (e
-
n
n(n+l))
-
nn
n(= It)
e
+ In(t)
-29-
where
p2t 2
t
In(t)
I V(Y)
= I
Y primitive p=l
2
1
n
4(
4(rt)2
2
s
4t
e
and
In W
e
(2n-2m-1) e (n-m)(n-m-l)t
= -2(2-2g)
O<m<n- 1
-t/
-b 2/4t
Co
4
- 47(2-2g)
2
(47t3/
f db
0
b e
ch (n-[n])b
sh b/2
They also introduced the two Selberg zeta functions
Zn(s) =
H
Y prim.
(1-v(X) 2n e -(p+s)l)
H
p=0
for n=0,
Let N- denote the number of zero modes of the Laplacian
n
A±-.
n
Note that N
1/2
=
N-
-1/2
corresponds to the number
of zero modes of the Dirac operator.
For g > 3, it not only depends
on the spin structure but also on the conformal class of the metric.
D'Hoker-Phong computed determinants of A- in terms of values of
n
Selberg zeta functions at half integer points except this Al case.*)
2
Notice that by considering
avoid those difficulties.
A1/2 + a
with a large, we can
The determinant of such operators can be
computed by the same technique as applied in [6] for A+, n > 1/2.
n
-
One uses the identity
v
f xV-
x
dx = 2 (
K (2/Vi ),
> 0,
X > 0
0
*)
Peter Sarnak's very recent preprint "Determinants of Laplacians"
completely settles this problem.
-30-
Also
Tr(e-t(a
=
1/2 + a2)
-z
Tr e-t(A1/
1/2
1/2
1/( (t) + 1 1/
e
2
+
1+ (a2 _
)
Finally we state
Theorem 2.2
det (A1/2 + a2
=
constant
- Z1/2(a + 1/2)
The proof is similar to that of Theorem 2.1.
We will finish this section by making an obvious remark, following
from the trace formula, that the length spectrum determines the
spectrum of the Dirac operator.
converse is true.
It is not clear to the author if the
-31-
CHAPTER 3
SOME ASPECTS OF CONFORMAL GEOMETRY
Consider the conformal Laplacian L = d*d + 4n-)
acting on
functions on a Riemannian manifold Mn with scalar curvature s.
is a conformally invariant operator.
to construct new conformal invariants.
It
Parker, Rosenberg [7] used 0
In particular they proved:
For odd dimensional manifolds det 0 is a conformal
Theorem 3.1
invariant.
We offer a different proof of this theorem as Corollary 2.
Here det E
=
(-1)
e
if ker U = {0} and there are v negative
eigenvalues (counted with multiplicity).
if ker 0 i {0}
0
Note that det E is nonzero on a conformal class admitting a metric
of positive scalar curvature.
Namely,
where c is a positive constant.
in this metric E = A + c,
Since A is nonnegative, the operator
on the right-hand side does not have a kernel,
justifying the above
remark.
For the sake of completeness,
(M ,g) be a C
R its
we state the Yamabe Problem.
Let
compact Riemannian manifold of dimension n > 3,
scalar curvature.
The problem is: does there exist a metric g',
-32-
conformal to g,
constant?
such that the scalar curvature R'
of the metric is
The positive answer has been recently proved by R. Schoen.
Therefore, in this metric
l
=
A + constant
,
where both sides act on functions.
Let B be a 0th order differential operator (or multiplication by
a matrix function B(x)).
Let
k(BIA) be the coefficients of the asymptotic expansion
for Tr(B e-tA ) when t
+0;
+
so that
k(BA)t-k
Tr(B e-tA)
(*)
We have
Sk(BIA)
f tr(B(x) Tk(xIA))dv(x)
=
where
Tk (xlA)t-k
e-tA (x)
From Shvarz [8] we have
Theorem 3.2
Suppose that for an elliptic, positive operator A
depending on the parameter T,
there is
a positive elliptic operator
T and 0th order differential operator B satisfying the relation
Tr e
'T=O = t i-Tr(B e-tT)
Then
d
-
- log det A(T)
IT=
4(BIT) - Tr(B P(T))
where P denotes a projection on the kernel.
-33-
Proof
Using (*) we can represent ?(SJA) for large Re s > 0 in the form
a
(**)
s
-
=
6
k
F'(s)
k>O sk0
6 s-k
Tr(P(A)) + f ts- 1 Tr(et
+ f ts- 1 p(t)dt
-
6
P(A))dt}
where
p(t) = Tr e-tA akt-k , a
k>0
= (Dk(11A)
,
(ak = 0 for the finite number of subscripts k defined by the order
of A and the dimension of the manifold), and 6 is an arbitrary
positive number.
The right-hand side of (**) is an analytic function for all Re s
> 0 (except poles).
