University of Florida Digital Collections Home |   Help

# Geometric quantization of symplectic tori

## Material Information

Title:
Geometric quantization of symplectic tori
Physical Description:
vii, 91 leaves : ; 29 cm.
Language:
English
Creator:
Chastain, Scott Gregory, 1970-
Publication Date:

## Subjects

Subjects / Keywords:
Mathematics thesis, Ph.D   ( lcsh )
Dissertations, Academic -- Mathematics -- UF   ( lcsh )
Genre:
bibliography   ( marcgt )
non-fiction   ( marcgt )

## Notes

Thesis:
Thesis (Ph.D.)--University of Florida, 1998.
Bibliography:
Includes bibliographical references (leaf 90).
Statement of Responsibility:
by Scott G. Chastain.
General Note:
Typescript.
General Note:
Vita.

## Record Information

Source Institution:
University of Florida
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 030004008
oclc - 41448250
System ID:
AA00018778:00001

Full Text

GEOMETRIC QUANTIZATION ON SYMPLECTIC TORI

By

SCOTT G. CHASTAIN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

For my wife Stacey and my entire family.

ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Paul Robinson, for his guidance

during preparation of this thesis and for his intellectual influence on my development

as a mathematician. I would also like to thank the members of my committee for

their attention and guidance during this process. Finally I would like to thank the

wonderful office staff, past and present, of the Department of Mathematics.

ACKNOWLEDGEMENTS ............................... iii

ABSTRACT .................................... vi

CHAPTERS

1 INTRODUCTION ............................... 1
1.1 Quantization . . . . . . . . . . . . . . 1
1.2 Kostant's Geometric Quantization . . . . . . . . 2
1.3 Notation ........................... .... 2
1.4 O outline . . . . . . . . . . . . . . . . 3
1.5 Prequantization ...................... ....... 3
1.6 Quantization with Real Polarizations ................. 5

2 LINE BUNDLES ...................................... 7
2.1 Definition of Line Bundles ................. ...... 7
2.2 Sections ...... .. ............................. 7
2.3 Connections ............................. 8
2.4 Metric Connections ......................... ... 9
2.5 Pullback Bundles ............................ 9
2.6 Equivalence of Line Bundles ........... ..... . 10
2.7 Construction of Line Bundle L on the Torus ........ ..10
2.8 Pulling a Section of L Back to a Complex Function ....... 14
2.9 Equivalence of Sections and Quasiperiodic Functions ...... 15
2.10 Curvature ............................... 16
2.11 Kostant Line Bundles ........................ 16
2.12 Connections on the Torus ................. ..... 17
2.13 Kostant Line Bundles on the Torus . . . . . . . ... 21
2.14 Classification of Kostant Line Bundles . . . . . . ... 23
2.15 Tensor Product of Line Bundles with Connection . . . ... 24
2.16 Dual Bundles with Connection . . . . . . ..... . 25
2.17 The Group of Flat Line Bundles . . . . . . ... 26
2.18 F(M) Action on K(M,w) ..................... 26
2.19 Flat Line Bundles on the Torus . . . . . . . . ... 26
2.20 The Classification of Kostant Line Bundles on the Symplectic Torus 29
2.21 Common Description . .. .. ............... 29

3 FOURIER ANALYSIS ... ... ................ .. 32
3.1 Trigonometric Series ... ... .... .... .......... . 32
3.2 Fejer's Kernel .. .. ....................... 33
3.3 Riemann-Lebesgue Lemma . . . . . . . . . ... 34
3.4 Convergence of Fourier Series . . . . . . . . ... 34

3.5 Decay of Coefficients ......................... 35
3.6 Fourier Series of Functions on R x T . . . . . . .... 37

4 COHOMOLOGY OF SHEAVES ....................... 38
4.1 Fmnctions Constant along a Subtangent Bundle ......... 38
4.2 Presheaves and Sheaves .................. . 38
4.3 Differential Complex Based on C . . . . . . ... . 40
4.4 Sheaf Cohomology ......................... 41
4.5 Sheaf of Sections of a Line Bundle ................. 43
4.6 F-Connection .................... ...... .. .. 44

5 COHOMOLOGICAL SOLUTIONS ................. .. . 46
5.1 A Kostant Line Bundle on the Torus . . ......... 46
5.2 Translation Invariant Real Polarizations on the Torus . . 47
5.3 --twisted Functions ......................... .. 47
5.4 Polarized Sections .......................... 48
5.5 Fourier Expansion of 7-twisted Functions . . . . . ... 50
5.6 No Constant r-twisted Functions in the Direction of X, . .. 52
5.7 No Nontrivial Polarized Sections . . . . . . . ... 54
5.8 Cohomology of the Sheaf of Polarized Sections . . . ... 54
5.9 Solving the PDE .......................... 55
5.10 The Change Of Spaces ....................... . 58
5.11 The Transform T, Is Onto ...................... 60
5.12 The Set Po ................... ....... . .. 65
5.13 L'H6pital Approach to Po ...................... 68
5.14 The Set P ......................... . . . 68

6 L'HOPITAL RESULT ............................. . 71
6.1 The Question . . . . . . . ... .. . . . . 71
6.2 A Generalized L'H6pital's Rule . . . . . . . . ... 71

7 DISCONTINUOUS AND DISTRIBUTIONAL SOLUTIONS ....... 76
7.1 Discontinuous Solutions . . . . . . . . . . 76
7.2 Bohr-Sommerfeld Set ....................... . 78
7.3 Distributional Solutions ...... .............. . 79

8 FURTHER QUESTIONS .. ................... .. .. .. 80
8.1 Other Kostant Line Bundles on the Torus . . . . .... 80
8.2 Asymptotic Solutions . ............ ... .. .... . 82
8.3 Pairing . . . . . . . . . . . . . . . . .88

REFERENCES ........ ...... .. .................. .. .. 90

BIOGRAPHICAL SKETCH

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

GEOMETRIC QUANTIZATION OF SYMPLECTIC TORI

By

Scott Gregory Chastain

December 1998

Chairman: Paul Robinson
Major Department: Mathematics

As a test case for the application of the methods of geometric quantization

the symplectic torus with affine real polarizations displays two important pathologies.

The first occurs at the prequantization stage. The symplectic form is not exact so

we cannot expect to have a trivial Kostant line bundle. Thus we must construct a

nontrivial Kostant line bundle. In this construction we see that a set of complex-

valued functions on the plane with a quasiperiodicity condition is the key to working

with sections of a Kostant line bundle. The second pathology arises as we pass from

the prequantization stage and attempt to form the quantum space. We find that there

are no nontrivial polarized sections. The quantum phase space is usually constructed

from the polarized sections. A common technique for overcoming this obstr action is to

use as elements in the quantum phase space, generalized sections of our Kostant line

bundle with support contained in the Bohr-Sommerfeld sets. This technique works

producing a one-dimensional quantlun phase space in the case of polarizations formed

by lines in the plane with rational slope, the rational polarizations. In the case of

vii

irrational polarizations problems with the Bohr-Sommerfeld sets preclude the normal

application of these ideas. We solve this problem by using the cohomology of the sheaf

of polarized sections and find our answer is the same for both rational and irrational

polarizations. Also we see our answer can be expressed in terms of distributional

sections matching the standard approach in the case of rational polarizations. Finally

we raise questions about the possibility of finding answers using asymptotic sections

and the need to define pairings between our quantiun phase spaces.

CHAPTER 1
INTRODUCTION

1.1 Quantization

Quantization is the association of a Hilbert space with a set of operators on

the Hilbert space to a symplectic manifold with its Poisson algebra of functions. This

association is to satisfy the four conditions of Dirac[1].

First, each fiction in the Poisson algebra of the symplectic manifold is asso-

ciated with a self-adjoint operator on the Hilbert space such that the association is

linear. Second, this association should preserve the Poisson bracket as the commu-

tator of operators, if f f denotes the association of a function with an operator

then fi f = ih {f, g} where {, } is the Poisson bracket. Next it is required that

constant functions quantize as multiplication operators, so if f a then f = al

where I is the identity operator.

Given A a finite dimensional subspace of the Poisson algebra of functions

that is closed under Poisson bracket such that each element of A has a complete

Hamiltonian vector field, we may consider A as the Lie algebra of some Lie group

that acts on our manifold. Dirac's fourth requirement is that if this group acts on

the manifold transitively then the representation of the group on the Hilbert space

obtained through quantization by integrating the image of its Lie algebra should be

irreducible.

It should be mentioned that Von Howe showed on R2" there is no way to
quantize all observables and satisfy the conditions of Dirac.

1.2 Kostant's Geometric Quantization

Broadly speaking Kostant's geometric quantization is an approach for find-

ing imitary representations for Lie groups. Underlying structures are a symplectic

manifold (M, w) with an associated Hermitian line bundle L over M with compatible

connection V having curvature a multiple of w, a complex involutory Lagrangian dis-

tribution F on M ( called a polarization), and the cohomology H (C) of the sheaf of

local sections of L constant along the polarization F ( called polarized sections). In

the most basic examples the full cohomology is not mentioned as there exist global

polarized sections on which to quantize a subalgebra of the Poisson algebra of ob-

servables. The question of computing the sheaf cohomology has been studied, for

example when the sheaf of local holomorphic sections is under consideration. In this

dissertation we are interested in a rather different situation and will consider the

example of the symplectic two dimensional torus with a real invariant polarization.

1.3 Notation

In this section we will establish some of the notational conventions we will

be using in what follows. No more than one manifold will ever be considered at a

time and therefore M will always be used for a manifold. Real valued functions on a

manifold, which will sometimes be called classical observables, will be denoted by f

and g. Complex valued functions on M will be denoted by 4 and Vi. Vector fields and

tangent vectors will be denoted by (, qj, and (. Alpha will be used for one-forms and

omega will always be a symplectic form, with capital f used for a purely algebraic

symplectic form on a vector space and lower case w when we are discussing a two-

form on a manifold. Sections of a line bundle on a manifold, which will always be L,

will be denoted by s and t.

1.4 Outline

In the remaining part of this chapter some illustrative examples of geometric

quantization are given. In Chapter 2, the theory of line bundles is developed to the

point that all Kostant line bundles on the torus are described and a convenient means

of representing sections of these bundles as quasiperiodic functions on the plane is

explained. Because of the appearance of periodic functions in this work, techniques

from Fourier analysis are used in the main results. Chapter 3 describes the effects

of smoothness on the rate of decay of Fourier coefficients. The sections of a line

bundle have the structure of a sheaf and the cohomology of the tons with sheaf co-

efficients will substitute for global polarized sections in our main result. Chapter 4

defines sheaves and describes a method of computing sheaf cohomology. In Chapter

5 we compute the cohomology of the sheaf of polarized sections. This computation

involves all of the machinery described in the previous three chapters. Some of the

proof in Chapter 5, which uses Fourier techniques, was first done with a generalized

L'H6pital's rule argument which is the content of Chapter 6. Another approach to

the absence of global polarized sections is the use of distributional solutions. The

usual approach takes distributional solutions supported on the Bohr-Sommerfeld set.

In Chapter 7 we describe the Bohr-Sommerfeld set on the symplectic torus and see

that for irrational polarizations the Bohr-Sommerfeld set approach does not provide

a solution. We then show that the results of our computation of the cohomology

of polarized sections can be viewed as distributional solutions and that these distri-

butional solutions make sense even for irrational polarizations. Finally in Chapter 8

directions for further research are described.

1.5 Prequantization

In this chapter geometric quantization in R2n will be reviewed. This is well

known in the literature, but as each author has his or her own views on some of the

incidental choices that are made while writing out these examples, I believe it will be
useful to have them written up in my notation and with my choices. Even in the basic
examples in this chapter we find instances where there exist no nontrivial polarized
sections and therefore it is useful to consider the two approaches to overcoming this
difficulty that are central to this paper: generalized functions and cohomology of po-
larized sections. Another advantage of working with this example is that no nontrivial
line bumdles exist so that we may just talk about complex functions and consider the
added detail of line bundles when we consider the torus. Consider M = R2n with
coordinates {qj,pi} and with symplectic form w = E dp, A dqi. This form is exact;
for example we may choose a = E 1 (pidq, q dp2) so that w = da. If n = 6 this can

be thought of as the phase space of a single particle where the qis label position and
the pis label momentum [2]. For f E CR(M), the smooth real valued functions on
M, we associate f '- f by specifying (f w = -df. The vector field f is called the
Hamiltonian vector field associated with f. In coordinates, ( = E -_- 9 l
OpI Opq, O9q Op
C7(M) becomes a real Lie algebra with {f,g} = /gf, called the Poisson bracket. In
coordinates

Of 09 Of 0g
Oqi Opi Opi 9Oqi

So for example {qi,pj} = 6ij. Since M is contractible all line bmudles on M are
isomorphic to the trivial line bundle[3]. So we may identify sections of a line bundle
with smooth complex functions on M. Define a connection by V = + a((. This
connection will have curvature RV = I Here we are considering h to be a real
constant. A Hilbert space structure is obtained from the smooth complex functions

COC(M) by specifying the inner product to be

<^^ >= f ",

for 4, 0 E C@O(M). Then let 'No be the set of functions with finite norm and let 7i
be the Hilbert space completion of -"o. We prequantitize as follows: f f where

ff = -ihV4f + ff

This yields ,i = qi/2 + ih- and pi = pi/2 ih--. For example [i,,pj] = ihfy. And
in general [f,g] = ih{f,g}. This prequantization does not satisfy the irreducibility
condition of quantization.

