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Topological field theories and fractional statistics Bergeron, Mario 1993

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TOPOLOGICAL FIELD THEORIES AND FRACTIONAL STATISTICSByMARIO BERGERONB. Sc. (Physique) Universite Laval, 1987M. Sc. (Physics) University of British Columbia, 1989A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF PHYSICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1993® MARIO BERGERON, 1993In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of^19(-051(5The University of British ColumbiaVancouver, CanadaDate scp-FEHOEgillirr7DE-6 (2/88)AbstractWe examine the problem of determining which representations of the braid group on aRiemann surface are carried by the wave function of a quantized Abelian Chern-Simonstheory interacting with non-dynamical matter. We generalize the quantization of Chern-Simons 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 functionscarry a representation of the braid group as well as a projective representation of thediscrete group of large gauge transformations. We find a fundamental constraint thatrelates 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 canonicalstructures of the model using Dirac bracket, accounting for the non-trivial constraintsof the sigma-model. We show that solutions to the field equation are represented bysolitons. We also recover braid group representations for the low energy limit of solitonexchange.Table of ContentsAbstractTable ar coiAte 1-5List of FiguresAcknowledgement1vii1 Introduction 12 Selected topics in topology 82.1 Manifolds, exterior calculus and vector bundles ^ 82.2 DeRham cohomology, homology and homotopy 152.3 Characteristic classes ^ 192.4 Two dimensional topology and Riemann surfaces ^ 222.5 Three dimensional topology and the braid group 273 Abelian Chern-Simons theory 323.1 The decomposition of the gauge field ^ 353.2 Quantization and the SchrOdinger equation 403.3 Large gauge transformations ^ 453.4 The braid group on a Riemann surface and Chern-Simons statistics 483.5 Path integral quantization ^ 514 The CP(1) model and Abelian Chern-Simons theory 554.1 The Hopf map and the CP(1) model ^ 571114.2 Canonical structure of the CP(1) model^  664.3 Quantization and the SchrOdinger equation  744.4 The braid group and fractional statistics . .  ^774.5 Discussion  ^865 Advanced topics^ 895.1^The WZW model . . . .^.....^................ ^ 895.2 Non-Abelian Chern-Simons theory  936 Conclusion^ 97Appendices 99A Theta functions^ 99Bibliography^ 103ivList of Figures2.1 Example of a genus 3 Riemann surface ^  232.2 Holonomy of a Riemann surface represented by loops am and bm intersect-ing at 13Th,  ^232.3 Example of a braid ^ 282.4 Example of the generator a-3 and cr;1  292.5 Equivalent braids representing the relation (2.5.1) for i = 2 and j = 6. . ^ 292.6 Equivalent braids representing the relation (2.5.2) for i = 3. . . . . . .  ^ 304.7 The solid torus T+ as a fiber bundle. ^  604.8 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 ^ 614.10 Two fibers on T. ^4.11 We deform one of the fibers to a loop at the boundary of T+ as shown. Inorder to do this a small loop linking the other fiber must appear. ^ 624.12 The fiber on the boundary of T+ in Figure 4.11 can be contracted to apoint inside T_. The remaining loop links the other fiber^ 624.13 The remaining fiber in Figure 4.12 can also be contracted to a point insideT, leaving another loop in T. We are left with two loops linked once inT+^  634.14 Soliton exchange braids neighboring ribbons^  824.15 Rotation of a soliton in the plane twists its ribbon ^ 834.16 Exchange of soliton constituents. ^  84v4.17 A positive or right crossing. ^ 844.18 A negative or left crossing85viAcknowledgementI owe a lot of gratitude to many people for the work of this thesis and more generally forall that I have learned in my stay at UBC. I would like to thank my supervisor GordonSemenoff for his constant support, supply of research projects and many discussions. Hisdeep 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 therigor of mathematics.I greatly benefitted from the many discussions and friendship of follow graduate andpost-graduate students. In particular, I would like to thank David Eliezer, Richard Szaboand Edwin Langmann.I am grateful to all the members of the physics department and its staff for makingmy stay at UBC as friendly and pleasant as possible. I wish to thank also the Niels BohrInstitute 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.viiChapter 1IntroductionIn recent years, the role played by topology in physics has led to the discoveries of interest-ing 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 thelocal details out. This is, by definition, not a complete or even very deep description ofa given space. The importance of topology comes from the fact that it is often the onlyinformation available. In physics, we are also often faced with a problem whose solutionis rather intractable (no algebraic solution and poor numerical calculations). In some ofthese cases, the topology might be a tool that will give us some information.More importantly, it has become clear that topological features of field configurationsplay a fundamental role in the quantum description of physical models. In particlephysics, the study of anomalies in gauge field theories [14, 76] has shown that they arerelated to characteristic classes, a relation best expressed by the celebrated Atiyah-Singertheorem [5, 15]. In this thesis, we will investigate a different application of topologyrelated to spin and statistics in quantum mechanics.Since the development of quantum mechanics, the spin of a particle has been classifiedin 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, forwhich the wave function changes by a minus sign under a full rotation of the systemof coordinate. The statistics of an assembly of identical particles, that is how the wave1Chapter 1. Introduction^ 2function 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 quantumlevel by generators, that transform the wave function, forming a representation of thebraid 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 ordinarybosons or fermions. More generally, the braid group on a two-dimensional surface is aninfinite, discrete, non-Abelian group and has many potentially interesting representations[26, 48].Anyons are sometimes described mathematically by coupling the currents of pointparticles to the gauge field of a Chern-Simons theory [24, 35, 50, 65, 73, 74, 78]. This isrepresented by the actionwhereS =^ftwAA0i1Ad3x + J Amj"d3x47r I (1.0.1)dr"j"(x)= Eqj 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 thefractionally quantized Hall effect [3, 4, 68, 70]. The Hall resistivity exhibits plateaus whenthe filling fraction v is an integer for the integer quantum Hall effect. At the same timethe conventional resistivity drops to a very low value indicating that the two-dimensionalelectron gas flows with no resistance. Later, it was shown that the Hall resistivity exhibitalso plateaus when v takes some fractional values. It is still unclear why the electron gashas this behavior at these fractional filling. It is conjectured that anyons, represented byChapter 1. Introduction^ 3a 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 theexact ground state of the quantum Hall effect [11, 44, 64, 77].The recently discovered high temperature superconductors have yet to be describedsuccessfully by any theoretical models. Since this particular type of superconductivity isevolving 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 isstill open.Fractional statistics can also be realized for soliton excitations. The 0(3) non-linearsigma 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 orderedphase of a quantum Heisenberg anti-ferromagnet in two dimensions and of a classicalNeel anti-ferromagnet in three dimensions. It has been suggested that a topological termsimilar to the Chern-Simons action, the Hopf invariant [8], might be responsible for someproperties of high temperature superconductivity [25, 81]. This is, again, a controversialissue [22, 33, 34, 43, 85].It became clear that a thorough understanding of Chern-Simons theory, especiallythe Abelian version, at the quantum level is essential if we are to consider such a theoryin 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 particleson the plane was found by Dunne, Jackiw and Trugenberger [24, 50, 53]. In this casethe Chern-Simons theory has no physical degrees of freedom, the Hilbert space is one-dimensional and the only quantum state is given by a single phase. In order to study thestatistics of a given system we need to have identical particles, which implies a non-zerototal charge. In return, if we have non-zero total charge, Gauss' law for Chern-Simonstheory requires a non-zero total magnetic flux, which on a compact manifold means thatChapter I. Introduction^ 4the 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 thanone 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 particleswhen the space is a Riemann surface and when k is an integer.In Chapter 2, we will describe several novel (at least for physicists) mathematicalstructures. Since these mathematical constructions are rarely explained in physics pa-pers, this Chapter should be useful for putting topological field theory in two and threedimensions on a firm mathematical basis. In Section 2.1 we will review exterior calcu-lus and the properties of vector bundles, which will be used extensively in the followingsections and chapters. In Section 2.2 we will describe the DeRham cohomology at somelength. Topological structures used in physics are almost always represented by someintegrals of DeRham cohomologies. For vector bundles, we can define particular coho-mology classes known as characteristic classes, this will be explained in Section 2.3. Inparticular, we will show the origin of the Chern-Simons action, as shown in (1.0.1), forma particular characteristic class. In Section 2.4, we will describe the classification of aRiemann surface of genus g, while in Section 2.5 we will give a detail description of thebraid group on such a surface and some of its representations.In Chapter 3, we will describe the quantization of the Abelian Chern-Simons theorycoupled to point charge current, see (1.0.1) and (1.0.2), on a Riemann surface of genusg. In particular, in Section 3.1 we will show how to decompose the gauge field on aRiemann surface and how to write down a consistent Chern-Simons action for a non-zerototal charge. In Section 3.2 we will solve the SchrOdinger equation and show that thecorrect geometrical description of Chern-Simons theory necessarily introduces a framingof particle trajectories. This framing corresponds to an additional U(1) gauge connection,Chapter 1. Introduction^ 5in fact the connection of the tangent space of the surface, and plays an essential role indetermining the consistency of the theory. This point has been largely ignored in theliterature. We will show that for a set of particles of total charge Q = Ei qi and for k arational number, the fundamental constraintseivq,(Q+qi(g_i)) 1 (1.0.3)have to be satisfied. In Section 3.3, we will described how the wave function transformedunder large U(1) gauge transformations, determining what is the periodicity conditionsof the wave function around non-trivial holonomy cycles (non-contractible loops) of thesurface. 