Because C(sIA) is defined by analytic
continuation from the domain Re s >> 0 then for small Re s , (sjA)
is also defined by (**).
Then from the definition of the regularized determinant
det A = exp
C(sIA)Is=
S 0
(-L
it follows, that
ak_
- log det A
k>0
6k
6
+ f t~
- TrP(A)) +
+ Y(a
k
o
CO
p(t)dt +
0
Here Y = F'(l), the Euler constant.
f t- 1 Tr(e-tA -P(A))dt
6
-34-
Taking into account that
d
=
dT -k T=0
=
-k (k(BIT)
and the assumptions of the theorem, we have
0,
log det A
-
=
(k(BJT)6-k
+
6 d
+
f
(Tr B e
-
-tT
f
(Tr(Be -tT))dt
-
6
k>O
dt
k(BIT)t-k )dt
-
Sdt
k>O
Finally, taking into account the asymptotic expansion of Tr B e-tT
for t
+
0 and the relation
lim
t+O
Tr B e-tT
=
Tr B P(T)
we complete the proof of the theorem.
The previous theorem has the following application:
Theorem 3.3
Let A(T) be the family of nonnegative selfadjoint
elliptic operators such that
dA(T)
=
L(T)A(T) + A(T)R(T) ,
dTc
where L(T), R(T) are differential operators.
log det A(T)
dT
Then
= (D (L(T) + R(T)IA(T))
0
- Tr((L(T) + R(T))PA(T))
-35-
Proof
We have
d
-tA(T)
edT
-t Tr
=
dTr e
dT
-tA(T)
tA(T))e
+ Tr (R e-tA(T)
( Tr(L e-tA(T)
t
Applying the previous theorem we finish the proof.
= d*d +
For the conformal Laplacian LI
(n-)
we have
n-2
n+2
-- 7=;- 09
Here g
(x)v(x),
function on M.
d
du
uh(x)
-e
where P(x)
, u E
[0,1] and h is a C
Then
_
Pg
n+2 h
4
g
+
2 .
4
g
h
V
Applying Theorem 3.3 to Og without zero eigenvalues we obtain
Corollary 1
6 log det
Ig =
0(h
)
.
Taking into account that for n odd %0(BIA) = 0, we have
Corollary 2
For n odd we have
6 log detL
= 0
,
-36-
This result was first discovered by Parker-Rosenberg [7] in a slightly
different setting.
anomaly effect.
coefficient
For n even, however, we encounter the conformal
For instance, for n=4, computing the Seeley
(h0Ih
), we obtain
Corollary 3
6 log det W
f h(x)(R
f472
=
- RRR
R6
+ As) dvol
Proof
For B of the zero order we obviously have
k(BIA) = f tr(B(x) ' k(xlA)) dvol
Also for the operator A + k(x), where k(x) is an arbitrary smooth
function,we have
< x lexp(-t(A+k))Ix>
=
1
+ t(-s
n
1
k) +
-
(4nt)2
+ t2 [- (
- k)2 -R
26s
180sv
Here Rp6
R
+
180
R
v
R
P6
+ 14-k)]+...
6 5
stands for the curvature tensor, RP=%O6
Ricci tensor, and s = Rfrtg
c
u
for the scalar
curvature
.
P
o
h
-37-
In order to derive the latter
expression in four dimensions one
should write g v(x) according to the powers of the curvature
tensor and its covariant derivatives expressed in the geodesic
coordinates.
Finally, take k(x) =
=1
s(x).
At the end of this chapter we will present some results on the
conformal aspects of the square of Dirac operators.
The following two facts can be found in Hitchin [9].
Theorem 3.4
For g < 3,
the dimension of the space of harmonic
spinors is independent of the metric.
For g
' 3,
this is no longer
the case.
Theorem 3.5
The dimension of the space of harmonic spinors on a
two-dimensional riemannian manifold varies with the choice of a metric.
but
Theorem 3.6
The dimension of the space of harmonic spinors is a
conformal invariant.
-38-
In order to discover other conformal invariants related to the
Dirac operator, let's
consider the zeta function of D2
CO
L(t)
-t
e
=
,
i=o
where eigenvalues are counted with their multiplicities.
The following fact is a consequence of the general theory of
elliptic operators and the evaluation of the universal constant in
particular cases like a sphere.
Theorem 3.7
IU) (4Trt)
t+0+
2. E d.tj
j=o a
where
do = dim S . vol(X)
d
dim S
= - d2
12
S fs(x) dx
X
Here S stands for the spin bundle.
There is also an expression for
the coefficient d 2 but it will not be needed for our 2-dimensional
case.
Note that in such a case the dimension of the positive spinors
is one and
1
1(0)f (-
1
s(x))dv
2-(2-2g)
48
g-d
e
dim ker D
2
_
-
1-dimker D
-
k
2
2
-39-
We will state it as:
Theorem 3.8
The value at zero of the zeta function of the Dirac
square operator is a conformal invariant.
