Determinants of laplacians

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 MathematicsJune 10, 1986 CERTIFIED BY ACCEPTED BY: Isadore M. Singer Thesis Supervisor A

U

Nesmith C. Ankeny

Chairman, Departmental Graduate Committee MA SSACHUSE TT S NSTITUTc

OF TECHNOLOGY

.FP

96

1986

IRARI 3

ARCHIVES

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. Applications to conformal geometry are presented. Finally, we discuss some obstructions to the existence of parallel spinors and symplectic structures.

Thesis Supervisor: Isadore M. Singer

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

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.

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. We also show that the determinant of 2 is not a conformal invariant.

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.

CHAPTER 1

DETERMINANTS OF LAPLACIAN AND DIRAC ON THE TORUS

Let W be a compact complex analytic manifold without boundary, of complex dimension N. If D 'q(W,L) denotes the space of C

complex (p,q) forms on W with values in a flat holomorphic vector bundle L, then the exterior differential d splits:

d = d' + d",

where

d':D 'q + Dp+1,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 defines for each choice of p is

usually called the a-complex. We are using the notation of [1].

Let Z p, denote the kernel of d" in D '9(WL). The Dolbeault L

cohomology groups H '9(WL) = Z 'q/d"Dp 'q (WL) are isomorphic to

, where 0 is the sheaf of germs of forms of type (p,O) with holomorphic coefficients.

Now let X be a finite dimensional unitary representation of the fundamental group 7 (W), and let L(X) be the associated complex vector

bundle. Let Dp'q(W,X) = 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'

= - ' +DP~lp

6" = -d* + D q

Let A be the corresponding Laplacian

pq -q

A =-(6@@dl + d"":

Let HP)9(W,X) be the space of harmonic forms in (X),

that is, satisfying Af = 0. 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, that is, when the 2-form

w = h dzi A dzk 2 jk

is closed, 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 - A where A is the Hodge Laplacian

3 2 dd* + d*d.

A generates a semi-group of compact operators exp(tA ), t > 0, on the Hilbert space completion of D p,(W,X). As t + + c, these

operators approach the projections P of D '9(W,X) onto the sub-space HPq (W,X). 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. Here

dtr (A-s) d -s -s

Formally, at s = 0, this would be Z log X= log det A. Hence, we define log det A = (S)

s=0

Consider the 2-dimensional torus, T = C/F, where r is the lattice generated by 1 and T, ImT > 0. Consider the corresponding Laplacian A5. We assume that our representation X is trivial and

p

= q =

0.

Theorem 1.1

det A7 = (IMT)2 1 rI(t) 4 where II li 2TrnT elT =R (1 - e2Wn n=1 Proof

The eigenvalues of A on D*(W,l) are

42 2

(I - n)

in,n

2

I-~

Applying the Poisson summation formula we obtain -lmit + nI2

tr(et 4tme

Now we have where E CO 1 1- -tA c(s) =

f

ts- tr(e- )dt 0 _ 1 InT + IMT F(s) 47(s-1) 4rF(s) 00 + fts-i (tr et)dt

= sum over all (m,n) 0 (0,0).

1

f

s-2 0 -I+n1 2 4t e dt

When Re s < 0, we apply the expression for tr(e-t ) in the last term on the right to obtain

-= IT ' ts-2 4TrF(s) 0 -mt+n1 2 4t e lInt F(1-s) ' j 4 1-s 4U F s) 2* ImT + ni Consider E(T,u) (- IMT u , Re u > 1 ImT+nI

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. At u=1 it has an expansion:

E(Tu)

=

+ 2R(Y - log2 - log((ImT) |T1(T)1 )) + a (u-1) +

It is called the first limit formula of Kronecker. dt =

Rewriting C(s) further we have lInT 4 1 F (l-s) C (S) = IT (I 1-s

r-

E (T,1-s) (ImT) lint Si1 2 1 2. 2 = ( ) [S.-2F'(l)s +..] [- + 2Tr(y-log(2(Imt) z

(t)I

))+...] =( -) [s-2F1(1)s+...] [- + 2 (Y-log(2(Imt) 2 2 1 = -(n)s [1 + 2log(2(ImT)2 If(T) 2) s + ... ] as F'(l) = -y Therefore 1 V(0) =-[log( I) + 2log(2(ImT) 2 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

A (f dx I....d x ) (A f)dx dx

P 11 1 p 0 1 i

shows that the p-forms spectrum is the same as the function spectrum, each repeated (n) times. Therefore the det A determines det A completely.

p 0 p

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, A C GL(n;R). The metric structure of Rn

projects to T such that vol(T) = Idet Al. The set L = {a C R n a a C Z,

-i t n

V a E L} is the dual lattice of L; L = (A ) Zn. The eigenvalues of T 2 ~2

are 42 Hal for a arbitrary in L where II II is the Euclidean norm.

