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Topological field theories and fractional statistics Bergeron, Mario 1993

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TOPOLOGICAL FIELD THEORIES AND FRACTIONAL STATISTICS By MARIO BERGERON B. Sc. (Physique) Universite Laval, 1987 M. Sc. (Physics) University of British Columbia, 1989  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  September 1993 ® MARIO BERGERON, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  ^ Department of  19(-05 1(5  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  scp-FEHOEgillirr7  Abstract  We examine the problem of determining which representations of the braid group on a Riemann surface are carried by the wave function of a quantized Abelian Chern-Simons theory interacting with non-dynamical matter. We generalize the quantization of ChernSimons theory to the case where the coefficient of the Chern-Simons term, k, is rational, the Riemann surface has arbitrary genus and the total matter charge is non-vanishing. We find an explicit solution of the SchrOdinger equation. We find that the wave functions carry a representation of the braid group as well as a projective representation of the discrete group of large gauge transformations. We find a fundamental constraint that relates the charges of the particles, q, the coefficient k and the genus of the manifold, g. We study the non-linear sigma model with a Chern-Simons term. We find the canonical structures of the model using Dirac bracket, accounting for the non-trivial constraints of the sigma-model. We show that solutions to the field equation are represented by solitons. We also recover braid group representations for the low energy limit of soliton exchange.  Table of Contents  Abstract  Table ar coiAte 15  1  -  List of Figures Acknowledgement  vii  1  Introduction  1  2  Selected topics in topology  8  3  4  2.1  Manifolds, exterior calculus and vector bundles  2.2  DeRham cohomology, homology and homotopy ^  15  2.3  Characteristic classes ^  19  2.4  Two dimensional topology and Riemann surfaces  2.5  Three dimensional topology and the braid group ^  ^  ^  8  22 27  Abelian Chern-Simons theory  32  3.1  The decomposition of the gauge field ^  35  3.2  Quantization and the SchrOdinger equation ^  40  3.3  Large gauge transformations ^  45  3.4  The braid group on a Riemann surface and Chern-Simons statistics  48  3.5  Path integral quantization ^  51  The CP(1) model and Abelian Chern-Simons theory  55  4.1  57  The Hopf map and the CP(1) model ^  111  4.2 Canonical structure of the CP(1) model ^  66  4.3 Quantization and the SchrOdinger equation ^  74  4.4 The braid group and fractional statistics . .  ^77  4.5 Discussion  ^86  5 Advanced topics^  89  5.1^The WZW model . . . .^.....^................ 5.2 Non-Abelian Chern-Simons theory ^ 6 Conclusion^ Appendices  Bibliography  89 93 97  ^  A Theta functions  ^  99  ^  99  ^  103  iv  List of Figures  2.1 Example of a genus 3 Riemann surface ^  23  2.2 Holonomy of a Riemann surface represented by loops am and bm intersecting at 13Th, 2.3 Example of a braid  ^23 ^  2.4 Example of the generator a-3 and cr;1 ^  28 29  2.5 Equivalent braids representing the relation (2.5.1) for i = 2 and j = 6. . ^ 29 2.6 Equivalent braids representing the relation (2.5.2) for i = 3. . . . . . . ^30 4.7 The solid torus T+ as a fiber bundle. ^  60  4.8 The fiber from Figure 4.7 deformed to the boundary of T. ^ 4.9 The fiber on T+, shown in Figure 4.8 becomes a contractible loop on T ^  61  4.10 Two fibers on T. ^ 4.11 We deform one of the fibers to a loop at the boundary of T+ as shown. In order to do this a small loop linking the other fiber must appear. ^ 62 4.12 The fiber on the boundary of T+ in Figure 4.11 can be contracted to a point inside T_. The remaining loop links the other fiber ^ 62 4.13 The remaining fiber in Figure 4.12 can also be contracted to a point inside  T, leaving another loop in T. We are left with two loops linked once in  T+ ^  63  4.14 Soliton exchange braids neighboring ribbons ^  82  4.15 Rotation of a soliton in the plane twists its ribbon ^ 83 4.16 Exchange of soliton constituents. ^ v  84  4.17 A positive or right crossing. ^ 4.18 A negative or left crossing ^  vi  84 85  Acknowledgement  I owe a lot of gratitude to many people for the work of this thesis and more generally for all that I have learned in my stay at UBC. I would like to thank my supervisor Gordon Semenoff for his constant support, supply of research projects and many discussions. His deep understanding of many aspects of physics has been an invaluable learning experience. I would like also to thank Roy Douglas for introducing me to algebraic topology and the rigor of mathematics. I greatly benefitted from the many discussions and friendship of follow graduate and post-graduate students. In particular, I would like to thank David Eliezer, Richard Szabo and Edwin Langmann. I am grateful to all the members of the physics department and its staff for making my stay at UBC as friendly and pleasant as possible. I wish to thank also the Niels Bohr Institute in Copenhagen for their hospitality during my short stay over there. Finally, I would like to thank my parents for their endless encouragement and support.  vii  Chapter 1  Introduction  In recent years, the role played by topology in physics has led to the discoveries of interesting new models, which might explain new features of physical systems. Mathematically, the topology is a tool used to obtain global information about various spaces, leaving the local details out. This is, by definition, not a complete or even very deep description of a given space. The importance of topology comes from the fact that it is often the only information available. In physics, we are also often faced with a problem whose solution is rather intractable (no algebraic solution and poor numerical calculations). In some of these cases, the topology might be a tool that will give us some information. More importantly, it has become clear that topological features of field configurations play a fundamental role in the quantum description of physical models. In particle physics, the study of anomalies in gauge field theories [14, 76] has shown that they are related to characteristic classes, a relation best expressed by the celebrated Atiyah-Singer theorem [5, 15]. In this thesis, we will investigate a different application of topology related to spin and statistics in quantum mechanics. Since the development of quantum mechanics, the spin of a particle has been classified in two distinct classes. On one hand there are bosons corresponding to an integer spin, for which the wave function is invariant under a rotation of the system of coordinate. On the other hand there are fermions corresponding to an integer and a half spin, for which the wave function changes by a minus sign under a full rotation of the system of coordinate. The statistics of an assembly of identical particles, that is how the wave  1  Chapter 1. Introduction^  2  function changes under permutations, is similarly classified by the spin-statistic theorem. These properties are valid for a three dimensional space, plus one dimension of time. However, about fifteen years ago it was pointed out by Lienaas and Myrheim [58] that for a two dimensional space the notion of statistics (and of spin) is very different. They have shown that the exchange of identical particles is represented at the quantum level by generators, that transform the wave function, forming a representation of the braid group [87, 92]. Particles whose wave function changes by a phase (not zero or pi) under permutation are called anyons [32, 45, 47, 82], they are a generalization of ordinary bosons or fermions. More generally, the braid group on a two-dimensional surface is an infinite, discrete, non-Abelian group and has many potentially interesting representations [26, 48]. Anyons are sometimes described mathematically by coupling the currents of point particles to the gauge field of a Chern-Simons theory [24, 35, 50, 65, 73, 74, 78]. This is represented by the action S =^ftwAA0i1Ad3x + J Amj"d3x 47r I  where j"(x)=  (1.0.1)  Eqj dr dr" cT;-83(x — ri(r))^ (1.0.2)  with r(r) and qi the trajectory and the charge of the i'th particle. The interest in anyons has been partially motivated by their conjectured role in the fractionally quantized Hall effect [3, 4, 68, 70]. The Hall resistivity exhibits plateaus when the filling fraction v is an integer for the integer quantum Hall effect. At the same time the conventional resistivity drops to a very low value indicating that the two-dimensional electron gas flows with no resistance. Later, it was shown that the Hall resistivity exhibit also plateaus when v takes some fractional values. It is still unclear why the electron gas has this behavior at these fractional filling. It is conjectured that anyons, represented by  Chapter 1. Introduction^  3  a Chern-Simons theory, could be responsible for the fractional quantum Hall effect [52]. In fact, it has been shown that the Laughlin wave function, representing anyons, is the exact ground state of the quantum Hall effect [11, 44, 64, 77]. The recently discovered high temperature superconductors have yet to be described successfully by any theoretical models. Since this particular type of superconductivity is evolving in two dimensions, it has been conjectured by some to be a superfluid anyon gas [13, 30, 59]. Such explanations have not produced positive results so far, but the issue is still open. Fractional statistics can also be realized for soliton excitations. The 0(3) non-linear sigma model, or the equivalent CP(1) model, in two dimensions is such a model [60, 73, 74]. This model is thought to describe the large wavelength behavior of the ordered phase of a quantum Heisenberg anti-ferromagnet in two dimensions and of a classical Neel anti-ferromagnet in three dimensions. It has been suggested that a topological term similar to the Chern-Simons action, the Hopf invariant [8], might be responsible for some properties of high temperature superconductivity [25, 81]. This is, again, a controversial issue [22, 33, 34, 43, 85]. It became clear that a thorough understanding of Chern-Simons theory, especially the Abelian version, at the quantum level is essential if we are to consider such a theory in the description of physical phenomena. This is what we intend to do in this thesis. The wave function of Abelian Chern-Simons theory coupled to classical point particles on the plane was found by Dunne, Jackiw and Trugenberger [24, 50, 53]. In this case the Chern-Simons theory has no physical degrees of freedom, the Hilbert space is onedimensional and the only quantum state is given by a single phase. In order to study the statistics of a given system we need to have identical particles, which implies a non-zero total charge. In return, if we have non-zero total charge, Gauss' law for Chern-Simons theory requires a non-zero total magnetic flux, which on a compact manifold means that  Chapter I. Introduction^  4  the gauge connection A in (1.0.1) is not a function but a section of a U(1) vector bundle. We will show how to overcome this difficulty. In previous literature, this complication has been avoided by considering more than one kind of particle so that their total charge adds to zero. In that case, Bos and Nair [9] solved the SchrOdinger equation for Abelian Chern-Simons theory coupled to particles when the space is a Riemann surface and when k is an integer. In Chapter 2, we will describe several novel (at least for physicists) mathematical structures. Since these mathematical constructions are rarely explained in physics papers, this Chapter should be useful for putting topological field theory in two and three dimensions on a firm mathematical basis. In Section 2.1 we will review exterior calculus and the properties of vector bundles, which will be used extensively in the following sections and chapters. In Section 2.2 we will describe the DeRham cohomology at some length. Topological structures used in physics are almost always represented by some integrals of DeRham cohomologies. For vector bundles, we can define particular cohomology classes known as characteristic classes, this will be explained in Section 2.3. In particular, we will show the origin of the Chern-Simons action, as shown in (1.0.1), form a particular characteristic class. In Section 2.4, we will describe the classification of a Riemann surface of genus g, while in Section 2.5 we will give a detail description of the braid group on such a surface and some of its representations. In Chapter 3, we will describe the quantization of the Abelian Chern-Simons theory coupled to point charge current, see (1.0.1) and (1.0.2), on a Riemann surface of genus  g. In particular, in Section 3.1 we will show how to decompose the gauge field on a Riemann surface and how to write down a consistent Chern-Simons action for a non-zero total charge. In Section 3.2 we will solve the SchrOdinger equation and show that the correct geometrical description of Chern-Simons theory necessarily introduces a framing of particle trajectories. This framing corresponds to an additional U(1) gauge connection,  Chapter 1. Introduction^  5  in fact the connection of the tangent space of the surface, and plays an essential role in determining the consistency of the theory. This point has been largely ignored in the literature. We will show that for a set of particles of total charge Q = Ei qi and for k a rational number, the fundamental constraints eivq,(Q+qi(g_i)) 1 (1.0.3)  have to be satisfied. In Section 3.3, we will described how the wave function transformed under large U(1) gauge transformations, determining what is the periodicity conditions of the wave function around non-trivial holonomy cycles (non-contractible loops) of the surface. We will show how a set of theta functions satisfy such conditions. In particular, it will be shown that for fractional k we will need a particular type of theta functions defined with fractional indices. In Section 3.4, we will show that the wave function acquires phases when the particles move on the surface. More precisely, we will show that the statistic of such particles is given by a representation of the braid group on the Riemann surface. The description of the Abelian Chern-Simons theory by using the path integral formalism, in contrast with the canonical formalism, will be discussed in Section 3.5. The consistency of the two formalisms will be shown with respect to the braid group representations and the framing regularization. A topological constraint like (1.0.3) might not be of immediate relevance, since most physical situations do not occur on a compact Riemann surface. On the other hand, often a field is considered to tend toward zero (or a constant for a unit vector) at large distances on a two-dimensional plane, which is usually equivalent to a physical system on a sphere. Alternatively, periodic boundary conditions, like for the Bloch wave function in a periodic potential of a lattice, is equivalent to a system on a torus. In particular, it is not certain at all if the constraint (1.0.3) is related to the filling fraction v of the fractional quantum Hall effect. This will have to be investigated further in future work. For the application  Chapter 1. Introduction^  6  of Chern-Simons theory on a Riemann surface to the fractional quantum Hall effect, see Wen in [83, 84]. In any cases, it is hard to believe that there is no system in nature that would be represented by an Abelian Chern-Simons theory, where this constraint would be satisfied. In Chapter 4, we will study the canonical structure of the CP(1) model with the addition of a Chern-Simons or Hopf term [8]. In Section 4.1, we will described the relation between the CP(1) variables and the 0(3) variables, achieved by the Hopf map. In particular, we will explain the precise meaning of the Hopf invariant as a Chern-Simons term. In Section 4.2, we will work out the canonical structure of the CP(1) model. This is particularly complicated by the constraints, involve in using the CP(1) variables, that has to be satisfied. We will make some novel observations about the canonical structure of the CP(1) model, in particular the existence of a local SU(2) x SU(2) algebra. In Section 4.4, we will show how the ground state (topological) excitations of the CP(1) model obey fractional statistics in the adiabatic limit. In particular, we find that the representation of the ground state solitons, which are extended field configurations over the whole surface, introduces intrinsically a natural framing of the solitons. This is in contrast to the framing of point particles described in Chapter 3. It is shown that the CP(1) field solitons obey the spin-statistic relation. In fact, the fractional statistics of the CP(1) model are characterized by a two-color braid group representation. This fact has not been fully appreciated in the past literature on fractional statistics generated by the Hopf map, and is still not widely known [18, 75, 80]. Also, its physical importance is still not clearly understood. We indicate a generalization of the constraint (1.0.3) for particles (or solitons) with spin. We conjecture that, even in the adiabatic limit, the thermodynamic properties of a many soliton system would differ from that of an ordinary anyons gas. Understanding this thermodynamics, and other physical properties, is an important problem for the future.  Chapter 1. Introduction^  7  In Chapter 5, we will described some topological models that have a non-Abelian group structure, in particular the Wess-Zumino-Witt en model and the non-Abelian ChernSimons model. The Wess-Zumino-Witten model (at a specific coupling), described in Section 5.1 is a bosonized version of a fermionic model, which was pivotal in the understanding of bosonization in two-dimensions [88]. This model has been used extensively in the description of exactly solvable two-dimensional models [21]. Some of the features of the non-Abelian Chern-Simons theory will be described in Section 5.2, this should be contrasted with the Abelian Chern-Simons theory studied in Chapter 3 and 4. We will show that the non-Abelian Chern-Simons theory produces more exotic (not pure 0-statistic) representations of the braid group. The thermodynamics properties of such systems is still unknown and is under study. It is believed that interesting physical properties might emerge from a system obeying a more general para-statistics. Currently, a lot of research is centered around quantum groups and W-algebras [23, 57]. The nonAbelian Chern-Simons theory is used as a toy model, if not a real physical model, to study and learn about these intricate structures.  