1.6 Quantization with Real Polarizations

Now we restrict our attention to R2. For each real number 7 let F, be the
polarization, a one dimensional subbundle of the tangent bundle of R2 in this case,
spanned by the constant vector field X, = - +7- Also, let F be the polarization
spanned by Xo = -. We wish to compute FFT = { E cm(R2) : VxQ = 0}.
Thus 4 must satisfy the differential equation XT( + -a(X,)4 = 0. Solving we can
write (p, q) = h(q pr)e- i (q-p)p where h E C"(R). In order to recover the usual
Schrhdinger representation we must take as the presymplectic form 0 = pdq, while
leaving the expression for the connection V + 27ri0. Then for the polarization F = -&
the polarized sections will be functions constant along the leave of the polarization. In
this case, the space of polarized sections is essentially the smooth complex functions of
position. The functions that quantize as multiplication operators are the real valued
functions constant along the leaves of the polarization. The full Poisson subalgebra
of functions that can be quantized are functions of the form a(q)p + b(q) where a
and b are both smooth real functions on R. An obvious inner product on the set of
polarized sections is available in the case of the Schr6dinger representation. That is
for p and 4 two fimctions of position let < p I >= f p (q)V) (q) dq. And then take
for the quantlun phase space the completion of the set of functions with finite norm.

This is not the same inner product structure as used in prequantization and the inner

product in prequantization would not work on polarized sections which are constant

on momentum fibres and thus would have infinite norm with respect to w". It is not

clear how the volume form for integration in the configuration space was obtained in

this representation. Some justification and insight is gained by viewing the polarized
sections not as functions of position but as half-densities. Thus the inner product is

the usual inner product of half-densities and the integral is the integral of a density.

Fumctions that quantize (most simply) as multiplication operators are those

whose Hamiltonian vector fields lie in the polarizations. In quantizing the one-

dimensional harmonic oscillator, it is therefore desirable to employ a polarization
containing the vector field associated to its energy, H = (p2 + q2). For the harmonic

oscillator representation we take M = R2 {0} since the Hamiltonian vector field

of the energy, XH = p q, is zero at the origin; take w the same as before.

Then polarization F where Fp,q) =span( ) and =- ), contains
M 010- vr2T p, -( q ,ap
XH. In polar coordinates w = rdr A dO; for computational convenience switch back

to using a = (pdq qdp) which becomes in polar coordinates a = 1r dO. Then

0 = V P implies 0 = + r2 so that (r,0) = y(r)e-~r2o. But this must

satisfy 27r-periodicity in 0 which it does only in the case r = J2n for some integer

n.

So there are no global polarized sections. We may then need to consider

generalized solutions of the form p~,(r, 8) = 6(r 2nh)e-\$ r2. When complex

multiples and linear combinations of these solutions are considered we are led to

interpret the quantum phase space as a space of square sunmable sequences. This

approach has in fact been pursued by Sniatycki[2].

CHAPTER 2
LINE BUNDLES

2.1 Definition of Line Bundles

A line bundle is an ordered triple (L, M, 7r) consisting of smooth manifolds

M and L, and a subjective smooth map r : L -* M. The map 7r has fibers,

1r-1 (x) for x e M, that are complex lines. Furthermore the following local triviality

condition is required: there exists an open covering {Ui} of M such that there exist

diffeomorphisms (i : Ui x C -* r-1 (Ui). These diffeomorphisms must preserve fibers

in the sense that 7r o (x, z) = x, and these diffeomorphisms must be linear on fibers

so each can be represented by a smooth nowhere vanishing function si : U -- L such

that

<& (x, z) = z Si (x).

Finally these maps glue together on overlaps of the open covering: there exist tran-

sition functions cj E C" (U, n Uj) so that cCjsj = si on Ui n Uj. Call a collection

{(Ui, si)} that satisfies the above condition a local system of the line bundle.

2.2 Sections

Denote by F (L) the set of smooth global sections of L. Thus s E F (L) if
s : M -+ L is smooth and 7r os (x) = x for all x E M. F (L) is a CO (M) -module

where

fs(x) =f (x)s(x)

for f E CO (M) and s E F (L).
The most elementary line bundle is the trivial line bundle. Form manifold
M x C, and define 7r: M x C M by n (x,z) = x. Then {(M, s)} where s : M -
M x C is the map s (x) = 1, is a local system. The set of sections F (M x C) is

naturally equivalent to the familiar set CO (M).

2.3 Connections

A connection on the line bundle (L, M, r) is a linear map

V: VecM -+ End (F (L))

satisfying

Vfx = fVx

for f E COO (M) and X eVecM, and

Vxfs = X (f)s + fVxs

when also s E F (L).

2.4 Metric Connections

A line bundle is said to possess a Hermitian metric if there is a smoothly
varying Hermitian inner product on its fibers. If s and t are sections of the line

bundle then < s I t > (x) = < s (x) I t (x) wheree s (x) and t (x) are elements of

r-1 (x) and < I >x is the inner product on the fiber 7r-1 (x). By smoothly varying
we mean < s I t > is a smooth function on M for any sections s and t.

Say V is a metric connection if

X < slt > = < Vxs t > + < s Vxt >

where X is a real vector field on M and s, t are sections of L.

2.5 Pullback Bundles

Given a line bundle (L, M, r) and a smooth map p : M -- N between smooth
manifolds M and N, the pullback line bmudle over N ( of line bundle L ) is the set

p*L = {(x,1) E N x L : p(x) = r()}

with projection j : p*L -- N given by r((x, 1)) = x. If {(Ui, s,)} is a local system

for L with maps Pi : Ui x C -- 7r -(Ui) given by 4i((y,z)) = z si(y), then define

P : p-'(UC) x C -- -l(p-1(U)) by Wi((x,z)) = (x,z si(p(x))). The fibres of p*L
are complex vector spaces with scalar multiplication

z (x, 1) = (x,z 1).

Thus we have a local system {(Vi, 3i)} for p*L where V = p- (Ui) and gi : Vi p*L
is defined by

Si(x) = (X, Si(p(x))).

Transition functions cij : Vi x Vj -- C are defined by Zj(x) = cij(p(x)) so that

cjij3(x) = aj (x)(x, Sj(p(x)))

= (x, Cij(P(x))Sj(p()))

= (x,Si(p())

and the c(j are clearly smooth.

2.6 Equivalence of Line Bundles

Let (L, M, 7r) and (L', M, 7') be line bundles. Then y : L -- L' is a line bundle
equivalence if y is a diffeomorphism that commutes with projection, 7r' o a = 7r, and
such that y is an isomorphism of complex vector spaces when restricted to fibres.
Notice y(s) = y o s E F(L') if s E F(L).

2.7 Construction of Line Bundle L on the Torus

Suppose we are given a collection of smooth maps

{E : R2 -- C}IEZ2

that satisfy the condition

eA+K(X) = EA(X + K.e,(x)

for all x E R2, A, K E Z2 where the right hand side is multiplication of complex

numbers. Then we use these functions to describe an action of Z2 on R2 x C. An

element A E Z2 acts on pair (x, z) E R2 x C via the prescription

X (x, z) H- (x + A, e\(x)z).

Using this action we form the quotient, which will be a smooth manifold:

L = (JR2 X C)/Z2.

An argument to this effect can be found in chapter five of Wells[4]. The torus M

is the quotient R2/Z2 where Z2 acts on R2 via A x = x + A. Form the subjective

smooth map

JR2 x C R2
7r Z2 Z2

by factoring the quotient the map (x, z) -+ x from R2 x C to R2.

Now we will verify that (L,M, r) is a line bundle. Let {Ui} be an open

contractible covering of the manifold M = R2/Z2. Any set Ui from this cover will

lift to disjoint components in the plane which are translatable onto each other by the

action of Z2. For a fixed Ui fix one of its components, call it U,. Define the map

s : U --+ L by

Si : [x] -+ [(x, 1)

where x E UV and [x] is the equivalence class of x in M. [(x, 1)] is an equivalence
class in L written with the representative of the first component taken from U'. In
this way form a function si for each of the Ui in the open cover, i.e. pick a particular
component associated with each Uj. Claim {(Ui, si)} is a local system for (L, M, r).
On connected components of Ui nUj the transition functions cij are defined as follows:

Cij ([x]) = E(x)

where the representative x of [x] in Ui n Uj is taken from U0 and A is such that x + A
is in Uj. Then

Si ([Z]) = [(x, 1)]

= [(x + A, EX (x) 1)]

= E (x) [(x + A, 1)]

= Cij ([a]) Sj ([x1) .

Let p : R2 R2/Z2 : x -+ [x] be the quotient map. Then we may form the
pullback bundle p*L. Define a map y : p*L --+ R2 x C by

(x, [(x, z)]) (, z).

By a standard result of differential topology since R2 is contractible, any line bundle

on the plane is equivalent to the trivial line bundle. Thus there is guaranteed to be

an equivalence between the line bundle p*L and the trivial bundle. The map 7 is

such an equivalence. The fact that this equivalence is not unique will be pursued

later.

In general if we are given a line bundle on the torus, we may pill it back

to the plane and form a bundle equivalence with the trivial bundle. If 6 is such

an equivalence, then given I C L, consider (x, 1); 6(x, I) = (x, z) for some z. Now

consider passing from the fibre over x in the trivial bundle over to the fibre over x in

p*L to the fibre over [x] in L to the fibre over x A in p*L to the fibre over x A

in the trivial line bundle. Since each step is an isomorphism of the complex line,

by composition we have produced an isomorphism of C, call this complex number

( ) (x). For a given A we can do this for all x. So we have a function ( : R2 -+ C.
We could pass from the fibre over x in the trivial bundle to the fibre over x K in

the trivial bundle and likewise we can pass from the fibre over x K in the trivial

bundle to the fibre over (x a) A in the trivial blmdle; the composition of these

two isomorphisms of the complex line will be the same as the isomorphisni obtained

by passing directly from the fibre over x in the trivial bundle to the fibre x (K + A)

in the trivial bundle. Written in terms of (, and (, this will provide exactly the

condition that makes {(j }EZ, a suitable collection of functions for producing a line

bundle.

For the line bundle L produced from the collection {eC} the particular trivial-

ization 7 we produced above will lead to exactly the {eI} if analyzed along the above

lines. It distinguishes itself from all the possible trivializations for this reason. Of

course, if we are given a line bundle without knowledge of functions used to construct

it, no trivialization is automatically distinguished.

2.8 Pulling a Section of L Back to a Complex Function

Let s be a section of the line bundle (L, M, 7r) formed using the collection of

functions {(E}. Then s pulls back to a section of p*s of p*L via

p*s(x) = (x, s(p(x))).

And we can push the section p*s over to a section of the trivial bundle with the

equivalence -y:

Y(*s)(xS) = (x,z)

where z depends on x. The section y (p*s) can be viewed as a function: 9 y 7 (p*s)

where p (x) = z if 7 (p*s) (x) = (x, z). Consider p(x + A) = z if 7 (p*s) (x + A) =

(x + A, z), which by our construction of p implies s (p (x + A)) = [x + A, z]. Now

p (x + A) = p(x) so s (p (x + A)) = s (p(x)). We know how to handle this in terms of
equivalence classes:

s (p (x + A)) = [x + A, z] = [x, e (a + A) z] = s (p (x)).

Passing back to ( this yields

S(tx) =a ( + A)z

that is

w (x) = e-x (x + A) ( (x + A).

Then by the properties of the es's,

E\ (x) o (x) = E6,x (x + A) (x) p (x + A)

x (X) i (X) = o (x) v (x + A).

That is

cP (x + A) = (,) W (a)

Call p : R2 -- C a e-quasiperiodic function if it satisfies

O(x + A) = (E ) W () .

