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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. TABLE OF CONTENTS 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 RiemannLebesgue 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 FConnection .................... ...... .. .. 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 7twisted Functions . . . . . ... 50 5.6 No Constant rtwisted 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 BohrSommerfeld 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 BohrSommerfeld sets. This technique works producing a onedimensional 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 BohrSommerfeld 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 selfadjoint 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 oneforms 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 BohrSommerfeld set. In Chapter 7 we describe the BohrSommerfeld set on the symplectic torus and see that for irrational polarizations the BohrSommerfeld 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 (qp)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 halfdensities. Thus the inner product is the usual inner product of halfdensities 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 27rperiodicity 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, 1r1 (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 * r1 (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 r1 (x) and < I >x is the inner product on the fiber 7r1 (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(p1(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) = ex (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 equasiperiodic 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 equasiperiodic 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 onetoone. Given an equasiperiodic 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 welldefined. Then s maps to 'p showing that our map is onto. We also have an isomorphism of C' (M)modules. The equasiperiodic 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 twoform 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 oneform 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 oneform 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 oneforms 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 oneform 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 (i1 (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 [711 (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 doublyperiodic 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 1forms {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, 16) 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 (6,16). 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 Trtwisted functions. It turns out that Gotay's sections are exactly our Ttwisted 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) e2iktdt. 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 RiemannLebesgue Lemma We now describe a proof of the RiemannLebesgue 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 Inloo 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 FConnection 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 Fconnection 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  dpjF, 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 Fconnection 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 equasiperiodic 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 oneform 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 rtwisted Functions For L a line bmudle constructed using a family of functions {E\}, in our de scription of the equivalence of the equasiperiodic 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 rtwisted 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 rtwisted 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 Frpolarized 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 rp)) 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 FTpolarized 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 rtwisted 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+qrpr) (yp) (p+ 1,q) = ei(ac(p,q)2rp+qr)eii(q) (y1) (p, q) e= riar(p,q) ri(2q2rpr) (1y ) (p q) gri(2(qTp)Tr)W (p, q) = e7ian(pq+l) (7y;') (p,q +1) = e7i(p(+lTp)) (') (p, + 1) = eri(p(qTp)+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
As before te the first cohomology group is isomorphic to the group P/Pr, but is