Chapter 2  Selected topics in topology  In this Chapter, we will describe various topics in mathematics, and in particular topology, that will be used in the next Chapters. It is not, by any extent, a complete and formal description of these subjects. For most physicists it should be considered as a fairly self-contained presentation that can be used as an introduction, while for most mathematicians it should be read more as a review. The reader interested in more details of any of these topics should refer to the cited references in the text.  2.1 Manifolds, exterior calculus and vector bundles Standard calculus is done on the well known open space TO. This space has an origin, and from there we can define vectors. The parallel transport on /in is path independent, it corresponds to a simple shift in the coordinates. Since 10 is made of copies of R and the real line is the fundamental set from which we do calculus, we call Rn a differential space. In abstract set theory, we construct spaces by taking a set of points, forming the space, and by choosing a system of subsets (or neighborhoods) satisfying some conditions (see [15, 54] for details). For Rn the subsets are the neighborhoods Ix — xo I < E around each point xo E R. In general the system of subsets does not have to be chosen this way, which can lead to very different spaces. We call manifolds the spaces where the usual calculus can still be performed. Not all topological spaces are manifolds. A manifold M, of dimension n, must have the property [15] that any small open 8  Chapter 2. Selected topics in topology ^  9  region U C M can be mapped into TV = (x1, , xn) by a one-to-one and continuous function 111 : U R'. The system of coordinates (x1, , xn) on lin is a representation of the manifold M, within the patch U. For two different patches U1 and intersection  U12  we obtain a transition function  4112 = 412 0 ‘111-1 that  U2  with  maps /in = I1i (U12)  into Rn 1112(U12), or (xi) into (yi). As a compatibility condition, the function 4112 must be one-to-one and continuous. The smoothness of the manifold, the number of allowed derivatives, is given by the smoothness of its transition functions On a triple intersection, we find  4123 0 4112 (1113 0 412-1) 0 (1ff2 0 1111-1) 1113 0 TV- = Wm,  so  when we move from region 1 to region 3 we find no difference if we pass by the region 2. We can perform calculus on M by performing the corresponding calculations on the  R" = W(U) space, where U C M is any relevant subspace of M to our calculations. A tangent vector at u0 E U, is represented on /in, at '11(u0), by  Ei cib. From a different  map 111 with coordinates (y1, , yn), we use the transformation = Ei(Z)047 . Alternatively, a differential element is represented on R by  Ei cidxi, with the transformation  dxi = Ej(e-;)0dy2. For the integration, we break the region of integration into small pieces belonging to small regions Uk from which we can integrate on the Rn space lifk(Uk). For a line integral along a curve C on M, made up of disjoint pieces Ck E Uk,  E  fidu ' ICEM  Lk(Ck)E(Rn)k  i(' fx)dx  The geometric language of forms [15, 72, 61] is very natural for performing calculations on a manifold. To every n-dimensional manifold M we associate a space of k-forms Ak (M) with basis dxi' A • • A dxik, for each k = 0, • • • , n. The indices (i1,. . ., ik) E (1,..., n) and are all distinct. The wedge A indicates that we consider the antisymmetric part of the product of dx's, for example du A dv = du 0 dv — dv 0 du = —dv A du. n! The dimension of Ak (M) is thus CT kqn—k)! • Under a change of coordinates, dxi  ^  Chapter 2. Selected topics in topology^  10  the volume form is transformed as dxlA•• • Ade = J•dyl A • Ade', where J is the Jacobian det(z), by the antisymmetric nature of the wedge product. The Jacobian factor is directly produced, showing the geometric nature of the forms. Forms are truly coordinate independent, which make them the perfect objects to use for calculus on a manifold. A general k-form has the decomposition A = Ai, dxil A • • A dxik, where we can choose i1 < • • <  ik to avoid redundancy. A 0-form is a scalar function.  We can define product of forms, using the wedge operator A : (Ak, A1)^Ak+1, and differentiation, using the differential operator d : Ak ^Ak+1. This algebra of forms is sometimes referred as exterior calculus. For A, B a k-form and an /-form respectively and f a function, we have the following properties A A B = (-1)k1B A A E A/1(M)^ (2.1.1) aAii,...  'ik dxk A dxil A • • • A dxik E Ak+1(M)^(2.1.2) dA = E ^ 01,•••,ik^  aXk  d(A A B) = d(A) A B (-1)k A A d(B)^(2.1.3) d(dA) = 0^  (2.1.4)  af d(f) E — =dx' df^(the ordinary differential)^(2.1.5)  ax'  Often, for simplification, the wedge A is dropped from formulas. Unless specified, there is always a wedge product in between forms. Stokes' or Green's theorem can be represented easily by using forms. The integration of the n-form dA, for a (n — 1)-form A, over a region V satisfies dA I A  ^av^  (2.1.6)  av is the boundary of V. When A is a 2-form and ay a closed region of volume V, (2.1.6) is Green's theorem, while it is Stokes' theorem for A a 1-form and av a closed  where  curve of surface V.  Chapter 2. Selected topics in topology ^  11  Given a point u0 E U C M, we represent a tangent vector at u0 on M as a tangent vector at 111(tt0) on  Rn =  111(U), written generally as Ei cib. For an n-dimensional  manifold, the tangent space forms a n-dimensional vector space. We call the inner product on the tangent space, for every point of the manifold, a metric , 0  " =^; I 5;7^(2.1.7) The cotangent space is the dual of the tangent space, (61dxj)^61, thus the inner product on the cotangent space is the inverse of the tangent space inner product  (dxildxj) = gii This should be viewed as an inner product on the Al(M) space. In general, on the Ak(M) space, we have the following (local) inner product, written in terms of its basis elements, (dxil • • • dxik 1dx^• • dxik)^det[(dxim Idxin)]^(2.1.8) The volume element on a manifold is given by dnx = 1/191dx1 • • dxn, where g det(9i3). The presence of 191 makes the n-form invariant under coordinate transformations, and we also find (dnxIdnx) = ( 191)2 det(9ij) = sign(9) ^±1. If we can extend this volume element uniquely over the whole manifold then we call the manifold orientable. For an non-orientable manifold, the volume element n-form changes sign after some coordinate transformation (the MObius strip is such a space), rendering the volume form defined only up to a sign. For an orientable n-manifold equipped with a metric, we can defined a new operator called the Hodge star operator. The Hodge star of a k-form B, written as *B, is the unique (n — k)-form defined for any k-form A by  (AIB)clnx = A A* B^  (2.1.9)  Chapter 2. Selected topics in topology^  12  Note that dnx = *1. It has the property that TA) = (-1)k(n-k)sign(g)A  (2.1.10)  (*Al*B) = sign(g)(A1B)  (2.1.11)  for k-forms A and B.  -  It is useful to define the co-derivative operator 8: Ac __, Ak i  8 = (-1)k(n+1-k)sign(g)*d* It has the properties that  (2.1.12)  S(8A) = (-1)k*d(d*A) = 0 by (2.1.4). The laplacian operator,  0, is represented by 0 = (d + 8)2 --= c18 + Sd. When the k-forms A and B on an orientable manifold M have compact support, or if M is closed, then we can integrate the local inner product of A and B in (2.1.8), which gives a global inner product of forms on M < AIB >.  ^ (AlB)dnx = I A A* B Im^.A4 (2.1.13)  Under this inner product, the co-derivative 8 is the adjoint of the derivative operator < AldB >=< (SAW >^  (2.1.14)  which also shows that the laplacian is self adjoint, while the Hodge star operator satisfies <* AB >. sign(g) < Al/3 >^ (2.1.15) We often have to consider a vector space on top of each point of a manifold, like the tangent space discussed above. In general, we call a vector bundle [15] E(M, 71- , L, G) the construction corresponding to a manifold M, the base of E by the projection map 7r : E --+ M, which is equipped with a vector space L, at each point of M, transformed by a representation of a group G. If instead of a vector space L we have a representation  Chapter 2. Selected topics in topology^  13  of the group G itself, from which a group transformation is represented by left multiplication, then we call this a principal fiber bundle P(M,r, G). Every vector bundle has an associated principal bundle. We refer to the vector space of L of E, or the group G of P, as the fiber of the bundle. We represent a group G by using its algebra [14, 37]  ^g(X)  =  exp (i EXaTa)^where^[Ta,ilb= ifabcTc^(2.1.16) a  The constants fabc are the structure constants of the group. Although we have maps 111 from a region of the manifold into a lin space, we have not specified how a vector could be (parallel) transported on a vector bundle. We say that a vector field x E L is parallel transported along a given vector, Vk , if its covariant derivative DkX vanishes along the direction of that vector, vkDkx = 0. We define the covariant derivative as ^Dk  -a  ak + Ak^Or^Thad+A  ^  (2.1.17)  The one-form A is the connection gauge field, or simply a connection, associated to the given vector bundle. The one-form A takes values in the Lie algebra of G  A = iETaAdXk a  A manifold is represented by mapping small region U --)• 111(U) = Rn of M, thus the connection A is defined only within this region U of the manifold. In the intersection U12 of two regions U1 and U2, the vector space xi, of the first patch, is related to the vector space x2, of the second patch, by an element of the gauge group Xi -4 X2 = Wi2X1^with^W E G^(2.1.18) To be consistent with ordinary differentials, the transformation of the covariant differential of this vector field has to transform like the field itself, such that it becomes  Chapter 2. Selected topics in topology ^  14  independent of the choices of patches coordinates and vector field basis. With the use of (2.1.18), Di Dx2 = Wi2DX1 D(Wi2X1) Note that the transformation of the basis dxk is implicit in the form notation. This indicates that the gauge connection of the region 1 and region 2 are related by  A1 A2 = W - 2--1W1-21 1 A W12dW1-21  (2.1.19)  If we transform the vector space L of a vector bundle by a gauge transformation W, as in (2.1.18), then we must also transform the gauge connection A following the equation (2.1.19). When we parallel transport a vector along a loop on M, the vector will not necessarily come back to its original value. To characterize this, we parallel transport a vector around an infinitesimal loop C, which is the boundary of a surface S, then it is found that the vector will be transformed by an infinitesimal group element exp(fs F), where  F iETaF4clxi A^= dAd AAA  ^  -  (2.1.20)  a  Or = AA; - 0.;14 — fabcAcIAI; We call F the curvature of the vector field L of the vector bundle. Under a gauge transformation (2.1.19), the curvature F is transformed covariantly as  —* F2 = WF1W-1^  (2.1.21)  When F 0, then the parallel transport is path independent. In fact, within a patch where A is defined, we can represent A as A W-idW  Chapter 2. Selected topics in topology^  15  where  W (x) = P exp ( f A)  EE 1+^ I x A+^du'^dv3Ai(u)Ai(v) + • • -  X0^  X0^ X 0^XO  The operator P means that we have to path order the integration when A belongs to a non-Abelian group. It might not be possible to extend W to the whole manifold, there might be some topological obstruction when we move from one patch to another. We will say more on this point in Section 2.3.  2.2 DeRham cohomology, homology and homotopy In Section 2.1 we defined manifolds and described the exterior calculus of forms, which are coordinate independent objects on a manifold. This coordinate independence of the forms makes them a powerful tool to investigate the topology of the manifold itself. We would like to find some properties common to a set of manifolds related to each other by a smooth change of the metric. In other words, can we put a set of manifolds related only by stretching and shrinking in some class. We will show how the existence of some forms on a manifold can actually determine such classes. Let us study in more detail the properties of the differential operator 'd'. To do so, it is useful to look at the sequence 0 a_.4 Ao a). Ai Ak_i ^Ak^Ak+i An-1  d  > An  d  0  Let us denote by Zk the subspace of Ac mapped into 0 by the differential operator 'd', called the kernel of 'd' on Ak. We will call forms belonging to Zk closed forms. Alternatively, let us denote by Bk the subspace of Ak which is the image of all the forms in Ak mapped into Ak by the differential operator 'd'. We will call forms belonging to Bk exact forms.  Chapter 2. Selected topics in topology^  16  The exact forms are a subset of the closed forms  Bk c zk since any exact form, dA, is necessarily closed, d(dA) = 0, by (2.1.4). The question that arises then is whether there are closed forms on Ak, that are not exact forms. Since a lot of closed forms might only differ by an exact form, we regroup the closed forms differing only by an exact form into one class. More precisely, we define the DeRham cohomology [61, 72] class H' (M) by Zk^ker dk  Hk — — — ^ Bk im dk_i  (2.2.1)  where the index on `dk' indicates that it is applied to the Ak space, and the division by Bk means we built an equivalence class of exact forms.  The calculations of the cohomology classes of a manifold are not trivial in general and sometimes involve rather advanced mathematical tricks. There are, however, some simple cases that can be easily computed. The lowest cohomology class, H°, is given by the 0-form, that is functions, whose derivative, df, vanishes, since B° = 0; there is no (-1)-form to produce an image on A° by applying `d'. The solution to df = 0 is given by the set of constant functions on M, and a constant is represented by a real number. This is assuming that M is path connected, otherwise H° will contain a copy of R for each path connected region of M,  H°(M) = R,^for M path connected The highest cohomology class (by definition Hk = 0 for k > n), kin, is given by the n-forms, A, that are necessarily closed since there are no (n+1)-forms, modulo any exact n-forms, B. If M is orientable without boundary, then we can use the generalization of Stoke's theorem to kill any exact forms,  Im(A - E dB) .---1 A+1 m=o B=I A m^a^m  Chapter 2. Selected topics in topology^  17  which gives a real number. So, the generator of Hn is the volume element dnx, which is the only form, up to normalization, not killed by the above integral. Thus  H' (M) = R,^if M is a closed and orientable manifold of n dimensions (2.2.2) Without proof, it is useful to give the DeRham cohomology of the following spaces  Hk (Rn)  0 otherwise  Hk(sn) _  IR  for k = 0, n  0 otherwise Hk 041 X M2) =  ^Hk(Tn) ^k  ETP(mi) x fra-k(m2)  n! = R ED^R^= ^ times) ^k!(fl — k)!  where Sn is the n-sphere and Tn = (511)n the n-torus. For a closed and orientable manifold, the Hodge star operator isomorphism can be extended to the differential operator `d', see the properties (2.1.14) and (2.1.15). We find that the cohomology Hk from the operator is isomorphic to the cohomology (in general different) on A/C from the operator '8', which is in this case the usual cohomology Hn-k  ^Hk(m)  = Hn-k(m)^for M orientable without boundary  This theorem indicates that there exist some k-forms h E Hk that have also the property that *h E Hn-k. We call such a form harmonic. The harmonic forms can be characterized as being closed and co-closed  h E Hk(M) and *h E Hn-k (M)  ^dh = 8k  =0  Chapter 2. Selected topics in topology^  18  This equation uniquely determines h, without having to introduce an equivalence class. This allows us to decompose any A E Ac into an exact, co-exact and harmonic form  A=dcr+8,3+h^  (2.2.3)  where a is a (k-1)-form, # a (k+1)-form and h an harmonic k-form. The decomposition (2.2.3) uniquely define a, # and h. It is called the Hodge decomposition theorem [15]. Given a space V, we represent its boundary by OV where we call '8' the boundary operator. Since the boundary of a space does not have a boundary, we find a(Ov) = 0. This is similar to the property (2.1.4) of the differential operator 'd'. We introduce homology by considering the space Sk(M) of maps f : Ak -- M of k-simplexes. The boundary operator produces a map O f: (OA) --) m that belongs to the space Sk-i(M). A 0-simplex is a point, a 1-simplex is a line, a 2-simplex is a filled triangle, and so on. The boundary of a k-simplex is a sum of (k — 1)-simplexes. From these maps we can build a topological class known as the homology class [72] ker ak  Hk(m) = i n m Uk+1  (2.2.4)  The homology class is represented by the Abelian group (under addition) of integers. A homology generator of H1, can usually be associated with a non-contractible loop on M. A general homology class, Hk, has several copies of Z, plus a torsion term. In this context, torsion means a sum of finite cyclic groups. We can tensor Hk with the real numbers, this has the effect of killing the torsion part, and each copy of Z is transformed into a copy of R. Under this tensor product, we have the theorem  Hk(M; R) = Hk(M) 0 R = H-L,1%Rham(A4 ) best illustrated by the duality of 'd' with '8' in the generalization of Stoke's theorem (2.1.6). Note that an algebraic definition of cohomology can be given, which is based on  Chapter 2. Selected topics in topology^  19  Z instead of R. For such a cohomology, the torsion part may differ from the corresponding homology. Finally, it is worth giving a short introduction to homotopy groups of spaces [72]. The n-homotopy group of M, represented as 7r(M), is the group formed by the maps: Sn —* M. The lowest homotopy 7r0, which is actually not a group, simply counts the  number of path connected components of a space. The next homotopy group, 7r1, called the fundamental group, indicates if there are non-trivial loops on M. For each nontrivial loop corresponds an element of the group, with appropriate multiplication and inverse. Sometimes, the fundamental group of a manifold may be non-Abelian. The other homotopy groups, 7r„ for n> 1, are always Abelian. The homology, cohomology and homotopy groups of simple spaces are often identical, but it is important to understand that they are very different mathematical structures.  2.3 Characteristic classes In Section (2.2) we constructed some topological classes that can (but not always) distinguish topologically different manifolds. For vector bundles, it is also possible to construct some topological invariants. We can then, in general, distinguish topologically different vector bundles. These invariants are known as characteristic classes. A good candidate, to represent such classes, must be invariant under coordinate transformations, thus we should use forms. In addition, it must especially be invariant under gauge transformations, as in (2.1.18), of the vector space. Since the curvature 2-form is transformed covariantly, see (2.1.21), under these gauge transformations, we can construct an invariant called the Chern classes [15] of a vector bundle, for a base space consisting of a manifold of even dimension 2k. The Chern classes are best expressed in term of the  Chapter 2. Selected topics in topology^  20  Chern characteristics —1 k^ tr(F A • • • A F) (k times) ck(E(M, , L, G)) = ( -7) 2ri JM  (2.3.1)  This is always an integer on closed and orientable manifolds. Note that in (2.3.1) and the following characteristic classes, we can replace a vector bundle with a principal bundle. The first one is trivial c0 (E) = 1. The next class is given for a 2-dimensional manifold equipped with a vector space, L, transformed by an Abelian group, U(1), 1^1 2F0 (E(M2, 7r, L, U(1))) =^im F = -----  (2.3.2)  where F0 is the total flux, a real number after eliminating a factor T. Let us explain in more detail what this invariant means. Since we have to consider the case where F0 is non-zero on M, the representation of the gauge field A can be done only on a set of patches covering the manifold. Let us consider the set of patches Ui as a good cover of the 2-dimensional manifold M. That is, the intersection of any two patches is a single contractible surface. We have a field A(1) on each patch Ui, with the transition functions (2.1.19) defined on the intersection of any two patches  0  11 tri,  — Aci )^idx (i i)^ (2.3.3) where x(ii) = —x(ii) by definition. On triple intersections Ui n Ua n Uk we can use (2.3.3) to find the relation x(ii)^x(jk)^x(ki) = c(iik) = constant^(2.3.4) The set of constants C(iik) are related to the total flux by [2] F0 = —ifm dA = —i E where the surfaces Vi C Ui n  IP.  0  i  dA(i) . E^dx(ij) . E c(iik)^(2.3.5) ii Li,pijk  and they are bounded by a line, Via, dividing the intersection  On the triple intersection, we let the three lines Via, Val' and Vki meet at one  point Pak.  