2.9 Equivalence of Sections and Quasiperiodic Fnmctions

We have seen that a section of L maps to an e-quasiperiodic function; now we
show this is a bijection. If s, t E F (L) and s 5 t, then s (y) 5 t (y) for some y E M.
Let x E R2 be such that p(x) = y. Then p*s(x) = (x, s (p(x))) (x, t (p(x))) =
p*t (x), that is p*s 5 p't. In general if y : L1 --+ L2 is a bundle equivalence and
s, t L1 are such that s f t, then y (s) will not be equal to y (t). There exists x in
the base manifold such that s (x) f t (x); then

() ( x) ( = (s (x)) Y (t (x)) = y (t) (x).

Applying this to our situation, we have 7 (p*s) 7y (p*t). Thus the map from sections
to quasiperiodic fictions is one-to-one.

Given an e-quasiperiodic function p, view 'p as a section of the trivial bundle
on the plane, W,, (x) = (x, p (x)). Then apply the inverse of our bundle equivalence

y- -P(x) = (x, [x, W(a)]).

Suppose 7r(x) = y and define a s E (L) by s (y) = [x, W (x)]. The quasiperiodic
property of 'p ensures s is well-defined. Then s maps to 'p showing that our map is

onto.
We also have an isomorphism of C' (M)-modules. The e-quasiperiodic fmuc-

tions are a C' (M)-module in the sense that if f E C" (M) and p is a quasiperiodic;
lift to a doubly periodic function f and write f p (x) = f(x) p (x).

2.10 Curvature

Suppose we have a line bundle with connection (L, V). We form the operator

Rv on a pairs of vector fields

RV (X, Y) = [Vx, Vy] V[x,Y]

where X, Y EVecM and [X, Y] is the Lie bracket of vector fields. This is the standard

curvature operator and is C' (M)-bilinear in the vector fields. The right hand side is

an endomorphism of F (L), which we can naturally identify with a complex function

on L. Thus we may view Rv as a two-form on M. This complex twoform is called
the curvature of the connection.

2.11 Kostant Line Bundles

Now consider the case where the manifold M has the additional structure of

a symplectic manifold with symplectic form w. Call a Hermitian line bundle with

metric connection a Kostant line bundle if its curvature is w/ih. In the case of the

torus we will be looking at Kostant line bundles with curvature -27rinw for some

integer n. The general theory of Chern classes shows that the Chern class [-_ ],

where R is the curvature form of line bundle L, is an integral class. Furthermore if

[w] is integral then the integral classes of H2 (M) will be integer multiples of [w].

2.12 Connections on the Torus

Let (M, w) be a symplectic torus with line bundle (L, V) created with functions

{e\}. Due to the isomorphism between F (L) and the quasiperiodic functions we
may consider our connection as taking values in the set of endomorphisms of the

quasiperiodic fimctions. By then identifying the elements of VecM with doubly

periodic vector fields on the plane, we can consider the connection as mapping these

doubly periodic vector fields on the plane to endomorphisms of the quasiperiodic

functions. Given doubly periodic vector field X denote by Vx the endomorphism

associated to X by V. Given also quasiperiodic function p denote by Vxy the result

of the endomorphism Vx acting on p. Thus our notation will exactly match the

normal notation for vector fields on M and sections.

We will express our endomorphism Vx in the form of a partial differential

operator

X ke (X)

where k is some complex constant and E is some operator on the doubly periodic

vector fields.

Suppose 0 is a one-form in the plane so that at a point it gives a functional
on tangent vectors:

e (X)(x) = OxXx

for x a point in the plane. Then we show the formal condition

Ex+, o (T,), = O +(d

where T(x) x +, ensures that
where T\(x) = x + X, ensures that

Vx = X () kO (X) P

will again be a quasiperiodic function for any quasiperiodic function (p.

Vx (x + A) = X(p)(x + A) kO(X)(x + A)o(x + A)

= X,,+A() kO(X)(x + A)p(x + A)

= X(ex( ) ke(X)(x + A)V(x + A)

= X(EXP)(x) ke(X)(x + A),(x + A)

= X(e() (x) kE (X) (x + A) eA(x)P (2)

= X (eip) (x) kEOx+Xx+A~E (x) (x)

= X (eAX) (x) kE)+A((TA).X.)ex (x) P (x)

= X (e) (X) (- k (eOX + X() EA () (\ (x)
keA (x),)

= X (e~) (x) ke (X) (x) e6 (x) P (x) X (EA) (x) c (2)

= X (eC) (x) p (x) + A (X) X (P) (x)

k )(X) () Ex (x) p (x) X (E) (x) V (x)

=e (x) (X (() (x) ke (X) (x) ( (x))

= x (2) Vx y (x).

The fact that Vx defined as the above partial differential operator involving
the given one-form 0 is a complex linear endomorphism follows by the linearity of
multiplication and differentiation. We must show the two conditions

Vfx = fVx

and

Vxf V = X (f) W + fVx-.

where f is a doubly periodic function, X is a doubly periodic vector field and p is a
quasiperiodic function. The first condition is true as one-forms are C' (R2) linear:

fX (p) kO (fX) 9p = fX (p) kfE (X) p

= f (X (p) kO (X)) .

The second is the product rule for vector fields:

Vxfy = X(f)-) kO (X) f

= X (f) + fX (p) k) (X) fy

= X (f) 9 + f (X () k9 (X) p)

SX (f) + fVxw.

Given a connection defined this way we check that its curvature is -kdO:

[Vx, Vx'] = [X k(X),X' ke (X')]

= (X ke (X)) (X' kQ (X')) (X' kOe (X') (X kO (X))

= x(x' ke (x')) ke (x) X' + k2 (X) e (x')

(X ke (X)) + ke (X') X k28 (x') e (x)

= x (') x (ko (X')) ke (x) x'

X' (X) + X' (k (X)) ke (X')X

= [X, X'] kX (e (X')) k8 (X') X ke (X) X' + kX'(e (X))

+ ke (X) ke (X') x

= [X, X'] + k (X' ( (X)) x (e (X')))

[Vx, Vx,] V[x,x' = k (X' ( (X)) X (0 (X'))) + k8 ([X, X'])

= -kd(X, X').

2.13 Kostant Line Bundles on the Torus

Let Q be a constant nonsingular antisymmetric bilinear form on the plane,
that will factor to a symplectic form on the torus, and let a be a one-form such that
da = Q. Fix h =1 for 1 < n < oo and as usual denote h = Define

A (X) = exp (r (A) + (A, ))}
fi 2

where A E Z2, x R2 and r : Z2 --- R satisfies

r(A) + r(n) r(A + K) + Q (A, ) E 2Z

where also K E Z2. Then

EX,+ (x) = exp (r (A ) + (A + K, ))

= exp{ (r(A + ) + (A, ) + 0 (,x a ))}
Zh

= exp (r (A) + r (r,) + Q (A, K,) + Q (A, x) + Q (K, x))

= exp (r (A) + 0 (A, x + ) + r (K) + Q (K, ))}

= E (x + K) E, () .

Thus the family {ex} is appropriate for constructing a line bundle on the torus. Let

Lr be this line bundle. Now define connection V on the quasiperiodic fictions by

Vxy = X () + -a (X) p
ih

where h = 1/ (27rn). By what was said above this really is a connection and this

connection will have curvature

1
R = -(- )d
ih

= -2Iriuw.

In order to show this is a Kostant line bundle we must specify a Hermitian

metric on L. As is now our custom we describe the metric in terms of the quasiperiodic

functions. For quasiperiodic p and i define the function (p~1) by

(W}10) (x) = ((x) W(x))

where (1) on the right hand side is the usual inner product of complex numbers. Then
given quasiperiodic ip and I and doubly periodic vector field X, we have

= (X ) + a (X) )I+

+ (xI(M) + ( jp a (X) 0
~ ix

= X( cp)

since

Thus we have a Kostant line bundle.

2.14 Classification of Kostant Line Bundles

Have we displayed the most general Kostant line bundle? In order to an-
swer this question we must delve into the classification of Kostant line bundles on
symplectic manifolds. In the end we will find that we have given the most general
Kostant line bundle on the torus.
Equivalence of line bundles was defined above. If we add the requirement
Y(Vxs) = Vx (ys) where 7 is our equivalence, X is any vector field and s a section

( ) x ) = (x) .
ih = i

then we have an equivalence of line bundles with connection. This is an equivalence
relation on line bundles with metric connection and equivalent line bundles with
metric connection will have equal curvature. We may consider the subset of all

equivalence classes of line bundles with metric connection consisting of the equivalence
classes of Kostant line bundles K (M, w). In our situation we will show that for a
fixed Q and n every Kostant line bmudle on the torus has a representative of the form
L, for some r satisfying the requirement laid out before.

In Kostant's description of prequantization, the theory of line bundles is de-

veloped and the result F (M) H' (M,T) Hom(nf (M) ,T) is proved [5].

2.15 Tensor Product of Line Bundles with Connection

Given line bundles (L1, M, 7r1) and (L2, M, r2) form the triple (Li ,?> L2, M, r)

as follows: L1 0 L2 = UEC6A (i-1 (x) 7"21 (x)), the union of tensor products of

complex lines, and let ir map elements of L1 0 L2 to the unique x such that the given

element is in 7ri1 (x) 0 ir21 (x) If {(Ui, si)} and {(Ui, ti)} are local systnis for L1
and L2 respectively, where without loss of generality the local systems have the same

open cover of M (obtainable by refinement), then define maps

U, x C -* Uxeu, (7r1 (x) 0 7 (x))

by

(X, 2) z (Si (x) (9 ti (x)) .

The overlaps between these maps will be smooth and we have a line bundle on M,

which we will denote in short by L1 0 L2.

Given two line bundles with connection (L1, V1) and (L2, V2) we form the
tensor product with connection (L1 0 L2, V1 V2) by specifying

(V1 V2)X (s t) = (V1s) x t + s (Vt).

If V1 has curvature form RV1 and V2 has curvature form RV2, then V' V2 will
have curvature form R"v + R2.

2.16 Dual Bundles with Connection

If V is a complex line, then denote by V* the dual of V. V* is again a complex
line. If z E V, then denote by z* the unique element of V* that satisfies z* (z) = 1.
Given line blmdle (L, M, r) define the set L* = U.EM [71-1 (x)]*. Since every element
of L* will be in some 7r- (x)* for a unique x, let 7r* be the map that picks out this
x. Given a local system { (U, si)}, define maps s* : Ui -- L* by specifying that
s( (x) = [si (x)]*. The transition functions Cj = cy1 will be smooth so {(UI, s;)} is a
local system. Denote the dual line bundle of L by L*. If a is a smooth section of L*
and s is a section of L then denote by a (s) the function a (s) : M -+ C defined by

a (s) (x) = a (x) (s (X)).

Given a line bmndle with connection (L, V), form the dual line bundle with
connection L* with connection V* by specifying

(V*a) (s) = -a (Vs).

If V has curvature RV, then V* will have curvature -Rv.

2.17 The Group of Flat Line Bundles

Say (L, V) is flat if its curvature form is the zero form. Denote by F(M)
the set of equivalence classes of flat line bundles. This set in fact can be endowed
with a group structure. The group operation is induced by the tensor product of
line bundles with connection [Lo] o [Lo] = [Lo 0 Lb]. The trivial line bundle with
connection given by regular differentiation of vector fields is a representative of the
identity class. Given an equivalence class of flat line bmudles, the inverse equivalence
class is represented by the dual of a representative of the given class.

2.18 T (M) Action on K (M,w)

Y (M) acts on K:(M, w) on the right by [Li] o [Lo] = [L1 Lo]. This action
is transitive as [Li] o [L 0 L2] = [L2]. The action is regular, that is the stabilizer of

any element of K: (M, w) is trivial. If the action of [Lo] on K (M, w) leaves [L1] fixed,

[L1] o [Lo] = [L1], then we must show the class [Lo] contains the trivial bundle. Using
the facts that [L* L] contains the trivial bundle and that tensoring by the trivial
bundle does not change the equivalence class of a bundle, we see:

[Lo] = [L L L1 Lo] = [L 9 Li].

As the action is transitive and regular we may fix [L1] E K (M,w) and form a

bijection % (M) -- K (M, w) by '-* [Li] o 0.