The Poisson summation formula states -a tA)a -tA det A 4t trke )L n n (41 t)2 aEZ e so ?(s) reads S)= detA _n (47)2

r()

acZ- 0 F(-n - s) det A 2 n 7 (S) TT 4 CO s- n_ f t 2

0

-attA)a 4t e n-s (at (A tA)a) -dt

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) =! I (ta Sa) ,

n 2

aeZn-O

where the sum is over all column vectors with integral coordinates, not all of which are zero. Define

n

k (S) lim {Z (SP) 2 1

n

n n m. n - 12

~

n

2 I'( )IdetSI 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]* Let n = n, + n₂, with 1 < ni < n-l. Then S can be represented as

S(n) (n) 22]

t

[T(nP) 0 I

1

t (n2 _ n 2)

1 12 2 -01 - 0 S'2

L

_

Let n 1 and T t.

1 1

=t

2 -l

2

Define z{b} = bQ + i t (S [b]) , Y - log 2 -c _n = < 1 -(n-3) y -E r=0 n r=1 (2r+l) , for n even, for n odd k(S) _{= Z n-_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. In the 2-dimensional case

s = s ] 1 q t

0 1

s12 s 2 0 1 0 s2 q

forts -q 2 2 1 2 12 > ,andqs _{1 2}/s ₂' Then

Then

1

lim {Z2(S,p) - r(det 5) p-i

p+1

TT

2

2

= 1 {X - log 2 - log(t TI(z)I

)}

(det S)2

1 11

2 2 -1

for z = q + i t s2 = 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 1

-.Pp)Z p 2 ~- 2n 1 _-11

- I(P)Z (S P) = (detS) 2i F( -p) Z (S ) -P)

together with the generalizations of Kronecker's limit formula included on the previous page.

We assume the reader knows about the Dirac operator so we discuss it only briefly.

Let X be a compact, oriented riemannian manifold. The bundle of orthonormal frames E is a principal SO-bundle. 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 bundle of spinors. E lifts to E iff w2 (x) = 0 and any two liftings

differ by a Z2 1-cocycle, so the number of inequivalent liftings

(the number of spin structures) is # H (X,Z2). In even dimensions, V splits into V + V so one can define the desired Dirac operator

: C (V) - C (V). In particular we get a decomposition of harmonic spinors H into the space of positive H and negative H_ harmonic 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 (W2

c 1 ()mod 2). Furthermore, the set of spin structures on X is in 1-1

2

correspondence with the [L:L = K,L a holomorphic line bundle]. Choosing

a spin structure means choosing an L, so

jf:V

+ V turns out to be the same as :C (AOeven d L) + C (AO,odd 3 L). For a compact KAhler manifold we have H+ ~ Heven(X,(L)), H~ Hodd(X,0(L)).

Consider now a 1-dimensional torus T and the square of the Dirac operator on T. There are four of them, corresponding to the underlying spin structures. The eigenvalues of these operators are

-41T2

1 1 2

m,n 4I2 2 m + 2 - t(n + f cE) 2

where (e1,E2 ), 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 1 1

det 2 e4 1 1 2 2 - 1 2

det 2 =Ie2

where (EE2) (0,0) and 01 is the theta function

Tri(w + 1) C 2ni(IkIT - akw)

(w-n(T) e 6 (l-e

1 2

ak sign (k + )