Chapter 2. Selected topics in topology^  21  If M is a filled circle, then c1 reduces to an integral on the boundary = -  1  27ri Li  A  This is an Abelian Wilson loop integral. If the curvature vanishes, then the gauge connection is trivially represented on the boundary A = U-idU -m(id0), for U = We find ci = m, an integer representing the winding number of the U(1) field U over S1. The next Chern class is particularly important in understanding three- and fourdimensional manifolds. Given a vector bundle with a 4-dimensional manifold as the base space and a vector space transformed by a non-Abelian group G, the Chern class is 1 c2(E(M,r,L,G)) =^tr(F A F) .A4 872 82 IM  (2.3.6)  If this 4-dimensional space has a 3-dimensional space as boundary then we can use the identity F A F = d(A A dA  2 - A A A A A) = d(AdA  3  2 - A)  3  (2.3.7)  to reduce the Chern class to Ss(A) =  1 2 tr(AdA^A') + 87r2^3 am  (2.3.8)  This integral is also known as the Chern-Simons integral. Note that it is not necessarily an integer. If the gauge field is trivial A = U'dU, then we find for a 3-dimensional manifold  1 Swz(U) = 24r2 tr(U-IdUU-1dUU-IdU)  (2.3.9)  which is an integer for a closed space. If the 3-dimensional manifold has a 2-dimensional manifold as boundary, then we call this integral the Wess-Zumino integral. Given a 2dimensional space with a group valued field U, we can always view this space as the boundary of a 3-dimensional space to which we can extend U and evaluate the WessZumino integral (2.3.9) uniquely, modulo an integer.  ^  Chapter 2. Selected topics in topology^  22  For a closed 3-dimensional manifold, the Chern-Simons integral (2.3.8) is not completely invariant under a gauge transformation, but instead it changes by the WessZumino integral (2.3.9) A ---* U-1AU  ^  Ss(A)^Ss(A)+ Swz(U)  ^  (2.3.10)  A gauge transformation corresponds to a map M ^G, and (2.3.9) is an integer representing the winding number of this map. We call such a non-trivial gauge transformation a large gauge transformation. If M = S3, then the classes of these maps are classified by r3(G), which is Z for any non-Abelian group. The Wess-Zumino integral can be recognized as the integration of a closed, but not exact, 3-form, thus representing the cohomology1/3(M). This cohomology comes from the cohomology of G, H3(G), brought (pullback) on M by the map M G of the gauge transformation. Finally, if we consider the tangent bundle T*M = E(M, r , V2n, GL(2n)), of an orientable manifold of even dimension 2n, then we can build another invariant by using the Riemann curvature 2-form Rii = gikRi;. We represent this class, known as the Euler class of M, as X(M) =  (-1)n (47r)nn!  Im  ^D.^  122  A • • • A Ri2n_ii-2n  Of particular interest is the Euler class of a 2-dimensional manifold 1 X=2 —7 m f R2 1 We will describe 2-dimensional surfaces in more detail in the next Section.  2.4 Two dimensional topology and Riemann surfaces Two dimensional manifolds are very important in physics and mathematics because this is the only dimension (except the trivial dimension one) where we have a thorough understanding of their classification as spaces. We will be particularly interested with orientable  Chapter 2. Selected topics in topology^  23  Figure 2.1: Example of a genus 3 Riemann surface  Figure 2.2: Holonomy of a Riemann surface represented by loops a, and bm intersecting at P„, and closed two dimensional manifolds, which are also called Riemann surfaces. The best way to represent them is by using a picture such as Figure 2.1. We see that we can classify these surfaces by using an integer g, called the genus of the surface .A4 9 In term of the Euler class the Riemann surfaces are classified as -  x(mg) = 2(1 - g)  (2.4.1)  The genus is basically the number of holes or handles of the surface. If we concentrate on one of the holes in particular, as seen in Figure 2.2, we find a pair of non-contractible loops, or homology cycles, which we call a, and V', m = 1, ,g. The intersection numbers, seen at the point Pm in figure 2.2, of these generators are given by v(ai, am) = v(b1 , bm) =  0, v(ai, Um) = —v(bm , ai) = ^(2.4.2)  where v(Ci, C2) is the signed intersection number (number of right-handed minus number  Chapter 2. Selected topics in topology ^  of left handed crossings) of the oriented curves C1 and  C2.  24  The first homology group  (with integer) of Mg is generated by these homology generators, which are independent, thus we find H1 (M9)= Z ED • - - e Z (2g times). There is one copy of Z for each homology cycle. Alternatively, it is very useful, for calculations in physics, to study the DeRham cohomology of a Riemann surface. The cohomology can be represented as the dual of the homology. For each homology cycle, we can find a non-trivial closed form belonging to the first cohomology class of  M9.  More precisely, given a homology cycle, we build a  function 0 that is multi-valued around this cycle, like the standard angle around a circle, but single-valued around all the remaining non-trivial cycles. Then the one-form h = — de 27r will represent the cohomology generator associated to the given homology generator. We will associate the cohomology generators hm to the homology generators am, and him to b. This construction can be summarized in hi = 81m, I h1 .-_- 0, I hi =-_- 0, I ii1 = sim am bra am bra  f  (2.4.3)  One of the important theorems in the theory of Riemann surfaces is Riemann's period relation. Using the cohomology generators (2.4.3), we can state this theorem as i III A km= 8 ni l , h1 A len = I Aiti = km 0 (2.4.4) Mg  fm, mg  This theorem is the counterpart of the intersection (2.4.2) of the homology generators. Two dimensional surfaces can be better described using a complex variable. To do so, we have to choose a complex structure on M. This amounts to a choice of basis dz, as dx + idy by making use of the real basis. The difficulty is to extend such a basis globally on the whole manifold, but this can always be done on two dimensional manifolds (unlike higher dimensional ones). For example, by using the theory of conformal mapping we find systematic choices of complex structures.  Chapter 2. Selected topics in topology^  25  The advantage of using a complex variable comes from the fact that if a one-form, A = Adz + AAi', is closed, dA = 0, then A, is a function of z only and AE is a function of i only. In other words, d[ck(z,..-)dz1 = 0 gives the Cauchy-Riemann equations that has to be satisfied to obtain a holomorphic function 0(z). If we want A to be real, then we also find that AE(i) = Az(z). In addition, we find that *dz = —idz and *di = idi, so that the co-derivative operator '5' is proportional to the derivative operator 'd'. This tells us that a holomorphic one-form A(z)dz is closed and co-closed, hence harmonic. The Hodge decomposition theorem (2.2.3) states that such a one-form can be decomposed as a sum of an exact, co-exact and harmonic form. Going back to our cohomology generators h and Ii, we can add an exact piece to these closed forms such that we obtain harmonic forms. This gives us a 2g dimensional real space of harmonic forms (isomorphic to the cohomology group). Alternatively, the space of harmonic forms can be described by complex harmonic forms, hence also holomorphic, that we will call col = col(z)dz. Since the homotopy cycles al are independent, see (2.4.2), we can use them to normalize col, but leaving no constraint along the b1 cycles. This gives the con/ the standard normalization [16, 51], for an arbitrary complex matrix film, wm = film  i Wm = 87in , al  ibl  The relation between the real and complex harmonic forms, using (2.4.3), is expressed by g  col =  h1  + E nimilm^  (2.4.5)  m=1  With the use of an arbitrary complex harmonic form L = Ef_lcicol and using the equations (2.4.5) and (2.4.4), we find that  0=1  Mg  LAL=  g E 1,711=1  cicm(12" — Sri)  Chapter 2. Selected topics in topology  26  and 0 <  ^IL rd2x ---= ±1 L A L ----= —1 E cic,Gim mg^2 mg^2 477.,  where dm =  ii  Mg  col Am = 2Im(nim), GiniG" = 87/1^(2.4.6)  This shows that the matrix fen is symmetric and its imaginary part, Glm, is positive definite. We can recognize the equation (2.4.6) as a metric in the space of holomorphic harmonic forms. If we write the basis of the harmonic forms as (co1, ... , cog, Col, ... , Cog), ( 0 G) then the global inner product of forms (2.1.13) gives the inner product ^011 —G 0 this harmonic space. We will use GI, and Glm to lower or raise indices when needed and use Einstein summation convention over repeated indices. Any linear relation, with integer coefficients, of al and I) that satisfies (2.4.2) is another valid basis for the homology generators. These transformations form a group, called the modular group, Sp(2g, Z):  (ab) S (ab)  where S =  (2.4.7)  (DC BA )  ( 01 ) with S EST = E and E =^. The g x g matrices A, B, C, D have integer entries. —1 0 For the real harmonic forms, the modular transformation (2.4.7) takes the form  ( iht) --+ ( ih't')=h ''j ( it)  where  =_  s-iT  (A— —C D B )  This preserves the duality (2.4.3) of the homology and cohomology generators. A modular transformation will transform the complex harmonic forms and the fi matrix  co --> co' = It' + frit' = (A — fl'C)h + (—B + CZ'D)it = (A — fi'C)co  Chapter 2. Selected topics in topology ^  27  from which we find the relation = (AQ B)(C12 D)IL^and^w = (CC/ + D) - 11-^(2.4.8)  It is easy to verify that the harmonic space inner product (2.4.6) is transformed as G^= (CO D) - 1T G(Cf2 D) - 1 2.5 Three dimensional topology and the braid group In recent years, the study of three dimensional manifolds has been greatly advanced by using knot theory. A knot is a (or a collection of) closed curve, or 51, embedded in some manifold, sometimes referred to as a mapping S1 ----> M. By definition, this mapping will represent the fundamental homotopy group of the manifold, or ri(M). On the other hand, if we restrict the homotopic deformation of S1, say represented by 0, such that no two points of this loop intersect on M, or x(0i) x(02) where x represent points on M, then we obtain a new mathematical structure. In effect we obtain knot theory. Such knots are trivial in dimension one and two, and for any dimension above three, since any knot can be turned into a trivial knot (a simple circle in an Rn neighborhood of the manifold) in dimension four or higher. This leaves the third dimension where, indeed, knot theory has a rich structure and plays a crucial role in the classification of three manifolds. We will study in this Section a related subject to knot theory, that is the braid group. Let us consider n identical particles, with coordinates X = (x1,• • • ,x) E Mn, on a two dimensional manifold M. As these particles move in time, they will trace a trajectory in space-time X(t). When these particles are not allowed to intersect, this eliminates a subspace A = {X E MnIxi xj, for any i j}. By considering all the n particles as identical, we must consider any configurations differing only by a permutation, Sn, as  Chapter 2. Selected topics in topology^  28  Ti Figure 2.3: Example of a braid. being identical. Thus, the configuration space of such a system is mn _ A  Qn(M) = The representation of the fundamental group of Q(M) is the braid group B(M) = 71(c2,-,(m)). A periodic trajectory on M, starting and ending at the same configuration Xo, up to a permutation, is a closed loop on Q(M). Such loops are elements of 13n, also called braids. We can represent a braid as n strings, the particle trajectories, in a three dimensional space-time. Since the particles are not allowed to coincide, the strings cannot pass through each other. See Figure 2.3 for an illustration of a braid. The composition law of two braid elements for this group corresponds to attaching the beginning of the second braid to the end of the first braid, on the common configuration X0, to form one new braid. The identity element is n non-braiding strings, or visualized as n strings going straight down. It can also be shown that the inverse of a braid exists (it corresponds to applying the inverse of each generator defined below in inverse order of the original braid). Let us study the case M2 = R2. It can be shown that we can represent an arbitrary braid in terms of n — 1 generators cr„ that represent the exchange of the string i and i  1. The string i can go around the string i 1 by going either in front or behind  it, we have to choose one of these moves (similar to the right hand rule) to represent ai, as pictured in Figure 2.4. The other move corresponds to cfV, since we do find that  Chapter 2. Selected topics in topology ^  29  1  and  Figure 2.4: Example of the generator 03 and a3-1.  Figure 2.5: Equivalent braids representing the relation (2.5.1) for i = 2 and j = 6. -1^-1  0-1Gri^  = 1.  All these generators are subject to the relations  cricri = crjcri^— jj > 2  ^  (2.5.1)  illustrated by the equivalence of the braids seen in Figure 2.5, and the relations cricri+iori  = cri±icricri+i^1 < i < n  —  2^(2.5.2)  which is represented by Figure 2.6. The braid group on an arbitrary Riemann surface Mg has more generators. In fact, by taking the string 1, we can associate to each homology generator al and b1, see (2.4.2), a corresponding braid group generator, that we will called cri and Oi. Now in addition to the relations (2.5.1) and (2.5.2), there are several additional relations as follows  jo-i,^= [cri, 13i] =0^2 < i < n — 1; 1 < 1 < g  Chapter 2. Selected topics in topology  30  =--  Figure 2.6: Equivalent braids representing the relation (2.5.2) for i = 3.  p > 1; 1 < I,p < g  criaparat = atoricepai. a^--1^a^--I. cripperi Pt a =-- i3/0-wperi aitaraiPI =  p > 1; 1 < l,p < g  1<I<g  A cri fircii  a -1^a  --1  p > 1; 1 < 1,p < g  al Rpal al = al(71PPal  criaperifil = Acricepai  p > I; 1 < 1,p < g  ai-ictiai /3/ = filo-ice/al  ^  1 < I < g^(2.5.3)  All these relations correspond to equivalent braids, which could be visualized as we did for the relations on the plane. To see these braids, please refer to reference [55J. There is one additional relation that follows from the fact that there always exists a trajectory of a particle encircling all other particles and tracing all homology generators of M in such a way that it forms a trivial loop. For the braid group, this trivial loop must be represented as the identity, leading to the relation 2^a  -1  a-1  N^  ai • - • an_i • • • Cri(Plai Pi al i • • •  a  -1  a-i N  (figa g lig ag ) = 1  (2.5.4)  The above generators and relations constitute the general abstract braid group. In _ most cases, we are interested in representations of this group, even finite dimensional ones.  31  Chapter 2. Selected topics in topology^  The representations obtained by the Abelian Chern-Simons theory are the so-called pure 0-statistics representations where the generator of an interchange of neighboring particles is represented by a phase, times a unit matrix. In this particular type of representation, the generators for particle exchanges ai and those for transport around handles satisfy a far less restrictive set of relations due to the Abelian structure of these o-i. They satisfy the relations (2.5.1) trivially while the relations (2.5.2) tell us that the o-i are equal, which we will refer as a = eie . The remaining relations (2.5.3) become [a all = [cr, 131] = [al, am] = [131 i3m] = 0 [ai, #771] = 0^for^/ m cei 131 = a2,31 • al^  (2.5.5)  and the global constraint (2.5.4) for a closed manifold is o- 2(n+g-1) = 1  ^  (2.5.6)  Given a possible 0-representation of the braid group, we have to verify that it satisfies the relations (2.5.5) and the constraint (2.5.6).  Chapter 3  Abelian Chern-Simons theory  In Section 2.3 we found that the Chern class of a vector bundle over a 4-dimensional space can be reduced to an integral known as the Chern-Simons integral Scs(A), see (2.3.8), over a 3-dimensional space. This was done for a vector space that is transformed by a non-Abelian group. We showed that a large gauge transformation changes this integral by an integer. Thus, an action with Scs(A) will be multi-valued. In the quantum theory we can use a multi-valued action if it is single-valued modulo 2r (or 27rh to be precise), since it would leave the wave function invariant, producing a factor e27ri.(an  integer)  1.  We find the same conclusion using the path integral formalism. Thus, a consistent theory can be defined with the integral Ss(A) with a coefficient 271-k, for an arbitrary integer constant k, which would be invariant under large gauge transformations. This is the famous Chern-Simons action. When coupled to a current source .PL = i Ea  Taja 14, it  is  represented as  2 k S = -- I tr(AdA + — A3) + tr(A,Md3x 4ir m^3  (3.0.7)  For an Abelian gauge, we can still use this action, even though the second Chern class, from which (3.0.7) is derived, is zero in Abelian case. The major difference for the Abelian theory corresponds to the fact that the Chern-Simons action is classically invariant under a gauge transformation (for a closed 3-dimensional space). In other words, the Wess-Zumino integral (2.3.9) is zero in the Abelian case, thus it is not present in the gauge transformation (2.3.10). There are no large gauge transformations (on the 3-dimensional space) since r3(U(1)) = 71-3(S1) = 0. So now the constant k, used above, 32  Chapter 3. Abelian Chern-Simons theory ^  33  has no reason to be an integer. We write the Abelian Chern-Simons action coupled to charge currents as = --k—it4 AdA 47r  IM A„_ed3x^ (3.0.8)  This action will be the center of our research in this Chapter. We shall examine the question of which representations of the braid group on a given Riemann surface are obtained from the wave functions of an Abelian Chern-Simons theory in the most general case where the constant k is a rational number, the Riemann surface has arbitrary genus g and the total charge of the particles is non-zero. We shall construct the wave functions of the quantum theory with action (3.0.8) explicitly and find that, depending on the coefficient k and the genus of the configuration space, the wave function carries certain multi-dimensional, in general non-Abelian representations of the braid group. The wave function of Abelian Chern-Simons theory coupled to classical point particles on the plane was found by Dunne, Jackiw and Trugenberger [24, 50, 53]. In this case the Chern-Simons theory has no physical degrees of freedom, the Hilbert space is onedimensional and the only quantum state is given by a single unimodular complex number. They show that the statistics of particles of charge q is represented by  a =  7r  2  (3.0.9)  When two identical particles are interchanged, the wave function changes by the phase (3.0.9) (or some power of a, depending on the exchange path). This yields a representation of the braid group on the plane. Because of the Gauss' law constraint (see ahead (3.2.4))  x = -k27?