2.19 Flat Line Bundles on the Torus

Consider the constant functions Ex (v) =exp{7rir (A)} where r : Z2 R satis-
fies r (A) + r (K) r (A + K) E 2Z. These functions satisfy the condition required to
construct a line bundle as described above. Call this line bundle L,. Make this a

line bundle with connection by specifying

Vx( = Xp

where V is a quasiperiodic fiction. Then (Lr, V) will be a flat line blundle. Let

L,, be another flat line bundle constructed as was Lr with r' : Z2 -- E satisfying

r' (A) +r' (K) r' (A + K) C 2Z and possibly different from r. Suppose L nad Lr, are
equivalent line bmudles. As noted before, equivalences of line bundles mavp sections

to sections so that we may consider the effect of the equivalence on the quasiperiodic

functions of each bumdle. Thus there will be a map 4 : R2 -- C* such that (1) maps

Lr quasiperiodic functions to L,, quasiperiodic functions by multiplication. This

constrains 4: if p is L,. quasiperiodic then 4 (V) should be Lr' quasiperiodic so

S() (x + A) = (x+ A)c(x + A)

= 4 (x+ A)exp {rir (A)} (x).

So we need

4( (x + A) exp {rir (A)} = 4 (x) exp {rir' (A)}

thus

( (x + A) = exp {rri (r' (A) r (A))} ( ( x).

If we further demand that 4 be an equivalence of line bundles with connections we

have the further restriction

4 (Vxo) = Vx (V0) ,

that is

4 (X ())= X (p)

but the Leibniz property of differentiation implies X (4) = 0. X was an arbitrary

doubly-periodic vector field so 4 must be constant. In order for the two restrictions

on 4 to coincide we must take

r' (A) r (A) e 2Z

for all A.

Let a : R -- be projection. Then a o r : Z2 -+ and ao r': Z2 -- are

homomorphisms. Our above argument implies that if Lr and Lr, are to be equivalent

line bundles with connection then these homomorphisms will be equal.

So far we have shown that every flat line bmudle constructed in the way Lr

was can be mapped to an element of Hom(Z2, T) and that equivalent line blmdles

with connection get mapped to the same element.

Now let 7 be an arbitrary element of Hom(Zi, T). Let f map T "linearly"
onto [0, 2). Then 3 o y : Z2 -- R satisfies

o 7 (A) + p o (K) p o 7(A + .) = P (7(A)) + P (7 (K)) ( (A).7 (/)) E {0,2}

and thus can be used to construct a flat line blmdle LOp Different elements in
Hom(Z2, T) correspond to different equivalence classes of flat line bundles with con-
nection so there is a bijection between the equivalences class of flat line bundles and
the elements in Hom(Z2, T).

2.20 The Classification of Kostant Line Bundles on the Symplectic Torus

We know HI1 (T2) j Z2 so that for M = 2,

F (M) 1 Hom (I1 (M),T) Hor (Z2, T)

so our technique for constructing flat line bundles has described all the flat line
bundles.

Now consider a particular Kostant line bundle as described above. To be very
particular, take r (A) = pv where A = (, v). Then tensoring all L,, in F (M) by

Lr,n we obtain all Kostant line bundles. In fact [Lr,n Lr,] = [Lr+,r,n]. In this way
we have described all Kostant line bundles.

2.21 Common Description

In practice it is often useful to work with the following data. A complex line
bundle L over M is represented by an open covering {U,} of M together with a family

{cij} of smooth complex functions with ci defined on Ui n Uj and satisfying ci, = 1

and cijcjk = Cik on Ui, Uj n Uk. A section of L is a family {si} of smooth nowhere

zero complex functions with si defined on Ui and satisfying si = cijsj on Ui n U,.

A connection V on the line bundle L is determined by a family of complex

1-forms {ai} with ai defined on Ui and satisfying

1
aj i a = --dincij
27ri

on Ui n U.

By the symplectic torus we mean (T2, w) where T2 is R2/Z2 and w = dp A dq.

We display a quantum line bundle: set

U1 = {(p,q) (-6, 1-6) x [0,1]}

U2-= {(p,q) E (6, 1 + 6) x [0, 1]}.

Set ai = pdq on Ui. Then

a1 a2 =

-dq,p E (-6,6)

0,pE (6,1-6).

Since al 02 = 'dlnc12 we must have

C12 = {

e2,p E (-6, 6)

1,P (6, 1- 6).

Thus the section can be identified with function ((p, q) with the quasiperiodicity

condition

Q(p + n, q) = e2 iqc(p,q)

4)(p, q + n) = 1D(p, q).

This is the form of the quantum line bundle and corresponding quasiperiodic func-

tions given in Mark J. Gotay's paper On a Full Quantization of the Tor Is[61. Our

quasiperiodic functions will have a different periodicity condition than Gotay's but

when we consider the ;syminpl tii torus with real polarizations parametrized by r we

will develop isomorphs to the quasiperiodic functions called Tr-twisted functions. It

turns out that Gotay's sections are exactly our T-twisted functions where 7 = 0.

CHAPTER 3
FOURIER ANALYSIS

3.1 Trigonometric Series

For our computation with quasiperiodic functions we find that some results

from Fourier analysis are useful. In this chapter we will describe the needed results.

The information needed is found in Katznelson [7]. We need the Riemann Lebesgue

lemma and some facts concerning the decay properties of Fourier coefficients. We

expand on the description of these decay properties given in Katznelson [7]

First some basic terminology and notation is established. We shall identify

the circle group with a half open interval in R, and then extend to periodic functions.

Thus we work with the complex valued fimctions with period one on the real line.

The most general subset of these functions we will consider is the integrable functions

L' (T): fE L' (T) if

SIf (x) \dx < oc.

Most of our functions will also be smooth, that is infinitely differentiable.

A trigonometric series is a formal expression

P ak2rikt
kEZ

The series consists of a complex sequence {ak}, no asstunption being made about
convergence. Given f E L' (T) we may form a trigonometric series with

ak = f (t) e-2iktdt.

This trigonometric series is called the Fourier series of f. If a trigonometric series is
the Fourier series of some function then call it a Fourier series. If only a finite number
of the terms ak of a trigonometric series are nonzero then call the trigonometric series
a trigonometric polynomial. Clearly trigonometric polynomials are functions, not just
formal expressions.

3.2 Fejir's Kernel

The density of the trigonometric polynomials in L1 (T) can be shown with
summability kernels. In particular use the Fejer kernel {K,} where

K., (t) x (E I '_') e27ikt.
k=-n +

Define

(n (f) (t) = (K, f) (t)

= Kn(t) f(t dx

n I(1- kJl ake2ikt
=-n n+

where {ak} are the Fourier coefficients of f. Then it can be shown that a, (f) -- f
in the L1 (T) norm as n cxo.

3.3 Riemann-Lebesgue Lemma

We now describe a proof of the Riemann-Lebesgue lemma, which says that
if {ak keZ are the coefficients of the Fourier series of a fiction in L' (T) then

limkij-,ooak = 0. This result follows from the fact that the trigonometric polynomials
are dense in the L' (T) norm, fI| = fo If (x)\dx.
To make the argument we must note a few properties of Fourier coefficients.
If f has Fourier coefficients {ak} and g has Fourier coefficients {bk} then f + g
has Fourier coefficients {ak + bk} If a is some complex number, and f has Fourier
coefficients {ak} then af has Fourier coefficients {fak}. If f has Fourier coefficients

{ak} then akI < f01 If (x) Idx for all k.
Now if we fix a L1 (T) function f there will be a trigonometric polynomial P
arbitrarily close to f in the sense of the L' (T) norm. Let f have Fourier coefficients

{ak} and denote the coefficients of P by {Pk}. There is some number N such that
for k with IkI > N then pk = 0. Then since jaki = \ak PkI < If PILI the norm of
ak is arbitrarily small.

3.4 Convergence of Fourier Series

So far in this discussion there has been no mention of when a Fourier series
converges and when a convergent Fourier series equals the fiction for which it is a
Fourier series. Fejer's theorem says that if the Fourier series converges at a point of
continuity of f then the value of the series at that point equals the value of f. Thus
if f is smooth, f e CO (T) then the Fourier series of f converges and is equal to f
as a fiction, that is pointwise.

3.5 Decay of Coefficients

Suppose {a,} is a sequence of numbers indexed by the integers. Furthermore
suppose this sequence has the property that

lim In'~ lan = 0
Inl-oo

for any natural number 1. For each natural munber j form sequence {(in)j an} in-
dexed by the integers. From the convergence property assumed for {an} we see

: nil anl < oo

for any natural number j. Thus the series

Y- (in)ja, 2"7e2

is a uniformly convergent series for any natural number j. Thus for each natural
number j we have a continuous function

(in) ane2inz

In Rudin the following result is proved:[8] If

CO

is given such that

N

-N

uniformly and

N

-N

converges at some point then

N

-N

and

(ZEf)'YZ f,

This result implies that

Sane27rin

is a smooth function with jth derivative

(in)j ane2 inx.

Now if we are given a smooth function f with Fourier coefficients then by

differentiation {(in)j n are all Fourier coefficients thus by the Riemann Lebesgue

37

lemma

lim n [n =0
17n--.oo

for any natural number j. Also

f U(x)= (in)j 7fe2rinx.

3.6 Fourier Series of Functions on R x T

In what follows we will have functions on the plane that axe periodic in one

of the coordinates and we will find it useful to apply Fourier analysis. Suppose f E

C (R x T). For a fixed x E R let {ak(x)} be the Fourier coefficient of f (x) E C7 (T)

and notice that since f (x) is smooth it is equal to its Fourier series. This is true for

all x E R so we have

f (x, t) = ak (x) e2ikt
kcZ

for x E R, t E T and {ak(x)} the Fourier coefficients of f(x,t) thought of as a

function of t.

CHAPTER 4
COHOMOLOGY OF SHEAVES

4.1 Functions Constant along a Subtangent Bundle

A subtangent bundle on the smooth manifold M is a smooth subbundle F of
the complexified tangent bundle TMC. As usual we will use the notation F (F, U)

when we speak of the smooth sections of F over the open subset U of M. Elements

of r (F, U) are vector fields on U; we may consider their action on smooth functions

defined on U. In particular we waut to pick out all smooth functions that look constant

to all the vector fields in F (F, U) Denote by CF(U) the set of p E C (U) that have

the property X (p) = 0 for all E F (F, U). If V is a subset of U then elements of

CF (U) when restricted to V are elements of CF(V); this is a map from CF (U) to

CF (V) which is called the restriction map. The sets CF (U) and CF (V) are both
complex vector spaces and the restriction map just described is a homomorphism

with respect to this algebraic structure.

4.2 Presheaves and Sheaves

What we are describing here is an example of a structure that has been gen-

eralized and given the name presheaf. One starts by assigning an object from an

algebraic category to each open subset of a manifold. We will use vector spaces for

the moment; that is what we used in the example above. Second, one needs mor-

phisms with respect to the relevant algebraic structure that are parametrized by the

subset relation on the open sets. in a transitive way. These maps, one from each sub-

set inclusion, are called the restriction morphisms of the presheaf. Not all restriction

morphisms are actually obtained by literal restriction, indeed not all objects assigned

to sets will be functions, but in the case of our example it happened to be literal

restriction. There are a few more requirements one may place on a presheaf. When

these requirements are satisfied we distinguish the presheaf among presheaves as a

sheaf. The requirements are that local information determines global information

and that local information demands corresponding global information. The sense of

global and local here comes from open unions U = U Ui; U is global, the U, are local.

By information we mean the elements of the objects assigned to open sets. Such

elements are commonly called sections. Explicitly a presheaf is a sheaf if for open set

U and open cover of U, U = Ui :

si = ti for all i implies s = t

where the si and t, are sections over the Ui and s and t are sections of U such that

they restrict to si and ti respectively over any Ui. And given a collection of si defined

over the Ui that agree when restricted to overlaps, there must exist a section s over

U that restricts to the si.

There are presheaves that are not sheaves, but there is no presheaf from which

a sheaf cannot be created. Given a presheaf the machinery of tale spaces is used

to generate a sheaf in a unique way. This means given the presheaf a new, possibly

different sheaf is created in such a way that the new presheaf is a sheaf. Importantly, if

the given presheaf already owned the title of a sheaf the generated presheaf is exactly

the same presheaf. In practice one will describe a presheaf and then immediately pass

to the sheaf generated by the presheaf without specifying whether the underlying

presheaf changed in the process.

We now do this with our example, the CF (U) for U open. In this case we

denote the generated sheaf by Cp. To recapitulate: sheaf CF assigns to open set U

the vector space of sections CF (U), also for any subset inclusion V C U of open sets
the sheaf CF produces a restriction mapping of the vector spaces CF (U) --* CF (V).