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, = - 42 1 u+m - T(v+n)2 m,n (ImT)2

and the eigenfunctions corresponding to ,n are 2Vi

Pmn(z) = exp Im(z(u + m - T(v + n))}. We have, of course,

x

t

tr(e ) = l 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

-Iz-z'1 2/4t

t4 t e

is the euclidean heat kernel on C. From this we obtain

tr(e t) = Z IMT -|mt+n 2/4te 2Ti(imu+nv) M,n 4nt

The above yields the analytic continuation of the zeta function. For Re s large we can write

c(s) = Z (-X )-s

m,n m,n

00

1 _{ft s-1 tr(e tA)dt}

1 IMT +IMT 2Tri(mu+nv) 1s-2 2-m+n12/t

(s) 4n(s-1) 4ur s 2 2 e f t0 etdt

in+n >0 0

+ s f 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

_ _4IMT 2Tri(mu+nv) f ts-2e-Imt+n1

/4tdt 411F(s) 2 2

m +n >0 0

IMT F(l-s) 27i(mu+nv)( 4 1-s

40 F(s) m2 2>0 Imt + n 12

Since the series does not converge absolutely when s 0, let us explain how to evaluate '(0). 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

c(s) =IT

r(-s)

n

+ Z

e2Timu zo M-YO n=-M 2lTinv 4 )1-s n e2 4 2rinv( 1S lMT + ni Introduce the Dirichlet kernel

A = erinv sin TI(n+1)v

n sin Tv

2s-2

and set b n linT + n1 Then

1 lmT+n2-2s -n(A - A)bl -n n-inb~ = z'

_.co

A (b n n+l - bn

)

< 1 Z' b -bi = sin rv -O n+l n < 1 2 = sinnv (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: co2Tnv n1 2 n 1 e2 imu c =-o 2ninv 2lmImTl e mT + n 2

The first series above is standard, and the first term on the right side of (*) is given by

2 1 TrImT(2v -2v +), 0 < v <. zo e 2minv n=--we =2 - Im T 'Tr (*) + ---2Tr M I

To compute the second series, define

F(x) = Z e-2n(imlImTlx+kl+imReT(x+k)) 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

f 1e-2 inxF(x)dx = f e-2ff(lmImTlxl+i(mReT+n)x)dx 0

1

ImIImT

|

ImT + n 2

So the second term on the right in (*) becomes

1 2nimu o -27(ImIImTlv+kl+imReT(v+k))

M O Im, k=--W

= -COC log - 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 , 2) are identical to

the special twisted case. Namely, instead of considering a general non-trivial character given by X(mT+n) = e2-i(um+vn) 0 < u,v < 1,

1 1

take u e₂ 2 el. It gives the formula of Theorem 1.4.

Remark: 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

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 SL2(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 and p = p > 0. We call

an element of F primitive if it is not a power in F. r is the union of disjoint conjugacy classes. PY = P , if Y,Y'

Selberg introduced the zeta function:

-2p Y(6+k) Z (6,x) = H H det (1-X(y) e )

{y} k=0

which is well defined by the above product for Re6 > 1. It can be extended to an entire function with zeros at the non-positive integers

and also at the points 1 + i -X- for each eigenvalue X of the

2 - 4

Laplacian. Z satisfies the following functional equation: 6-1/2

Zp (1-6,X) = Zr(6,X) exp(-4n(g-1) f r tanh Tr dr)

0

Let F(T), F(T') be discrete subgroups of SL(2,R) such that H/F(T) and H/F(T') have the same genus g > 1. Consider the

standard -1 curvature metric on those Riemann surfaces induced from H. Let X : Fr(T) +* U(l), X' : r(T') + U(l) be

nontrivial representations. Following [1], consider s-1 tA(T)

e dt if X is nontrivial;

(s F(S) ₀ ts-1 (tr etA(T) -l)dt, if X is trivial.

Remember that Co,0 = C0,1. Notice then in both cases

00?

(S)- ) =

f

S 1 (tr e A(T) -tr e tA(v ))dt Th c r rAe r Id 0 ts

The Selberg trace formula applied to the heat kernel on D 0(H/]F(tE),X) states and

t

t0 ( + r2)t tr(e ) = (2g-2) f r tanh (Trr) e

0

(*) + I I {Y} k=1 tr(Xk Y 1 sinh(kp Y) -dr + 2 2 t k _P 4 t

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 (*) F x (t). We have IF (t)

I

< Ce -at-(b/t) with a,b > 0 and

CO (S) -AV,(S) = Is) f ts-1(F .C(t) - F

0()

dt r~)0

When Re s < 1, we can write

CO

ts-1 _{t =} 1 (6(6-1))-se-6(6-l)t(26-1)d6

___0

Using this and interchanging integrations

CO (S ) 1- f (6(6-1))-s(G (6)-G (6))d6 '(s)'(l-s) 1XT where G)

Gx (

6

)

= _Xj~(~d (26-1) fe _0~- 6

(

6

l1)t

F (t, d =

dlo

_{log-rd r} MX

So we conclude

det A(T) - P(-) lX)

det A(T)

that is, Zet (,X) is independent of the complex structure and

det A X(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. Therefore,

d

detLA(T) = constant * 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.