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 functionsdefined with fractional indices. In Section 3.4, we will show that the wave functionacquires phases when the particles move on the surface. More precisely, we will showthat the statistic of such particles is given by a representation of the braid group on theRiemann surface. The description of the Abelian Chern-Simons theory by using the pathintegral formalism, in contrast with the canonical formalism, will be discussed in Section3.5. The consistency of the two formalisms will be shown with respect to the braid grouprepresentations and the framing regularization.A topological constraint like (1.0.3) might not be of immediate relevance, since mostphysical situations do not occur on a compact Riemann surface. On the other hand, oftena field is considered to tend toward zero (or a constant for a unit vector) at large distanceson 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 periodicpotential of a lattice, is equivalent to a system on a torus. In particular, it is not certainat all if the constraint (1.0.3) is related to the filling fraction v of the fractional quantumHall effect. This will have to be investigated further in future work. For the applicationChapter 1. Introduction^ 6of Chern-Simons theory on a Riemann surface to the fractional quantum Hall effect, seeWen in [83, 84]. In any cases, it is hard to believe that there is no system in nature thatwould be represented by an Abelian Chern-Simons theory, where this constraint wouldbe satisfied.In Chapter 4, we will study the canonical structure of the CP(1) model with theaddition of a Chern-Simons or Hopf term [8]. In Section 4.1, we will described therelation 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-Simonsterm. In Section 4.2, we will work out the canonical structure of the CP(1) model. Thisis particularly complicated by the constraints, involve in using the CP(1) variables, thathas to be satisfied. We will make some novel observations about the canonical structureof the CP(1) model, in particular the existence of a local SU(2) x SU(2) algebra. InSection 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 therepresentation of the ground state solitons, which are extended field configurations overthe whole surface, introduces intrinsically a natural framing of the solitons. This is incontrast to the framing of point particles described in Chapter 3. It is shown that theCP(1) field solitons obey the spin-statistic relation. In fact, the fractional statistics ofthe CP(1) model are characterized by a two-color braid group representation. This facthas not been fully appreciated in the past literature on fractional statistics generated bythe Hopf map, and is still not widely known [18, 75, 80]. Also, its physical importanceis 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, thethermodynamic properties of a many soliton system would differ from that of an ordinaryanyons gas. Understanding this thermodynamics, and other physical properties, is animportant problem for the future.Chapter 1. Introduction^ 7In Chapter 5, we will described some topological models that have a non-Abeliangroup structure, in particular the Wess-Zumino-Witt en model and the non-Abelian Chern-Simons model. The Wess-Zumino-Witten model (at a specific coupling), described inSection 5.1 is a bosonized version of a fermionic model, which was pivotal in the under-standing of bosonization in two-dimensions [88]. This model has been used extensivelyin the description of exactly solvable two-dimensional models [21]. Some of the featuresof the non-Abelian Chern-Simons theory will be described in Section 5.2, this shouldbe contrasted with the Abelian Chern-Simons theory studied in Chapter 3 and 4. Wewill show that the non-Abelian Chern-Simons theory produces more exotic (not pure0-statistic) representations of the braid group. The thermodynamics properties of suchsystems is still unknown and is under study. It is believed that interesting physical prop-erties might emerge from a system obeying a more general para-statistics. Currently, alot of research is centered around quantum groups and W-algebras [23, 57]. The non-Abelian Chern-Simons theory is used as a toy model, if not a real physical model, tostudy and learn about these intricate structures.Chapter 2Selected topics in topologyIn this Chapter, we will describe various topics in mathematics, and in particular topol-ogy, that will be used in the next Chapters. It is not, by any extent, a complete andformal description of these subjects. For most physicists it should be considered asa fairly self-contained presentation that can be used as an introduction, while for mostmathematicians it should be read more as a review. The reader interested in more detailsof any of these topics should refer to the cited references in the text.2.1 Manifolds, exterior calculus and vector bundlesStandard 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 andthe real line is the fundamental set from which we do calculus, we call Rn a differentialspace.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 eachpoint 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 usualcalculus can still be performed. Not all topological spaces are manifolds.A manifold M, of dimension n, must have the property [15] that any small open8Chapter 2. Selected topics in topology^ 9region U C M can be mapped into TV = (x1, , xn) by a one-to-one and continuousfunction 111 : U R'. The system of coordinates (x1, , xn) on lin is a representationof the manifold M, within the patch U. For two different patches U1 and U2 withintersection U12 we obtain a transition function 4112 = 412 0 ‘111-1 that maps /in = I1i (U12)into Rn 1112(U12), or (xi) into (yi). As a compatibility condition, the function 4112must be one-to-one and continuous. The smoothness of the manifold, the number ofallowed derivatives, is given by the smoothness of its transition functions On atriple intersection, we find 4123 0 4112 (1113 0 412-1) 0 (1ff2 0 1111-1) 1113 0 TV- = Wm, sowhen 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 theR" = W(U) space, where U C M is any relevant subspace of M to our calculations. Atangent vector at u0 E U, is represented on /in, at '11(u0), by Ei cib. From a differentmap 111 with coordinates (y1, , yn), we use the transformation = Ei(Z)047 . Alter-natively, a differential element is represented on R by Ei cidxi, with the transformationdxi = Ej(e-;)0dy2.For the integration, we break the region of integration into small pieces belonging tosmall regions Uk from which we can integrate on the Rn space lifk(Uk). For a line integralalong a curve C on M, made up of disjoint pieces Ck E Uk,fidu ' E fx)dxi('ICEM Lk(Ck)E(Rn)kThe geometric language of forms [15, 72, 61] is very natural for performing calculationson a manifold. To every n-dimensional manifold M we associate a space of k-formsAk (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 antisymmetricpart of the product of dx's, for example du A dv = du 0 dv — dv 0 du = —dv A du.The dimension of Ak (M) is thus CT n!kqn—k)! Under a change of coordinates, dxi•Chapter 2. Selected topics in topology^ 10the volume form is transformed as dxlA•• • Ade = J•dyl A • Ade', where Jis the Jacobian det(z), by the antisymmetric nature of the wedge product. The Jacobianfactor is directly produced, showing the geometric nature of the forms. Forms are trulycoordinate independent, which make them the perfect objects to use for calculus on amanifold. 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, anddifferentiation, using the differential operator d : Ak^Ak+1. This algebra of forms issometimes referred as exterior calculus. For A, B a k-form and an /-form respectivelyand f a function, we have the following propertiesA A B = (-1)k1B A A E A/1(M)^ (2.1.1)aAii,...dA = E '^ik dxk A dxil A • • • A dxik E Ak+1(M)^(2.1.2)01,•••,ik^aXkd(A A B) = d(A) A B (-1)k A A d(B)^(2.1.3)^d(dA) = 0^ (2.1.4)ad(f) E — f=dx' df^(the ordinary differential)^(2.1.5)ax'Often, for simplification, the wedge A is dropped from formulas. Unless specified, thereis always a wedge product in between forms.Stokes' or Green's theorem can be represented easily by using forms. The integrationof the n-form dA, for a (n — 1)-form A, over a region V satisfiesdA I A^av^ (2.1.6)where av is the boundary of V. When A is a 2-form and ay a closed region of volumeV, (2.1.6) is Green's theorem, while it is Stokes' theorem for A a 1-form and av a closedcurve of surface V.Chapter 2. Selected topics in topology^ 11Given a point u0 E U C M, we represent a tangent vector at u0 on M as a tangentvector at 111(tt0) on Rn = 111(U), written generally as Ei cib. For an n-dimensionalmanifold, the tangent space forms a n-dimensional vector space. We call the inner producton 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 innerproduct on the cotangent space is the inverse of the tangent space inner product(dxildxj) = giiThis should be viewed as an inner product on the Al(M) space. In general, on theAk(M) space, we have the following (local) inner product, written in terms of its basiselements,(dxil • • • dxik 1dx^• • dxik)^det[(dxim Idxin)]^(2.1.8)The volume element on a manifold is given by dnx = 1/191dx1 • • dxn, where gdet(9i3). The presence of 191 makes the n-form invariant under coordinate transforma-tions, and we also find (dnxIdnx) = ( 191)2 det(9ij) = sign(9)^±1. If we can extendthis volume element uniquely over the whole manifold then we call the manifold ori-entable. For an non-orientable manifold, the volume element n-form changes sign aftersome coordinate transformation (the MObius strip is such a space), rendering the volumeform defined only up to a sign.For an orientable n-manifold equipped with a metric, we can defined a new operatorcalled the Hodge star operator. The Hodge star of a k-form B, written as *B, is theunique (n — k)-form defined for any k-form A by(AIB)clnx = A A* B^ (2.1.9)Chapter 2. Selected topics in topology^ 12Note that dnx = *1. It has the property thatTA) = (-1)k(n-k)sign(g)A(*Al*B) = sign(g)(A1B)for k-forms A and B.It is useful to define the co-derivative operator 8: Ac __, Ak - i8 = (-1)k(n+1-k)sign(g)*d*(2.1.10)(2.1.11)(2.1.12)It has the properties that 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 ifM is closed, then we can integrate the local inner product of A and B in (2.1.8), whichgives a global inner product of forms on M< AIB >. Im^.A4(AlB)dnx = I A A* B^(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 thetangent 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 map7r : E --+ M, which is equipped with a vector space L, at each point of M, transformedby a representation of a group G. If instead of a vector space L we have a representationChapter 2. Selected topics in topology^ 13of the group G itself, from which a group transformation is represented by left multipli-cation, then we call this a principal fiber bundle P(M,r, G). Every vector bundle hasan associated principal bundle. We refer to the vector space of L of E, or the group Gof 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, bil = ifabcTc^(2.1.16)aThe 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 havenot specified how a vector could be (parallel) transported on a vector bundle. We saythat a vector field x E L is parallel transported along a given vector, Vk , if its covariantderivative DkX vanishes along the direction of that vector, vkDkx = 0. We define thecovariant 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 thegiven vector bundle. The one-form A takes values in the Lie algebra of GA = iETaAdXkaA manifold is represented by mapping small region U --)• 111(U) = Rn of M, thus theconnection A is defined only within this region U of the manifold. In the intersection U12of two regions U1 and U2, the vector space xi, of the first patch, is related to the vectorspace x2, of the second patch, by an element of the gauge groupXi -4 X2 = Wi2X1^with^W E G^(2.1.18)To be consistent with ordinary differentials, the transformation of the covariant differ-ential of this vector field has to transform like the field itself, such that it becomesChapter 2. Selected topics in topology^ 14independent 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. Thisindicates that the gauge connection of the region 1 and region 2 are related byA1 A2 = W1 A- 2--1W1-21 W12dW1-21 (2.1.19)If we transform the vector space L of a vector bundle by a gauge transformation W, asin (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 necessarilycome back to its original value. To characterize this, we parallel transport a vector aroundan infinitesimal loop C, which is the boundary of a surface S, then it is found that thevector will be transformed by an infinitesimal group element exp(fs F), whereF iETaF4clxi A^= dAd- AAA^(2.1.20)aOr= AA; - 0.;14 — fabcAcIAI;We call F the curvature of the vector field L of the vector bundle. Under a gaugetransformation (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 patchwhere A is defined, we can represent A asA W-idWChapter 2. Selected topics in topology^ 15whereW (x) = P exp ( f A) EE 1+^XI A+^du'^dv3Ai(u)Ai(v) + • • -xX0^ X0^0^XOThe operator P means that we have to path order the integration when A belongs to anon-Abelian group. It might not be possible to extend W to the whole manifold, theremight be some topological obstruction when we move from one patch to another. Wewill say more on this point in Section 2.3.2.2 DeRham cohomology, homology and homotopyIn Section 2.1 we defined manifolds and described the exterior calculus of forms, whichare coordinate independent objects on a manifold. This coordinate independence of theforms makes them a powerful tool to investigate the topology of the manifold itself. Wewould like to find some properties common to a set of manifolds related to each otherby a smooth change of the metric. In other words, can we put a set of manifolds relatedonly by stretching and shrinking in some class. We will show how the existence of someforms 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 sequence0 a_.4 Ao a). Ai Ak_i^Ak^Ak+i An-1  d > An  d 0Let 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 formsin Ak mapped into Ak by the differential operator 'd'. We will call forms belonging toBk exact forms.Chapter 2. Selected topics in topology^ 16The exact forms are a subset of the closed formsBk c zksince 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 notexact forms. Since a lot of closed forms might only differ by an exact form, we regroupthe closed forms differing only by an exact form into one class. More precisely, we definethe DeRham cohomology [61, 72] class H' (M) byZk^ker dkHk — — — ^ (2.2.1)Bk im dk_iwhere the index on `dk' indicates that it is applied to the Ak space, and the division byBk means we built an equivalence class of exact forms.The calculations of the cohomology classes of a manifold are not trivial in generaland sometimes involve rather advanced mathematical tricks. There are, however, somesimple cases that can be easily computed.The lowest cohomology class, H°, is given by the 0-form, that is functions, whosederivative, 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 onM, and a constant is represented by a real number. This is assuming that M is pathconnected, otherwise H° will contain a copy of R for each path connected region of M,H°(M) = R,^for M path connectedThe highest cohomology class (by definition Hk = 0 for k > n), kin, is given by then-forms, A, that are necessarily closed since there are no (n+1)-forms, modulo any exactn-forms, B. If M is orientable without boundary, then we can use the generalization ofStoke's theorem to kill any exact forms,Im(A - E dB) .