4.3 Differential Complex Based on Cp

We will be occupied with the task of reformulating the structures such as
the tangent bundle, cotangent bundle and the differential forms on our manifold
in a way that focuses on the subbundle F. In particular we develop a differential
complex that operates on sections of F instead of all vector fields. Later we shall
generalize the notion of a connection on a line bundle: instead of taking vector fields to

endomorphisms of sections of the line bundle, taking sections of F to endomorphisms
of the sections of the line bundle.

Along the lines of the usual construction of the differential forms fP as the
sets of sections of the bundle APT*M, we form for an open set U the set PF (U)
of sections of the bundle APF* over U, where F* is the dual bundle of F. Thus an
element of QP (U) is an alternating C(U) multilinear form on p vector fields from F.
In order to include p = 0 we say 0 (U) = C (U).

The space of vector fields on M is a Lie algebra with the usual Lie bracket
of vector fields. Say that subvector bundle F is involutive if whenever the bracket

of two vector fields that are sections of F is taken, a vector field that is a section of
F is obtained: X, Y E F (F) implies [X, Y] E F (F). Involutivity is necessary for a

differential of elements of Qf (U) to be defined. Define then

dF : F (U) '-*Q+(U)

by

p+i
(df') (XI, ...,Xp+l)= Z(-1)i+X. [a (X1, ...,Xi, ..., Xpi)
i=1

+ E (-1)+J a ([Xi, Xj],Xl,..., X,..., Xj,...,X l)
i

where the Xi are sections of FIU, a is an element of Qf (U) and the notation Xi means

leave out this element of the list. For the set, St (U) this gives the usual differential

of a fiction but restricted to sections of F : for E C (U), dFp = dWpr(F,U). By
definition dR o dF = 0. From all of this we have the differential complex

0 dF M dF dF dF
(M)(M) ...-- - (M) 0

where n is the rank of the subbundle F. From this differential complex we may define

cohomology groups, HP (Q* (M)) in the customary manner.

4.4 Sheaf Cohomology

There are also cohomology groups associated with a sheaf on a manifold. We
have already alluded to this sort of sheaf cohomology as a key aspect of our overall

plans in the introductory material. The cohomology theory of sheaves can be given

in a long list of the axioms of the cohomology followed by existence and uniqueness

proofs for the cohomology. The axioms are given in Wells, [4], and the proofs are

given in Gunning and Rossi [9]. For manifold M and sheaf S on M there is a sequence

of vector spaces HP (M, S) for p > 0, called the cohomology groups of the space M
of degree p with coefficients in S.

The cohomology group of degree zero is the space of global sections of the
sheaf S, Ho(M, S) = (M, S). Here F (M, S) is the space of global sections of the

sheaf which we may also denote as S (M) meaning the vector space the sheaf assigns
to the whole space M.
Now let us consider an example; we associate to the sheaf CF we intro-
duced earlier the cohomology group HP (M, CF) of M with coefficients in CF. Then
H (M, C) = r (M, CF) = CF(M). Is CF (M) different from CF (M)? That is, did we
have to add any global sections when we passed to the generated sheaf. In fact the
presheaf consisting of the CF (U) and the restriction maps is already a sheaf so there
is no difference between CF (M) and CF (M). Thus

H (M, CF) = (M, CF)

=C (M)

= CF (M).

Remember CF (M) = {o EC (M) : dplF = 0} = { E C (M) : dF = 0} Thus for
the differential complex

n (M) ( M) ... -, (M) M 0

which has cohomology groups HP (Q (M)) we have

HO(Q* (M)) = ker de : Q (M) --+ 1 (M)

= ker dF: C (M) -*1F (M)

= CF (M)

= Ho(M, CF).

So far, we have given no way of computing the cohomology groups H1p (M, CF)

for p > 0; we do know how to compute the cohomology groups HP(QP (M)). Encour-

aged by the equality Ho(M, CF)-=H (*f (M)) we may wonder if the equality holds for
higher cohomology so that computation of HP (f* (M)) provides a practical means
of calculating the HP (M, Cp). John Rawnsley shows that the equality holds when F
and dF satisfy certain condition that do hold in our case [10].

Moving closer to the problem at hand suppose we are dealing with a symplectic

manifold (M,w), a Kostant line bundle (L,V), and a polarization F. The global

sections of the sheaf CF are the functions on the manifold that are constant in the

direction of F in the sense of the usual differential. If we were dealing with a trivial

line bundle with connection given by differentiation in the direction of a vector then

CF (M) would be the polarized sections and the cohomology groups HP (M. CF) would
be where we would look if there were no polarized sections. On the torus our Kostant

line bundle is not trivial and the connection is not just differentiation in the direction

of a vector, so we need to modify our constructions to handle this new situation.

4.5 Sheaf of Sections of a Line Bundle

Let FF (L, U) be the space of sections of L over U that satisfy Vxs = 0 for

all X E Flu. For V C U map F (L, U) to FF (L, V) by restriction. This then is

the description of a presheaf and the global sections of the presheaf are the polarized

sections.

Let CF be the sheaf generated by the presheaf just described. It is important

here to note that the presheaf is already a sheaf so that L (M) = r (M,) =

rF (L, M) the polarized sections. Again we note that from the cohomology of M
with coefficients in F we have Ho (M, C) = F (L, M) the polarized sections. The

cohomology groups HP (.1, F) are what we need to look at but at this point we are

unable to compute them.

4.6 F-Connection

In the CF case the differential complex was formed by restricting the usual
differential to F and then extending to higher orders. By analogy we may consider
restricting our connection to F and extending to higher order.

First consider the restriction of connection V to vector fields from F. An

F-connection on L is a map from sections of F to endomorphisms of sections of L :

V: r(F) -+ End(r (L)).

When we were developing the Of,. we started with QRO = C (M). The differ-

ential taking an element from 02 to Q2 was dF : p -- d-pjF, and we viewed dplfF as

a section of the bundle F*. By analogy set S = r (L) and define a map from S --

Sk by s VFs where VF is the F-connection obtained from the connection V by

restricting V to sections of F. The object VFs is a F (L) -valued linear form on the

sections of F; this can be viewed as an element of F (F* L).

Now extend this new differential to higher orders by defining

S = r (APF* L)

and defining FO: SF S+ by

p+ 1
(aoF) (X,,..., XP,) = -(-1) +'Vx.(o(Xl,...,Xi,..., xl))
i=1

+ (-1) i+a OX X] X, ..., Xi, ..., X, A...,X,+)
i

for p > 0 and in the p = 0 case change notation VF = F.

Thus we have the differential complex

+ S, +...+S O0

where n is the rank of F. Note that H0 (S;) =kerOF : SF -, S1 =
{s E L : Vxs = O,Vx E r (L)} is the set we call the polarized sections of L.
Denote by FF (L, U) the sections of L over the open set U that are polarized
in the sense that s E FF (L, U) implies Vxs = 0 for all X E F (F, U). These form a
presheaf with restriction mapping given by the usual restriction mapping. Let SF
denote the sheaf generated by this presheaf and note that passing to the sheaf adds no
global sections: SF (M) = FF (L, M) = FF (L). Then we have the cohomology of M
with coefficients in SF, the polarized sections are the H (M, SF) and Ho (M, SF) =

H (S;) .
Rawnsley proves that HP (M, SF) = HP (S;) for all p, therefore we have a way
of computing the cohomology groups HP (M, SF) [10]. The proofs of these theorems
equating cohomology groups involve extensions of the Poincare Lemma for the usual
differential forms. This may be expected, as if in the first case we set F = TMc then
the fF are the usual differential forms and the sheaf CF is the constant sheaf denoted
C. The equating of cohomology groups is then the de Rham theorem which says
the differential complex of differential forms can be used to compute the cohomology

groups HP(M, C).

CHAPTER 5
COHOMOLOGICAL SOLUTIONS

5.1 A Kostant. Line Bundle on the Torus

Using the template developed in the chapter on line bundles we shall now

pick a particular Kostant line bundle on which the calculations in the rest of this

chapter shall be based. Let {p, q} be coordinates on R2 and let Q be the form

Q (v1, 12) = p1q2 qiP2 where v7, = (pi, q,) for i = 1,2. Define r (A) == pv where
A = (p, v) E Z2, and fix h = --. Then

E (v) = exp {iri (r (A) + Q (A, x))}

= exp {ri (pv + pq vp)} .

is a suitable family of fimnctions from which we can form a line bundle on the torus.

Since we will work with this line bundle throughout this chapter we will simply call

it line blmdle L on the torus M. The set of e-quasiperiodic functions on the plane

we shall denote by F and we will omit the reference to the functions E, j ist calling

F the quasiperiodic functions. In order to describe the connection on L we need to

specify a one-form 0 on the plane such that dO = Q. We use 0 = ( pdq qdp).

Then given a doubly periodic vector field X on the plane we define

Vxy = X (p) 2riO (X) op.

5.2 Translation iv'ariant. Real Polarizations on the Tonus

For a given real number r consider the doubly periodic vector field + 7r

on the plane. This spans a translation invariant real polarization on the plane with

leaves parallel to the line through the origin with slope r. We shall not consider

the polarization spanned by as its properties will be similar to the 7 = 0 case.

Otherwise we are considering all translation invariant real polarizations of the plane.

As doubly periodic vector fields on the plane the + T- correspond to vector fields

on the torus. These vector fields span translation invariant real polarizations on

the torus. Thus we have a polarization F, of the torus whose lift to the plane is

spanned by 9 + 7r. Since the connection we have defined with our Kostant line

bundle is described in terms of doubly periodic vector fields on the plane, we will be

using the vector field + 7 when computing the connection in the direction of the

polarization; denote this vector field by X,.

5.3 r-twisted Functions

For L a line bmudle constructed using a family of functions {E\}, in our de-

scription of the equivalence of the e-quasiperiodic functions with the sections of the

line bundle we used a line bundle equivalence y between the line bundle p*L and the

trivial line bundle R2 x C. This choice was not unique and we find that by choosing

other maps in y's place we may find other useful families of functions which can serve

as the sections of our line bundle.

A line bundle equivalence of trivial line bundles in the plane is representable

by a nowhere zero complex valued function on the plane, where sections are mapped

to sections by multiplication by the fiction. Define a, : R2 -- R by c, ((p, q)) =

p (q Tp) and define y, : R2 C by y7, = e7"i. We take the line bundle whose
sections are represented by quasiperiodic functions, elements of F, and look at the

equivalent bundle obtained by the equivalence represented by 7,, call the sections of

this new bundle the r-twisted functions and denote this collection of functions by
r, where we subscript the letter r because the equivalence depends on the parameter
r. Thus working with the r-twisted functions presumes a choice of polarization F,.

5.4 Polarized Sections

Given polarization F, on M, a section s of L is called a polarized section if
the connection in the direction of the polarization acting on s yields the zero section.
In terms of quasiperiodic functions we have

Vx, = X, (p) + ri (q 7p)

where X, is the doubly periodic vector field + 7r that spans the lift of F,, and

yp E r. The quasiperiodic function io represents an Fr-polarized section if

X, (W) + ri (q rp) = 0.

We compute

X1 (eFisOp) = X. (e rio) + eriaX, (X )

= 7riX, ((r)) e ior, + e"a'X, (,) .

We see

X, ()= + 7- (p (q -r-p))

p ( Oq p 5
= (q-- rp)-+-p -+r- (q-- rp)

= (q rp) + p (- + r)

= (q rp).

So

X, (e""op) = e"iT (X, (c) + 7ri (q rp) V)

e7ria, Vx.O.

Thus Vx, = 0 == ee ia, x,p 0 = X, (e"'" p) = 0. Thus the qiusiperiodic
function o corresponds to a FT--polarized section if and only if the corresponding
T- twisted function is constant in the direction of X,. This brings out the formal
utility of introducing the r- twisted functions.

5.5 Fourier Expansion of r-twisted Fmnctions

For E F,

5 (p,q + 1)

= 7 (p+ 1,q) (-r 1p) (p+ 1,q)

= eicr'(p+',q) (-y-'p) (p + 1,q)

S l( -rp+( +l))) (1.17)(1 (p + 1, q)

= ,,,',- rp)-rp+q-rp-r) (-yp) (p+ 1,q)

= ei(ac(p,q)-2rp+q-r)eii(q) (y-1) (p, q)

e= riar(p,q) ri(2q-2rp-r) (-1y ) (p q)

gri(2(q-Tp)-Tr)W (p, q)

= e7ian(pq+l) (7-y;') (p,q +1)

= e7i(p(+l-Tp)) (') (p, + 1)

= eri(p(q-Tp)+p) -1) (pq +)

= e7ria,(p,q)e.iprwi(-p) (T) (p, q)

= ~i, (Y ) (p, q)

W (, 9q).