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 22g 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. The covariant derivative V sends Tn into Tn e (T1 'iT), and it can be decomposed accordingly as V =V n + Vz, with V": TX'+ Tne TTf +

z n z

Vz Tn + Tn 0 T1 T n- (that is V = 3 9 on sections of Tn). The

n

spaces of L 2-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

A+ = -V ZV 'M

n n Z

- n z

In local isothermal coordinates z, we can write 2 ds = 2g - dz dz, n zz Vzf = gZZ n zz z <fig> f dzdz g - (g )nf*g. T n M zz zz

Our M = H/F, F is a discrete subgroup of SL(2,R)/{+1},

all of whose elements are hyperbolic. Let P be the subgroup of SL(2,R) containing -I which projects to P, and let Yi,. .,Y2g be a

fixed set of generators for P. 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 - n(n+l)) nn

Tr (e n - n(= It) + In(t) e

where In(t) = I I V(Y) 2_n Y primitive p=l t

1

4 ( s 2 4(rt)2 and In W = -2(2-2g) (2n-2m-1) e (n-m)(n-m-l)t e O<m<n- 1 -t/ 4 Co -b2/4t - 47(2-2g) f db b e ch (n-[n])b (47t3/2 0 sh b/2 They also introduced the two Selberg zeta functions

Zn(s) = H H (1-v(X) 2n e -(p+s)l) Y prim. p=0

for n=0,

Let N- denote the number of zero modes of the Laplacian n

A±-. Note that N = N- corresponds to the number

n 1/2 -1/2

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

Notice that by considering avoid those difficulties. The computed by the same technique One uses the identity

f xV- x dx = 2

0

integer points except this Al case.*) 2

A1/2 + a with a large, we can

determinant of such operators can be as applied in [6] for A +, n > 1/2.

n

-v

( K (2/Vi ), > 0, X > 0

*) Peter Sarnak's very recent preprint "Determinants of Laplacians" completely settles this problem.

p2t 2 4t e

Also

Tr(e- 1/2 + a2) Tr e-t(A1/ 2 + 1+ (a2 _

-t(a -z 1/2 1/2

= 1/( (t) + 1 1/ ) e

Finally we state Theorem 2.2

det (A 1/2 + 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. It is not clear to the author if the converse is true.

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. It is a conformally invariant operator. Parker, Rosenberg [7] used 0

to construct new conformal invariants. In particular they proved:

Theorem 3.1 invariant.

For odd dimensional manifolds det 0 is a conformal

We offer a different proof of this theorem as Corollary 2.

Here det E = (-1) e

0

if ker U = {0} and there are v negative eigenvalues (counted with multiplicity).

if ker 0 i {0}

Note that det E is nonzero on a conformal class admitting a metric of positive scalar curvature. Namely, in this metric E = A + c,

where c is a positive constant. 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, we state the Yamabe Problem. Let (M ,g) be a C compact Riemannian manifold of dimension n > 3,

conformal to g, such that the scalar curvature R' of the metric is constant? 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

(*) Tr(B e-tA) k(BA)t-k

We have

Sk(BIA) = f tr(B(x) Tk(xIA))dv(x) where

e-tA (x) Tk (xlA)t-k

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.

Proof

Using (*) we can represent ?(SJA) for large Re s > 0 in the form

a

6

(**) s = _{F'(s) k>O sk0}k 6s-k + f ts-1 p(t)dt

- Tr(P(A)) + f ts-1 Tr(et - P(A))dt}

6

where

p(t) = Tr e-tA - a kt-k , a = (Dk (11A) , k>0

(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 (-L C(sIA)Is=S ₀

it follows, that

a

k_

- log det A - 6k + Y(a - TrP(A)) +

k>0 k o

6

CO

+ f t~ p(t)dt + f t-1 Tr(e-tA -P(A))dt

0 6

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 +

f

- (Tr(Be -tT))dt

k>O 6 dt 6 d _-tT +

f

- (Tr B e Sdt - k(BIT)t-k )dt k>O

Finally, taking into account the asymptotic expansion of Tr B e-tT for t + 0 and the relation

lim Tr B e-tT = Tr B P(T) t+O

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. Then

log det A(T) = (D (L(T) + R(T)IA(T)) - Tr((L(T) + R(T))PA(T))

Proof We have d -tA(T) = -t Tr -tA(T) dTr e tA(T))e dT edT t ( Tr(L e-tA(T) + Tr (R e-tA(T)

Applying the previous theorem we finish the proof.