-- 1 A+1 m=o B=I Am^a^mChapter 2. Selected topics in topology^ 17which gives a real number. So, the generator of Hn is the volume element dnx, which isthe only form, up to normalization, not killed by the above integral. ThusH' (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 spacesHk (Rn)0 otherwiseIR for k = 0, nHk(sn) _Hk 041 X M2) = ETP(mi) x fra-k(m2)n! ^Hk(Tn) = 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 beextended to the differential operator `d', see the properties (2.1.14) and (2.1.15). Wefind that the cohomology Hk from the operator is isomorphic to the cohomology (ingeneral different) on A/C from the operator '8', which is in this case the usual cohomologyHn-k^Hk(m) = Hn-k(m)^for M orientable without boundaryThis theorem indicates that there exist some k-forms h E Hk that have also the propertythat *h E Hn-k. We call such a form harmonic. The harmonic forms can be characterizedas being closed and co-closedh E Hk(M) and *h E Hn-k (M)  ^dh = 8k = 00 otherwiseChapter 2. Selected topics in topology^ 18This 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 formA=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 boundaryoperator. 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 introducehomology by considering the space Sk(M) of maps f : Ak -- M of k-simplexes. Theboundary operator produces a map Of:  (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 canbuild a topological class known as the homology class [72]ker ak Hk(m) = i nm 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 onM. A general homology class, Hk, has several copies of Z, plus a torsion term. In thiscontext, torsion means a sum of finite cyclic groups. We can tensor Hk with the realnumbers, this has the effect of killing the torsion part, and each copy of Z is transformedinto a copy of R. Under this tensor product, we have the theoremHk(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 onChapter 2. Selected topics in topology^ 19Z instead of R. For such a cohomology, the torsion part may differ from the correspondinghomology.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 thenumber of path connected components of a space. The next homotopy group, 7r1, calledthe fundamental group, indicates if there are non-trivial loops on M. For each non-trivial loop corresponds an element of the group, with appropriate multiplication andinverse. Sometimes, the fundamental group of a manifold may be non-Abelian. Theother homotopy groups, 7r„ for n> 1, are always Abelian. The homology, cohomology andhomotopy groups of simple spaces are often identical, but it is important to understandthat they are very different mathematical structures.2.3 Characteristic classesIn Section (2.2) we constructed some topological classes that can (but not always) distin-guish topologically different manifolds. For vector bundles, it is also possible to constructsome topological invariants. We can then, in general, distinguish topologically differentvector bundles. These invariants are known as characteristic classes. A good candidate,to represent such classes, must be invariant under coordinate transformations, thus weshould use forms. In addition, it must especially be invariant under gauge transfor-mations, as in (2.1.18), of the vector space. Since the curvature 2-form is transformedcovariantly, see (2.1.21), under these gauge transformations, we can construct an invari-ant called the Chern classes [15] of a vector bundle, for a base space consisting of amanifold of even dimension 2k. The Chern classes are best expressed in term of theChapter 2. Selected topics in topology^ 20Chern characteristics—1ck(E(M, , L, G)) = (-7) k^tr(F A • • • A F) (k times)2ri JM(2.3.1)This is always an integer on closed and orientable manifolds. Note that in (2.3.1) and thefollowing 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 manifoldequipped with a vector space, L, transformed by an Abelian group, U(1),1^1(E(M2, 7r, L, U(1))) =^im F = -----2F0 (2.3.2)where F0 is the total flux, a real number after eliminating a factor T. Let us explainin more detail what this invariant means. Since we have to consider the case where F0is non-zero on M, the representation of the gauge field A can be done only on a set ofpatches covering the manifold. Let us consider the set of patches Ui as a good cover ofthe 2-dimensional manifold M. That is, the intersection of any two patches is a singlecontractible 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 (ii)^ (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 relationx(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 i dA(i) . E^dx(ij) . pijkE c(iik)^(2.3.5)ii Li, where the surfaces Vi C 0 and they are bounded by a line, Via, dividing the intersectionUi n IP. On the triple intersection, we let the three lines Via, Val' and Vki meet at onepoint Pak.Chapter 2. Selected topics in topology^ 21If M is a filled circle, then c1 reduces to an integral on the boundary1= - 27ri Li AThis is an Abelian Wilson loop integral. If the curvature vanishes, then the gauge con-nection 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 four-dimensional manifolds. Given a vector bundle with a 4-dimensional manifold as the basespace and a vector space transformed by a non-Abelian group G, the Chern class is182 IMc2(E(M,r,L,G)) =^tr(F A F)72 .A4 (2.3.6)If this 4-dimensional space has a 3-dimensional space as boundary then we can use theidentity2 2F A F = d(A A dAto reduce the Chern class to-3A A A A A) = d(AdA -3A) (2.3.7)1Ss(A) = tr(AdA^A')+ -2(2.3.8)87r2^3amThis integral is also known as the Chern-Simons integral. Note that it is not necessarilyan integer. If the gauge field is trivial A = U'dU, then we find for a 3-dimensionalmanifold1 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-dimensionalmanifold as boundary, then we call this integral the Wess-Zumino integral. Given a 2-dimensional space with a group valued field U, we can always view this space as theboundary of a 3-dimensional space to which we can extend U and evaluate the Wess-Zumino integral (2.3.9) uniquely, modulo an integer.Chapter 2. Selected topics in topology^ 22For a closed 3-dimensional manifold, the Chern-Simons integral (2.3.8) is not com-pletely invariant under a gauge transformation, but instead it changes by the Wess-Zumino 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 repre-senting the winding number of this map. We call such a non-trivial gauge transformationa large gauge transformation. If M = S3, then the classes of these maps are classifiedby r3(G), which is Z for any non-Abelian group. The Wess-Zumino integral can berecognized as the integration of a closed, but not exact, 3-form, thus representing thecohomology1/3(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 ori-entable manifold of even dimension 2n, then we can build another invariant by using theRiemann curvature 2-form Rii = gikRi;. We represent this class, known as the Euler classof M, asX(M) = (-1)n^D.^122 A • • • A Ri2n_ii-2n(47r)nn! ImOf particular interest is the Euler class of a 2-dimensional manifold1X = —27 f R2m 1We will describe 2-dimensional surfaces in more detail in the next Section.2.4 Two dimensional topology and Riemann surfacesTwo dimensional manifolds are very important in physics and mathematics because thisis the only dimension (except the trivial dimension one) where we have a thorough under-standing of their classification as spaces. We will be particularly interested with orientableChapter 2. Selected topics in topology^ 23Figure 2.1: Example of a genus 3 Riemann surfaceFigure 2.2: Holonomy of a Riemann surface represented by loops a, and bm intersectingat P„,and closed two dimensional manifolds, which are also called Riemann surfaces. The bestway 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 ofthe surface .A4 9 - In term of the Euler class the Riemann surfaces are classified asx(mg) = 2(1 - g) (2.4.1)The genus is basically the number of holes or handles of the surface. If we concentrateon one of the holes in particular, as seen in Figure 2.2, we find a pair of non-contractibleloops, or homology cycles, which we call a, and V', m = 1, ,g. The intersectionnumbers, seen at the point Pm in figure 2.2, of these generators are given byv(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 numberChapter 2. Selected topics in topology^ 24of left handed crossings) of the oriented curves C1 and C2. 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 eachhomology cycle.Alternatively, it is very useful, for calculations in physics, to study the DeRhamcohomology of a Riemann surface. The cohomology can be represented as the dual ofthe homology. For each homology cycle, we can find a non-trivial closed form belongingto the first cohomology class of M9. More precisely, given a homology cycle, we build afunction 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 = —de27rwill represent the cohomology generator associated to the given homology generator. Wewill associate the cohomology generators hm to the homology generators am, and him tob. This construction can be summarized inf hi = 81m, I h1 .-_- 0, I hi =-_- 0, I ii1 = sim (2.4.3)am bra am braOne of the important theorems in the theory of Riemann surfaces is Riemann's periodrelation. Using the cohomology generators (2.4.3), we can state this theorem asi III A km= 8 nil , h1 A len = I iti =Mg fm, mg A km 0 (2.4.4)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, asdx + idy by making use of the real basis. The difficulty is to extend such a basis globallyon the whole manifold, but this can always be done on two dimensional manifolds (unlikehigher dimensional ones). For example, by using the theory of conformal mapping wefind systematic choices of complex structures.Chapter 2. Selected topics in topology^ 25The 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 functionof i only. In other words, d[ck(z,..-)dz1 = 0 gives the Cauchy-Riemann equations thathas to be satisfied to obtain a holomorphic function 0(z). If we want A to be real, thenwe 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'. Thistells 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 decom-posed as a sum of an exact, co-exact and harmonic form. Going back to our cohomologygenerators h and Ii, we can add an exact piece to these closed forms such that we obtainharmonic forms. This gives us a 2g dimensional real space of harmonic forms (isomorphicto the cohomology group).Alternatively, the space of harmonic forms can be described by complex harmonicforms, hence also holomorphic, that we will call col = col(z)dz. Since the homotopycycles al are independent, see (2.4.2), we can use them to normalize col, but leaving noconstraint along the b1 cycles. This gives the con/ the standard normalization [16, 51], foran arbitrary complex matrix film,i Wm = 87in ,al iblwm = filmThe relation between the real and complex harmonic forms, using (2.4.3), is expressedbygcol = h1 + E nimilm^ (2.4.5)m=1With the use of an arbitrary complex harmonic form L = Ef_lcicol and using the equa-tions (2.4.5) and (2.4.4), we find thatg0=1 LAL= E cicm(12" — Sri)M g 1,711=1Chapter 2. Selected topics in topologyandwhere0 < ^IL rd2x ---= ±1 L A L ----= —1 E cic,Gimmg 2 mg^2 477.,26dm = i i col Am = 2Im(nim), GiniG" = 87/1^(2.4.6)MgThis shows that the matrix fen is symmetric and its imaginary part, Glm, is positivedefinite. We can recognize the equation (2.4.6) as a metric in the space of holomorphicharmonic 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 0this harmonic space. We will use GI, and Glm to lower or raise indices when needed anduse Einstein summation convention over repeated indices.Any linear relation, with integer coefficients, of al and I) that satisfies (2.4.2) is anothervalid basis for the homology generators. These transformations form a group, called themodular group, Sp(2g, Z):(ab) S (ab)where S =(DCBA )(2.4.7)( 01 )with S EST = E and E =^. The g x g matrices A, B, C, D have integer entries.