(p+ 1,q)

And

Thus V E r, if and only if

y (p + 1, q) = eri{2(-rp)-}(p,q),

and

Y (p,q + 1) = p(p,q).

In particular

variable.

As discussed in the section on Fourier analysis, smooth periodic functions are

equal to their Fourier expansions. It will later be useful to use the periodicity in the

second variable to write p E Fr as

V (P) e2rinq
nEZ

where the cup's will be smooth functions of p.

Previously in our discussion of polarized sections we have demonstrated that

ye F, represents a F,-polarized section if it is constant in the direction of X,. In

this case we can write p (p, q) = h (q rp) for some smooth function h : R -- C so

that p is invariant under shifts of the form (n, rn). Now using the quasiperiodicity

condition for 7-twisted functions we have

h ((q Tp) + 1) = h ((q + 1) rp)

= (p, q+ 1)

= (p, q)

= h (q Tp)

thus h is a smooth periodic function with period one and is equal to its Fourier

expansion

h (t) = hke2wikt
kEZ

5.6 No Constant Tr-twisted Fumctions in the Direction of X,

Claim: If ( E FT and X, (o) = 0 then cp = 0.

If Xr (p) = 0 then p (p, q) = h (q rp) for some h E CO (R). A- noted in

the previous section

h (t) = hke2ikt.
keZ

53

The 7--twisted quasiperiodicity has another effect: if t = q rp then

h(t T)= h(q- (p+ 1))

= (p+ 1,q)

= eri(2(q-TP)-7)V (p, q)

= e7(2t-T)h(t)

Applying this property of h to its Fourier expansion we have

ZhkC27k(t-r) = ei(2t-r) hke2ikt
kEZ kEZ

Shk.e- 27ike27ikt = hke-ir e2ri(k+1)t
kEZ kEZ

k-2rikr 27rikt hke -WiT e27rikt
ke-e e E hk--le e
kEZ kEZ

Now equate coefficients

h e-27rikT kle-riT

What is important to note here is Ihkl = hk-1 Therefore if any coefficient is

nonzero, all of the coefficients are nonzero with the same norm. This would contradict

the Riemarm-Lebesgue lemma ( hk --- 0 as k --+ 0 ) unless hk is zero for all k in which

case h, and thus p, is trivial.

5.7 No Nontrivial Polarized Sections

In view of our correspondence between the r-twisted functions F, and the

quasiperiodic functions F and furthermore the correspondence between and the

sections of L we have: given 7 real, for the real translation invariant polarizations

F, whose lifts to the plane are spanned by X, = + there are no nontrivial

polarized sections.

In the usual formulation of geometric quantization the set of polarized sections

is the underlying object from which the quantum Hilbert space is constructed. In

cases where this set is trivial there are two lines of thought: one is to look for

distributional polarized sections, and the other is to look to the sheaf cohomology of

sections. In a later chapter we will consider the distributional approach. Now we

develop the cohomological approach.

5.8 Cohomology of the Sheaf of Polarized Sections

Let F be the sheaf generated from the polarized sections of L on open sets

where polarization is with respect to F = F, for some real T. Then HO (M, CF) is

the set of global polarized sections which in this case we have shown to be trivial,

H (M, F) = {0}. When there are no global polarized sections Bertram Kostant

suggested that one should look in the higher cohomology groups. By dimensional

considerations for the two torus HP(M, f) = {0} for p > 1. We must consider the

group H1 (M, CF), which we may compute using the differential complex

OF S1 0

where the first aF is 0F = V1' the F-connection obtained by restricting the connection

V to vector fields that are contained in F. The second 'F is the extension of 'F = VF

to higher orders which is zero here. We see H1 (M, CF) = H1(S*) is

VF(S)'

We make use of the fact that F* is one dimensional. Let X, be our usual basis
vector for F, and let p E F(F*) be such that p (X,) = 1. Then every element of S}
can be written in the form p : t for some t e r (L), Vfs is expressible as p Vxs.
So the F* part of the tensor on top and bottom does not effect the computation of

the cohomology group. That is

F(L)
H1 (M, LF) r (L)
ImVx,

We need to find what condition on t E F (L) implies t = Vxs for some s E r (L).

5.9 Solving the PDE

Given t E F (L), when does the equation Vx,s = t have a solution? Because of

the form Vx, takes, we will work with T-twisted functions. Given r-twisted function
V), when can we find a r-twisted function p that satisfies Xo = V?

Using the method of characteristics we have

S(p,q) = V(t, ,(p,q) +t)dt + h(Q,(p,q))

where h is a smooth complex valued function of a single real variable and Or is the
function Q, (p, q) = q rp.

The method of characteristics does not account for the periodicity conditions,
so we must impose these conditions as a second step. The function O must satisfy

P (p,q + 1) = p (p, q).

We have

p(p,q+ l)= q +(t,,q(p,q+ )+rt)dt+h(Q,(p,q+ ))

= p (t, rT (p, q) + rt) dt + h ( (p, q) + 1)

because V was assumed to be r-twisted. Thus if o is to also be r-twisted

h (, (p,q)+ l) = (p, q+ 1)- i? (t, r, (p, q) +rt)dt
oP
= p(p, q) (t, r, (p, q) + rt) dt

= h(, (p,q));

and hence h must be periodic with period one. Assume then that h is peri odic.
We also need

P (p + 1, q) = e'ri(2((p,q)-7)V (p, q).

We have

p(p+ 1,q)=
= /p+1

(t, Q, (p + 1, q) + Tt) dt + h (Q, (p + 1, q))

*0 (t, Or (P, q) T + Tt) dt + h (Q, (p, q) r)

If

p (p + 1, q) ei(2(q)-) (p, q)

= eii(2Q,(pq)-) (1) (t, Q (p, ) + ) dt + h (, (p, q))

Using the T-twisted nature of 0 we see

V (t + 1, Q, (p, q) + 7t) = e7i(2Q,(t,,(pq)+t)-7) (t, QT (p, q) + t).

And

2QS (t, Q, (p, q) + rt) 7r 2, (p, q) + rt -- vt T

= 2, p,q) -r.

So if we pass the exi(21"(pq)-r) through the integral we have

P (p + 1, q) = / (t + 1,0, (p, q) + rt) dt + e (20(pq)-)h (p, )).
J0

And then a change of variable in the integral gives

p+ 1
p (p + 1, q) = (t, QT(p, q) + rt r) dt + e"i(2ar(p,q)-r)h ( (p, q))
,1

Now subtract the two expressions for p (p + 1, q) :

0 = 0 (t, Qr (p, q) + Tt T) dt

+ h (Qr (p, q) T) e'i(2 p,)-r) h (, (p, q)).

Here p and q occur only in the function Q5 (p, q) which maps the plane onto the real
line. So we can treat Q, (p, q) as a single real variable, say z.

(t, z + t r) dt = e(2z-)h (z) h (z ) ,

or by taking w = z T we have

S(t, w + Tt) dt = ei"(2'+)h (w + r) h (w).

Given 0 this then is a statement about h. If h is periodic with period one and h
satisfies this last condition then p given by the method of characteristics is a solution
to X,~ = '.

5.10 The Change Of Spaces

The integral in the last expression in the previous section is key to oiur thinking

x F- 0 (t, x + rt) dt

is a periodic, with period one, complex valued fiction of one real variable. Let P
denote the complex valued functions of one real variable that are periodic with period

one. Then define as a transformation of functions T, : F, -- P given by

y 1 f (t, + rt) dt.

Now we define a subset P, of P to be those functions in P that satisfy the right hand

side of the last equation of the previous section: that is

P, = g E P : g () = ei(2x+T) f (x + 7) f () for some f E P}.

The results of the last section can be stated in terms of P, and T,. Given V) e r,

there exists o( Fr such that Xc = 40 if and only if T~, E P,. That is

T, oX,(r,) C P,

and

Tr-1 (PT) 9XT (,) .

The first of the above two statements means we can factor the transformation T,;

there exists TT such that

r, P
x,r, P,

The second fact in the above statement means the map T, is one-to-one. We will

show T, is an isomorphism. For this we show T, is onto so that also Tr is onto.

5.11 The Transform T, Is Onto

As before we represent the r-twisted function Wo as a sum:

p(p, q) = Z, (p) e2""in

We ask then, how does the T-twisted nature of the fiction express itself in the
functions pn? The --twisted periodicity condition is

p (p + 1, q) = e"i(2"(Pq)-T)o (p, q).

Applying this to the series we see

( (p + 1) e "in = (p) e2wi"eri(2(q-rp)-7)

= in (P) e2i(n+l)q e-ri2TP- iT

= - (p) e-iT (2p+l) 2ni(n+l)q

= n- ( e-ir(2p+)e27rin

Thus

. (x- + 1) = .-1 (x) e-wir(2x+l)

And for k an integer greater than 1

n, (x + k) = n-, (x k 1) e-ir-(2(x+k-1)+1)

= n-2 (x k 2) e-ir(2(x+k-l)+l)e-iTr(2(x+k-2)+l)

n- k ( -) i-riT(2(x+k-1)+l)e-7rir(2(x+k-2)+l) -rit(2x+ 1)

S_nk(X)(,-riT(2kx) e- r((2(k-1)+l)+(2(k-2)+l)+-+1)

= nk(X), -Tir(2kx)e -rir((2k-1)+(2k-3)+-.+l)

Then since

(14 3+5+...(2k-1))= k2

we have

4n (x + k) = ,n-k(x)e -ri(2kZ+k2)

When k = n this becomes

S(, (+ n) = 0()e-

So given io we can define ,n by

n (x) = (x n)e-rir(2n(x-n)+n2)

that is

n (x) = o(x n)e r(n2-2fx)

Next we consider how the n, should be defined so that we can recover the gn.

We see what the operator T, looks like on the Fourier representation of p. Consider

01
Tr[](x)-= J y(t,X+rt)dt

= 1 n(t)e21in(xs+t)dt

Thus we need to set up a T--twisted y with <, such that

n e(t)e2in dt = gn.
.0

Thus

S= o(t nr)eir(n2 -2nt)e27riTtdt

= o(t rn)e"7" 2dt

= 0 (t)eriT2dt.
J --?

That is, we need

/n 1o(t)dt = -e-~in2
.n

Let g E P. We build a p as follows: let B, be in Co'((n, n + 1)) such that

Jn n+1
Jn

B,(t)dt = 1

for all integers n and define

S^(x)= Bm() lx) e-T2

Then o is smooth as the B1, are smooth and have disjoint supports. Finally we
define the yn for all n in terms of 0o such that p defined by

p = Z n (q) e2"inp

will be 7--twisted. As before, this is done by setting

,, (r) = eoi(22nx)0 ( n).

Each of the n are smooth as a0 is smooth. Then we must show that the sum is
a smooth function. Note that for a given x we have ~ (x) = 2e-2',i7rT (B ) where
k depends on the interval of x and on n in a straightforward manner, in particular
n + 1 will correspond to k + 1 etc. Thus

I\ (x)I = Igkl IBk (x)

where K is the maximum value of Bk. Thus by the Weierstrass test 'p is a continuous
fiction.

We consider the higher derivatives now to show p is smooth. Consider

O I (p) e27 in {&gke-27"rinTBk (p) } (27in)b e2n'in
apa t9q' bI

(e27inTp a-j )
(e-- )- (Bk (P))

Ca) 7nbe2").

i )

Thus

a= qj

j=O

Ypa .Jqb

(27rnr)k Ma

where Ma = max sup -' (B (p)) And since
O

a
(2rn)b gkI
j=0

(a)

[)

(27rnr)k Ma < (27rn)b icl (1 + 27lrn)a Ma

Igkl Na+bha+b

for some constant Na+b, we have

I O O, (p) e2&in l ,N.+bna+b

Hence all derivatives converge and are smooth again by Weierstrass since 1 nd \gn <
oo for any d.
The r-twisted periodicity was built into the definition of the coeffcients.

a
(27rn)b 1kI j
j=0

a

J

We show that T, [p] = g.

T, [] (x) = f o(t,x+rt)dt

= (t) e27in(X+Tt)dt

= e27in I (t) e 2intdt

using dominated convergence to switch the summation and the integral. Continuing:

e27i / erir(n2-2nt)r^ (t "n) e27rifltdt

= e2i nx4 itn2t -) e2( irtdt

= e2in- 7- i-2 (t nt) dt

= e2rinx+Ti2 / B_ (t n) e- i"2dt

n e2nixn

5.12 The Set Po

We first consider the T = 0 case, where

Po = {g E P I 3f E P s.t. g(x) = (e2"i 1)f (x)}.