For the conformal Laplacian LI = d*d + (n-)

n+2 n-2

-- 7=;-

09

Here g (x)v(x), where P(x) -e uh(x) _{, u E [0,1] and h is a C} function on M. Then

d _ _{n+2 h + 2 .}

du Pg 4 g 4 g h

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 det L = 0 ,

we have

This result was first discovered by Parker-Rosenberg [7] in a slightly different setting. For n even, however, we encounter the conformal anomaly effect. For instance, for n=4, computing the Seeley

coefficient (h0 Ih ), we obtain

Corollary 3

6 log det W =

f472

f h(x)(R R6 - RR R + 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 = 1 1 < x lexp(-t(A+k))Ix> n + t(-s - k) + (4nt)2 + t2 [- ( - k)2 -- R R + R R + 14-k)]+... 26s 180 sv 180 v P6 6 5

Here Rp6 stands for the curvature tensor, R P=%O6 P o h

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

=1

coordinates. Finally, take k(x) = 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.

In order to discover other conformal invariants related to the Dirac operator, let's consider the zeta function of D2

CO -t

L(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) 2. E d.tj t+0+ j=o a where do = dim S . vol(X) dim S d = - d2 S fs(x) dx 12

X

Here S stands for the spin bundle. There is also an expression for the coefficient d2 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 2 1(0) f (- s(x))dv - dim ker D 2-(2-2g) _ 2 48 - k g- d e 2 1-dimker D

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. We have

D2 = (1+E)- D2 (l+E)g _g and as a consequence d _{v ' (0) =n(1+d)} 20) + dE =0 D

2

(ln:)d 6 0D ₂ ( (l+E)g E=Q g + V D2 D2 (0) g 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, so we expect an analogy with the 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.

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 S 1) 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

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

c (M)

We obviously have c1(M) = c1(K) and c(

)

= 12 . Suppose s

is a nonzero parallel spinor. Then s+, s_ are also parallel and at least one of them must be nontrivial. If s+,

1

0, then s+ is a never

vanishing section of /~Kso by the Gauss-Bonnet theorem c1

(/K )

= 0. 0,1

If s_,

1

0, then s_ is nonvanishing section of A 0.1 K, and the first

Chern class of this bundle is + c(M) + 2 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.

Let M be any nonsingular algebraic surface of degree 4 in

3 4

CP . Then M is simply connnected, has signature -16 and Euler characteristic 24. Its intersection form has the form

2 1 0 1 1 2 1 E + E + 3 where E 1 2 1 8 8 1 0 8 1 2 1 1 2 1 1 1 2 1 1 2 1 2

and all other entries are 0. 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 CP3. 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. It is a simply connected spin manifold with

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:*)

1 1

index

a

(+M) = X(+M) + I(+m=

= (2X(M

)

-

2)

+ 2T(M4)) 4

=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.

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 2k

2-form a such that a is nonsingular. 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.

We will combine those facts into

Theorem 4.5: If Thurston's Conjecture is true, then Weinstein's Question has a positive answer.

Proof: Consider (2k+l)M , k > 0. 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.

Question: M as the nonsingular hypersurface in CP has an

3

induced symplectic structure from CP . Does the Darboux theorem and convexity considerations (fwl + (1-f)w2) yield the existence of a symplectic structure

on (2k+l)M4?

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, let Wlm$n be a connected sum of 1 copies of +CP 2, 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, Vol. 155, 1966), for some infinite number of 1,m,n; W1,M,n, A and B have almost complex structure but no complex structures which implies they have no K~hler structures.

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

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 Forms of Symmetric Spaces of Rank One," Ill J. Math., 21, 1977, 403-423.

[6] E. D'Hoker, D. Phong, "On Determinants of Laplacians on Riemann Surfaces," preprint.

[7] T. Parker, S. Rosenberg, "Invariants of Conformal Laplacians," preprint.

[8] A.S. Shvarz, "Elliptic Operators in Quantum Field Theory," Modern Problems in Mathematics," Vol. 17, (in Russian).

[9] N. Hitchin, "Harmonic Spinors," Adv. in Math, 14, 1974, 1-44. [10] , "Compact Four-Dimensional Einstein Manifolds," J.