(x) the case when the space is compact is somewhat more complicated than that of the  Chapter 3. Abelian Chern-Simons theory ^  34  plane. In order to have an assembly of identical particles, it is necessary to have nonzero total charge. If we have non-zero total charge, Gauss' law requires a non-zero total magnetic flux, which on a compact manifold means that the gauge connection A is not a function but a section of a U(1) vector bundle. This requires some modifications of the Chern-Simons action, which we shall discuss in more detail in Section 3.1 of this Chapter. In previous literature, this complication has been avoided by considering more than one kind of particle so that their total charge adds to zero. In that case, Bos and Nair [9] solved the SchrOdinger equation for Abelian Chern-Simons theory coupled to particles when the space is a Riemann surface of genus g and when k, the coefficient of the ChernSimons term, is an integer. They found that the wave functions carry a representation of the braid group on the Riemann surface. Although they use a different polarization than we used, they found a kg dimensional space represented by a set of theta functions. In a previous work, [7] we found a generalization of their quantization to the case where  k is a rational number. We will show that the correct geometrical description of Chern-Simons theory on a Riemann surface necessarily introduces a framing of particle trajectories. Framing is a standard part of the study of the relationship between the Chern-Simons theory and knot polynomials in the path-integral approach, which was first introduced by Witten [78, 79, 89], and which we discuss in Section 3.5. Variants of framing (such as the point splitting discussed by Bos and Nair [9]) have also appeared in the literature on the Hamiltonian approach to the quantization of Chern-Simons theory. Here, we shall find that our geometrical approach to framing plays an important role in the consistency relations between the parameters k, g and the values of the charges of particles qi.  Chapter 3. Abelian Chern-Simons theory ^  35  3.1 The decomposition of the gauge field Our space will be an orientable 2-dimensional Riemann surface, M, of genus g, while our space-time will be a 3-dimensional manifold M x 1i, formed as the Riemann surface M times a real line for the time direction. In other words, the space-time metric is goo = 1, go1 = gin = 0 and the remaining components form the metric on M. We will decompose the degrees of freedom of A in its various components. To separate the effect of the non-zero total flux (2.3.5) we will break it in two parts. First a fixed field Ap with a total flux Fo on M localized at a reference point zo. This is an "imaginary" field without a direct physical meaning, its purpose is to take care of the total flux. This field has to be defined on patches, as was explained in (2.3.3)-(2.3.5). The second field, Ar, is the remaining degree of freedom of A on M, a globally well defined 1-form. So we have A = Ap + A,^  (3.1.1)  We decompose A, (without the Ao part) into its exact, co-exact and harmonic parts. More precisely, the Hodge decomposition, see (2.2.3) and section 2.4, of A, on M is given by (d and * act on the two dimensional surface M in this Chapter) g 1 1 A, ,--- d(- *d*Ar) + *d(- *dAr) + 2ri Eey-lcol — 7i})^(3.1.2)  of^of  i=.1.  where 1/0' is the inverse of the laplacian (0) acting on 0-forms where the prime means that the zero modes are removed. With our decomposition (3.1.1), dA, does not have a zero mode. Also we will set the zero mode of *d*A, = '.0 • iii., to zero, using a time independent gauge transformation. We can define  k 1  e = -- — *d*A, Fr = *dA,  Chapter 3. Abelian Chern-Simons theory ^  36  So we then have the complete decomposition of the gauge field, with the Ao part, A, = Aodt —  1 ^*d( —'Fr) + 27ri(y-ico1 — 71W1) D  (3.1.3)  Similarly we can write the current two-form j in terms of the one-form *j = j4de  3, using the 3-dimensional star operator * and Hodge decomposition of .3 on M  jodt +  j„ = gi„ju. We can use again the  —^*clik^i(i11^iiwi)  ^  (3.1.4)  The continuity equation dj (f/ • 3.)d3x =^d3x + d*3 A dt = 0 shows that j is a closed two-form that can be used to solve for 7/)  1 ajo = -o, at We shall consider a set of point charges moving on M, with trajectories zi(t) and charge qi, where zi(t) z3(t) for i L j. The current is represented by jo(z,  = E qib(z — zi(t)), 3(2•,t)^E qi8(z — zi(t))(i1(t)cl-2 Ei(t)dz)^(3.1.5)  Integrating (3.1.5) with the harmonic forms col, we find the topological components of the current in (3.1.4) j1(t)^  E qi,ii(t)col(zi(t)) ,^At)^E q 406)1 i(t))^(3.1.6)  This is just telling us that integrating the topological currents ji(t) over time is equivalent to a sum of the integral of the harmonic forms w1 over each charge trajectory. To solve for x, it is best to use complex notation R = 71) + ix = R(z,"i)  37  Chapter 3. Abelian Chern-Simons theory ^  where we find *dx chi) = OzRdz (92.Rdi'. From (3.1.4), (3.1.5) and using (3.1.6) we find  azR jicol(z) =^— zi(t)) jic-D1()^E^— zi(t))^(3.1.7) The solution of (3.1.7) is found using the prime form E(z, w), see Appendix A, with the result that pz E(z,zi(t)) )]^ j1(t) (C,-)1 — wi) R = 121— 27rEi '13g(E(zo, zi(t» ^ at^1^ zo  (3.1.8)  where we have chosen R such that R(zo, 20) = 0 for an arbitrary point zo, which we choose to be the same as the zo in the definition of Ap (We can choose zo = oo for genus zero). The important fact about R is that it is a single-valued function. If we move z around any of the homology cycles, R returns to its original value. In fact, this is also true for windings of zo, an important relation since it is only a reference point. So  x —a  E(z' Mt)) )] —z [(ji(t)ji(t)) jz (6)1 — c.o1)] (3.1.9) o Ot 27r i E(zo, zi(t)) 2  The action (3.0.8) is written for a trivial 15(1) bundle over M, corresponding to a zero total flux. Every integral of the gauge field, which is invariant under a gauge transformation of A, can be extended uniquely into an integral using the A(i), defined on the set of patches, that is patch independent by adding appropriate terms. We will represent the set of 3-dimensional patches as V'. Then Vi and Vj will share a common boundary, a 2-dimensional surface Vii. Finally, three surfaces Vu, Vik and Vki will intersect along a line Lijk , and four of these lines will terminate at a point Pijk 1. This might be best visualized as a triangulation of M x R in term of 3-simplexes (or tetrahedrons), the with 2-simplex boundaries (or triangles), the Vii, which in term has 1-simplex boundaries (or lines), the Pik, and finally those have 0-simplexes (or points) as boundaries, the  ^  Chapter 3. Abelian Chern-Simons theory^  Pijkl.  38  The consistent topological extension of (3.0.8), see [2, 66], is  k k k S = -- E vi A(i)dA-1-- E^x(odA-- E f coik)A(k)>^c<(ijk)x(k1)> 47r^47r^47r Liik^47r ij ijk ^  pi3ki  Ev Acoi _ E^y-^c<(ijk)w(k)> E copox-(ko>(p) Lix .^vii^ piiki ijk^ where the one-form W is defined by j  (3.1.10)  dW locally since j is a closed two-form. Since  fm j = Q, W can only be defined on patches. In between patches we find W(i) — W(j) dVij) in the same way as we did for the gauge field A. The <^> means put the indices in increasing order (with appropriate sign) and set the repeated index according to position, see [66]. It will be useful to do the same decomposition of j as we did for A, by having j =^jr, where jp is a term corresponding to a single particle of charge Q at the reference point zo. The complicated expression (3.1.10) for the action ensures that the total expression is independent of the triangulation of the manifold used for the evaluation of each integral. For example, if we change the patches Vi, the integrand in the first term will change by a total derivative, d(x(ij)dA), leading to a correction term integrated over the boundaries of Vi by the generalization of Stokes' theorem (2.1.6). The boundaries of Vi is a combination Vij for all possible j, thus the second term, in return, will change in such a way as to cancel the change generated by this first term. The total action is thus invariant under deformation of the Vi. A similar analysis can be done for the variation of Vii leading to another correction term, the third term in (3.1.10). Finally we end up with all the terms in (3.1.10), producing a topologically invariant action. Using (3.1.1), the decomposition of j and performing several integration by parts brings (3.1.10) as  S  —^ArcIA,^ArdA  f  Ar(ir  jp)^WrdAp  47 MxR^MxR^P M xR^MxR  ^[  — —47T  ApdAp^Apjp] -I- Surface terms^(3.1.11)  Chapter 3. Abelian Chern-Simons theory ^  39  The terms in brackets, involving Ap and jp, has to be performed using the extended decomposition (3.1.10), by replacing A with A. For our case, we extend the triangulation of M trivially through the time direction. The surface terms, appearing at the time boundaries (t = 0 and t = if), are not important for the quantum theory or the braid group representation that we will find later on. They can't be avoided since the action is not invariant under gauge transformations at the time boundaries. Thus there is no term to cancel the triangulation dependent term. This will not be a problem since under a periodic configuration, we are effectively working on .M x SI-, so there is no surface term, or alternatively the surface terms are equal and cancel each other. Also, surface terms do not affect the dynamics or quantization of the system. We also represent Ap such that dAp = F08(z — zo)d2x, which implies that zo must stay within one patch at all time, and similarly for jp since it is equal to QS(z — zo)dt. After a quick calculation, we find that the terms inside the brackets are all zero, except for the integral, j c<(iik)Wk>, which is defined modulo c(ijk)Q (for periodic motion). This is because W is defined on patches also, due to the total charge Q. At the quantum level, we are left with a phase eic('", but since the C(iik) are arbitrary except for the constraint (2.3.5), the real ambiguity is eiQF°. Actually, the integral f Aj is equal to Eiq fc, A, the Wilson line integral for a set of charges qi following the curves C. In this case for each of these Wilson line integrals, corresponding to the charge qi, we find a phase ei") instead. To resolve these ambiguities, we impose these phases to be equal to unity as constraints on our system. On the other hand, if in addition to the gauge field A, we had a second independent Abelian gauge field, say I', then a similar phase ambiguity, eihsxE, would arise. Here  hi will be the charge attached to the particle i corresponding to this new field, and xE = fm dr is the total flux. The important fact, now, is that the phase ambiguity from both gauge fields would appear at the same time, thus we would have to impose the  Chapter 3. Abelian Chern-Simons theory^  40  constraint eigiFb-ihixE = 1 ^  (3.1.12)  to obtain a consistent quantum theory (The minus sign has been added to simplify the notation later on). At this stage, the new field r seems artificial, but it turns out that it is necessary to introduce such a field for Chern-Simons theory. In fact, it corresponds to a connection on the tangent space of M. We will need it because for each charge trajectory we will attach a framing (a unit vector on M). Such a framing has to be defined in relation to the basis of the tangent space, so r does not have to be the associated metric connection, but it will enjoy the same global properties. It is well known that  = x(M) = 2(1 - g), the Euler class of M. Note that we will assume  that the field r does not have any flux around the particles (an effect similar to cosmic strings), this would lead to a change in the statistics of these particles. The charges hi will be equal to e/2k, this will appear quite naturally in the next Section. Like we did for the field A, we will concentrate all the flux, xE, of  r around the point zo. This will  allow us to assume a constant framing on M, except when we cross the point z0, in which case the constraint (3.1.12) will be used to fix any phase ambiguity. The term fm.RWrdAp = Fo fRWro(zo)dt, but a simple calculation shows that Wro = -x. Since x(zo) = 0, we set it up this way by definition, this term vanishes. If we had not used our freedom in the definition of x to set it up this way, we would have to take care of its effects on the hamiltonian and ultimately the wave function.  3.2 Quantization and the SchrOdinger equation Now we are ready to solve for the action. By putting (3.1.3) and (3.1.4) back into (3.1.11) we find  S=1 - (V. - P-1,.)d3x + irk 1(7% - ;y111)dt + Ao(io — 2— 7r F)d3x 2J  Chapter 3. Abelian Chern-Simons theory ^  it 27r ‘.ajo J k at  41  + Frx)d3x + 27ri (j/11 — 31-yi)dt + Surface terms  (3.2.1)  From this we obtain the equal-time commutation relations of the quantum theory  Wz), Fr(w)] = —iP8(z — w)^or^Fr(z) = iP and  4(z)  (3.2.2)  1 1^a^10 (3.2.3) [11,-rmi= 27rku-b.^or^=^cirm^= 271-k^0-ym^27rk a71^  The projection operator, P, in (3.2.2) changes the delta function to S(z—w)-1/Area(A4), this is needed since Fr does not have a zero mode (fm Fr = 0). The functional derivative must also be defined using this projection operator. With this holomorphic polarization [9] it is convenient to use the following measure in -y space 1412)  =  f  e-2/rk-vmG,a-y1 AFT(1)412 (7) !GI Ild,rnd  M  where 1G1 = det(Gnin). With this measure, we find that 7t =1 as it should be. Ao is a Lagrange multiplier that enforces the Gauss' law constraint 27r^S^27r^27r  F(z) — —1-c- i0(z) = iP  g(z)  d- F06(z — A)) — — j,.0(z) + — QS(z — zo)•=s-', 0  k  k  from which we extract Fo = Q. Since Fo and Q are not quantum variables, this is a strong equality, thus leaving 2r  27r .  Fr(z) — — (3.2.4) k r°(z) j = . 3 4(z)^Tc Jr°(z) °^ Under a modular transformation, the basis 71, 11 will be transformed accordingly. This will not change the choice of polarization, since the modular transformations do not mix -y and 1. From (3.2.1), (3.2.2) and (3.2.3), we find that the hamiltonian, in the Ao =- 0 gauge, can be separated into two commuting parts (where we used 19-61- =  AA*3=Ho+HT  ^  Chapter 3. Abelian Chern-Simons theory ^  where  I27r air°^a Ho = m( k at +ixPT4,)d2x  42  (3.2.5)  while the additional part, which takes care of the topology, is -^1 .1 a HT 07*Y - i -  57)  (3.2.6)  To solve the SchrOdinger equation, we will use the fact that the hamiltonian separates, thus writing the wave function as  t) = wo(^) T ( , t ) with the Gauss' law constraint (3.2.4) 27r — — jr0)x110(,t) 0 k which is solved by t)^(fm (z).iro(z, t)d2x)](t)^(3.2.7) Note that in (3.2.7) there is a term —CK(zo) out of the integral, since jr0(z) jo(z) —  QS(z — zo) gives an extra contribution at z = zo. This shows the presence of an "imaginary" charge at zo, with a flux Fo. The first SchrOdinger equation is  ^(9410(4., t) = How^t)^[ f (271 (9 jro + ix p 6 d2^F0(e t)  ^z  at^k at  which has the solution [9, 24, 50, 89]  (z,e)iro(z,e)d2xdti ^(3.2.8) W(t) exP^foi fmx For a system of point charges, the use of (3.1.9) allows us to write (3.2.7) and (3.2.8) as 27ri^ ^[ ‘110(, t) = exp --Tc--(E ^ g (zi(t)) — W(zo))  t^.  + yk- E qiqj f dtOij(t)+ (I)(t)^(3.2.9) i  ^  Chapter 3. Abelian Chern-Simons theory ^  where  43  7r^t [lo^ j _^t _^t'  i(e)de^ji(e)dt" —^ji(e)de i0 j1(t")dt1 Jo  7r  4-  a {lot j1(e)de Ij1(e)de — j1(e)de fji(e)dt1  (3.2.10)  and  Oij(t)^Im log [  E(zi(t),za(t)) ^1 E(zi (t), zo)E(zo, z3(t)).1  hni fzs(o) col iz,(t)(wi^fzi(o)^iz,(t)(wi^too] 1.1.0^L.70)^izo^Jzs(o)^  (3.2.11)  is a multi-valued function defined using the prime form. We will need the phase (3.2.10) for the topological part of the wave function. The function Oii(t) is the angle function for particle i and j. For i = j, we find a self-linking term of the form Imlog(zi — zi) = Imlog(0) which is an undetermined expression, although not a divergent one. One way to solve the problem is to choose a framing  zi(t) = z3(t) Efi(t) (3.2.12) which leads to the replacement of E(zi(t), zi(t)) by fi(t). This corresponds to a small shift in the position of the charges in jr0, but not in x. In effect, this leads to a small violation of the continuity equation. Alternatively, we can view this term as the additional gauge field F introduced in the last Section. With the framing (3.2.12), we find that (72  4•.dt -= hi^F  2k  "^Ci  (3.2.13)  where Ci is the trajectory of qi on M, representing a coupling of the particles, of charges  hi, to an Abelian gauge field F, as claimed in the last Section. We also recover these charges as hi =-- q.?/2k, which actually are the conformal weights of the underlying two dimensional conformal field theory [89].  Chapter 3. Abelian Chern-Simons theory^  44  The angle function (3.2.11) depends on zo, but it should be invariant if we move zo either infinitesimally or around a homology cycle. For a small displacement there is no change unless one of the charge trajectories, zi(t), passing by zo from one side is now moving from the other side. Looking at the denominator of Oij, we see that this will change To by eiakl", while looking at the numerator, we find a phase ei2-rq(g-1) due to the flux xE of F. Or alternatively, the framing of zi is subject to a rotation of xE/271- = 2(1 — g) turns as we go around M, an effect that we concentrated around zo here. The total phase shift is constrained as 1  (3.2.14)  since it is nothing but the constraint (3.1.12), with the use of the Gauss' law constraint  2irQ and our choice of xE. The equation (3.2.14) will represent a fundamental constraint that has to be satisfied by all charges if we want a consistent solution to Chern-Simons theory. Looking at (3.2.11) shows that we can write  ai  qiqiImlog[l/E(zi(t), zo)/E(zo, MO)]  as Ei q2(—Q)Imlog[E(zi(t), zo)] >(—Q)qiImlog[E(zo,zi(t))], thus representing an additional charge —Q at zo. The constraint (3.2.14) indicates that this is indeed an "imaginary" charge and that it should not be seen by any real charge. For the displacement of  zo around a homology cycle, we find that the angle function changes only by a constant, thanks to the second term in (3.2.11), which will be canceled when we take the difference in (3.2.9). This point is actually more complicated; we will come back to it later. So, at this stage, the wave function (3.2.9), with the angle function (3.2.11), accurately forms a representation of the braid group on a plane [9, 66, 67]. We will cover the full braid group in more detail later. Now, the topological part of the hamiltonian is used to find the part of the wave function affected by the currents going around the non-trivial loops of M. The SchrOdinger  ^  Chapter 3. Abelian Chern-Simons theory  45  equation for (3.2.6) is .OTTe-y, t) .^/^1 ./ z HTAFT(7, t) = z (z7rJry — ^--)111T(7, t) at^ k a-y1  which has the solution WT(7, t) = exp 1271-71/ j/(e)de —^i t ji(e)de I ji(e)dt" +7,(7,t) ^(3.2.15) k o Note that with the phase (3.2.10), the double integral above will turn into fotji(e)de • f0t31(e)de, a topological expression. The remaining equation for iirT(y,t)  ^alifT(7,t) ^1 .1,941(7,t)  at^— c^a-yi  (3.2.16)  is easily solved in the form  1 lifT(71,t)=417,(71--k  ft  ji(e)de)  (3.2.17)  3.3 Large gauge transformations The wave function (3.2.17) is not arbitrary, but must satisfy the invariance of the action (3.0.8) under large gauge transformations, when there is no current. So let us set j4 = 0 for this Section and find the condition on xifT. In general, the large U(1) gauge transformations are given by the set of single-valued gauge functions, with sm and trn integer-valued vectors, U.,,t(z) = exp (27ri(tmrim(z) — snaijm(z))  where nm(z) = i^  (frn lwi _ nnac-,5 1 ) ZO  (Z) =(COm, COm ZO  If we change the endpoint of integration by z^z aiul bmvm with u, v integer and a, b defined in (2.4.2), we find rim —+ 1m + um, ijni --*^vni and Us,t  Chapter 3. Abelian Chern-Simons theory ^  us,te2tmum-smv,n)^  46  us,t. The transformation of the gauge field (3.1.2) under U,,t is  given by 7n1^7m +  Sm  ^ 9171  1m _3^  (3.3.1)  The classical operator that produces the transformation (3.3.1) cs,t(7,1) = exp {(sm ^(sm arym^arym must be transformed into a proper quantum operator acting on the wave function By using the commutation (3.2.3) to replace T. by —271k-ym we find the operators  T  C3,-t  which implement the large gauge transformations [56] exp {-271-k(sm +1/miti)-yn, —  k (sm  Dmiti)Gmn(sn^ti)]  e(sm+Cimiti)8,°.  (3.3.2) The quantum operators  Cs,t  From now on, we will set k =  do not commute among themselves for non-integer k. k21 for integer kj, and k2. Now, in contrast with their  classical counterparts, the operators C,,t satisfy the clock algebra C31 C32,2 = e-27,-ik(3yzi_m2 - trr,  „...1  ,t2  ^  (3.3.3)  Their action on the wave function is Cs,t(7)14(7m) = exp 1-27k(sm fralti)7m —rk(sm +  firniti)Gmn (371 +  friti)]^(7m Sm^tl)^(3.3.4)  On the other hand Ck23,k2t commutes with everything and must be represented only by phases eigss,t. This implies, using (3.3.4),  + k 2 (sm + f2nati)) = exp [—i08,t + 2irki (snl + f2miti)7n, -Orkik2(sm + fritz)Gmn(Sn nnito] 1JJT(7-)^ (3.3.5)  Chapter 3. Abelian Chern-Simons theory^  47  The only functions that are doubly (semi-)periodic are combinations of the theta functions (A.0.1). After some algebra, we find that the set of functions a÷ki.  a^  qfp,r^(71n) =  k2  em"m (  (  1471k 1 k 2 0)^(3.3.6)  where p = 1, 2, ... , k2 and r = 1, 2, . . . , ki with the vectors am, iSm E [0, 1] solve the above algebraic conditions (3.3.5). Their inner product is given by ^ (1 f^p2,r2) =  e-21rk-enGm1-itth ^  \ th  k7  1G1  P2 ,r2 (7)  IG1-1 H d7md-Im  (3.3.7)  141,P2 Sri '1'2  The integrand is completely invariant under the translation (3.3.1), thus we restrict the integration to one plaquettes P (7m = um + fimivi with u, v E [0, 1]), the phase space of the -y's. Under a large gauge transformation  ^ a 2ri ^(a C.,,tillp,r^(7) = exp [27rikpmsm + irksmt, + --,—(amsni — #intm)] klip+t,r^(7) /C2 ^# ^ #  = E[c.,,t]pp4p,,r ( a^ ) (7)  (3.3.8)  101  The matrix [C,,tipm, forms a (k2)9 dimensional representation of the algebra (3.3.3) of large gauge transformations. The parameters a and 9 appear as free parameters, but in fact they may be fixed such that we obtain a modular invariant wave function. The modular transformation (2.4.7) on our set of functions (3.3.6), see Appendix A for more detail, is  (142)^ICS2  a'  Chapter 3. Abelian Chern-Simons theory ^  48  where -y' =--- (C52 + D)-11-7,^=^+ B)(C9 D) ' and 0 is a phase that will not -  concern us here. Most important are the new variables = Da — C/3 where (M)d means  Mdd,  k1k2 k1k2 (ABT)d (CDT )d^= Ba Ai3 2^ 2  the diagonal elements.  A set of modular invariant wave functions {26, 48, 56, 67] can exist only when k1k2 is even, where we set a = 0 (and also 0 = 0). For odd k1k2, we can set a, 13 to either 0 or which amounts to the addition of a spin structure on the wave functions. This will increase the number of functions by 49, which will now transform non-trivially under modular transformations.  3.4 The braid group on a Riemann surface and Chern-Simons statistics Considering a set of point charges leads to the set of wave functions t_ a^ 27i (E qi(zi(t)) I Um im)cle ^( Wp>r^(', 7, tit-) = exp In-k-ynz7n, + 277m o^k^i ^i3^ —  --(A(z0))^qiqa  I  —  —  7r^_  (0 (t) — 0 (0)) + ^  Um — j ,i)de • I Um — 3 "lcit'  cx-1-kip-Fk2r  k1 k2  )t (ki-ym — k2 I jrndelkik2f2) o  (3.4.1)  The wave function depends on charge positions through the integrals over the topological components of the current jm, jm, and through the function 013 (t)  —  0 ij (0). Consider  for a moment motions of the particles which are closed curves, and are homologically trivial. We focus first on the integrals over jm,3771. If, for example, a single particle moves in a circle, we find that the integral of these topological currents vanishes. We  Chapter 3. Abelian Chern-Simons theory ^  49  conclude that these currents contribute nothing additional to the phase of the wave function under these kinds of motions. The function O(t) — O(0) must be treated differently here, because it has singularities when particles coincide, and thus, while motions that encircle no other particles may be easily integrated to get zero, this is not true when other particles are enclosed by one of the particle paths. The result is non-zero in this case, in fact it is 271- (with appropriate sign depending on the loop orientation). Nevertheless, this function is still independent of the particular shape of the particle path. The definition of Oij in terms of the prime form E(z, w) is just the generalization to an arbitrary Riemann  surface of the well known angle function on the plane, that is as the angle of the line joining the particle i and j relative to a fixed axis of reference, determined by zo here. Thus, we may conclude that under the permutation of two identical particles of charge  q, the wave functions defined here acquire the phase aevirq2^  (3.4.2)  For homologically non-trivial motions of a single particle on M , the current integral fot ii(e)de will in general change as fot ji(e)cle _+fot  ii(e)de + Si + S/lmtm, where sI and  tin are integer-valued vectors whose entries denote the number of windings of the particle  around each homological cycle. However, for multi-particle non-braiding paths, there is no contribution coming from Oii. Thus, for a closed path on M, the wave functions become [ 27i^27ri WThr(t) = exp --rmsm — — E qi((a — k2a0)ms'in — 63 — kArtirn) — U k^ki ^ • 1117,,r-f-t(0)^E[Bs,t]r,rillip,r/(0)  (3.4.3)  where a() and 00 are defined by Ei qi go) col^ithnOom and with J = Ei f1(t) — f2(0)) the self-linking term. The matrices (3.4.3) satisfy the cocycle relation B31 ,t1 B32^=  e-  (8itm2_s;ntni1)  32,t2B3,,t,  ^(3.4.4)  Chapter 3. Abelian Chern-Simons theory ^  50  This cocycle has to be contrasted with the large gauge transformations cocycle (3.3.3). They are very similar except that k is now i and the operator act on the wave function Wp,r  on the other index. In this sense, these two cocycles play a dual role on the wave  function. The self-linking contribution, J, in (3.4.3), plays an important role here. For homologically trivial closed particle trajectories, we find J = 0 if the path does not enclose zo in the patch that we are working on, since we choose to put all the flux of F around zo. Otherwise, we find a contribution--LYF q22k  q12,  g) to J, for the particle i. This can be  illustrated by checking for independence of the braiding (3.4.3) on zo. In the definition of the angle function Oij in (3.2.11), we argue that by moving zo along a homology cycle, the angle function is changed by a constant that should cancel out in (3.4.3). Now the function (3.4.3) changes by eiVrQqieiVq=(g - 1) to an integer power. Fortunately this is one, being our fundamental consistency condition (3.2.14). The first phase comes from the shift in ce0 and #0, while the second phase comes from the fact that each charge trajectory crosses zo, which produces a shift in J. To study the permuted (identical particles) braid group, we will consider n particles of charge q, so Q = nq. The representation of the braid group is characterized by its generators, the permutation phase a in (3.4.2), and the braid matrices 138,t in (3.4.3). These generators are the result of the action of elements of the permuted braid group on the particles, which form the external sources in our theory. In fact, let the integer vectors %a denote vectors that are 0 in all entries except for the /th and mth, respectively,  and 1 at the remaining position. Then with the identifications al =Bt,0, 13,„ 130,im, it is easy to check that we recover all of the necessary relations of the braid group on the Riemann surface, given in (2.5.5). In particular, we recover the global constraint (2.5.6), this is just our fundamental constraint (3.2.14), using (3.4.2), applied to this case. We have quantized Abelian Chern-Simons theory coupled to arbitrary external sources  Chapter 3. Abelian Chern-Simons theory^  51  on an arbitrary Riemann surface, and solved the theory. We find that the presence of nontrivial spatial topology introduces extra dimensionality to the Hilbert space separately for the large gauge transformations and the braid group. We find a set of fundamental constraints (3.2.14), relating the charges, k, and g such that we recover a consistent topological field theory representing a general (with some identical and non-identical particles) braid group on M. In particular, we recover the permuted braid group on M.  3.5 Path integral quantization The path integral for the Abelian Chern-Simons action can also be explicitly calculated (on the sphere here). It can be used to check the consistency of the calculations in the canonical formalism. Using the action (3.0.8) on an Euclidean closed 3-dimensional space M, we need to fix the gauge symmetry A --* A + dx. Let us work first in the Lorentz gauge (9AP = 0. The standard Faddeev-Popov procedure [14, 4, 89] gives rise to the gauge fixed path integral  Z[J] = f DADaMexp {—ill f AdA + i f A„j"d3x + i f 471- m^m^m  (.9„24)2 (  + -613c)d3x} (3.5.1)  where c and are the ghost fermionic variables. Gathering the bilinear terms together in (3.5.1) gives iA„(TkirevAaA •+ arv )A, = iApKiLl'Av. The propagator of the gauge field is then GPI' = (K - ')Av .^  0-A____ °A + 00_1 av  k^0^0  (3.5.2)  Thus, integrating the A field and the ghost field (which is decoupled from the gauge field and produces only a constant) is straightforward, resulting in 49), I.,d3x} cAvA 1•0_0_ Z[J] = constant • exp{ — ii L j„G"uji,d3x} = constant exp{ -i7 k- 14 (3.5.3)  Chapter 3. Abelian Chern-Simons theory ^  52  where we used the continuity equation Omj„ = 0. It was expected that the final result would be independent of the gauge fixing parameter. For point charges following closed curves C. (for the zero charge sector on M x R, the curves have to be closed), the current is given by j„ =  Eq J  dri dr— d:S(3)(x — ri(r)) =  E  c 6(3)(x — ri(r))dri,, ,  Using this decomposition of the current and the identity 10 S(3)(x)^—^(remember 4rITI the Coulomb potential), the expectation value of Wilson lines (the integration of A along the loop Ci) are [71, 89] < C >=  Z[J]^.7r Z[0]  = expfz— k  Egigir(ci, can  (3.5.4)  ii  where  (x_ dy,,e,,Ay)A /(Ci,Ci)^dxm^ Jci c _  (3.5.5)  is the Gaussian linking number of the closed loops Ci and Ca, a three dimensional version of the intersection number (2.4.2). Every time a particle of charge qi encircles another particle of charge qi once, /(C2, will jump by 1 (or -1 depending on the direction), which will produce a phase eilgol• For the exchange of two identical particles, of charge q, the phase is given instead by • ir 2  cr = etV , as we found in the last Section (3.4.2). More interestingly, we also find a phase for the self-linking of a curve W(C) = dx„ f dy,E"" (x Y)A c^c^ill3  (3.5.6)  Even though the integrand is divergent when x y, the integration measure cancels the divergence for sufficiently smooth (up to 3 derivatives) curves. The quantity W(C), called the writhe of a closed loop, is not a topological invariant. It is not an integer in general and under a small deformation of the loop C, the writhe W(C) will change by  Chapter 3. Abelian Chern-Simons theory ^  53  a small amount. Let us define a framing of the loop C in terms of a unit vector 71, not parallel to the loop. We can construct a second loop a by deforming C infinitesimally along this framing. The linking of C and 0 is an integer, which can be broken into two pieces [78, 79]  I(C, 0) = W(C) T(C)^  (3.5.7)  where T(C) is called the torsion of the curve C. By using the framing re, the torsion is 1 „ 27r c dx • (n x (Ts )  T(C) = —  (3.5.8)  where ds = (di di)1. A different choice of framing would only change T(C) by an integer since W(C) is independent of framing and the linking of C and 0 is an integer. To obtain a topological field theory, we have to add the torsion (3.5.8), for each particle, to the action. This amount to the addition of the phase q?^. E -2k e d t^ 3  (3.5.9)  where 0 =^2) for a framing il of a particle trajectory. For a 3-manifold Mg x R, a choice of framing with ñ perpendicular to the time would produce the expression (3.2.13) for 03 found in Section 3.2. This shows that the addition of this framing integral is a fundamental requirement, that seems independent of the approach used to solve the theory. We also could solve the path integral in the Coulomb gauge, where Ao = 0 and = 0 for i = 1, 2. This is the same gauge used in the canonical formalism. The gauge  fixed path integral becomes, with A =  Z[J] = DADcDeexp {—izTr k^AdA^A,j"cPx  (OAT  +i (— +^+ 6A c)d3 x} .A4 6^4.2 -  (3.5.10)  Chapter 3. Abelian Chern-Simons theory  It is easier to work in the  6 -*  CXD gauge,  54  in which case the propagator is  GA, = ___ 47 (peov.), _ cv _oALA)OA 1 Ay _ + t Al, uo t^s2,, z12 •-• k 0^ A  (3.5.11)  This time, using the decomposition (3.1.5) of the current, we find that the path integral (3.5.10) reproduces exactly the phase (3.2.10) of the wave function.  Chapter 4  The CP(1) model and Abelian Chern-Simons theory  In Euclidean space the 0(3) non-linear sigma model or the equivalent CP(1) model represents classical ferromagnetic and antiferromagnetic spin systems at their critical points [20]. This is particularly interesting in 1+1 and 2+1 dimensions where certain topological terms can be added to the standard sigma model action. A representative of 72(S2) = Z gives instanton number in 1+1 dimensions and soliton number in 2+1 dimensions and the Hopf invariant 7r3(S2) = Z gives instanton number in 2+1 dimensions. In fact, in 1+1 dimensions the antiferromagnetic quantum spin chain is represented by the quantized 0(3) non-linear sigma model only when a particular topological term is added to the action [1, 42]. In 2+1 dimensions the sigma model is believed to give a good phenomenological description of the 2 dimensional quantum Heisenberg antiferromagnet [12]. It was suggested that a topological term, the Hopf invariant, is important for some of their quantum properties, particularly for those systems exhibiting high 7', superconductivity [25, 81]. Whether or not a topological term appears there is controversial [22, 33, 34, 43, 85]. Since the Hopf term can be used to represent the linking number of soliton histories in 2+1 dimensions, it has been argued [87] that its presence in the action of the sigma model is associated with fractional spin and statistics of solitons when the model is quantized. Their argument is semi-classical in that it depends on the relative importance of classical Hopf instantons [93] in the path integral measure and should apply to the model in the ordered phase. We shall review the canonical approach, which establishes  55  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^56  that the presence of the Hopf term in the action of the CP(1) model is equivalent to the requirement that the wave functionals of the theory without the Hopf term satisfy certain boundary conditions rendering them multi-valued. This multi-valuedness will be responsible for the fractional statistics of the quantum excitations of the CP(1) field. Although canonical quantization of the 0(3) non-linear sigma model is familiar and straightforward [10, 74], the CP(1) model is not so widely studied. For an exception in a slightly different model, see [63]. This is partially due to the complicated nature of the constraints that must be solved in order to go from the variables appearing in the action to the physical phase space of the quantum theory. Also, the CP(1) model is widely viewed as being completely equivalent to the 0(3) non-linear sigma model, which has a simpler canonical structure. However, the CP(1) model has a distinct advantage over the sigma model. The Hopf invariant has a local integral representation in terms of CP(1) model fields and the action, with the Hopf term added, has the desirable feature of manifest locality [93]. In particular, manifest locality in time is important for a consistent canonical quantization of the model. In this Chapter we will explain the relation between the variables of the non-linear sigma model and of the CP(1) variables. In particular, we will describe the Hopf map and the Hopf invariant. We shall explain the canonical quantization of the 2+1 dimensional CP(1) model where a topological term, as a functional of the CP(1) variables, representing the Hopf invariant is added to the action. We will extract the canonical commutation relations of the constrained theory, using the Dirac bracket, in the functional SchrOdinger picture and find that the local U(1) gauge invariance of the model is represented projectively. We construct the 1-cocycle associated with this projective representation explicitly. We argue that the cocycle is non-trivial in the sense that it cannot be removed by a canonical transformation with a single-valued generator. We will show that it can be  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^57  removed using a multi-valued generator and thus obtain multi-valued quantum states carrying a 0-statistic representation of the braid group for multi-soliton configurations. We argue that topological solitons obey the conventional spin-statistics relation. For recent work on this subject, see [6]. We shall also solve the constraints and demonstrate the equivalence of the CP(1) model and 0(3) non-linear sigma model at the canonical level.  4.1 The Hopf map and the CP(1) model Before we begin we shall review some of the features of the Hopf map and of sigma model fields in two and three dimensions. The two-sphere is described by the set of unit vectors na, a = 1, 2, 3 with  Dna)2 a___ 71 2 = 1^ a  (4.1.1)  An explicit parameterization of 77 is given in term of spherical coordinates 77 = (sin 0 cos 0, sin 0 sin 0, cos 0)^ (4.1.2) On the other hand, the coordinates of CP(1) are represented as the manifold of complex rays obtained by taking all complex vectors (zi, z2) with the identification z2)^(Az1, Az2) for all complex numbers A. We will show that the CP(1) manifold is identical to S2. Let us first rescale A such that we obtain a three-sphere, represented by the pair of zi complex variables z = ( with the constraint  Z2  t — t 4 z2 = 1 =^  Z Z  (4.1.3)  Then CP(1) is represented by S3, the z in (4.1.3), with the remaining identification of z with Az where .\ is a pure phase. A more elegant picture of CP(1), or 82, is given as the  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^58  base of a fiber bundle, the Hopf bundle, represented by the constrained z variable PH P (S2 7r H U (1)) S3  where U(1) is the fiber of PH represented by the phase A above. The identification of z with Az corresponds to the map 7rH :^S2, called the Hopf map, ria = zto-az  (4.1.4)  where o-a are the Pauli matrices 0. 1  =  (10  2  01 )  Cr :=  t o^  (1  —  03  0  -  0^-1  )  It is easy to see that ii is uniquely determined within a U(1) fiber,  for 0 <  < 27r.  To show that we do obtain 82, we use an explicit parameterization of z Z+  =e  z= e  •  •  (^) cos ei° sin 2 e-i(k cos sin  , for^0 0 7r  , for^0 0 0  (4.1.5)  which gives the parametrization (4.1.2) of 71, by using the Hopf map (4.1.4). The two-sphere is conformally flat. It is convenient to take advantage of this fact by using complex coordinates on S2. For this we need two coordinate patches, the disc D+, that excludes the point at the south pole, with coordinates  0^z2  = e ick tany =^0 < 0 < z^ 2  (4.1.6)  —  and the disc D_, that excludes the point at the north pole, with coordinates  0^7F 77_ = e-icb cot (-) =^- < 0 < 712 z2 2 -  (4.1.7)  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^59  In the overlap region D+  n D_ Si,  the equator of 82, the coordinates are related by  the holomorphic transition function  71+^0, i_  when  (4.1.8)  0  The upper hemisphere D+ is covered by the unit disc 177+12 < 1, and on the lower hemisphere D_ by 177_12 5_ 1. On the equator 177+12 = 1 = 171_12. Specifying an image point of the Hopf map, 77±, determines the relative phase and moduli of z1 and z2 but is invariant under multiplication of z1 and z2 by a common phase. The pre-image of a point on D+ is given by a circle on S3 , the fiber of the Hopf map, Z+ —  1^ei4 Z+ 1 + 171+12^7 2  =._  1 1 + 19412 71+ ei4 for ^  <^< 2 ^(4.1.9)  The set of all fibers for each point of the disc D+ : 17412 < 1 , 0 <^<27r is a solid torus with coordinates 77+, e+ denoted as T+. Similarly, the pre-image of D_ is the solid torus T-, represented as  1 zj — ^ 1 + 177-i2  ei^=  e2^for 0 <^< 27r 177_ 12 < 1 (4.1.10) 1 1 + 177_12  The Hopf bundle S3 is obtained by sewing T+ and T  -  together on their boundaries by  using the U(1) gauge transformation, see (4.1.5), — = Ci95Z^ Or  =—0  (4.1.11)  We did not specify a connection, A, on the Hopf bundle PH. Such a connection must properly represent the first Chern class (2.3.2) of PH  ci(PH) = —  1^1^ 1 dA = . (A+ = . I^= 1 27ri fs2^2R- 1^ 27rt —  —^  —  (4.1.12)  where the integration on 52 has been reduced to an integration on the equator, the boundary of the two patches D±, where A+ = A_ —id0 by the transition function (4.1.11).  