Now consider the metric (1+e)g together with the metric g.
= (1+E)D2
(l+E)g
and
We have
D2
g
as a consequence
d
dE
v'
=0
(0)
D 2(ln:)d
(l+E)g
+ V D2
g
=n(1+d)
20)
6 0D
E=Q
+
2 (
g
D2 (0)
g
This number is not equal to zero except in very special cases.
We
will state it as
Theorem 3.9
The determinant of D2 is not a conformal invariant.
I.M. Singer suggested one should consider the difference of
determinants related to two different spin structures rather than a
single determinant.
Remember that spin structures can be thought
as Z2 characters of the fundamental group,
with the
so we expect an analogy
torsion (for the twisted case) on a Riemann surface.
Compare"Theta Functions, Modular Invariance, and Strings",
L. Alvarez-Gaum6, G. Moore, C. Vafa, HUTP-86/A017.
-40-
CHAPTER 4
PARALLEL SPINORS AND SYMPLECTIC STRUCTURES
We have already seen in the previous chapter that the dimension of
the harmonic spinors space was important when computing the determinants
of the Dirac operator.
We include below a few results relevant to this
issue.
If X is a spin manifold with a zero scalar curvature, then every
harmonic spinor is parallel.
The following two theorems (due to
Hitchin [9,10]) show that parallel spinors are not very common.
Theorem 4.1.
Let X be a compact simply connected spin manifold which
admits a parallel spinsor.
Then if dim X is even
(resp.odd), + X (resp. + X x S1) is a KAhler manifold
with a vanishing Ricci tensor.
There are no known
examples of odd dimensions.
Theorem 4.2.
On a compact 4-dimensional manifold, the existence of a
non-trivial parallel spinor field implies that the
manifold is either the flat torus or a K3 surface.
-41-
Now we are ready to state:
Theorem 4.3
A Riemann surface M of genus 0 or genus at least two
does not admit a nonzero parallel spinor.
Proof
)
We obviously have c1 (M) = c1 (K) and c(
is a nonzero parallel spinor.
c (M)
12
.
Suppose s
Then s+, s_ are also parallel and at
least one of them must be nontrivial.
vanishing section of /~Kso
=
If s+,
1
0, then s+ is a never
by the Gauss-Bonnet theorem c1 (/K
) = 0.
0,1
If s_,
1
0 1
0, then s_ is nonvanishing section of A .
Chern class of this bundle is + c(M) +
2
K, and the first
Therefore,
again
c1 (M) = 0 so M is a torus.
We will close this chapter with some remarks related to zero Ricci
and scalar curvatures in 4-dimensions.
I thank I.M. Singer for improving my originally weaker version of
this theorem.
-42-
Let M
3
CP .
be any nonsingular algebraic surface of degree 4 in
Then M
4
is simply connnected, has signature -16 and Euler
characteristic 24.
Its intersection form has the form
0
E
+ E + 3
8
8
1
1
0
where E
8
2
1
1
2
1
1
2
1
1
2
1
1
2
1
1
2
1
1
1
2
1
and all
other entries are 0.
2
Its second Betti number is 22.
As every diagonal entry is even, M
is a spin manifold (those two
facts are equivalent for simply connected 4 manifolds (Milnor)).
A K3
surface is a complex surface with the first Betti and first Chern class
zero.
Kodaira's theorem states that all K3 surfaces are diffeomorphic
to a quartic surface in CP 3 .
Moreover,
for a K3 surface the
admitting of a riemannian metric of zero scalar curvature is equivalent
to the admitting of a Ricci-flat Kghler structure (see [10]).
Yau's proof of the Calabi's conjecture yields the existence of a
Ricci flat metric on the K3 surface (the simple-connectivity of a K3
surface guarantees that it does not admit any flat metric).
Therefore M
carries a metric with a zero scalar curvature.
Consider M = M4 # M4.
a nonzero A genus.
It is a simply connected spin manifold with
-43-
Theorem 4.4
M does not admit a metric with zero scalar curvature.
Since the scalar curvature is the trace of the Ricci tensor, M does not
carry a metric with zero Ricci curvature.
Proof
If M admits a metric of a nonnegative scalar curvature, then by a
theorem of Kazdan and Warner, it admits a metric of scalar curvature
identically zero.
But then by the Lichnerowicz vanishing theorem there exists a
parallel spinor,**) so by Hitchin's theorem +M is a Khler manifold
with a vanishing Ricci tensor.
But according to the Atiyah-Singer index theorem:*)
index
a
(+M)
=
1
1
X(+M) + I(+m=
=
(2X(M
)
2) +
-
4
2T(M4))
=1 l- + 8 / Z
Therefore +M is not a complex Kghler manifold.