—1 0For the real harmonic forms, the modular transformation (2.4.7) takes the form( iht) --+ ( ih't')=h''j ( it)where = _ s - i T (A— B— C D )This preserves the duality (2.4.3) of the homology and cohomology generators. A modulartransformation will transform the complex harmonic forms and the fi matrixco --> co' = It' + frit' = (A — fl'C)h + (—B + CZ'D)it = (A — fi'C)coChapter 2. Selected topics in topology^ 27from 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 asG^= (CO D) -1T G(Cf2 D) - 12.5 Three dimensional topology and the braid groupIn recent years, the study of three dimensional manifolds has been greatly advanced byusing knot theory. A knot is a (or a collection of) closed curve, or 51, embedded in somemanifold, sometimes referred to as a mapping S1 ----> M. By definition, this mapping willrepresent the fundamental homotopy group of the manifold, or ri(M). On the otherhand, if we restrict the homotopic deformation of S1, say represented by 0, such thatno two points of this loop intersect on M, or x(0i) x(02) where x represent points onM, then we obtain a new mathematical structure. In effect we obtain knot theory. Suchknots are trivial in dimension one and two, and for any dimension above three, sinceany knot can be turned into a trivial knot (a simple circle in an Rn neighborhood of themanifold) 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 threemanifolds.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 twodimensional manifold M. As these particles move in time, they will trace a trajectoryin space-time X(t). When these particles are not allowed to intersect, this eliminates asubspace A = {X E MnIxi xj, for any i j}. By considering all the n particles asidentical, we must consider any configurations differing only by a permutation, Sn, asChapter 2. Selected topics in topology^ 28 TiFigure 2.3: Example of a braid.being identical. Thus, the configuration space of such a system isQn(M) = mn _ AThe 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 configurationXo, 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 athree dimensional space-time. Since the particles are not allowed to coincide, the stringscannot 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 thebeginning of the second braid to the end of the first braid, on the common configurationX0, to form one new braid. The identity element is n non-braiding strings, or visualizedas 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 orderof the original braid).Let us study the case M2 = R2. It can be shown that we can represent an arbitrarybraid in terms of n — 1 generators cr„ that represent the exchange of the string i andi 1. The string i can go around the string i 1 by going either in front or behindit, we have to choose one of these moves (similar to the right hand rule) to representai, as pictured in Figure 2.4. The other move corresponds to cfV, since we do find thatChapter 2. Selected topics in topology^ 29 and 1Figure 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^-10-1Gri = 1.All these generators are subject to the relationscricri = crjcri^— jj > 2^(2.5.1)illustrated by the equivalence of the braids seen in Figure 2.5, and the relationscricri+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 tothe relations (2.5.1) and (2.5.2), there are several additional relations as followsjo-i,^= [cri, 13i] =0^2 < i < n — 1; 1 < 1 < gChapter 2. Selected topics in topology =--Figure 2.6: Equivalent braids representing the relation (2.5.2) for i = 3.30criaparat = atoricepai.a^--1^a^--I.cripperi Pta =-- i3/0-wperiaitaraiPI = Acri firciia -1^a --1al Rpal al = al(71PPalcriaperifil = Acricepaip > 1; 1 < I,p < gp > 1; 1 < l,p < g1 < I < gp > 1; 1 < 1,p < gp > I; 1 < 1,p < gai-ictiai /3/ = filo-ice/al^1 < I < g^(2.5.3)All these relations correspond to equivalent braids, which could be visualized as we didfor 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 atrajectory of a particle encircling all other particles and tracing all homology generatorsof M in such a way that it forms a trivial loop. For the braid group, this trivial loopmust be represented as the identity, leading to the relation2^a -1 a-1 N^a -1 a-i Nai • - • an_i • • • Cri(Plai Pi al i • • • (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 dimensionalones.Chapter 2. Selected topics in topology^ 31The representations obtained by the Abelian Chern-Simons theory are the so-calledpure 0-statistics representations where the generator of an interchange of neighboringparticles is represented by a phase, times a unit matrix. In this particular type ofrepresentation, the generators for particle exchanges ai and those for transport aroundhandles satisfy a far less restrictive set of relations due to the Abelian structure of theseo-i. They satisfy the relations (2.5.1) trivially while the relations (2.5.2) tell us that theo-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^/ mcei 131 = a2,31 • al (2.5.5)and the global constraint (2.5.4) for a closed manifold iso-2(n+g-1) = 1^(2.5.6)Given a possible 0-representation of the braid group, we have to verify that it satisfiesthe relations (2.5.5) and the constraint (2.5.6).Chapter 3Abelian Chern-Simons theoryIn Section 2.3 we found that the Chern class of a vector bundle over a 4-dimensional spacecan 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 anon-Abelian group. We showed that a large gauge transformation changes this integralby an integer. Thus, an action with Scs(A) will be multi-valued. In the quantum theorywe 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 theorycan be defined with the integral Ss(A) with a coefficient 271-k, for an arbitrary integerconstant k, which would be invariant under large gauge transformations. This is thefamous Chern-Simons action. When coupled to a current source .PL = i Ea Taja14, it isrepresented asS = --k I tr(AdA + —2 A3) + tr(A,Md3x4ir m^3(3.0.7)For an Abelian gauge, we can still use this action, even though the second Chernclass, from which (3.0.7) is derived, is zero in Abelian case. The major difference forthe Abelian theory corresponds to the fact that the Chern-Simons action is classicallyinvariant under a gauge transformation (for a closed 3-dimensional space). In otherwords, the Wess-Zumino integral (2.3.9) is zero in the Abelian case, thus it is not presentin the gauge transformation (2.3.10). There are no large gauge transformations (on the3-dimensional space) since r3(U(1)) = 71-3(S1) = 0. So now the constant k, used above,32Chapter 3. Abelian Chern-Simons theory^ 33has no reason to be an integer. We write the Abelian Chern-Simons action coupled tocharge currents as= --k—it4 AdA47r 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 agiven Riemann surface are obtained from the wave functions of an Abelian Chern-Simonstheory in the most general case where the constant k is a rational number, the Riemannsurface has arbitrary genus g and the total charge of the particles is non-zero. We shallconstruct the wave functions of the quantum theory with action (3.0.8) explicitly andfind that, depending on the coefficient k and the genus of the configuration space, thewave function carries certain multi-dimensional, in general non-Abelian representationsof the braid group.The wave function of Abelian Chern-Simons theory coupled to classical point particleson the plane was found by Dunne, Jackiw and Trugenberger [24, 50, 53]. In this casethe Chern-Simons theory has no physical degrees of freedom, the Hilbert space is one-dimensional 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 bya = 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 representa-tion of the braid group on the plane.Because of the Gauss' law constraint (see ahead (3.2.4))x = -k-27?(x)the case when the space is compact is somewhat more complicated than that of theChapter 3. Abelian Chern-Simons theory^ 34plane. In order to have an assembly of identical particles, it is necessary to have non-zero total charge. If we have non-zero total charge, Gauss' law requires a non-zero totalmagnetic flux, which on a compact manifold means that the gauge connection A is not afunction but a section of a U(1) vector bundle. This requires some modifications of theChern-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 thanone 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 particleswhen the space is a Riemann surface of genus g and when k, the coefficient of the Chern-Simons term, is an integer. They found that the wave functions carry a representationof the braid group on the Riemann surface. Although they use a different polarizationthan 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 wherek is a rational number.We will show that the correct geometrical description of Chern-Simons theory on aRiemann surface necessarily introduces a framing of particle trajectories. Framing is astandard part of the study of the relationship between the Chern-Simons theory andknot 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 thepoint splitting discussed by Bos and Nair [9]) have also appeared in the literature onthe Hamiltonian approach to the quantization of Chern-Simons theory. Here, we shallfind that our geometrical approach to framing plays an important role in the consistencyrelations between the parameters k, g and the values of the charges of particles qi.Chapter 3. Abelian Chern-Simons theory^ 353.1 The decomposition of the gauge fieldOur space will be an orientable 2-dimensional Riemann surface, M, of genus g, whileour space-time will be a 3-dimensional manifold M x 1i, formed as the Riemann surfaceM times a real line for the time direction. In other words, the space-time metric isgoo = 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 separatethe effect of the non-zero total flux (2.3.5) we will break it in two parts. First a fixed fieldAp 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. Thisfield 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 wehaveA = 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 givenby (d and * act on the two dimensional surface M in this Chapter)gA, ,--- d(-1*d*Ar) + *d(-1*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 meansthat the zero modes are removed. With our decomposition (3.1.1), dA, does not havea zero mode. Also we will set the zero mode of *d*A, = '.0 • iii., to zero, using a timeindependent gauge transformation.We can definee = --k —1 *d*A, Fr = *dA,Chapter 3. Abelian Chern-Simons theory^ 36So we then have the complete decomposition of the gauge field, with the Ao part,1A, = Aodt — ^*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 = j4dejodt + 3, using the 3-dimensional star operator * and j„ = gi„ju. We can use again theHodge decomposition of .3 on M—^*clik^i(i11^iiwi)^(3.1.4)The continuity equationdj (f/ • 3.)d3x =^d3x + d*3 A dt = 0shows that j is a closed two-form that can be used to solve for 7/)1 ajo= --o, atWe shall consider a set of point charges moving on M, with trajectories zi(t) andcharge qi, where zi(t) z3(t) for i L j. The current is represented byjo(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 ofthe 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 equivalentto a sum of the integral of the harmonic forms w1 over each charge trajectory.To solve for x, it is best to use complex notationR = 71) + ix = R(z,"i)Chapter 3. Abelian Chern-Simons theory^ 37where we find *dx chi) = OzRdz (92.Rdi'. From (3.1.4), (3.1.5) and using (3.1.6) wefindazR 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 theresult that1^pzE(z,zi(t))  )]^(C,-)1 — wi)R = 121— E '13g(E(zo, zi(t»^j1(t) zoat^27r i (3.1.8)where we have chosen R such that R(zo, 20) = 0 for an arbitrary point zo, which wechoose to be the same as the zo in the definition of Ap (We can choose zo = oo for genuszero). The important fact about R is that it is a single-valued function. If we move zaround any of the homology cycles, R returns to its original value. In fact, this is alsotrue for windings of zo, an important relation since it is only a reference point. Sox —a E(z' Mt))  )] —z [(ji(t)ji(t)) jz (6)1 — c.o1)] (3.1.9)Ot 27r i E(zo, zi(t)) 2 oThe action (3.0.8) is written for a trivial 15(1) bundle over M, corresponding to azero total flux. Every integral of the gauge field, which is invariant under a gauge trans-formation of A, can be extended uniquely into an integral using the A(i), defined on theset of patches, that is patch independent by adding appropriate terms. We will representthe 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 alonga line Lijk , and four of these lines will terminate at a point Pijk 1. This might be bestvisualized as a triangulation of M x R in term of 3-simplexes (or tetrahedrons), thewith 2-simplex boundaries (or triangles), the Vii, which in term has 1-simplex bound-aries (or lines), the Pik, and finally those have 0-simplexes (or points) as boundaries, theChapter 3. Abelian Chern-Simons theory^ 38Pijkl. The consistent topological extension of (3.0.8), see [2, 66], isS = --k E vi A(i)dA-1--k E^x(odA--k E f coik)A(k)>^c<(ijk)x(k1)>47r^47r 47r ijk Liik^47rij pi3kiEv Acoi _ E^y-^c<(ijk)w(k)> E copox-(ko>(p) (3.1.10).