Chapter 4. The CP(1) model and Abelian Chern-Simons theory  ^  60  )  ^4  1  Figure 4.7: The solid torus T+ as a fiber bundle. It was expected that ci (PH) was not zero, since otherwise PH would be equal to 52 x S', a trivial U(1) bundle. A consistent representation, independent of any exterior parameters, of the connection A of PH is given by the condition 0 = zt Dz = ztdz Aztz = ztdz + A, that is A  =  -ztdz  (4.1.13)  The transition function of z does agree with the transformation of A. The curvature 2-form F = dA = -dztdz = - sin(0)dcbd0 2  is gauge invariant. Since sin(0)d0d0 is the volume element on S2, we find that F is constant on the two-sphere. The first Chern class (4.1.12) shows that the fibers of the PH bundle, or .53, are nontrivially related from each others. The importance of the Hopf map, and its relation with the Hopf invariant explained later, resides in the fact that any two distinct fibers on S3 are linked exactly once. To show this, let us view T1 : (77±,e±) as cylinders. The variable e goes along the axis of symmetry and jij< 1 forming the disk where it is understood that (g, 0) is identified with (7/, 27r). A fiber on D+ : (0,0 < e+ < 21r) is depicted in Figure 4.7. Since 7ri (S3) =- 0, the solid torus T+ and T- must be sewn together in such a way that the fiber, which is not contractible in T+, is contractible on S3. That this is the  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^61  ....  T*: I  /  /  /  /  /  Figure 4.8: The fiber from Figure 4.7 deformed to the boundary of T.  T-  I I  •  F  l  e)  1i  I s I \  t  Figure 4.9: The fiber on T+, shown in Figure 4.8 becomes a contractible loop on T. case can be seen by deforming the fiber on T+ to the loop on ST+ (its boundary) shown in Figure 4.8. The transition of this loop from D+ to D_, using the transition function (4.1.11), produces the contractible loop on T  -  shown in Figure 4.9. Now let us consider  two fibers on T+ as depicted in Figure 4.10. When T+ and T are sewn together to make S' these two fibers turn out to be linked. To see this, deform one of the fibers to -  the boundary as shown in Figure 4.11. Note that in order to do this it is necessary to  T _  +  + Figure 4.10: Two fibers on T.  t _ I4--. 1  ‘  j  Chapter 4. The CP(1) model and Abelian Chern-Simons theory  e  114  40  ... ....^  .4 1^ ".  ''''  -  -  7/- °- r  - t ------  ^  I  62  )  I 4  Figure 4.11: We deform one of the fibers to a loop at the boundary of T+ as shown. In order to do this a small loop linking the other fiber must appear.  T  +  l  e-  I  I  i  Figure 4.12: The fiber on the boundary of T+ in Figure 4.11 can be contracted to a point inside T. The remaining loop links the other fiber. leave behind a small loop linking the other fiber. Now the part of the first fiber, which lies on the surface of T+, can be shrunk to a point in the interior of T. This leaves the configuration in Figure 4.12. Then, once again we can deform the second fiber to the boundary of T+ so that it forms a loop similar to that in Figure 4.8 and then move it into T- so as to unlink the hole in T+ to get the knot in Figure 4.13. Thus, under the Hopf map (4.1.4), the pre-image of any point on S2 is a great circle on S3. Furthermore, the pre-image of any two points on S2 are two great circles on S3, which link each other exactly once. Now, let us study the general properties of a CP(1) field on a closed two-dimensional _ space M. For the moment, we will consider time-independent fields z(x) or ri.(x) on M.  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^ 63  T:  oo  Figure 4.13: The remaining fiber in Figure 4.12 can also be contracted to a point inside T-, leaving another loop in T. We are left with two loops linked once in T+. These fields and the Hopf bundle are related by the diagram  P(A4 , , U(1))^---0 S3 = P(S2 , H, U(1)) z^7rH  ^  (4.1.14)  S2 The map il : M^S2, representing the iI(x) field, has non-trivial classes (if M = then the class of this map is determined by a class of 7r2(S2) = Z). In other words, the configuration ii(x) represents a particular soliton configuration. The class of this configuration is given by the Chern class of the bundle P(M, U(1)) obtained from the PH = P (S2 lrH , U(1)) Hopf bundle by the map M^S2. The diagonal map of the diagram (4.1.14), z : M^53, represents the z(x) field. The connection of the  P(M, U(1)) bundle is obtained (pullback) from the connection on PH A(x) = —zt(x)dz(x) With this connection we can calculate the Chern class of this bundle, or the charge sector of the soliton configuration. It is an integer for a closed and orientable manifold  ci(P(M,i,U(1)) = — 271i fjo dA(x) = q This means that the manifold M covers S2 exactly q times, thus the Chem class of PH in (4.1.12), which is unity, is counted q times.  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^64  For dynamical fields, where the space-time is M x R with R the dimension of time, we can generalize the diagram (4.1.14) to the following diagram  P(M x R, U(1)) --)• S3 = P(S2,7H,U(1)) ^ Fr^z M x R^  (4.1.15)  52  Even if the fields ft' and z are now time dependent, the charge sector will be conserved for smoothly varying fields. We can define a topological local current, that integrates to the total soliton charge q, by  j(x) =^(x)dz(x) = -1---ii(x) • (dil(x) x clii(x)) 27r^87r  (4.1.16)  The conservation of this current is also a consequence of the continuity equation dj = 0. Alternatively, under a variation i + 6 '72 which preserves the normalization of n , that is such that -4 • 6-4 0, the variation of the 2-form j is an exact 2-form  1 Sj = d (- 72 • a 72 X clii) 47r  (4.1.17)  This again shows that q is a topological invariant  1^1^ 1 Sq=8-1^• dn' x dii^8(7-1 •^=^d (ft, 871 x d71) = 0 (4.1.18) 87r m^87r m^47r m With the parametrization (4.1.2) we have 1^  1  jd0(x)d(cos 0(x)),^q = — I dOd(cos 0) 4ir m2  (4.1.19)  The Hopf invariant is the Chern-Simons three-form integral (2.3.8) evaluated with the gauge connection A of the bundle P(M x R,51-,U(1)), with a slightly different normalization,  1^1 N =^f AdA^ztdzdztdz 4.2 .m3^47r2 .m3  (4.1.20)  Chapter 4. The CP(1) model and Abelian Cbern-Simons theory ^65  Using the diagram (4.1.15), we find that the integer N represents the class of the mapping 71 : M x R --* 52• For S3 instead of M x R, this mapping is represented by 71-3(S2) = Z. On the other hand, the diagonal map z:MxR —* S3 also has non-trivial classes. Again for a space-time S3, this map is represented by 71-3(S3) = Z. The power of the Hopf map 71H is that it is an isomorphism between these two classes; the class of the map M x R --* 52 is the same as the class of the map M x R --, S3, which is nothing but the Hopf invariant (4.1.20). In gauge field theory, the Chern-Simons term is not a topological term in the sense considered here. Its variation by the gauge field A, 8 f AdA = 2 f 6AdA — f d(ASA), depends on the gauge invariant quantity dA locally in M x R. In the present situation, where A --= — ztdz, the Chern-Simons term is a topological invariant. Under local variations of the fields, which preserve the constraint (4.1.3), 8(ztz) = zt6z + Sztz --= 0,  8 (AdA) = 6 (ztdzdztdz) = d (zt6zdztdz — ztdz6ztdz + ztdzdzt6z) (4.1.21) so that the integral of this quantity is not sensitive to the local profile of the field configuration but depends only on global features. With the coordinates (4.1.5) of z, N has the form  1 N = — I dad(cosO)d0 8r2 m3  (4.1.22)  Here, for each fixed a, 0 and 0 are the spherical coordinates of 52• Furthermore, for each fixed 0 and 0, the a coordinate parameterizes a great circle 51 C S', the fiber of the Hopf bundle. The integrand in (4.1.22) is the volume element on 53• In later Sections we shall show that the Hopf invariant N gives a linking number of soliton trajectories on M x R.  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^66  4.2 Canonical structure of the CP(1) model In this Section we shall study the canonical quantization of the 2+1-dimensional CP(1) model with a Hopf term. Most of our arguments about canonical quantization apply to the CP(1) model in any dimension if we set the coefficient of the Hopf term to zero. It is also straightforward to generalize them to consider CP(N) or other Grassmannian sigma models [94]. Since the Hopf term is a topological term its presence in the action does not influence the classical equations of motion of the CP(1) model. Its presence therefore makes only a superficial difference to the classical phase space. However, as we shall see, it has an important effect at the quantum level. We shall work on the Minkowski signature open space M x R where R parametrizes the time dimension and the spatial manifold M is two dimensional and closed. For the purposes of canonical quantization it is sufficient to treat M as if it were R2 and impose suitable boundary conditions. The Lagrangian density with a topological term is  ^=  D*ZtRIZ  ^47r where  0  Ho, A  ci A  (4.2.1)  -^Uvf 2^M  DA = O,. + A. Let us define, for the moment, A as 1^t^. 7r A =^dz — dzt • z) A  where A is a gauge invariant globally well-defined 1-form. In principle, it could be present in the definition of A since it does not affect the gauge transformation or the value of the Chern class of this U(1) bundle. Since we don't want such an external field in the action, we will shortly set it to zero. With this field, the Lagrangian (4.2.1) produces the action  0 f DztD"zd3x + —  472  k  ztdzdztdz + JA„jAd3x — — f AdA 47r  where k^5. From the last Chapter, we can expect that the exchange generator of soliton field excitation, of charge one, will be represented by a phase eif = eie.  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^67  Now, let us set A = 0, thus using 1 t A =^dz — dzt z)  (4.2.2)  which satisfies the constraints  ztz = 1,  —0 — D* ztz ztD z^— —  (4.2.3)  These are constraints that must be properly satisfied in the canonical formalism. The presence of the Hopf term in the action does not influence the classical equation of motion,  DoD"z(x) — (zt(x)DAD"z(x))z(x) = 0^(4.2.4) Also, it does not enter the classical energy momentum tensor, which can be obtained as a variation of the action by the space-time metric,  T,„(x) (Doz)tDuz (D,,z)tDoz — gi(DAz)tDAz^(4.2.5) which is gauge invariant, symmetric and conserved,  = T^ vp,^aTo, " =0  (4.2.6)  Under the U(1) gauge transformation  z(x)^eix(x)z(x) , z(x)^zt(x)e-ix(s)  (4.2.7)  (4.2.1) changes by an exact term,  0  Zd3x Cd3x d(i- xdA) 47r2  (4.2.8)  If either x or dA vanishes on the boundaries of space-time, the action formed by integrating over space-time will be invariant.  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^68  The canonical momenta, found using Z' aoz and the Lagrangian L = f d2x.C, are given by 7r =  Doz — ° ej(2(ztaiz)ajz (Azt jz)z)^(4.2.9) 6i4^872  = (51 =^zt^(2(zt aiz)a jzt (aiztajz)zt)^(4.2.10) °^87r2 (where fii = Oij). The non-vanishing Poisson brackets are' { (x), 7r1b.(y)lp = 80,8(x — y) =  {4(x), 7b(y)IP^(4.2.11)  Equations (4.2.9) and (4.2.10) determine the time derivative of the fields D5zt and Doz as functions of the phase space coordinates. As we shall see shortly, not all components of Do*zt and Doz are determined. This leads to constraints in the Hamiltonian formalism, similar to the constraints encountered in canonical quantization of a gauge field theory. The canonical momenta in (4.2.9) and (4.2.10) are not gauge covariant, This is a result of the fact that the Lagrange density is not strictly gauge invariant but transforms by a derivative term. Since the covariant velocity operators in (4.2.9) and (4.2.10) are invariant, we see that under the gauge transformation  z(x)^eixMz(x) , z(x)^zt(x)e-ix(')^(4.2.12) the canonical momenta transform as  r(x)^ _4^ eiX(x)(7(x) _Leij aixajz)^71-t(x)^(rt(x)^° Eija.xa.zt)e-iX(S) (4.2.13) 471-2^ 47r2^3 Their gauge transformation can be made covariant with a further canonical transformation  1  7r -4 r + {F, '}p + — {F, {F,r}p}p + • • 2!  + P  2!  (4.2.14)  1^P) P  1Here and in the following the Poisson bracket is denoted denoted by  {...,...}p  and the Dirac bracket will be  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^69  with the generator  0 F = 472 im2 dx(x)zt (x)dz(x)  (4.2.15)  Thus the canonical variables are covariant under a combination of a gauge transformation and canonical transformation. Also, the canonical transformation shifts the Lagrangian by the time derivative of F in (4.2.15) which compensates the gauge transformation of the action in (4.2.8). This is sufficient for gauge invariance of the model in the canonical formalism. In the quantized theory, the transformation law for the action in (4.2.8) and the canonical momenta in (4.2.14) lead to a projective representation of the gauge symmetry. If we impose the CP(1) model constraint ztz — 1 0 2, the equations (4.2.9) and (4.2.10) do not determine all components of the velocity fields i and it and as a result the canonical momenta and coordinates are constrained by two identities, Do*zt(x)z(x) 0, zt(x)Doz(x) 0. One combination of these constraints, Dzt(x)z(x)  —  zt(x)Doz(x) = rt(x)z(x) — zt(x)r(x) — i 2197Tp(x) 0 (4.2.16)  will turn out to be the generator of U(1) gauge transformations. We find here that  p(x) =  i0.zt5.z(x) is the topological charge density. This constraint requires that 2ir^ 2^3  a particular combination of the U(1) charge density of the z fields and the topological charge density is set to zero. It will be treated as a weak relation and will later be imposed as a physical state condition in the quantum theory. It has vanishing Poisson bracket with the CP(1) model constraint, {z(x)z(x) — 1, I:4 zt (y)z(y) — zt (y)Doz(Y)1P = 0  The other constraint is  13; zt(x)z(x) zt(x)Doz(x) 0 2Here we denote equations of constraint by weak equalities^These cannot be made strong equalities until the bracket structure is suitably modified [19J.  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^70  which generates dilatations of Doz(x) and z(x). The constraint (4.2.18) is equivalent to  7rt(x)z(x)d zt(x)r(x) Rz. 0 when zt(x)z(x) cs--2. 1. It has the Poisson bracket -  -  {Dzt(x)z(x) + zt(x)Doz(x),zt(y)z(y) — 1}p =  —  2zt(x)z(x)5(x  —  y).c:_,-  —  245(x  —  y)  (4.2.19) In Dirac's terminology [19] (4.2.18) and ztz —1:::.,- 0 are second class constraints. We shall use them to eliminate redundant variables of the classical theory before quantization, that is we now impose the conditions ztz = 1, irt z + ztr = 0. The remaining degrees of freedom obey Dirac brackets3 {z(x),zt(y))D = lz(x),z(Y)1D = {zt(x),zt(Y)}D = 0^(4.2.20) {z,,,(x), rb(y)ID  = --.12za(x)zb(x)(5(x — y)^(4.2.21)  {zta(x), 4(y)1D = —1.-4(x)zt(x)6(x — y)^(4.2.22) {za(x), 4(Y)}D — (aab — za(x)4(x)) 6(x — y)^(4.2.23) {4(x), rb(Y)}D = (Sab — -4(x)zb(x)) S(x  —  y)^(4.2.24)  {a(x), irb(Y)}D =- 1-2- (ra(x)zb(x) — za(x)rb(x))6(x — y)^(4.2.25) frta(x),7rt(y)1D =^(rta(x)zt,(x) — zta(x)irt,(x)) S(x  —  y)^(4.2.26)  3Suppose we have a set of independent constraints (c, :--- 0, such that det{,0}p 0 0 so that the matrix {, O}p is invertible. When these constraints are used to eliminate variables the canonical brackets must be modified so that the remaining variables have vanishing brackets with the constraints. This is achieved by using the redefined variables x' = x — {x, Calla (-R., Cjp)c—, Co ..,:', x The new variables have the Poisson bracket {xi, V}p ,--- {x, Op — {x, Ca}i- ({C, C}P)al {0, OP fx, OD which defines the Dirac bracket of the original variables x and y. It is these brackets that must be used if the constraints are to be imposed in a strong sense [19].  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^71  ^fra(x)77t(y)}D = i2± (7„(x)4(x) — za(x)711;(x))6(x — y)  ^  (4.2.27)  The velocity fields Doz(x) and Dt)zt(x) differ from the canonical momenta r(x) and rt(x), respectively, by functionals of z(x) and zt(x). It is straightforward to show that the Dirac brackets of Doz(x) and Do*zt(x) with each other and with z(x) and zt(x) are identical with those of the canonical momenta 7r(x) and irt(x) with each other and with  z(x) and zt(x). Thus we expect and will later confirm that, at the classical level, the variables Doz(x), Do*zt(x) can be mapped to r(x), rf(x) by a canonical transformation. Furthermore, under this canonical transformation the action changes by a surface term that cancels the topological term. This is in accord with the fact that the classical theory, as given by the classical field equation and the energy-momentum tensor, is unaffected by the presence of the topological term in the action. Let the generating functional of the canonical transformation be [z, zt , 7r, irt]. It is necessary that z and zt do not transform,  ^5z(x) = {, z(x)}D = 0 , 8z (x) = {4., zt (x)}D = 0^(4.2.28) This holds if is independent of ir and rt. Also' r^1^t) ^6 ^,i.^t,^1^8 ,r^, 87ra(x) —^, r a(x)} D — (vat. — —zazb ^ciz , z j — z zb ^ i.z , zi.. i^(4.2.29) -V^ 2^Szt(x)^2 a 84(X)  8 1 t^)^ 87rta(x) = -- {.rta(x)} =(Sab — zazb ^, z, [ zt]^-_1- zta4 t8 ^[z, zl]^(4.2.30) Szb(x)^2^Szb (x)  4Note that the canonical transformation 7r1= 7 +  {,  7}D + yl M {e, 7}D)D + • • •  7re = +  {e,  1 rt}D + — 2! {, fC 7t1D}D + • • •  in (4.2.29) and (4.2.30) truncates at the first order.  