Claim g (0) = 0 is a necessary and sufficient condition for g E Po. Necessity is clear
from the definition of Po: given g E PO compute g (0) = (1 1)f (0) = 0.

66

For sufficiency start with g E P having Fourier coefficients g and assume

that g (0) = 0. In order for f E P with Fourier coefficients f/ to satisfy the equation

g (x) = (e21ix 1) f (x) the coefficients of f must be related to the coefficients of g
by the relationship j, = fn- fn. Define

f = g Ek for n > 0.
k>n

We could define the f/ this w\'-;I for all n and satisfy the relation b = - fn but

to be able show f with these coefficients is smooth we separately define J, when f is

negative:

f= k for n < 0.
k

This clearly guarantees the relationship gn = n- fn when n < 0; we must consider

the case of n = 0 for then we will be using both definitions of the fn. We would have

- fo = 9 k :9k-
k<-l k>0

Our assumption that g (0) = 0 implies that E n = 0 and thus

gk 9k = go
k<-l k>0

Now show fn defined thus are the Fourier coefficients of a fiction f in P. We

must show f with these Fourier coefficients is infinitely differentiable. Referring back

to the chapter on Fourier analysis we see we must show for all N,

nNfn 0
n-ooc

and

nNf -- 0.
n- -00

In the first case this is

n" k 0 as n -+ oo.
k>n

We know ,N+2gn -- 0 as n -- oo for any integer N. Thus we can choose n, so that

for n > nr we have nN+2;n1 < E (i.e. nNn < ). Then for n > n

2
fn n 'i k>n k>n k>n

And similarly for the n < 0 case.

By defining f in this way we have fn-1 fn = n so

2inx = ( 1 ) e2in

S- -1C2rinx E fne2: = Egne2lrin

E .27.i(n+l)x f2inx = e2in

(e2ix 1) Z fe27rinx = n e27inx

Since the definition of f required only that g (0) = 0 we see the sufficiency of this
condition.
This then means that Po =ker(evaluation at 0) so dimP/Po = 1.

5.13 L'H6pital Approach to Pn

The result Po =ker(evaluation at 0) was first derived by the author as an
application of an extended L'Hopital rule. That is, pose the question whether

g (x)
f () e2ix 1

can be defined at x = 0 as a smooth function. This method was completed but was
computationally hard. The work is shown in a separate chapter as it does have some
interesting twists. The simplified proof which was later found is given above.

5.14 The Set P,

Thus Po = {g E P g (0) = 0} For the general case

P, = {g E P: g (X) = eK(2x+)f (x + T) f (x) ,for some f E P}.

As both g and f are elements of P they are equal to their Fourier series and

g (x) = ei(2x+r) f (x + T) f (x)

becomes

g e27rinxz fe2rin(x+r)eri(2z+r) f27rinx

Then 2rin (x + r) + -i (2x + r) = 2irinT + ii- + 2ri (n + 1) x so we have

-n 2rinx = f 27ri(n+l) e 2inT+i- ne21ri

= le27rinzx 27i(n-1)rT+2ir- fn 2Krinx
S _--l e Je2 .

Here

2ri (n 1) r + 7ti7 = 27rin7 27rir + 7ritr

= 2rint 7rir

= 7ti (2n 1).

So the relationship of the Fourier coefficients of g and f needs to be

SriT(2n-1)
Hn -*- n-1 Jfn

We see

P, = g E P 3f e P s.t. n-,le"'(2"-) f = }.

Define a map S, : P P by

Sr (n = e-riTfn.2

We check

S (f)n-1 = e- r(n-2n+l) _1

S-_ STi n =e- [ei(2n1 l 1 f- f
Sr (f) n- ST (f)n = e- 2 [e( ) fn-1 rJ

S-win2 -
e gn

= g, {9)n.

S really is a linear automorphism as the coefficients fn and Sr (f) have the same

decay properties. Thus g E P, if and only if S, (g) E Po. So

P, = {g E P IS, (g)(0) = 0}

= gP e-7rin = 0 .
n=-oo

In particular P/P, = C for any r R.

CHAPTER 6
L'HOPITAL RESULT

6.1 The Question

Consider the equation

(e27x 1)f(x) = g(x).

We would like to apply L'H6pital's rule repeatedly to define f in the obvious way

and to see that it will be smooth. This is exactly what we do in the next section. We

will show that we must require simply that g(0) = 0. Since g is smooth and periodic

we can represent it by a Fourier series, g(x) = ZE,-o' ~ne2n" Thus the condition

for the r = 0 case is that E Coo = 0. We point this out to anticipate the general

condition.

6.2 A Generalized L'HMpital's Rule

We prove a lemma using the nth derivative product rule:

n

i=O

Let f and g be smooth such that f(l) (0) = g(') (0) = 0 for I < n and
for fixed integer n,

then:
1. (g2)() (0) = 0 for < 2n and (g2)2n (0) = (2n) (g() (0))2

2. (xg2)(') (0) = 0 for I < 2n + 1 and (xg2)(2n+1) (0) 5 0.
3. (gf- g'f)(0) () = 0 for 1 < 2n and (gf fg)(2n) (0)
= ( ) [g() (0) f(n+l) (0) gn( (0) f(n (0)].
For "3."

(gf'' g'f)(2n)() =

[(n ) (n)(0)f + [(20 [2n) g(n+)(0) f() (0).

And

[(n) 2n 2n 2n)
-n (n --1 1n+ n

2n 1

Now we use the lemma to prove the extension of L'H6pital's rule: let f and g
be smooth functions such that f (0) = 0 = g (0). Suppose h (x) defined by h (x) = f(
for x 5 0 and by L'H6pital's rule for x = 0 is continuous, and there is a positive integer
i such that (xg)(i) (0) 0, (xg)(" = 0 for 1 < i, and f(-1) (0) h (0) g(- 1) (0) = 0,
then h (x) is smooth.
We will proceed by induction with the following inductive hypothesis: the
following hold for positive integer n:
1. h(") exists and is continuous.

2. For y : 0, h(n) is of the form h(") = f/g where f and g are smooth functions
with f(O) = 0 = g(0), and f, g dependent on n.
3. There exists a unique integer in so that (xg)(i")(0) 5 0 and (xg)()(0) = 0
for this in we have

f(In-1)(0) h(n)(0)g(in-1)(0) = 0.

Note: (xg)(n) = xg(n) + ng('-1) so (xg)(n)(0) = 0 implies g(n-1)(0) = 0 and

(xg)(n)(0) \$ 0 implies g(n"-)(0) # 0. So for in in the inductive hypothesis g "-')(0)
0 and g(')(0) = 0 for I < in 1. Also since h(n) is defined and continuous a, x = 0 we
must also have f(')(0) = 0 for I < in 1.
Clearly h(') satisfies the inductive hypothesis. Suppose h(k) satisfies the in-
ductive hypothesis. h(k+1) exists and is defined by the quotient rule for 3- = 0. For
x = 0:

h(k+1)(0) = lim f/g (x) h(k) (0)
x-h( X
S f () -h ) (0) g (x)
= lim
-.o xg (z)
S f'(ik) () hk (0) g(k) (x)
x-o (xg (xi))

which exists.
For x 5 0, h(k+')(x) = (gf' g'f)/(g2) and (gf' g'f)(0) = 0 = (g2)().
It remains then to show that h(k+1) is continuous and that "3" from th( inductive
hypothesis holds.

From "2" of the Lemma we see ik+1 exists and ik+1 = 2ik 1. For "3" show

(gf' g'f)(ik+,-)(0) h(k+l)(0)(g2)(ik+l-1)(0) = 0.

For readability let m ik 1. By "1" and "3" of the Lemma:

(2m 1 g(m)f(+) g(m+l)f(m)](0) h(k+l)(0)[(2m)(g(m)(0))2] =
m+1

g(m)(0)(2m) [( )(f(m+l)
9 M M + (f

(m+l) ) (0) h(k+'l)(O)g(m)() =
g(M)

g(m) (0) ( )(f(m+) g(m+)h(k))(0) h(k+l)(0)g(m)(0)]
9 m Sm +-1

(since h(k) is continuous)

(m)g(m)((0)[ 1 (f(m+l)(0) g(m+1)(0)h(k)(0)) -
7:a + 1

f(m+-) h(k)(0)g(m+l)
h(k)()g(+l) )(0)g()(0)] = 0,
(xg)(m+')

since (xg)(m+l)(0) = (m + 1)g(l')(0).
Now show h(k+l) is continuous:

lir (gf' gf)/(g2) () h(k+l)(0)
x--O

.gf' (x) g'f (x) h(k+)(O)g2 (x)
= lihm r
x 0 xg2 (x)

Now (xg2)()(0) = 0 for 1 < ik+l,[(gf'-gf)() -h(+l)(0)(g2)(1)](0) = 0 for I < ik+-1,

and [(gf' glf)('k+l-1) h(k)0+)(0g)(ik+l-1)](0) = 0 by the argument given in the
justification of "3" above. So continue l'H6pital's rule until

= r (gf' glf)(ikf) (x) h(k+l)(0)(g2)(ik+1) ()
x-t (xg2 (X))(ik+)

which exists.

Using then this extended L'H6pital's rule we see if g(0) = 0 then f, defined

by

f( x) = i
e21ix 1

and by L'H6pital's rule for x an integer, is smooth.

CHAPTER 7
DISCONTINUOUS AND DISTRIBUTIONAL SOLUTIONS

7.1 Discontinuous Solutions

When the doubly-periodic vector F, = + Tq in the plae spans a line with

slope T rational, then the distribution spanned by Fr projected on the torus will have

leaves that are closed curves wrapping around the torus. Suppose r = ;./m where

gcd(n, m) = 1. Then F, spans the lines

{lb}C = q - b = 0
m

in the plane.

If we then consider --twisted fimctions on the plane, being covariantly con-

stant means being constant along every line lb. We know there are no smooth non-zero

covariantly constant fimctions, so we consider discontinuous functions. Certainly a

T-twisted function can be constant zero along any of the lines lb. We ask then are

there lines on which we can define constant non-zero functions that still are r-twisted

in the sense restricted to the line. For fixed real number b, on lb a T-twistcd fimetion

p would need to satisfy

S(m, n + b) = c(m, b)

= ei(2(b-r(m-1))-7) (m 1, b)

= eri(2b-r(2m-1))p (m 1, b)

= e2"7ibe -rir((2m-1)+(2m-3)+..+1)+ (0, b)

= e27imb e-Trirm2((0, b)

== e27r"ib e-inmp(O, b)

= ei(2mb-nm) (0, b).

Thus for p to be non-zero and covariantly constant along lb we would need

2mb nm E 2Z.

Solving for b we see

k n
b= -+-
m 2

for some integer k.

The lines are projected onto the torus so we must ask how many distinct closed

curves on the torus satisfy this condition. Claim the set {lb}0b<- covers all the

leaves of the polarization. Since by assumption gcd(n, m) = 1 we know there exists

an integer j such that jn = 1 (mod m). We see that the point (a, ') on the line lb

projects to the same point on the torus as 0, ) on the line l1. Thus 10 and l_

cover the same closed curve on the torus.

There is only one line lb, with b of the form

k n
b = + ,
n 2

in the set of lines {lb}0o,
termined by its value at any one point on lb as the value at any other point will be

determined by parallel translation. Thus the set of covariantly constant discontinuous

sections is one-dimensional.

7.2 Bohr-Sommerfeld Set

We may define a map Q from the set of closed curves on the torus to the

complex numbers with unit modulus by sending y : [a, b] -*- M to the complex

number Q (7) that described the linear isomorphism of Ly(a) = Ly(b) obtained by

translating points of Ly(a) about 7 by parallel translation. All Q (7) are of muit

modulus since our connection is compatible with the Hermitian metric. Also Q (7)

is independent of parametrization and independent of the initial point chosen on 7.

When we have a polarization the set of leaves of the polarization for which

Q (y) = 1 is called the Bohr-Sommerfeld set. It is usual when there are no smooth

global polarized sections to expect the quantum phase space to be C cross the space

of connected components of the Bohr-Summerfield set. This is exactly what we have

seen to be the case on the symplectic torus with translation invariant real polarization

defined along lines in the plane with rational slope. It is well known that lines in the

plane with irrational slope do not project to closed curves on the torus: they project

onto lines that wrap densely on the torus. Thus there is no Bohr-Somnrerfield set

and the usual method of discontinuous functions does not work.