^vii Lix^ijk piikiwhere the one-form W is defined by j dW locally since j is a closed two-form. Sincefm 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 theindices in increasing order (with appropriate sign) and set the repeated index accordingto 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 Qat the reference point zo.The complicated expression (3.1.10) for the action ensures that the total expression isindependent 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 atotal derivative, d(x(ij)dA), leading to a correction term integrated over the boundaries ofVi by the generalization of Stokes' theorem (2.1.6). The boundaries of Vi is a combinationVij for all possible j, thus the second term, in return, will change in such a way as tocancel the change generated by this first term. The total action is thus invariant underdeformation of the Vi. A similar analysis can be done for the variation of Vii leading toanother correction term, the third term in (3.1.10). Finally we end up with all the termsin (3.1.10), producing a topologically invariant action.Using (3.1.1), the decomposition of j and performing several integration by partsbrings (3.1.10) asS —^ArcIA,^ArdA f Ar(ir jp)^WrdAp47 MxR^MxR^P M xR^MxR^[ — —47T ApdAp^Apjp] -I- Surface terms^(3.1.11)Chapter 3. Abelian Chern-Simons theory^ 39The terms in brackets, involving Ap and jp, has to be performed using the extendeddecomposition (3.1.10), by replacing A with A. For our case, we extend the triangulationof M trivially through the time direction. The surface terms, appearing at the timeboundaries (t = 0 and t = if), are not important for the quantum theory or the braidgroup representation that we will find later on. They can't be avoided since the action isnot invariant under gauge transformations at the time boundaries. Thus there is no termto cancel the triangulation dependent term. This will not be a problem since under aperiodic 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 termsdo not affect the dynamics or quantization of the system. We also represent Ap such thatdAp = F08(z — zo)d2x, which implies that zo must stay within one patch at all time, andsimilarly for jp since it is equal to QS(z — zo)dt. After a quick calculation, we find thatthe terms inside the brackets are all zero, except for the integral, j c<(iik)Wk>, which isdefined modulo c(ijk)Q (for periodic motion). This is because W is defined on patchesalso, 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 ambiguityis eiQF°. Actually, the integral f Aj is equal to Eiq fc, A, the Wilson line integral fora set of charges qi following the curves C. In this case for each of these Wilson lineintegrals, corresponding to the charge qi, we find a phase ei") instead. To resolve theseambiguities, 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 independentAbelian gauge field, say I', then a similar phase ambiguity, eihsxE, would arise. Herehi will be the charge attached to the particle i corresponding to this new field, andxE = fm dr is the total flux. The important fact, now, is that the phase ambiguity fromboth gauge fields would appear at the same time, thus we would have to impose theChapter 3. Abelian Chern-Simons theory^ 40constrainteigiFb-ihixE = 1^ (3.1.12)to obtain a consistent quantum theory (The minus sign has been added to simplify thenotation later on). At this stage, the new field r seems artificial, but it turns out thatit is necessary to introduce such a field for Chern-Simons theory. In fact, it correspondsto a connection on the tangent space of M. We will need it because for each chargetrajectory we will attach a framing (a unit vector on M). Such a framing has to bedefined in relation to the basis of the tangent space, so r does not have to be theassociated metric connection, but it will enjoy the same global properties. It is wellknown that   = x(M) = 2(1 - g), the Euler class of M. Note that we will assumethat the field r does not have any flux around the particles (an effect similar to cosmicstrings), this would lead to a change in the statistics of these particles. The charges hiwill be equal to e/2k, this will appear quite naturally in the next Section. Like we didfor the field A, we will concentrate all the flux, xE, of r around the point zo. This willallow us to assume a constant framing on M, except when we cross the point z0, in whichcase 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 hadnot used our freedom in the definition of x to set it up this way, we would have to takecare of its effects on the hamiltonian and ultimately the wave function.3.2 Quantization and the SchrOdinger equationNow we are ready to solve for the action. By putting (3.1.3) and (3.1.4) back into (3.1.11)we findS = -1 J (V. - P-1,.)d3x + irk 1(7% - ;y111)dt + Ao(io — —27r F)d3x2 Chapter 3. Abelian Chern-Simons theory^ 41it 27r ‘.ajoJ+ Frx)d3x + 27ri (j/11 — 31-yi)dt + Surface termsk at (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 (3.2.2)and4(z)1 1^a^10[11,-rmi= 27rku-b.^or^=^cirm^=271-k^0-ym^27rk a71^(3.2.3)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 derivativemust also be defined using this projection operator. With this holomorphic polarization[9] it is convenient to use the following measure in -y space1412) = f e-2/rk-vmG,a-y1 AFT(1)412 (7) !GI Ild,rnd Mwhere 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 constraint27r^S^27r^27rF(z) — —1-c- i0(z) = iP g(z) d- F06(z — A)) — —k j,.0(z) + —k QS(z — zo)•=s-', 0from which we extract Fo = Q. Since Fo and Q are not quantum variables, this is astrong equality, thus leaving27r .Fr(z) — —2r jk r°(z) = . 3 4(z)^Tc Jr°(z) °^(3.2.4)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 notmix -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+HTChapter 3. Abelian Chern-Simons theory^ 42whereI27r air°^aHo = m( k at +ixPT4,)d2xwhile the additional part, which takes care of the topology, is-^1 .1 aHT 07*Y - i -57)(3.2.5)(3.2.6)To solve the SchrOdinger equation, we will use the fact that the hamiltonian separates,thus writing the wave function ast) = wo(^) T ( , t )with the Gauss' law constraint (3.2.4)27r— —k jr0)x110(,t) 0which 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 "imagi-nary" 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 atwhich has the solution [9, 24, 50, 89]W(t) exP^foi fmx(z,e)iro(z,e)d2xdti^(3.2.8)For a system of point charges, the use of (3.1.9) allows us to write (3.2.7) and (3.2.8) as27ri^ i^[t^.‘110(, t) = exp --Tc--(E g (zi(t)) — W(zo)) + yk- E qiqj f dtOij(t)+ (I)(t)^(3.2.9)Chapter 3. Abelian Chern-Simons theory^ 43wherej^[lo^_^t _^t'i(e)de^ji(e)dt" —^ji(e)de i0 j1(t")dt1Jo7r^t 4- a {lot j1(e)de Ij1(e)de — j1(e)de fji(e)dt1 (3.2.10)7rE(zi(t),za(t)) ^1Oij(t)^Im log [E(zi (t), zo)E(zo, z3(t)).1hni 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 functionfor particle i and j.For i = j, we find a self-linking term of the form Imlog(zi — zi) = Imlog(0) which isan undetermined expression, although not a divergent one. One way to solve the problemis to choose a framingzi(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 shiftin the position of the charges in jr0, but not in x. In effect, this leads to a small violationof the continuity equation. Alternatively, we can view this term as the additional gaugefield F introduced in the last Section. With the framing (3.2.12), we find that(724•.dt -= hi^F2k "^Ci(3.2.13)where Ci is the trajectory of qi on M, representing a coupling of the particles, of chargeshi, to an Abelian gauge field F, as claimed in the last Section. We also recover thesecharges as hi =-- q.?/2k, which actually are the conformal weights of the underlying twodimensional conformal field theory [89].andChapter 3. Abelian Chern-Simons theory^ 44The angle function (3.2.11) depends on zo, but it should be invariant if we movezo either infinitesimally or around a homology cycle. For a small displacement thereis no change unless one of the charge trajectories, zi(t), passing by zo from one side isnow moving from the other side. Looking at the denominator of Oij, we see that thiswill 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 ofxE/271- = 2(1 — g) turns as we go around M, an effect that we concentrated around zohere. The total phase shift is constrained as1 (3.2.14)since it is nothing but the constraint (3.1.12), with the use of the Gauss' law constraint2irQ and our choice of xE. The equation (3.2.14) will represent a fundamentalconstraint that has to be satisfied by all charges if we want a consistent solution toChern-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 ad-ditional charge —Q at zo. The constraint (3.2.14) indicates that this is indeed an "imag-inary" charge and that it should not be seen by any real charge. For the displacement ofzo 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 differencein (3.2.9). This point is actually more complicated; we will come back to it later. So, atthis stage, the wave function (3.2.9), with the angle function (3.2.11), accurately formsa representation of the braid group on a plane [9, 66, 67]. We will cover the full braidgroup in more detail later.Now, the topological part of the hamiltonian is used to find the part of the wave func-tion affected by the currents going around the non-trivial loops of M. The SchrOdingerChapter 3. Abelian Chern-Simons theory45equation for (3.2.6) is.^/^1 ./z.OTTe-y, t) HTAFT(7, t) = z (z7rJry —^--)111T(7, t)at^ k a-y1which has the solution^WT(7, t) = exp 1271-71/ j/(e)de —^it ji(e)de I ji(e)dt" +7,(7,t)^(3.2.15)k oNote 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)(3.2.16)at^— c^a-yiis easily solved in the form1 ftji(e)de)lifT(71,t)=417,(71--k3.3 Large gauge transformations(3.2.17)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 = 0for this Section and find the condition on xifT.In general, the large U(1) gauge transformations are given by the set of single-valuedgauge functions, with sm and trn integer-valued vectors,U.,,t(z) = exp (27ri(tmrim(z) — snaijm(z))where(Z) =(COm, COmZOIf we change the endpoint of integration by z^z aiul bmvm with u, v inte-ger and a, b defined in (2.4.2), we find rim —+ 1m + um, ijni --*^vni and Us,tnm(z) = i^ (frn lwi _ nnac-,5 1 )ZOChapter 3. Abelian Chern-Simons theory^46us,te2tmum-smv,n)^us,t. The transformation of the gauge field (3.1.2) under U,,t isgiven by7n1^7m + Sm 9171^1m _3^(3.3.1)The classical operator that produces the transformation (3.3.1)cs,t(7,1) = exp {(sm^(smarym^arymmust be transformed into a proper quantum operator acting on the wave functionBy using the commutation (3.2.3) to replace T. by —271k-ym we find the operatorswhich 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 do not commute among themselves for non-integer k.From now on, we will set k = 1 for integer kj, and k2. Now, in contrast with theirk2classical counterparts, the operators C,,t satisfy the clock algebraC31 C32,2 = e-27,-ik(3yzi_m2 - trr, „...1 ,t2^(3.3.3)Their action on the wave function isCs,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 byphases eigss,t. This implies, using (3.3.4),+ k2 (sm + f2nati)) = exp [—i08,t + 2irki (snl + f2miti)7n,-Orkik2(sm + fritz)Gmn(Sn nnito] 1JJT(7-)^(3.3.5)TC3,-tChapter 3. Abelian Chern-Simons theory^ 47The only functions that are doubly (semi-)periodic are combinations of the theta func-tions (A.0.1). After some algebra, we find that the set of functionsqfp,r^(71n)a^= em"m (a÷ki. k2(1471k1 k20)^(3.3.6)where p = 1, 2, ... , k2 and r = 1, 2, . . . , ki with the vectors am, iSm E [0, 1] solve theabove algebraic conditions (3.3.5). Their inner product is given by(1 f^p2,r2) =^e-21rk-enGm1-itth^\  thk7 P2 ,r2 (7) IG1-1 H d7md-Im (3.3.7)1G1 141,P2 Sri '1'2The integrand is completely invariant under the translation (3.3.1), thus we restrict theintegration to one plaquettes P (7m = um + fimivi with u, v E [0, 1]), the phase space ofthe -y's.Under a large gauge transformation^(a^ 2ri aC.,,tillp,r^(7) = exp [27rikpmsm + irksmt, + --,—(amsni — #intm)] klip+t,r^(7)# ^/C2 #= E[c.,,t]pp4p,,r ( a^ (3.3.8) ) (7)101The matrix [C,,tipm, forms a (k2)9 dimensional representation of the algebra (3.3.3) oflarge gauge transformations.The parameters a and 9 appear as free parameters, but in fact they may be fixedsuch 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, isa'(142)^ICS2Chapter 3. Abelian Chern-Simons theory^ 48where -y' =--  (C52 + D)-11-7,^=^+ B)(C9 D) -' and 0 is a phase that will notconcern us here. Most important are the new variables= Da — C/3 k1k2 (CDT )d^= Ba Ai3 k1k2 (ABT)d2^ 2where (M)d means Mdd, the diagonal elements.A set of modular invariant wave functions {26, 48, 56, 67] can exist only when k1k2is even, where we set a = 0 (and also 0 = 0). For odd k1k2, we can set a, 13 toeither 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-triviallyunder modular transformations.3.4 The braid group on a Riemann surface and Chern-Simons statisticsConsidering a set of point charges leads to the set of wave functions^(a^ t _Wp>r^(', 7, tit-) = exp In-k-ynz7n, + 277m I Um — im)cle — —27i(E qi(zi(t))i3 o^k^i7r^_--(A(z0))^qiqa (0 (t) — 0 (0)) +^I Um — j ,i)de • I Um — 3 "lcit'cx-1-kip-Fk2rk1 k2 )t(ki-ym — k2 I jrndelkik2f2)o(3.4.1)The wave function depends on charge positions through the integrals over the topolog-ical components of the current jm, jm, and through the function 013 (t) — 0 ij (0). Considerfor a moment motions of the particles which are closed curves, and are homologicallytrivial. We focus first on the integrals over jm,3771. If, for example, a single particlemoves in a circle, we find that the integral of these topological currents vanishes. WeChapter 3. Abelian Chern-Simons theory^ 49conclude that these currents contribute