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^72  We are therefore required to find a generating functional with the properties ( Sa b^ °Zb  0  z t ] — lzazbl-[z,zt] = — 2^Szb 871-2  Eij  (2(ztaiz)ôza _1_ (aiztajz)za)  (4.2.31)  1 (Sa l, --1.ztazb)^zt] —^t^ SZb z aSzb  , zt] =  0 8r2  (2(zta1z)ajzta (aiztajz)zta) (4.2.32)  Indeed, there is no single-valued functional of z and zt with this property. In Section 4.4 we shall, however, find a multi-valued functional solution of (4.2.31) and (4.2.32). In the classical field theory this provides a satisfactory canonical transformation. However, in the quantum theory the canonical transformation is represented by a unitary transformation on the Hilbert space. If the generating function is multi-valued the resulting wave functions are also multi-valued. The transformation therefore changes the boundary conditions of the quantum theory. This is a well-known feature of field theories with topological terms in the action, the most noteworthy example being the 0-angle term in 4-dimensional QCD [49]. We will return to this point in Section 4.4. There are six generators of linear transformations on the phase space sa(x) =^(gz(x)caz(x) — zt(x)o-aDoz(x))^(4.2.33) t+(x) = zt(x)o-2Dz*(x) , t(x) = Dozi(x)a2z(x) , (t±^t1 it2)^(4.2.34) t3(x) = -2i (gzt(x)z(x) — zt(x)Doz(x))^(4.2.35)  whose Dirac brackets form the Lie algebra su(2) su(2) = so(4). These generate the isometries of S3. The generators contain only those components of the velocity fields Doz and Do*zt determined by the relations (4.2.9) and (4.2.10) as well as one of the expressions (4.2.35), which is constrained to zero. The spin operators in (4.2.33)-(4.2.35) have the property sa(x)sa(x)^ta(x)ta(x)  ^  (4.2.36)  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^73  This identity can be demonstrated without use of the constraints. It is useful to recall the Pauli matrix identity ir : at. • (7 - - cd = 28 ad& — SabScd. They have the further property that sa (x)sa( x) =  (zt(x)z(x)) (Ho'zt(x)Doz(x)) — il- (D(Izt(x)z(x)-1- zt(x)D0z(x))2 = RoKzt(x)Doz(x)^(4.2.37)  where in the first equality no constraints are used and in the second equality the second class constraints ztz — 1 = 0 and /4ztz + ztDoz = 0 have been used. The Hamiltonian is  H = jr d2x (D'zt(x)Doz(x) + 15* zt(x) • Bz(x))^(4.2.38) where a term proportional to t3(x) has been dropped. Using the identities in (4.2.36) and (4.2.37) the kinetic energy term in the Hamiltonian can be written in terms of the generators of either one of the su(2) algebras,  H = I d2 x (sa(x)sa (x) + 17)* zt (x) • liz(x)) ^(4.2.39) The Hamiltonian has a global SU(2) invariance generated by the su(2) charges sa  =  I  d2x sa(x)  ^  (4.2.40)  Furthermore it commutes with t3(x) and has a local U(1) gauge invariance, which is generated by a u(1) subalgebra of the local su(2) algebra. The Hamiltonian (4.2.39) together with the brackets (4.2.20)-(4.2.27) and  { s a (x ) , s b () }D = f  a bcs c  (x ) a ( x _ 0^(4.2.41)  {sa(x), z(YilD = 7az(x)8(x — y) , {sa(x), zt(Y)}D = —zt(x)craS(x — y)  ^  (4.2.42)  and the first class constraint t3( x) p--- 0  ^  (4.2.43)  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^74  form a complete dynamical system on the constrained phase space. To completely specify the constrained phase space it would be necessary to impose a further gauge fixing condition that has a non-vanishing bracket with t3(x). Alternatively, we could impose it as a constraint on initial data for the initial value problem in classical field theory. This is similar to imposing the constraint as a physical state condition in quantum field theory. We shall discuss this possibility in Section 4.3. Neither the local su(2) generators t±(x) nor the global generators 71± = f d2xt±(x) commute with the Hamiltonian so that, even though the Hamiltonian has a local U(1) gauge symmetry generated by t3(x), and a global SU(2) symmetry generated by Sa -----  f d2xsa, it does not have a local SU(2) symmetry.  4.3 Quantization and the SchrOdinger equation To quantize, it is necessary to associate the Dirac bracket algebra of the coordinates with canonical commutation relations  {x,y}D -- -i-41[x,y}  ^  (4.3.1)  In the functional SchrOdinger picture of field theory the states are functionals of the field configurations, 111 a-_- xlik,z1-]^  (4.3.2)  The coordinates are represented by functional multiplication z0p(x)41 [z, ztj = z(x)11/[z , zt]^ (4.3.3) 4p(x)xF[z,zt] = zt (x)xF[z, ztj^ (4.3.4) The canonical momenta are to be represented by functional derivative operators. The constraint zt(x)z(x) —1 ',---- 0 is imposed by restricting our attention to classical configurations in (4.3.2) that have this property. However, the constraint (4.2.18) on the canonical  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^75  momentum operators is not so easy to impose due to operator ordering ambiguities. We can avoid this problem by realizing that, once the constraints zt(x)z(x) — 1 0 and (4.2.18) are imposed, the remaining relevant components of the canonical momentum operators are contained in the generators of the su(2)esu(2) algebra (4.2.33)-(4.2.35). They can be represented by'  1 (6.:(x)craz(x) zt(x)ua 8 azt(x)) sa(x)=^ ^2 87r2  Eli ((ztaiz)aa (zto.az)^(ôiztajz)(zto.az))  1^ t+(x) = zt kxicr2 6z(x)  r2 fii -47-  (4.3.5)  ((ztaiz)(zto-2aje))  (4.3.6)  1 ^a ^2z(x) —^Eij ((zta1z)(ajzto-2z)) az.(x)a 8 \ +. 0^ _ 1 . ^6 a.zta•z t3(X)^ • (8 Z(X) z(x) zt(x)szt(x)) •  (4.3.7) (4.3.8)  j8r2^3  The notation : : indicates a specific choice of ordering of the product of noncommuting operators. Here we shall assume that they are symmetrized, : z : (  +  Normal ordering is not necessary in (4.3.5)-(4.3.7) since tr(Gra) = 0. It Is straightforward to check that (4.3.5)-(4.3.8) represent the canonical commutation relations derived from the Dirac brackets. The physical state condition is t3(x)4iphys[z, it]  =o  ^  (4.3.9)  which, using (4.3.8), implies that the gauge symmetry is represented projectively:  r i^-i x z, z t e xi exp -° f d2xx(x)Eijaiztajz iliphys[z, zt] 472  illphys Le  (4.3.10)  'This is a result of the fact that they have vanishing Poisson brackets with the constraints zt (x)z(x)— 1 0 and (4.2.18) so that the components of the variables z, zt, r, irk, which are eliminated by these constraints, don't appear in sa and ta. Therefore, any projection of the functional derivative operators one would do to impose the constraints actually cancels out of the su(2)esu(2) generators in (4.3.5)(4.3.8).  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^ 76  Furthermore, since the topological charge density fijAztajz is invariant under local U(1) gauge transformations, the projective phase is a 1-cocycle of the local U(1) group.' This is consistent with the algebra [0(x), t3(y)] = 0 where we have converted the Dirac bracket to a commutator. The one cocycle is non-trivial in the sense that it cannot be removed by redefining the phase of the wave functionals by any single-valued functional of z(x) and zt(x). The quantum mechanical problem reduces to seeking simultaneous solutions of the physical state condition (4.3.10) and the eigenvalue problem for the functional Hamiltonian operator f d'x fsa (x)sa (x) n* zt (x) • 13z(x)}111pEhys[z, zt] pEhys[z , zt] (4.3.11) These equations depend on the coefficient of the Hopf term 0 both through the physical state condition (4.3.10) and through the 0-dependence of the su(2) generators sa(x) given in (4.3.5). The inner product for wave functionals is defined by functional integration,  <  1  1  2 >— TT IPZ(X)dZt(X)6 (Zt(X)Z(X) — 1) Tpthys,1[Z, Z141PhYs,2{Z, VU(1) x  (4.3.12) In order to make the inner product of physical state wave functionals finite (since the product of physical state wave functionals does not depend on the overall phase of z 'Symmetry of a wave function requires invariance up to a phase [86], O(x9) = exp{iw(x, g)}0(x)  The composition law for group operations induces a composition law for the phases, 0(x9.91) = exp{iw(x, g • g1)}0(x) x exp ifw(x° , g') + cv(s , g) — co(x, g - g')}  If the latter phase vanishes, w(xfi , g') + w(x , g) — w(x, g • g')^0 the projective phase w(z, g) is a onecocycle of the symmetry group. For a detailed discussion of the role of Lie group cohomology in quantum mechanics see [49].  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^ 77  and zt, there is a redundant integration in (4.3.12)) we have divided the integral by the infinite volume of the local U(1) gauge group.  4.4 The braid group and fractional statistics In this Section we shall construct the generator of the canonical transformation (4.2.29) and (4.2.30), which maps the velocity fields onto the canonical momentum fields and maps the Hamiltonian of the CP(1) model with Hopf term onto the Hamiltonian of the model without Hopf term. It also removes the projective phase in the gauge transformation (4.3.10) of the physical states. The field configuration z(x) and zt(x) are smooth functions where x takes values either on a compact two dimensional space (which we shall call M) such as 82 or on the open space Ir where we require the boundary condition that z go to a constant at infinity (obeying the constraint ztz -= 1) with covariant derivatives vanishing sufficiently rapidly at infinity. Here we shall assume that the space has been one-point compactified. The field configurations are distinguished into disconnected classes by their topological charge q = -i I dztdz 27M2 Explicit classical field configurations with non-zero topological charge will be constructed later in this Section. Let us consider the sector of the configuration space with fixed topological charge q. We consider a reference field configuration zo, 4 which has topological charge q and also an open three dimensional Euclidean manifold M3, which has as boundary two 2dimensional caps S.A43 = M ED M. In addition, let us consider a field configuration "Z", on M3 whose restriction to one cap coincides with the reference configuration 2,0, 4 and whose restriction to the other cap is the field configuration of interest z, zt.  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^ 78  Then, we define the functional  ztj -  (4.4.1)  41-2 it43  Equation (4.1.21) implies that a it] depends locally only on the field configurations z, zt and zo,  4  which reside on the boundaries of M3: Under a variation of it such that  = 0 and such that 8,Z,^8z, Szt or 6z0, 64 on the two boundaries of M3, respectively, =  1  4m3 r2  d 6^— "itcE(5.&"tcti^dicEt  1^(zt6zdztdz — ztdzSztdz ztdzdzt6z) 471-2 fm 1 — 2 (Z-01.8ZodZPZO — 4PZ084dZo 4dZod4SZO) m  This leaves the functional (4.4.1) depending on the embedding of the field configuration into M3 only. Using this formula, we can take the functional derivative of e by z(x), zt(x). We find that 1^.6^r^fi (5ab^Za4)[2. Z1.1  2^az1b.  (6a b^z "4) 1:27-b[Z  2^Szb  Z•  _1 ,t,t_ ,er,, zti 2^,zti  _ 1 eij (2(ztaiz)ôiza (aiztajz)za) 8r2  (4.4.2) 87r12Eij (2(ztaiz)ajzta^(aiztaiz)zta) (4.4.3)  and the canonical transformation generated by e is z/(x)^e-iqz(x)eiq^z(x) , zt'(x)^e-iKzt(x)eiK = z(x) (D0z(x))/ = e-iqD0z(x)eq r(x) , (D*zt(x))/  ^  (4.4.4)  e-iKD*zt(x)eiK^rf(x) (4.4.5)  that is, when expressed in terms of the original variables the operators of the canonically transformed field theory make no reference to the Hopf term in the original action and  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^79  are in fact independent of O. Furthermore, the Dirac bracket of the classical theory and the commutator brackets of the quantum theory are invariant under the canonical transformation and are also independent of O. However, in the quantum theory there is a further transformation of the quantum state vectors zt] = e-iek'zfhlf[z, zt]^  (4.4.6)  This phase transformation changes the boundary conditions on the quantum mechanical problem. [z, z1.] is a multi-valued functional of z and zt. If we had chosen a different (or the same) three dimensional manifold with the same boundaries .A4 e M and a different field configuration it to fill in .A-43 we would have arrived at a functional 4"[z,zt] which differs from ,[z,ztl: e[z, Yr] = 4.[z, zt] r^3em3^dit  (4.4.7)  The last term is an integer, the Hopf invariant (4.1.20) for the mapping of the extended field configurations and it to the manifold obtained by sewing M3 and ICC together on their common boundary M ED M. The holonomy of the multi-valued wave function is characterized by the topological term on the right-hand side of (4.4.7). For a periodic field configuration i(x, 7), it(x, 7), which corresponds to an integration on M x 511, [z, z1] changes by  1 it 4R-2 Jm2xsi didit Since this integral is a topological invariant, we can evaluate it by deforming i, it to a configuration where the integral can be done analytically. Two topological invariants, which characterize the configuration, are the soliton number, q, and the winding number n=  27r Js1  —  .itcti  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^ 80  We must find a configuration with a fixed q and n. One such configuration is a periodic interpolation of a gas of solitons. Solitons are static solutions of the classical equations of motion that have a fixed topological charge q. They are minima of the classical energy functional for static fields (where we use the complex coordinates, ri = x iy, on M which here we assume is R2) E = Jdx2 (n*zt(x)- z(x)).i chick* (DztD_z DztD+z)  ^  (4.4.8)  with the constraint that q = —^dztdz^  (4.4.9)  Here we define a+ -.7E- (9/(977 and (9_ '9/ô and D± = a, - zta,z, D^a, + zta,z. Without loss of generality, we assume that q> 0. Then, (4.4.8) can be rewritten as E 27rq + 2i I drithi* D_z12^ (4.4.10) which is minimized by functions obeying the condition  D_z(77,77*) = 0^  (4.4.11)  This equation has a unique solution in the sector with topological charge q given by [39, 90, 91, 94]  1 z(q) = I^J^(4.4.12) —^P■12^(77 — bi)) The parameters bi are the spatial points where the sigma-model field is oriented in the third direction, that is on the north pole of s2,^(0, 0,1) and ai are points where = (0,0, —1). A is a constant. Other solutions where z1 and z2 have differing degrees can be seen to have the same winding number (given by the maximum degree) and can be considered limits where some of the ai or bi 00 in (4.4.12) above accompanied by  ^  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^81  a singular rescaling of the parameter A. The function na  = ZtCraZ  is a mapping from the  Riemann sphere with coordinates zi^1^— ai Z2^A^77 — bi —  (4.4.13)  ^  on the upper hemisphere and  H  Z2^q  — = A  — bi  1^—  (4.4.14)  ai  on the lower hemisphere, to the manifold CP(l) :-- S2, which covers the latter exactly q times. This can be seen through the fact that it is represented as a qth order rational fraction. We could view each pair of parameters ai, bi as specifying the positions of two points on a spread-out soliton configuration [28, 29, 36]. For any value of these parameters, (4.4.12) saturates the lower bound of the energy (4.4.10), E = 2irq. We can therefore think of the solitons as non-interacting classical particles. As a first approximation to a theory of dynamical, rather than static solitons, time dependence of z is introduced by making the parameters ai, bi, A in (4.4.12) time-dependent. This approach has been used extensively by Zakrewski and collaborators to study the dynamics of many-soliton states in the low energy limit [18, 75, 80, 94]. Here, we shall consider a periodic interpolation of the soliton solution (4.4.12) ^1  ^z(9,r)  ^VIII  ^  2riwer)^  — ai(7)12 + 1A(7)12 III 19 — biNi2^AN  (9 — ai(T))  HI(— bi(r)) (4.4.15)  where no two of a(7), b1(r) are equal for any T E [0,1], and w(1) — w(0) = n is an integer. For this interpolation, (4.4.7) is given by an integer [17, 94] pl^  zt] —^zt] s qn — — q^ch-linA(r)  +^  d d —  E Imln(ai(r) — Mr))  2r o^A(r) 2r o^dr 3=1  (4.4.16)  Chapter 4. The CP(1) model and Abelian Chem-Simons theory ^82  4t4k4  Figure 4.14: Soliton exchange braids neighboring ribbons. Multi-valuedness of wave functions encountered here gives rise to fractional spin and statistics of many-soliton quantum states. Fractional statistics results from a projective representation of the exchange symmetry of quantum wave functions for identical particles [31, 32, 58]. The holonomy of the multi-valued wave function (4.4.6) which results from the last term in (4.4.16) implies that (4.4.6) carries a unitary representation of a particular version of the braid group on M [71]. Since there are 2 labels in (4.4.16), the a and b variables, the braid group representation would correspond to a 2 color braid group. We will describe this representation in more detail. A subset of the generators of this group can be viewed as the braid group generators a, acting on a set of framed strings corresponding to trajectories of the pairs of points (a„ bi) shown in Figure 4.14. In term of these generators, we expect to recover a pure 0-representation of the braid group of order q. The unitary representation carried by exp(—i0) associates a phase (xi = exp(i0) = a for each generator and exp(—i0) for each inverse generator. These phases are associated with fractional exchange statistics for solitons. The phase of a is exactly the phase that we expected to find back in the beginning of Section 4.2. Fn addition, each soliton can twist, as shown in Figure 4.15. Each such twist is accompanied by a phase si = exp(i0) = s which we associate with fractional spin. The  Chapter 4. The CP(1) model and Abelian Chem-Simons theory^83  Figure 4.15: Rotation of a soliton in the plane twists its ribbon. spin and statistics phases are equal, a^s^eie, indicating the validity of the spinstatistics theorem for the CP(1) field. A similar braid group structure for framed links has recently been studied in the context of lattice Chern-Simons theory [27]. The fact that the braid group elements have 2 colors is the source of the spin. The braid group for the permutation of particles with spin is different from the braid group discussed in Section 2.5. When the system of coordinates rotates by one turn the wave function, and the braid group representation, is transformed by an extra phase s. This phase is in addition to the phase obtained from the framing. The generators a, satisfy the usual braid group relations of a pure 0-statistic representation described in Section 2.5. In addition, on an arbitrary Riemann surface the additional generators would satisfy the relations (2.5.5). On the other hand, the global constraint (2.5.6) will be changed into Cr 2(q+g-1)  =  2(g-1)  ^  (4.4.17)  because of extra phase due to the spin .s. Since we found that the spin-statistic theorem  Chapter 4. The CP(1) model and Abelian Chem-Simons theory  84  Figure 4.16: Exchange of soliton constituents.  Figure 4.17: A positive or right crossing. is satisfied, the global constraint for the CP(1) model is 0.2q  es0-2q  =  1  (4.4.18)  The possible statistics are then given by one of the 2q'th roots of unity. If we are working on the plane then there is no such constraint, although imposing the requirement that the field vanishes, or ii, constant, at infinity is equivalent to a field defined on the sphere. The general 2-color braid group has more moves. One such move corresponds to solitons that exchange constituents, shown in Figure 4.16. This gives a structure more complicated than the usual braid statistics. The phase for a general exchange can be computed by associating exp(i0/2) to positive crossings, see Figure 4.17, and exp(-0/2) to negative crossings, see Figure 4.18. In the quantum mechanical system we are considering the exchange symmetry of the wave functionals, which is therefore represented projectively by eie[z'zt] and this projective phase is responsible for the fractional spin and statistics of solitons. In addition,  ^  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^85  Figure 4.18: A negative or left crossing. there are twists corresponding to the exchange of constituents between solitons, the twist n and the winding of A. Putting back the metric in the static energy function (4.4.8) shows that it is conformally invariant. The static solutions (4.4.12) can be reduced to a solution inside a small circle within a larger space. More than one solution can be considered for a given time slice. Choosing a proper orientation for the solutions (4.4.12), with the time direction going up, corresponds to a soliton of positive charge q, while solutions with the time direction going down, corresponding to a change of y to —y or 77 to 77*, means that we have a soliton with negative charge —q. For an arbitrary closed 3-manifold, M3, let us pick a number of oriented links, Ci, and a field configuration as in (4.4.12) within a small tube around each link, with charge  qi, where we use the link as the time direction. We will set Ai = A = constant, so the topological current ji = 0 outside the  ith  tube. The Hopf invariant (4.1.20) is an integer,  which stays constant if we decrease the tube diameter to an infinitesimal value, this will simplify the following calculation. The gauge fields outside the links are then given by u — x)), 2 A" = --(4"^ 1133/(17)d3u^  2 qic""  ^  Y  ..  