7.3 Distributional Solutions

We look for distributional solutions, where by distribution we mean a linear

fictional on the set of sections of L. Polarized sections would be solutions to the
equation VFS = 0. Therefore we should look for a linear function u such that

VFU = 0. As usual this means u (VFs) = 0 for all s E F (L) This specifics u on the
subset of F (L) consisting of sections t such that VFS = t for some section s; in fact

for such t, u (t) = 0. But we have already seen this subset of F (L); we previously saw
it as the isomorph of the image of VF in F (SF(L)) =Sec(F* 0 L). Likewise we have

already considered a linear function on F (L) whose kernel contains this subset. This

linear function would appropriately be called a distributional solution. Since we also

saw that the kernel of this linear function was exactly F (S]i (L)) we see the space of

distributional solutions is one-dimensional.
Now we shall describe our distributional solutions more explicitly. Define

S: r (L) -+ C

as the composite linear map

F(L) F T P P4 C.

Here t, is the map that associates sections with T-twisted functions,

T, ()(x) = o(t,x+r t)dt,

ST (f), = e- f

and eo is evaluation at zero.

CHAPTER 8
FURTHER QUESTIONS

8.1 Other Kostant Line Bundles on the Torus

Our calculations of the cohomology of the local polarized sections were based

on the line bundle constructed with h = -1. In the chapter on the general theory of

constructing Kostant line bundles on the torus we noted that h could take values

for any positive integer n. Here we consider how choosing n > 1 would affect our

calculations. The line bundle we use is created with transition functions

e\ (x) = exp {rrin (r(A) + f (A, x)}

where r (A) = AA2 and we use connection:

Vxy = X (p) 2rin0 (X) y,

expressed in terms of quasiperiodic functions.

The T-twisted functions were introduced so that when given a polarization

F,, covariant differentiation in the direction of the polarization Vx, would take the

simpler form X,. For h with the integer n introduced, the biuidle equivalence pro-

ducing the r-twisted functions must be changed to multiplication by the function

erina. This modification will produce the same simplification of derivation in the

direction of F,.

With this modification for p E r, we have

o (p + 1, q) = e"ri(2(q-'p)--) (, q).

And

o (p, + 1) = p (p,q).

In particular must be periodic with period one in the direction of q an([ as before

the Fourier expansion for p can be used.

As before p e Fr represents a F,-polarized section if it is constant in the

direction of X,. We can write p (p, q) = h (q Tp) for some smooth function h :

R -+ C so that (p is invariant under shifts of the form (m, rm). Now this h is

periodic as before

h ((q- rp) + 1) = h(q -rp)

thus h is a smooth periodic function with period one and is equal to its Fourier

expansion

h (t) = E hke27rikt
keZ

Claim: If y E F, and X, (p) = 0 then p 0.

The T-twisted quasiperiodicity has the effect: if t = q rp then

h (t r) = ein(2t-)h (t) .

Applying this property of h to its Fourier expansion we have

hke--27ikre27rikt = hk-e-ire 2rikt.
kEZ kEZ

Now equate coefficients hke-2'ikT hk-ne-i
Again we have hk = hk-n, and the Riemann-Lesbesgue lemma

As before te the first cohomology group is isomorphic to the group P/Pr, but is

this case

P, = {g E P | 3f s.t. rin(2x+r)f (X + T) f () = g()) .

In the case of T = 0 this becomes

Po = {g E P 13f s.t. (e2inx 1)f(x) = g(x)}.

We can see quickly from the L'Hopital's rule argument that this means we must

require g (m) = 0 for xm = -,, m = 0, 1,... ,n 1. Thus the quantum module will
be n-dimensional.

8.2 Asymptotic Solutions

In this work we have seen that we cannot find non-trivial solutions to the

linear differential equation Vps = 0. In the chapter, "Differential Operators and

Asymptotic Solutions", Guillemin and Sternberg describe techniques that may have
relevance in our situation [11]. Following their development we consider Isymptotic

solutions to our equation.

Guillemin and Sternberg describe the Luneberg-Lux-Ludwig technique for

finding asymptotic solutions to linear differential equations. We apply this tech-

nique as far as possible to our situation. In future work we hope to modify the

technique so that an asymptotic solution can be found. For computational reasons

the search for a solution is restricted to finding a solution from among the simple

asymptotic solutions. For our purposes we take the simple asymptotic sections of

line bundle L to be the formal expressions of the form

00

n= (it

where the u, are sections of L,

variable in (0, oo).

By differential operator of degree zero on F (L) we mean a linear map

L: r (L) F (L)

that commutes with multiplication by functions, that is

[L, f] = Lof foL = 0

for any smooth function f on M. Say linear map

L : F (L) F (L)

is a differential operator of degree one if [L, f] is a differential operator of degree zero

for any function f. For example the linear map Vx is a differential operator of degree

one, since for any function f :

(Vx o f)(s) (f o Vx)(s)
Vx (fs) fVxs

X (f) s + fVxs- fVxs

X(f)s

and thus for any two functions f and g :

[[Vx,f] ,g] (s)

([Vx,f] o g)(s) (go [Vx,f])(s)

[Vx, f] (gs) g [Vx, f](s)

X (f) gs gX(f)s

By induction say map

L: r (L) -- r (L)

is a differential operator of degree k if [L, f] is a differential operator of degree k 1
for any function f.
Differential operator L induces a map on the set of simple asymptotic sections
by

Lu = L(eitvun(it)-n).

[Vx,f] (s)

By an aymptotic differential operator we mean the formal expression

o Lk
S= O (it)k

where the Lk are differential operators from F (L) to F (L). It is usual to also require

that deg Lk < k and that for any positive integer I we have deg Lk < k 1 for all

but finitely many k.

As an example we consider the reduced wave equation:

1 + Au=0.

The operator 1 + (1/t2)A fits the definition of an asymptotic operator since multipli-

cation operators are degree 0 and the operator A is degree 2.

In our work we are considering covariant differentiation in the direction of a

given polarization; denote X = X,. Since Vx is a degree 1 operator we consider the

asymptotic operator L with

L Vx
L-
it

In analogy to our polarization condition we look for simple asymptotic solu-

tions to the equation

Lu = 0.

86

In our particular example we see

V (et" S = ezti X () uo+ + x (X ( p) Un +VXu,-1
it (i)- (it

Notice that the right hand side is again a simple asymptotic section Lu = v where
we denote

(it)"

Thus we must find a u so that the v, are all zero.
Notice that 'o depends only on an expression involving o and on uo. This is the
case in general and the expression involving cp is called the characteristic equation.
In general the characteristic equation will be in the form

vo = ( (L) (dp) uo

where a (L) is the symbol of [L] which is defined to be a (Lk) and

a (LE) (d() = [[Lk,(],... ,]
k! k brackets

In the examples mentioned above, for the wave equation

(L) (dp)= 1- |1 V|2
2

so that

o = 1 I|V )| uo.

On the symplectic torus with polarization F

Vo '-= (Xp)uo.

We will return to the characteristic equation for the torus in a moment.

First we consider what remains after solving the characteristic equation. We

see from Xp = 0 that

7vn = Vxun-1

for all n > 0. This is an example of what is called the sequence of transport equations.

Thus here solving the transport equations, that is solving

VXUn-1 = 0,

is the same as finding polarized sections. Hence even if we can solve the characteristic

equation we are back to a familiar impasse with the transport equations.

Nevertheless the characteristic equation is of interest because it reacts to T

being rational or irrational. If p is to be a solution of the characteristic equation

it must be a smooth function on the torus that is constant in the direction of the

polarization. Pulling back to the plane we see V must be a doubly periodic fiction

that is constant in the direction of X,. So

'(p,, q) = h(q Tp)

for some smooth function h. As before we use Fourier representation of the function.
Since p is periodic in q, h must be periodic with period 1 so we may equate h with
its Fourier series

h(x) = .^ne2inx.

The fact that V (p, q) is also periodic in p means that h must be periodic with period
7 as well as being periodic with period 1. Then

h(x + r) = h (x)

&C 2-rin(x-r) 27in

Se2 inrfne2rinx = e27rinx

27rinhn hn

This implies that hn = 0 for all n except n such that nrT Z. Thus the chluiacteristic
equation has non-constant solutions only in the case that 7 is rational.

8.3 Pairing

Given any translation-invariant real polarization on the torus we have seen
how to associate the the system a vector space, coming from the cohomology of the

sheaf of polarized sections. This vector space will have dimension n whel h = 1/n.

Further study is required to answer the question of what is the appropriate way to

put an inner product structure on this vector space and the question of how to relate

the vector spaces obtained from different translation invariant real polarizations.

The idea is to find a unitary transformation between the different vector spaces

that commutes with the operators on the vector spaces coming from observables on

the torus. If Vk for i = 1,2 are the vector spaces corresponding to two different

polarizations on the torus and f E CR (M) induces operator bi on Vi then we wish to

define a unitary transformation

Ui: Vi V2

such that

61 = U-162U.

This can be acheived by defining a pairing in the sense that there is a sesquilinear

map from V1 x V2 -* C for any two polarizations such that the transformation induced

by the map is unitary and commutes with operators coming from observables.

If such a pairing exists then it can be used to enlarge the set of observables

that can be quantized for a particular polarization. For example using I he pairing

we could quantize observables that preserve any one of the affine polarizations and

also observables that take one affine polarization to another.

REFERENCES

[1] P.A.M. Dirac, "The Fundamental Equations of Quanitum Mechanics,"
Proc.Roy.Soc.London ser A. 109, 1926, pp. 642-653.
[2] Jedrzej Sniatycki, Geometric Quantization and Quantum Mechanics, Springer-
Verlag, New York, 1980.
[3] Morris Hirsch, Differential Topology, Springer-Verlag, New York, 1]!&8.
[4] R.O. Wells, Jr. Differential Analysis on Complex Manifolds, Springer-Verlag,
New York, 1980.
[5] Bertram Kostant, "Quantization and Unitary Representations Part I." Lectures
in Modern Analysis and Applications III, Springer-Verlag, New York. 1970.
[6] Mark J. Gotay, "On the Full Quantization of the Torus," In Quantization, Co-
herent States, and Complex Structures, Plenum, New York, 1995.
[7] Yitzhak Katznelson, An Introduction to Harmonic Analysis, Dover, New York,
1976.
[8] Walter Rudin, Principle of Mathematical Analysis, McGrew-Hill, New York,
1976.
[9] R.C. Gumning and Hugo Rossi, Analytic Functions of Several Complex Variables,
Prentice-Hall, Englewood Cliffs, NJ, 1965.
[10] J.H. Rawnsley, "On the Cohomology Groups of a Polarisation and Diagonal
Quantization," Transactions of the American Mathematical Society, 230, pp235-
255, 1977.
[11] Victor Guillemin and Shlomo Sternberg, Geometric Asymptotics, American
Mathematical Society, Providence, RI, 1990.

BIOGRAPHICAL SKETCH

Scott Chastain was born in Danville, Indiana, on April 4, 1970. He graduated

high school from Cypress Lake High School in Fort Myers, Florida, in 1!1.-8 and was

named a famous Cypressonian. He received a Bachelor of Science degree with high

honors from the University of Florida in 1991. Upon graduation he was elected a

member of Phi Beta Kappa. He then completed a Master of Science degree from

the University of Florida in 1993 after which he began his further studies under the

direction of Paul Robinson.

I certify that I have read this study alnd that in my opinion it conforms ,o
acceptable standards of scholarly presentation and is fully adequate, in scope alid
quality, as a dissertation for the degree of Doctor of Philosophy.

dL4Q
Paul Rolbinion. Chairman
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.

Gerard Enich
Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms :o
acceptable standards of scholarly presentation and is fully adequate, in scope ardl
quality, as a dissertation for the degree of Doctor of Philosophy.

.lohi Ilauder
Pro essor of Mathematics

I certify that I have read this study and that in my 'opinion it (conilfrms o
acceptable standards of scholarly presentation and is fully adequate, in scope ai d
quality, as a dissertation for the degree of Doctor of Philosophy.

2,1W -rr^y[ _. _
Scott McCu'lloiugh
Associate Professor of Mathematics

I certify that I have read this study and that in my opinion it conforms 0
acceptable standards of scholarly presentation and is fully adequate, in scope ai ,1
quality, as a dissertation for the degree of Doctor of Philosophy.

Pierre a r
Professor of Physics

This dissertation was submitted to the Graduate Facuilty of the Department of
Mathematics in the College of Liberal Arts and Sciences and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Docror
of Philosophy.

December 1998

LD \
1780
1999

UNIVERSITY OF FLORIDA
I 1 III IIII IIIIIIII ll51111 l II
3 1262 08555 1041