nothing additional to the phase of the wave func-tion under these kinds of motions. The function O(t) — O(0) must be treated differentlyhere, because it has singularities when particles coincide, and thus, while motions thatencircle no other particles may be easily integrated to get zero, this is not true when otherparticles are enclosed by one of the particle paths. The result is non-zero in this case, infact it is 271- (with appropriate sign depending on the loop orientation). Nevertheless, thisfunction is still independent of the particular shape of the particle path. The definition ofOij in terms of the prime form E(z, w) is just the generalization to an arbitrary Riemannsurface of the well known angle function on the plane, that is as the angle of the linejoining 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 chargeq, the wave functions defined here acquire the phaseaevirq2^(3.4.2)For homologically non-trivial motions of a single particle on M , the current integralfot ii(e)de will in general change as fot ji(e)cle _+fot ii(e)de + Si + S/lmtm, where sI andtin are integer-valued vectors whose entries denote the number of windings of the particlearound each homological cycle. However, for multi-particle non-braiding paths, thereis no contribution coming from Oii. Thus, for a closed path on M, the wave functionsbecome[ 27i^27riWThr(t) = exp --rmsm — — E qi((a — k2a0)ms'in — 63 — kArtirn) — Uk 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 relationB31 ,t1 B32^= e- (8itm2_s;ntni1) 32,t2B3,,t,^(3.4.4)Chapter 3. Abelian Chern-Simons theory^ 50This 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 functionWp,r on the other index. In this sense, these two cocycles play a dual role on the wavefunction.The self-linking contribution, J, in (3.4.3), plays an important role here. For homo-logically trivial closed particle trajectories, we find J = 0 if the path does not enclose zoin the patch that we are working on, since we choose to put all the flux of F around zo.Otherwise, we find a contribution--LYFq22k q12, g) to J, for the particle i. This can beillustrated by checking for independence of the braiding (3.4.3) on zo. In the definitionof 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 thefunction (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 theshift in ce0 and #0, while the second phase comes from the fact that each charge trajectorycrosses zo, which produces a shift in J.To study the permuted (identical particles) braid group, we will consider n particlesof charge q, so Q = nq. The representation of the braid group is characterized by itsgenerators, 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 onthe 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, itis easy to check that we recover all of the necessary relations of the braid group on theRiemann 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 sourcesChapter 3. Abelian Chern-Simons theory^ 51on an arbitrary Riemann surface, and solved the theory. We find that the presence of non-trivial spatial topology introduces extra dimensionality to the Hilbert space separatelyfor the large gauge transformations and the braid group. We find a set of fundamentalconstraints (3.2.14), relating the charges, k, and g such that we recover a consistenttopological field theory representing a general (with some identical and non-identicalparticles) braid group on M. In particular, we recover the permuted braid group on M.3.5 Path integral quantizationThe 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 thecanonical formalism. Using the action (3.0.8) on an Euclidean closed 3-dimensional spaceM, we need to fix the gauge symmetry A --* A + dx.Let us work first in the Lorentz gauge (9AP = 0. The standard Faddeev-Popovprocedure [14, 4, 89] gives rise to the gauge fixed path integralZ[J] = f DADaMexp {—ill f AdA + i f A„j"d3x + i f ((.9„24)2 + -613c)d3x}471- m^m^m(3.5.1)where c and are the ghost fermionic variables. Gathering the bilinear terms togetherin (3.5.1) gives iA„(TkirevAaA •+ arv )A, = iApKiLl'Av. The propagator of the gauge fieldis thenGPI' = (K- ')Av .^ 0-A____°A + 00_1 avk^0^0(3.5.2)Thus, integrating the A field and the ghost field (which is decoupled from the gauge fieldand produces only a constant) is straightforward, resulting inZ[J] = constant • exp{ — ii L j„G"uji,d3x} = constant exp{ --k-i7 14 cAvA 1•0_0_49), I.,d3x}(3.5.3)Chapter 3. Abelian Chern-Simons theory^ 52where we used the continuity equation Omj„ = 0. It was expected that the final resultwould 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 bydrij„ = Eq J 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)^—^(remember4rITIthe Coulomb potential), the expectation value of Wilson lines (the integration of A alongthe loop Ci) are [71, 89]whereZ[J]^.7rk< C >= Z[0] = expfz— Egigir(ci, canii(3.5.4)/(Ci,Ci)^dxm^(x_Jcidy,,e,,Ay)A_cis the Gaussian linking number of the closed loops Ci and Ca, a three dimensional versionof 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 2cr = etV , as we found in the last Section (3.4.2).More interestingly, we also find a phase for the self-linking of a curveW(C) = dx„ f dy,E"" (x Y)Ac^c^ill3(3.5.6)Even though the integrand is divergent when x y, the integration measure cancelsthe 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 ingeneral and under a small deformation of the loop C, the writhe W(C) will change by(3.5.5)Chapter 3. Abelian Chern-Simons theory^ 53a small amount. Let us define a framing of the loop C in terms of a unit vector 71, notparallel to the loop. We can construct a second loop a by deforming C infinitesimallyalong this framing. The linking of C and 0 is an integer, which can be broken into twopieces [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 is1 „T(C) = —27r c dx • (n x (Ts ) (3.5.8)where ds = (di di)1. A different choice of framing would only change T(C) by aninteger 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 eachparticle, to the action. This amount to the addition of the phaseq?^.E - e3 d t (3.5.9)2k where 0 =^2) for a framing il of a particle trajectory. For a 3-manifold Mg x R, achoice 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 isa fundamental requirement, that seems independent of the approach used to solve thetheory.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 gaugefixed path integral becomes, with A =Z[J] = DADcDeexp {—izTrk^AdA^A,j"cPx(OAT +i (— +^+ -6A c)d3 x}.A4 6^4.2(3.5.10)Chapter 3. Abelian Chern-Simons theory54It is easier to work in the 6 -* CXD gauge, in which case the propagator is1GA, = ___47 (peov.), _ cv _oALA) _OA + t Al, Ayuo 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 4The CP(1) model and Abelian Chern-Simons theoryIn Euclidean space the 0(3) non-linear sigma model or the equivalent CP(1) modelrepresents classical ferromagnetic and antiferromagnetic spin systems at their criticalpoints [20]. This is particularly interesting in 1+1 and 2+1 dimensions where certaintopological terms can be added to the standard sigma model action. A representativeof 72(S2) = Z gives instanton number in 1+1 dimensions and soliton number in 2+1dimensions 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 representedby the quantized 0(3) non-linear sigma model only when a particular topological term isadded to the action [1, 42]. In 2+1 dimensions the sigma model is believed to give a goodphenomenological description of the 2 dimensional quantum Heisenberg antiferromagnet[12]. It was suggested that a topological term, the Hopf invariant, is important for someof their quantum properties, particularly for those systems exhibiting high 7', super-conductivity [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 historiesin 2+1 dimensions, it has been argued [87] that its presence in the action of the sigmamodel is associated with fractional spin and statistics of solitons when the model isquantized. Their argument is semi-classical in that it depends on the relative importanceof classical Hopf instantons [93] in the path integral measure and should apply to themodel in the ordered phase. We shall review the canonical approach, which establishes55Chapter 4. The CP(1) model and Abelian Chern-Simons theory^56that the presence of the Hopf term in the action of the CP(1) model is equivalent tothe requirement that the wave functionals of the theory without the Hopf term satisfycertain boundary conditions rendering them multi-valued. This multi-valuedness will beresponsible 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 andstraightforward [10, 74], the CP(1) model is not so widely studied. For an exception ina slightly different model, see [63]. This is partially due to the complicated nature ofthe constraints that must be solved in order to go from the variables appearing in theaction to the physical phase space of the quantum theory. Also, the CP(1) model iswidely viewed as being completely equivalent to the 0(3) non-linear sigma model, whichhas a simpler canonical structure. However, the CP(1) model has a distinct advantageover the sigma model. The Hopf invariant has a local integral representation in terms ofCP(1) model fields and the action, with the Hopf term added, has the desirable feature ofmanifest locality [93]. In particular, manifest locality in time is important for a consistentcanonical quantization of the model.In this Chapter we will explain the relation between the variables of the non-linearsigma model and of the CP(1) variables. In particular, we will describe the Hopf mapand the Hopf invariant. We shall explain the canonical quantization of the 2+1 dimen-sional 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 canonicalcommutation relations of the constrained theory, using the Dirac bracket, in the func-tional SchrOdinger picture and find that the local U(1) gauge invariance of the modelis represented projectively. We construct the 1-cocycle associated with this projectiverepresentation explicitly.We argue that the cocycle is non-trivial in the sense that it cannot be removed bya canonical transformation with a single-valued generator. We will show that it can beChapter 4. The CP(1) model and Abelian Chern-Simons theory^57removed using a multi-valued generator and thus obtain multi-valued quantum statescarrying a 0-statistic representation of the braid group for multi-soliton configurations.We argue that topological solitons obey the conventional spin-statistics relation. Forrecent work on this subject, see [6]. We shall also solve the constraints and demonstratethe equivalence of the CP(1) model and 0(3) non-linear sigma model at the canonicallevel.4.1 The Hopf map and the CP(1) modelBefore we begin we shall review some of the features of the Hopf map and of sigma modelfields in two and three dimensions. The two-sphere is described by the set of unit vectorsna, a = 1, 2, 3 withDna)2 a___ 71 2 = 1^ (4.1.1)aAn explicit parameterization of 77 is given in term of spherical coordinates77 = (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 ofcomplex rays obtained by taking all complex vectors (zi, z2) with the identificationz2)^(Az1, Az2) for all complex numbers A. We will show that the CP(1) mani-fold is identical to S2.Let us first rescale A such that we obtain a three-sphere, represented by the pair ofzicomplex variables z = ( with the constrainttt —Z Z =^4z2 = 1 (4.1.3)Z2Then CP(1) is represented by S3, the z in (4.1.3), with the remaining identification of zwith Az where .