x)A dl'^(4.4.19)  We find that the field configurations z outside the links are given by 1^1^g  z = ^IV( ) exp( E iAi) i 50 V1 +^A^ IA  (4.4.20)  Chapter 4. The CP(1) model and Abelian Chern-Simons theory ^86  while the field within one link, Ck, is  z exp(E Ai)zi iok so  (4. 4.2 1)  where io is an arbitrary reference point and zi given by (4.4.15). The Hopf invariant is then easily computed as  N = ik  —  ji Aj"d3x  27r^k  + /(c2, iok  co  (4.4.22)  where  si =^A jfid'x 27r  is given by (4.4.16), this is the self-linking number of a loop. It is interesting that the selflinking number is well defined, and an integer, without any regulation as was necessary in Section 3.2 and 3.5. For i k, I(Ci,Ck) is the Gaussian linking number (3.5.5) of the links Ci and Ck. These are not exact static solutions, but are a low energy limit of soliton interactions as the curvature of the representing links tends to zero.  4.5 Discussion We have demonstrated how, in the canonical formalism, the presence of the Hopf invariant in the action of the three-dimensional CP(1) model leads to fractional spin and statistics of quantum solitons. We conclude by discussing the equivalence of the CP(1) and 0(3) non-linear sigma models at the quantum level. The canonical transformation constructed in Section 4.4 can be used to remove the 0-dependence of the canonical velocity operators, which appear in the quantum Hamiltonian (4.2.39). In the canonically transformed system the  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^ 87  Hamiltonian is now represented by the operator (4.2.39) where the su(2) generators are sa(x) .1 ( ^ 2 Y.z(x)aaz(x) zt(x)°"a6zta(x))  (4.5.1)  and the physical state condition now reduces to Wphys[eixz,  zte^= Tphysk,  (4.5.2)  Neither of these equations refer to O. Instead, the wave functionals must be multi-valued in the sense that if we consider a continuous periodic interpolation it with initial and final configurations z, zt the wave function must change phase illphys[Z,  e iON Tphys[Z, 21] (4.5.3)  where N is the Hopf invariant of the interpolation. Thus the 0-dependence of the quantum mechanical problem has been transferred from the Hamiltonian and physical state condition to the boundary condition, which the multi-valued wave functionals should satisfy. On multi-soliton states we would interpret these wave functionals as possessing fractional spin and exchange statistics for the solitons. It is straightforward to solve the physical state condition: It implies that the physical states can only be functions of z and zt in the combinations zt(x)z(x) or zt(x)o-az(x). The first of these is constrained to one and the second is just the unit vector field na(x) defined in (4.1.4). Furthermore the Hamiltonian can be rewritten as 2  H = Jd3x {--12 (77(x) x aft,(x)) --Ctia(x) • t' fla(x)} (4.5.4) and where we must still take into account the boundary condition (4.5.3). This is the Hamiltonian of the 0(3) non-linear sigma model. Also, the normalization integral for wave functionals (4.3.12) can be written as < kif 1412^dii(x)6 (71 2(x) - 1)^(ft) 412(71)^(4.5.5)  Chapter 4. The CP(1) model and Abelian Chern-Simons theory^88  The CP(1) model and the 0(3) non-linear sigma model are thus equivalent at the Hamiltonian level. The additional boundary condition (4.5.3) gives solitons fractional spin and statistics in both cases. The boundary condition (4.5.3) is trivial when 0 = integer x 27r. The quantum mechanical problem is therefore periodic in 0. Physical quantities such as the energy spectrum should also be periodic.  Chapter 5  Advanced topics  The work done in Chapter 3 and Chapter 4 is based on the Abelian Chern-Simons theory. Alternatively, for a non-Abelian group we can also define topologically related integrals, as was shown in section 2.3. We would like to describe two models, the WZW model and the non-Abelian Chern-Simons theory, where topological features play an important role. These models have been at the forefront of a lot of research in theoretical physics in the last few years. Since a non-Abelian group has more intricate structures than an Abelian group, these models have more elaborate solutions and unfortunately are more difficult to solve.  5.1 The WZW model Some physical problems are better described by a group-valued field. The CP(1) field discussed in Chapter 4 can be viewed as a SU (2) S' valued field, gauged by a U(1) subgroup (the Hopf map). The standard dynamical action of a group-valued field U E G in a two-dimensional space M is given by 1.84 tr(ai,U-18"U)d2x However, there is another term that exist in two dimensions that should be considered, this is the Wess-Zumino integral (2.3.9)  J  1 c4tr(11-1dUU-idUU-ldU) 247r2 89  (5.1.1)  Chapter 5. Advanced topics ^  90  where .A-4 is a three-dimensional space that has M as boundary. We showed in section 2.3 that this integral is multi-valued under large gauge transformations. We can built a consistent theory if we use this term in an action with an integer coefficient k times 27r. We will, therefore, study the two-dimensional model, known as the Wess-Zumino-Witten model (or WZW for short),  Swzw =--8A7r —2^tr(011-104U)d2x  -1tr((U-idU)3)^(5.1.2) -12cr  The parameter A is dimensionless and is a priory arbitrary, but we will show that we obtain a critical theory when this parameter takes a specific value. The equation of motion for the WZW action can be calculated by performing an infinitesimal variation of the field variable U. We must insure that SU is such that  U SU still belongs to the group. The best way to achieve this is by doing a right gauge transformation U -4 USW, with SW around the identity in G, so U-1SU (SW - 1) ,--  iTaStva 0(8w2) is in the Lie algebra of G. Alternatively, we could also do a left gauge transformation U SVU, in which case SUU =- (SV -1) =- iTaSva 0(8v2). Under a right transformation, the variation of the WZW action is given by  6Swzwu = A2 1 (9,2(v^ k ^ ),7-18,0 g^A2 SU^47r^ \F---.4  (5.1.3)  while for a left variation of U we find A2 1^k egu SSwzw ^=\r___:g_ a„w g(g"' + A2 „v==q)(9UU-1)^(5.1.4) 4r SU When A2 k something special happens. The term -1(V---,4(e' -^becomes a A2 Nr_ti 2^ projection operator p/211 ^  1^1 - (g4v ^ €4u)  2  (5.1.5)  NI-L-.4  with the properties 131-LagoPl3v Pga,813"13 =  ^  gcoP4aPvie  = goPagPfiv = 0  Chapter 5. Advanced topics^  91  In an orthonormal basis, where gm, = eybriab with the local metric given by 71 ab (01)  1 0  corresponding to the local coordinates x± -1-kt ± X), we find the relation v2  L  (5.1.6)  -La uu-i. 2ir  The equation of  L  P"'AaB, = PabAaBb = A_B+^where Let us define the currents J+ = —k u -1 a +u and J_ = 2ir  = eati a  motion (5.1.3) becomes  sswzwu = a_j+= 0  (5.1.7)  SU  Furthermore, we can use the identity  a_(U-la+u)= -u-ia_uu-la+u +u-la+a_u =u-'(-a_uu-la+uu-l+a+a_uu-l)u =u-la+(a_uu--1)u which shows that the equation of motion (5.1.7) found from a right transformation is equivalent to the equation found from a left variation  6.5ivzw = a+L= 0^ SU For A2 =  —  (5.1.8)  k, the equation of motion (5.1.4) takes the form (5.1.8), which is equivalent  to (5.1.7), but with the change  4= tö+uu-1  and J_ =  tru-ia_u instead. The only  difference being in the ordering of U and dU in the definition of the non-Abelian currents (naively, we would expect the current to be defined as J  = u--149±u). The equation  (5.1.7) and (5.1.8) shows that the current Ja and the dual current  = &V, are both  conserved. For A2 ±k this would not be possible and only one type of current could be conserved, representing a very different theory. In fact, under renormalization the parameter A will be renormalized to the critical point A2 = k (for k positive) [88]. The solution to the two equations (5.1.7) and (5.1.8) is easily found  U = V(x)U01/1/(x)^  (5.1.9)  Chapter 5. Advanced topics^  92  where U0 E G is a constant matrix. For this solution, the currents becomes J+  J+(x) = trV-10+V and J_ = J_(x) = 4:Ta_WW-1. The solution (5.1.9) can be viewed as the non-Abelian version of non-interfering right and left moving plane wave solutions of electromagnetism. For an arbitrary field U, the WZW action is invariant under the transformations  U V(x+)UW(x-). In fact, for the classical solution (5.1.9), the WZW action is identically zero. It can be shown that the generators of these right and left symmetries are the currents <I± itself. At the quantum level, these symmetries leads to the Gauss' laws constraint on the ground state  < 4, 0 141410 >=< *01J_ >= o To find the structure of the states of the WZW model, we need to know the commutation relations of the currents, as operators, among each other. This can even be carried out at the classical level by using Poisson bracket, after properly defining the symplectic structure of the WZW model [88]. It is found that the right and left sectors commute with each other (as expected) and each sector has the same algebra. With the decomposition J = iTaJa, the commutation algebra is  [Ja(x),Jb(y)] = ifabcJc(x)8(x —) + Sabe(x — y) (5.1.10) which is known as the Kac-Moody algebra [57, 38]. The major difference of the KacMoody algebra, compared to a Lie algebra, is the presence of the Schwinger term 2/cir 6ab (x —  y). Also, since it depends on a one-dimensional space variable, usually consid-  ered a circle, it is an infinite dimensional algebra. The Kac-Moody algebra corresponds to the algebra of loop group [69] since V(x) corresponds to a map 511 G (for x representing a circle). The study of the algebra (5.1.10) shows that we can separate J as J T +Tt where  Chapter 5. Advanced topics^  93  T is a destruction operator and Tt is a creation operator. The ground state is defined by^>, 0, while Tt is used to create all the states of the theory. The WZW model can be used to solve exactly various two-dimensional models. For example, the well known Ising model or the 3-state Potts model [21]. 5.2 Non-Abelian Chern-Simons theory The non-Abelian Chem-Simons action (3.0.7), derived in Chapter 3,  k S --  1  tr(AdA —2 A3) + tr(AmMd3x 47r JM^3^m  (5.2.1)  is more complicated to solve than the Abelian version. For simplification, we will work on le and set the current j° = 0 for now. We can separate the time part from the space part of (5.2.1), which brings the action into  k k I Oiltr(AiA^ i)d3x +^tr(A0(-- B))d3x 47r n3 n3^27r  —  where B = OijFii. This indicates that the commutation relation are given by 2krisaboija(x _ y)^(5.2.2) [A7(x),Abi(y)J ___  Tb)^sab where we assume the normalization tr(T a^We will choose the polarization defined by the Al_ variables (with the same notation for x± as in Section 5.1), thus  A'  27ri  k SA  (5.2.3)  By working in the Ao 0 gauge, we find the Gauss' law generators  6^k Ga 27r^6A+a^27r^+  (5.2.4)  They satisfy the commutation relations  [Ga(x),Gb(y)]^ifabaGa(x)8(x — y)^ (5.2.5)  ^  Chapter 5. Advanced topics^  94  which is identical with the algebra of the group G. This allows us to consistently define the ground state by the Gauss' law constraint  Ga(x)41° = o  ^  (5.2.6)  It is possible to defined a group-valued field U by the condition  A+(x) = iTaA4(x) = U-1(x)8+U(x)^(5.2.7) The equation (5.2.7) can't be generalized to A = (I-1(1U since A might have non-zero curvature. From this, we can build the ground state wave function in terms of the WZW action (5.1.2) (at A2 = k), see [40, 41], = exP(—iSwzw(U)) ^  (5.2.8)  Using (5.1.3) we find, for M = R2 and SU 0 as IA --* oo, that  8Swzw = „ tr(a_(U-10+U)U"SU)d2x = zr 2^ R  zr R2  tr(S(U"a+U)U-1,9_U)d2x  From (5.2.3), (5.2.7) and (5.2.8) this means A_‘110 = iTa  SA1  1110 = u-ia_uwo  ^(5.2.9)  In other words, A = U'dU is a flat gauge (in the space dimensions) as an operator on the wave function 410. It follows that the Gauss' law constraint (5.2.6) is satisfied. To consider the effect of particles, we need to calculate Wilson lines. For one particle of unit charge the Wilson line is given by  ^W(C)  =  P  exp(i  fc, A)^  (5.2.10)  Since A in (5.2.10) is an operator and it belongs to a non-Abelian group, the evaluation of Wilson lines is fairly complicated. For multiple charges this becomes even more difficult. We would like to give some of the results of such calculations [40, 41, 57].  ^  Chapter 5. Advanced topics^  95  For a curve C starting at x1 and ending at x2, it can be shown that  w(c )m w o = ti71 (x i )uK.1 (x 2)T o^(5.2.11) where the indices I, J, K labels the groups element of U. The equation (5.2.11) can be pictured as the creation of a charge at x1 represented by U./1- (xi), which is later destroyed at x2 represented by U.KJ(x2). The contraction of the index K corresponds to the curve C from x1 to x2. In general, to each particle of unit charge corresponds a representation of G. For two charges, we can associate the state  IL0 >. II > IJ >. Is > +la > where Is > and la > are the symmetric and antisymmetric component of IL0 >. The exchange operator of the two charges and its inverse, after some non-trivial algebra, are given by IL_ >=.- EEIL0 >^and^IL+ >= EE-11L0 >^(5.2.12) Where El/ > IJ >. IJ > II > is the permutation operator and the matrix E is given by  E = exp(iirRa 0 Ra) (5.2.13) We should view E in (5.2.13) as the generalization of the permutation phase a = see (3.0.9), to a non-Abelian theory. For example, for the fundamental representation of  SU(N) we have the identity TbricL from which  1cc  -= SIL SJK — i—\7 °IJ°KL  (N — ___ IL_ >,_ exp(i Nk  )I> exp(  i  (N + 1)71-1),1 Nk  a >^  (5.2.14)  and  IL+ >,_ exp(—i (N — 1)7 )ls > exp(i (N + 1)7,1  Nk^Nk )1a >  (5.2.15)  Chapter 5. Advanced topics^  96  The equations (5.2.14) and (5.2.15) forms a two particle representation (not a pure 0-statistic representation) of the braid group. This can be generalized (not trivially) to an arbitrary number of particles. Recently a lot of work has been done on quantum groups [23, 57] (an extension of the standard Lie group algebra). The algebra of braiding of quantum states of the WZW model produces interesting quantum group representations.  Chapter 6  Conclusion  In Chapter 2, we described a number of advanced mathematical subjects. These subjects are not very well known in the physics community in general. Once understood, these mathematical tools give a deeper understanding of what really is a gauge theory, and what are the possible topological structures of the gauge field. Hopefully, this new kind of mathematical formalism will be used more often for a good understanding of physical models when needed. The study of Chern-Simons theory seems to confirm this trend. The solutions given for Chern-Simons theory in Chapter 3 are not necessarily a perfect model for a representation of the quantum Hall effect, or other physical models under study. Instead, we concentrated our efforts into building a consistent model of a ChernSimons theory, by working out its general solutions and properties. The fundamental constraints (3.2.14) e'2irq1(Q+Mg-1))  =1  of Chern-Simons theory imposes a rather interesting set of restrictions among the different parameters of the model. In principle, these restrictions might be used to check the validity of a Chern-Simons model as the representation of a physical system. Recent works [52] on the fractional quantum Hall effect, where v is the filling fraction, has been concentrated around a variation of Chern-Simons theory S  1 = _-  kif A1dAJ IJ  where  km  is a matrix with integer entries. The indices I labels different 'levels' of 97  Chapter 6. Conclusion^  98  the quasiparticles. For one level only, this is the standard Chern-Simons theory, it is conjectured that the filling fraction is related to the permutation phase by a = ei", thus we find the relation v = Alternatively, the Laughlin wave function [3, 4, 70], that describes phenomenologically the fractional quantum Hall effect, is very similar to the wave function (3.2.9) found in Chapter 3 (by working on S2 where E(z,w) = z — w). The work on the CP(1) model, or 0(3) non-linear sigma model, in Chapter 4 has shown how a dynamical field theory coupled to a Chern-Simons action is consistent with the results found in Chapter 3 about Chern-Simons theory. In particular, we found that the soliton field configurations obey a certain type (two-colors) of braid group. It is not clear if this modified CP (1) model might be related to the ordered phase of a quantum Heisenberg anti-ferromagnet in two dimensions; more study is still required at this point. It is our hope that a good understanding of such models, exhibiting fractional statistics, will be useful in future work on these recently discovered physical problems, and possibly new ones, whose dynamics are primarily two dimensional.  Appendix A  Theta functions  The theta functions [51, 62] are defined by  ao)  (z p)^E  ei7r(711-Faigem(nm-Fam,)+2/ri(nii-cei)(z1-1-#1)  (A .0.1 )  ni  where a, E [0, 1]. They satisfy the following properties  (zm^  ,30 nml) = e2iriaisi_iirt,ornit,_27ritm(zm+7-1  a )  (zin)  ( a P ) for integer-valued vectors .sm and t1, while for a non-integer constant c ^ ao (zm^= c i 2tm1mt tt2iriCtm(zm+13m)e  a + ct  (42)  Under the modular transformation (2.4.7), the theta functions are transformed as a a (z1D) --*^ o^  (z'10') = e-i1r(4. det (CD -I- D)1 eilrz(cf2+D)-1 '0^a^(z ID)  where z' = (Cu +^z, see (2.4.8), and q is a phase (that we will ignore). The new variables a',^are given by 1^ 1 a' = Da -^- - (CDT)d, ^,3/ = - Bad- AP - - (ABT)d  2  where (M)d are the diagonal elements  2  Mdd.  99  Appendix A. Theta functions ^  100  Closely related to the theta functions is the prime form  E(z,w) (h(z)h(w)) - i 0 (112 ) (jz co151) 1/2  where h(z) =  a ( 1/2 ) (02)1,, 0  7u^  • col(z). The prime form is antisymmetric in the  1/2  variables z and w and behaves like z — w when z w (the h(z), which appear in the denominator, are for normalization). This formalism can also be extended to include the sphere, where there are no harmonic 1-forms at all (the space of cohomology generators has dimension zero) by properly defining the prime form. We use stereographic projection to map the sphere onto the complex plane and use E(z, w) = w — z as the definition of the prime form. The prime form is particularly useful when solving the differential equation  ôR = 45(2)(z — zo) ^  (A.0.2)  on a Riemann surface of genus g. Since on the Riemann sphere 1 ^1^1 8(2)(z — zo) = az( )=— 7r .9..(az log(z — zo))  z — zo  the solution to the differential equation (A.0.2) is given by  R = 1— az(log(E(z, zo))) r  ^  (A.0.3)  Explicit formulas can be worked out for the special case of genus one. In this case it is customary to choose co(z) = 1 and 12 =  T,  a complex number with Im(r) > 0. The  torus is represented as the complex plane, parametrized by z, restricted to z = u  TV,  with u, v E [0, 1], and imposing the boundary condition that z = u is identified with  z = u +T and z =  TV  is identified with z =1+  TV.  Alternatively, we can view the torus  Appendix A. Theta functions^  101  as the complex plane where z is identified, with z m nr with m and n integer. Using the variables q = e21rir , thus lql < 1, and y = e21riz, we find (not trivially) the identities  0 o (zIT) E ynqfr f(q) fl(/ + ye+1)(1 + y-1gk-F1) (  k=0  0 ( 0  (ziT) = E(-1)nynqin2 f(q) H (1 - yqk+1)(1 - y-Y+1) k=0  1/2  ( 1/2)  = Eyn-F5e+112/2 =^f(q)  (*)  0^  k=0  (1 + yqk+1)(1 Y-lqk)  0. ( 1/2 )^ 0^(zir) = iE(-1)nyn+lq(n+1)2/2 = iyiqkf(q) 11(1 — yqk+1)(1 — y-igk) k=0 1/2^n^ where f(q) = 111°,11(1 — qk) = En(-1)nqin(3n+1) and ii(r) = q* f (q) is the Dedekind function. We also find that h(z) = —271-773(r) is independent of z. Thus the genus one prime form is E(z, where now y =  w  )  (1 - ye)(1 y-le) k=1^(1 - q92  —± ,„1^fi (  27r \a^`Y  e27r1(z-").  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