One should also mention, that (2k+l)M
= M4#M4#...#M4
2k+1 times
has another interesting property.
It has nonzero second and fourth
*) See Peter Gilkey, "The index theorem and the heat equation,"
Publish or Perish, Inc., 1974.
**) One can also use Theorem 4.2 as was pointed out to the author by
I.M. Singer.
-44-
Betti numbers.
Moreover,
it
is an almost complex manifold which does
not carry a complex structure (Van de Ven).
A symplectic manifold is a manifold of dimension 2k with a closed
k
2-form a such that a
is nonsingular.
2k
If M
is a closed
symplectic manifold, then the cohomology class of a is nontrivial,
and all its powers up to k are nonzero.
M also has an almost complex
structure associated with a, up to a homotopy.
The existence of such an element of H 2(M 2kR) is a necessary
condition for the compact manifold to admit a symplectic structure.
Now let us recall
Thurston's Conjecture:
Every closed 2k manifold which has an almost
complex structure T and a real second cohomology class a such that
ak
0 has a symplectic structure.
See his paper "Some Simple Examples of Symplectic Manifolds," Proc.
AMS, Vol. 55, (2), 1976,pp. 467-468.
Weinstein's Question:
Are there any compact simply connected
symplectic manifolds with no underlying Kdhler structure?
(Compare his article, "Fat Bundles and Symplectic Manifolds," Adv.
Math, 37, 1980.)
*) See also Dusa McDuff, "Examples of simply-connected symplectic
non-Kghlerian manifolds", J. Diff. Geom. 20, 1984.
-45-
We will combine those facts into
Theorem 4.5:
If Thurston's Conjecture is true, then Weinstein's
Question has a positive answer.
Consider (2k+l)M , k > 0.
Proof:
It satisfies all the
assumptions of Thurston's conjecture which would imply that it is a
symplectic manifold.
At the same time it is non-KAhler and simply
connected.
M
Question:
has an
as the nonsingular hypersurface in CP
3
induced symplectic structure from CP .
Does the
Darboux theorem and convexity considerations (fwl +
(1-f)w 2 ) yield the existence of a symplectic structure
on (2k+l)M 4 ?
Let Q denote the naturally oriented underlying differentiable
manifold of the product of a Riemann surface of genus 2 and one of
genus 0.
Finally, for integers l,m,n > 0, 1+m+n > 0,
be a connected sum of 1 copies of +CP 2,
let Wlm$n
m copies of -CP
, and n
copies of Q.
Also let A = CP2 # 2(SIxS ), B = (S xS2) # 2(S1xS ).
Following Van de Ven ("On the Chern numbers of certain complex and
almost complex manifolds," PNAS,
number of 1,m,n; W1,M,n,
Vol.
155,
1966),
for some infinite
A and B have almost complex structure but no
complex structures which implies they have no K~hler structures.
-46-
Since homology groups of a connected sum of oriented manifolds are
easy to express in terms of homology groups of summands; Wi,m,n, A,B
have nonzero second and fourth Betti numbers.
So we can always find
the element a in the second real homology class such that
a2 t 0.
If Thurston's Conjecture is true, then all those manifolds have a
symplectic structure.
So we have found new examples of compact
symplectic manifolds which admit no Kghler structures.
-47-
BIBLIOGRAPHY
[1]
D.B. Ray, I.M. Singer, "Analytic Torsion for Complex Manifolds,"
Ann. of Math. (2) 98, 1973, 154-177.
[2]
A. Terras, "Bessel Series Expansions of the Epstein Zeta
Function and the Functional Equation," Trans. of the AMS, Vol.
183, September 1973.
[3]
Th. Friedrich, Zur Abhgngigkeit des Dirac-Operator von der
Spin-Struktur, Coll.Math 47, 1984, p. 61.
[4]
E. D'Hoker, D. Phong, "Multiloop Amplitudes for the Bosonic
Polyakov String," preprint.
[5]
R. Gangolli, "Zeta Function of Selberg's Type for Compact Space
J. Math., 21, 1977,
Forms of Symmetric Spaces of Rank One," Ill
403-423.
[6]
E. D'Hoker, D. Phong, "On Determinants of Laplacians on Riemann
Surfaces," preprint.
[7]
T. Parker,
preprint.
[8]
A.S. Shvarz, "Elliptic Operators in Quantum Field Theory,"
Modern Problems in Mathematics," Vol. 17, (in Russian).
[9]
N.
[10]
Hitchin,
S.
Rosenberg,
"Invariants of Conformal Laplacians,"
"Harmonic Spinors," Adv. in Math,
14,
1974,
1-44.
, "Compact Four-Dimensional Einstein Manifolds," J.
Diff. Geom., 9, 1974, 435-441.