\ is a pure phase. A more elegant picture of CP(1), or 82, is given as theChapter 4. The CP(1) model and Abelian Chern-Simons theory^58base of a fiber bundle, the Hopf bundle, represented by the constrained z variablePH P (S2 7r H U (1)) S3where 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 Hopfmap,ria = zto-azwhere o-a are the Pauli matrices(4.1.4)0 t o^— 00.1 = Cr2 :=(1)010 - 3(10^-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• cosZ+ = e , for^0 0 7r(^)ei° sin 2•z = ee-i(k cos, for^0 0 0sin(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 byusing complex coordinates on S2. For this we need two coordinate patches, the disc D+,that excludes the point at the south pole, with coordinates0^z2= e ick tany =^0 < 0 <z — 2and the disc D_, that excludes the point at the north pole, with coordinates0^7F77_ = e-icb cot (-) =^- < 0 < 71-2 z2 2 -(4.1.6)(4.1.7)1 e2^for 0 <^< 27r 177_ 12 < 1 (4.1.10)1 + 177_12zj — ^ ei^=1 + 177-i21Chapter 4. The CP(1) model and Abelian Chern-Simons theory^59In the overlap region D+ n D_ Si, the equator of 82, the coordinates are related bythe holomorphic transition functionwhen 71+^0, i_ 0 (4.1.8)The upper hemisphere D+ is covered by the unit disc 177+12 < 1, and on the lowerhemisphere 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 andmoduli 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^ei41 + 171+12^7 Z+2 =._ 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 toruswith coordinates 77+, e+ denoted as T+. Similarly, the pre-image of D_ is the solid torusT-, represented as1The Hopf bundle S3 is obtained by sewing T+ and T- together on their boundaries byusing the U(1) gauge transformation, see (4.1.5),i95 —= C Z^Or = — 0 (4.1.11)We did not specify a connection, A, on the Hopf bundle PH. Such a connection mustproperly represent the first Chern class (2.3.2) of PHci(PH) = — 1^1^ 1dA = — . (A+ —^= — . I^= 127ri fs2^2R- 1 27rt (4.1.12)where the integration on 52 has been reduced to an integration on the equator, theboundary 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^60Figure 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', atrivial 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 isA = -ztdz (4.1.13)The transition function of z does agree with the transformation of A. The curvature2-formF = dA = -dztdz = -2 sin(0)dcbd0is gauge invariant. Since sin(0)d0d0 is the volume element on S2, we find that F isconstant on the two-sphere.The first Chern class (4.1.12) shows that the fibers of the PH bundle, or .53, are non-trivially related from each others. The importance of the Hopf map, and its relation withthe Hopf invariant explained later, resides in the fact that any two distinct fibers on S3are linked exactly once. To show this, let us view T1 : (77±,e±) as cylinders. The variablee goes along the axis of symmetry and jij< 1 forming the disk where it is understoodthat (g, 0) is identified with (7/, 27r). A fiber on D+ : (0,0 < e+ < 21r) is depicted inFigure 4.7.Since 7ri (S3) =- 0, the solid torus T+ and T- must be sewn together in such a waythat the fiber, which is not contractible in T+, is contractible on S3. That this is the)^4 1• 1itT -FIIIsI\le)Chapter 4. The CP(1) model and Abelian Chern-Simons theory^61 T*:..../////IFigure 4.8: The fiber from Figure 4.7 deformed to the boundary of 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) shownin 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 considertwo 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 tothe boundary as shown in Figure 4.11. Note that in order to do this it is necessary to+T+4--.t _I1‘ j_Figure 4.10: Two fibers on T.Chapter 4. The CP(1) model and Abelian Chern-Simons theory^62114... ....^ I )40e.4 1^ I". '''' - -7/- °- r - t ------ 4Figure 4.11: We deform one of the fibers to a loop at the boundary of T+ as shown. Inorder to do this a small loop linking the other fiber must appear. T +le-IIiFigure 4.12: The fiber on the boundary of T+ in Figure 4.11 can be contracted to a pointinside T. The remaining loop links the other fiber.leave behind a small loop linking the other fiber. Now the part of the first fiber, whichlies on the surface of T+, can be shrunk to a point in the interior of T. This leaves theconfiguration in Figure 4.12. Then, once again we can deform the second fiber to theboundary of T+ so that it forms a loop similar to that in Figure 4.8 and then move itinto 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 circleon 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: ooFigure 4.13: The remaining fiber in Figure 4.12 can also be contracted to a point insideT-, 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 diagramP(A4 , , U(1))^---0 S3 = P(S2 , H, U(1))z 7rH^ (4.1.14)S2The 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 thisconfiguration is given by the Chern class of the bundle P(M, U(1)) obtained fromthe PH = P (S2 lrH , U(1)) Hopf bundle by the map M^S2. The diagonal map ofthe diagram (4.1.14), z : M^53, represents the z(x) field. The connection of theP(M, U(1)) bundle is obtained (pullback) from the connection on PHA(x) = —zt(x)dz(x)With this connection we can calculate the Chern class of this bundle, or the charge sectorof the soliton configuration. It is an integer for a closed and orientable manifoldci(P(M,i,U(1)) = — 271i fjo dA(x) = qThis means that the manifold M covers S2 exactly q times, thus the Chem class of PHin (4.1.12), which is unity, is counted q times.Chapter 4. The CP(1) model and Abelian Chern-Simons theory^64For 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 diagramP(M x R, U(1)) --)• S3 = P(S2,7H,U(1))Fr^z^ (4.1.15)M x R 52Even if the fields ft' and z are now time dependent, the charge sector will be conservedfor smoothly varying fields. We can define a topological local current, that integrates tothe total soliton charge q, byj(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 , thatis such that -4 • 6-4 0, the variation of the 2-form j is an exact 2-form1Sj = d (-47r72 • a 72 X clii)This again shows that q is a topological invariant(4.1.17)1^1^ 1Sq=8-1^• dn' x dii^8(7-1 •^=^d (ft, 871 x d71) = 0 (4.1.18)87r m 87r m 47r mWith the parametrization (4.1.2) we have1^ 1jd0(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 withthe gauge connection A of the bundle P(M x R,51-,U(1)), with a slightly different nor-malization,1^1N =^f AdA^ztdzdztdz4.2 .m3^47r2 .m3(4.1.20)Chapter 4. The CP(1) model and Abelian Cbern-Simons theory^65Using the diagram (4.1.15), we find that the integer N represents the class of the mapping71 : 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. Againfor a space-time S3, this map is represented by 71-3(S3) = Z. The power of the Hopfmap 71H is that it is an isomorphism between these two classes; the class of the mapM x R --* 52 is the same as the class of the map M x R --, S3, which is nothing but theHopf invariant (4.1.20).In gauge field theory, the Chern-Simons term is not a topological term in the senseconsidered 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 situa-tion, where A --= —ztdz, the Chern-Simons term is a topological invariant. Under localvariations 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 con-figuration but depends only on global features.With the coordinates (4.1.5) of z, N has the form1N = — I dad(cosO)d08r2 m3(4.1.22)Here, for each fixed a, 0 and 0 are the spherical coordinates of 52• Furthermore, for eachfixed 0 and 0, the a coordinate parameterizes a great circle 51 C S', the fiber of theHopf 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 ofsoliton trajectories on M x R.Chapter 4. The CP(1) model and Abelian Chern-Simons theory^664.2 Canonical structure of the CP(1) modelIn 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 tothe CP(1) model in any dimension if we set the coefficient of the Hopf term to zero. It isalso straightforward to generalize them to consider CP(N) or other Grassmannian sigmamodels [94].Since the Hopf term is a topological term its presence in the action does not influencethe classical equations of motion of the CP(1) model. Its presence therefore makes onlya superficial difference to the classical phase space. However, as we shall see, it has animportant effect at the quantum level.We shall work on the Minkowski signature open space M x R where R parametrizesthe time dimension and the spatial manifold M is two dimensional and closed. For thepurposes of canonical quantization it is sufficient to treat M as if it were R2 and imposesuitable boundary conditions. The Lagrangian density with a topological term is0 Ho, A ci A^= D*ZtRIZ -^Uvf47r 2 Mwhere DA = O,. + A. Let us define, for the moment, A as1^t 7rA =^dz — dzt • z) . Awhere A is a gauge invariant globally well-defined 1-form. In principle, it could be presentin the definition of A since it does not affect the gauge transformation or the value of theChern 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 actionf DztD"zd3x + —472 ztdzdztdz + JA„jAd3x — —k f AdA047r(4.2.1)where k^5. From the last Chapter, we can expect that the exchange generator ofsoliton field excitation, of charge one, will be represented by a phase eif = eie.Chapter 4. The CP(1) model and Abelian Chern-Simons theory^67Now, let us set A = 0, thus using1 tA =^dz — dzt z) (4.2.2)which satisfies the constraintsztz = 1, ztD z^ —— 0 D* ztz — (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 equationof 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 asa 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, = 0Under the U(1) gauge transformationz(x)^eix(x)z(x) , z(x)^zt(x)e-ix(s)(4.2.1) changes by an exact term,0Zd3x Cd3x d(i-47r2xdA)(4.2.6)(4.2.7)(4.2.8)If either x or dA vanishes on the boundaries of space-time, the action formed by inte-grating over space-time will be invariant.Chapter 4. The CP(1) model and Abelian Chern-Simons theory^68The canonical momenta, found using Z' aoz and the Lagrangian L = f d2x.C, aregiven by7r = 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 Dozas functions of the phase space coordinates. As we shall see shortly, not all componentsof 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 aresult of the fact that the Lagrange density is not strictly gauge invariant but transformsby a derivative term. Since the covariant velocity operators in (4.2.9) and (4.2.10) areinvariant, we see that under the gauge transformationz(x)^eixMz(x) , z(x)^zt(x)e-ix(')^(4.2.12)the canonical momenta transform asr(x)^eiX(x)(7(x) _Leij_4 aixajz)^71-t(x)^(rt(x)^° Eija.xa.zt)e-iX(S)471-2^ 47r2^3(4.2.13)Their gauge transformation can be made covariant with a further canonical transforma-tion7r -4 r + {F, '}p + —1 {F, {F,r}p}p + • •2!+P 2! 1^P) P(4.2.14) 1Here and in the following the Poisson bracket is denoted {...,...}p and the Dirac bracket will bedenoted byChapter 4. The CP(1) model and Abelian Chern-Simons theory^69with the generator0F = 472 im2 dx(x)zt (x)dz(x) (4.2.15)Thus the canonical variables are covariant under a combination of a gauge transformationand canonical transformation. Also, the canonical transformation shifts the Lagrangianby the time derivative of F in (4.2.15) which compensates the gauge transformation ofthe action in (4.2.8).This is sufficient for gauge invariance of the model in the canonical formalism. Inthe quantized theory, the transformation law for the action in (4.2.8) and the canonicalmomenta 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 thecanonical 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 thatp(x) = i0.zt5.z(x) is the topological charge density. This constraint requires that2ir^2^3a particular combination of the U(1) charge density of the z fields and the topologicalcharge density is set to zero. It will be treated as a weak relation and will later beimposed as a physical state condition in the quantum theory. It has vanishing Poissonbracket with the CP(1) model constraint,{z(x)z(x) — 1, I:4 zt (y)z(y) — zt (y)Doz(Y)1P = 0The other constraint is 13; zt(x)z(x) zt(x)Doz(x) 02Here we denote equations of constraint by weak equalities^These cannot be made strong equalitiesuntil the bracket structure is suitably modified [19J.Chapter 4. The CP(1) model and Abelian Chern-Simons theory^70which generates dilatations of Doz(x) and z(x). The constraint (4.2.18) is equivalent to7rt(x)z(x)d- zt(x)r(x) R-z. 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 shalluse them to eliminate redundant variables of the classical theory before quantization, thatis 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 thematrix {, O}p is invertible. When these constraints are used to eliminate variables the canonicalbrackets must be modified so that the remaining variables have vanishing brackets with the constraints.This is achieved by using the redefined variablesx' = x — {x, Calla (-R., Cjp)c—, Co ..,:', xThe new variables have the Poisson bracket{xi, V}p ,--- {x, Op — {x, Ca}i- ({C, C}P)al {0, OP fx, ODwhich defines the Dirac bracket of the original variables x and y. It is these brackets that must be usedif 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) andrt(x), respectively, by functionals of z(x) and zt(x). It is straightforward to show thatthe Dirac brackets of Doz(x) and Do*zt(x) with each other and with z(x) and zt(x) areidentical with those of the canonical momenta 7r(x) and irt(x) with each other and withz(x) and zt(x). Thus we expect and will later confirm that, at the classical level, thevariables 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 termthat 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 unaffectedby the presence of the topological term in the action.Let the generating functional of the canonical transformation be [z, zt , 7r, irt]. It isnecessary 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-V^,r^,87ra(x) —^, r a(x)} D — (vat. — —zazb ciz , z j — z zb^ i.z , zi.. i^(4.2.29)2^Szt(x) 2 a 84(X)1 t^ 8^ [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 transformation7r1= 7 + {, 7}D + yl M {e, 7}D)D + • • •17re = + {e, rt}D + —2! {, fC 7t1D}D + • • •in (4.2.29) and (4.2.30) truncates at the first order.(Sa l, --1.ztazb)^zt]SZb1—^t^ , zt] =z aSzb08r2Chapter 4. The CP(1) model and Abelian Chern-Simons theory^72We are therefore required to find a generating functional with the properties(Sa b^ z t ] —°Zblzazbl-[z,zt] = —2^Szb0 Eij (2(ztaiz)ôza _1_ (aiztajz)za)871-2(4.2.31)(2(zta1z)ajzta (aiztajz)zta)(4.2.32)Indeed, there is no single-valued functional of z and zt with this property. In Section 4.4we shall, however, find a multi-valued functional solution of (4.2.31) and (4.2.32). In theclassical field theory this provides a satisfactory canonical transformation.However, in the quantum theory the canonical transformation is represented by aunitary transformation on the Hilbert space. If the generating function is multi-valuedthe resulting wave functions are also multi-valued. The transformation therefore changesthe boundary conditions of the quantum theory. This is a well-known feature of fieldtheories with topological terms in the action, the most noteworthy example being the0-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 spacesa(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 theisometries of S3. The generators contain only those components of the velocity fields Dozand 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 propertysa(x)sa(x)^ta(x)ta(x)^(4.2.36)Chapter 4. The CP(1) model and Abelian Chern-Simons theory^73This identity can be demonstrated without use of the constraints. It is useful to recallthe Pauli matrix identity ir: at. • (7- - cd = 28 ad& — SabScd. They have the further property thatsa (x)sa(x) = (zt(x)z(x)) (Ho'zt(x)Doz(x)) — li- (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 secondclass constraints ztz — 1 = 0 and /4ztz + ztDoz = 0 have been used.The Hamiltonian isH = 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 thegenerators 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) chargessa = I d2x sa(x)^ (4.2.40)Furthermore it commutes with t3(x) and has a local U(1) gauge invariance, which isgenerated 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 constraintt3(x) p--- 0^(4.2.43)Chapter 4. The CP(1) model and Abelian Chern-Simons theory^74form a complete dynamical system on the constrained phase space. To completely spec-ify the constrained phase space it would be necessary to impose a further gauge fixingcondition that has a non-vanishing bracket with t3(x). Alternatively, we could imposeit 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 fieldtheory. 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 equationTo quantize, it is necessary to associate the Dirac bracket algebra of the coordinates withcanonical commutation relations{x,y}D -- -i-4-1[x,y}^(4.3.1)In the functional SchrOdinger picture of field theory the states are functionals of thefield configurations,111 a-_- xlik,z1-]^ (4.3.2)The coordinates are represented by functional multiplicationz0p(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. Theconstraint zt(x)z(x) —1 ',---- 0 is imposed by restricting our attention to classical configura-tions in (4.3.2) that have this property. However, the constraint (4.2.18) on the canonicalChapter 4. The CP(1) model and Abelian Chern-Simons theory^75momentum operators is not so easy to impose due to operator ordering ambiguities. Wecan 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 momentumoperators 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=^ zt(x))sa(x) 2Eli ((ztaiz)aa (zto.az)^(ôiztajz)(zto.az))87r21^t+(x) = zt kxicr2 6z(x)1 ^a ^2z(x)az.(x)a8 \ +. 0^ a.zta•z_ 1 . ^6  z(x) zt(x)szt(x)) • j8r2^3t3(X)^• (8 Z(X)-47-r2 fii ((ztaiz)(zto-2aje))—^Eij ((zta1z)(ajzto-2z))(4.3.5)(4.3.6)(4.3.7)(4.3.8)The notation : : indicates a specific choice of ordering of the product of noncommutingoperators. 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 straightforwardto check that (4.3.5)-(4.3.8) represent the canonical commutation relations derived fromthe Dirac brackets.The physical state condition ist3(x)4iphys[z, it] = o^(4.3.9)which, using (4.3.8), implies that the gauge symmetry is represented projectively:i^-iillphys rLe x z, z t e xi exp --° f d2xx(x)Eijaiztajz iliphys[z, zt]472(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 theseconstraints, don't appear in sa and ta. Therefore, any projection of the functional derivative operatorsone 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^ 76Furthermore, 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.' Thisis consistent with the algebra [0(x), t3(y)] = 0 where we have converted the Dirac bracketto a commutator.The one cocycle is non-trivial in the sense that it cannot be removed by redefiningthe 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 thephysical state condition (4.3.10) and the eigenvalue problem for the functional Hamilto-nian operatorf 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 physicalstate condition (4.3.10) and through the 0-dependence of the su(2) generators sa(x) givenin (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 theproduct 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 one-cocycle of the symmetry group. For a detailed discussion of the role of Lie group cohomology in quantummechanics see [49].Chapter 4. The CP(1) model and Abelian Chern-Simons theory^ 77and zt, there is a redundant integration in (4.3.12)) we have divided the integral by theinfinite volume of the local U(1) gauge group.4.4 The braid group and fractional statisticsIn 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 mapsthe Hamiltonian of the CP(1) model with Hopf term onto the Hamiltonian of the modelwithout 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 valueseither on a compact two dimensional space (which we shall call M) such as 82 or onthe open space Ir where we require the boundary condition that z go to a constant atinfinity (obeying the constraint ztz -= 1) with covariant derivatives vanishing sufficientlyrapidly at infinity. Here we shall assume that the space has been one-point compactified.The field configurations are distinguished into disconnected classes by their topologicalchargeq = --i I dztdz27M2Explicit classical field configurations with non-zero topological charge will be constructedlater in this Section.Let us consider the sector of the configuration space with fixed topological chargeq. We consider a reference field configuration zo, 4 which has topological charge q andalso an open three dimensional Euclidean manifold M3, which has as boundary two 2-dimensional 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 andwhose restriction to the other cap is the field configuration of interest z, zt.Chapter 4. The CP(1) model and Abelian Chern-Simons theory^ 78Then, we define the functionalztj -41-2 it43(4.4.1)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 d 6^— "itcE(5.&"tcti^dicEtr2 1^(zt6zdztdz — ztdzSztdz ztdzdzt6z)471-2 fm—1 2 m (Z-01.8ZodZPZO — 4PZ084dZo 4dZod4SZO)This leaves the functional (4.4.1) depending on the embedding of the field configurationinto M3 only.Using this formula, we can take the functional derivative of e by z(x), zt(x). We findthat(5ab^Za4)[2. Z1.12^az1b.(6a b^z "4) 1:27-b[Z1^.6^r^fiZ •2^Szb_1 ,t,t_ ,er,, zti2^,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 isz/(x)^e-iqz(x)eiq^z(x) , zt'(x)^e-iKzt(x)eiK = z(x)^(4.4.4)(D0z(x))/ = e-iqD0z(x)eq r(x) , (D*zt(x))/ e-iKD*zt(x)eiK^rf(x) (4.4.5)that is, when expressed in terms of the original variables the operators of the canonicallytransformed field theory make no reference to the Hopf term in the original action andChapter 4. The CP(1) model and Abelian Chern-Simons theory^79are in fact independent of O. Furthermore, the Dirac bracket of the classical theoryand the commutator brackets of the quantum theory are invariant under the canonicaltransformation and are also independent of O.However, in the quantum theory there is a further transformation of the quantumstate vectorszt] = e-iek'zfhlf[z, zt]^ (4.4.6)This phase transformation changes the boundary conditions on the quantum mechanicalproblem. [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 sameboundaries .A4 e M and a different field configuration it to fill in .A-43 we would havearrived at a functional 4"[z,zt] which differs from ,[z,ztl:re[z, Yr] = 4.[z, zt] ^3em3^dit (4.4.7)The last term is an integer, the Hopf invariant (4.1.20) for the mapping of the extendedfield configurations and it to the manifold obtained by sewing M3 and ICC togetheron their common boundary M ED M.The holonomy of the multi-valued wave function is characterized by the topologicalterm 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 by1 it4R-2 Jm2xsi diditSince this integral is a topological invariant, we can evaluate it by deforming i, it to aconfiguration where the integral can be done analytically. Two topological invariants,which characterize the configuration, are the soliton number, q, and the winding numbern = — .itcti27r Js1Chapter 4. The CP(1) model and Abelian Chern-Simons theory^ 80We must find a configuration with a fixed q and n.One such configuration is a periodic interpolation of a gas of solitons. Solitons arestatic solutions of the classical equations of motion that have a fixed topological chargeq. They are minima of the classical energy functional for static fields (where we use thecomplex 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 thatq = —^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 asE 27rq + 2i I drithi* D_z12^ (4.4.10)which is minimized by functions obeying the conditionD_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 thethird 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 degreescan be seen to have the same winding number (given by the maximum degree) and canbe considered limits where some of the ai or bi 00 in (4.4.12) above accompanied byChapter 4. The CP(1) model and Abelian Chern-Simons theory^81a singular rescaling of the parameter A. The function na = ZtCraZ is a mapping from theRiemann sphere with coordinateson the upper hemisphere andzi^1^— ai— ^Z2^A^ 77 — bi(4.4.13)Z2^Hq — bi (4.4.14)— = A 1 — aion the lower hemisphere, to the manifold CP(l) :-- S2, which covers the latter exactly qtimes. This can be seen through the fact that it is represented as a qth order rationalfraction.We could view each pair of parameters ai, bi as specifying the positions of two pointson 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 thereforethink of the solitons as non-interacting classical particles.As a first approximation to a theory of dynamical, rather than static solitons, time de-pendence 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 thedynamics 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 ^ 2riwer)^(9 — ai(T))^z(9,r)  VIII — ai(7)12 + 1A(7)12 III 19 — biNi2^AN 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^ d^zt] —^zt] s qn — —q^ch-linA(r) +^o d — E Imln(ai(r) — Mr))2r o^A(r) 2r ^dr 3=1(4.4.16)4t4k4Chapter 4. The CP(1) model and Abelian Chem-Simons theory^82Figure 4.14: Soliton exchange braids neighboring ribbons.Multi-valuedness of wave functions encountered here gives rise to fractional spin andstatistics of many-soliton quantum states. Fractional statistics results from a projec-tive representation of the exchange symmetry of quantum wave functions for identicalparticles [31, 32, 58].The holonomy of the multi-valued wave function (4.4.6) which results from the lastterm in (4.4.16) implies that (4.4.6) carries a unitary representation of a particular versionof the braid group on M [71]. Since there are 2 labels in (4.4.16), the a and b variables, thebraid group representation would correspond to a 2 color braid group. We will describethis representation in more detail. A subset of the generators of this group can beviewed as the braid group generators a, acting on a set of framed strings correspondingto trajectories of the pairs of points (a„ bi) shown in Figure 4.14. In term of thesegenerators, 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) = afor each generator and exp(—i0) for each inverse generator. These phases are associatedwith fractional exchange statistics for solitons. The phase of a is exactly the phase thatwe 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 isaccompanied by a phase si = exp(i0) = s which we associate with fractional spin. TheChapter 4. The CP(1) model and Abelian Chem-Simons theory^83Figure 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 spin-statistics theorem for the CP(1) field. A similar braid group structure for framed linkshas recently been studied in the context of lattice Chern-Simons theory [27]. The factthat the braid group elements have 2 colors is the source of the spin. The braid groupfor the permutation of particles with spin is different from the braid group discussed inSection 2.5. When the system of coordinates rotates by one turn the wave function, andthe braid group representation, is transformed by an extra phase s. This phase is inaddition to the phase obtained from the framing. The generators a, satisfy the usualbraid group relations of a pure 0-statistic representation described in Section 2.5. Inaddition, on an arbitrary Riemann surface the additional generators would satisfy therelations (2.5.5). On the other hand, the global constraint (2.5.6) will be changed intoCr 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 theoremChapter 4. The CP(1) model and Abelian Chem-Simons theoryFigure 4.16: Exchange of soliton constituents.84Figure 4.17: A positive or right crossing.is satisfied, the global constraint for the CP(1) model is0.2q es0-2q = 1 (4.4.18)The possible statistics are then given by one of the 2q'th roots of unity. If we are workingon the plane then there is no such constraint, although imposing the requirement thatthe 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 tosolitons that exchange constituents, shown in Figure 4.16. This gives a structure morecomplicated than the usual braid statistics. The phase for a general exchange can becomputed 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 thewave functionals, which is therefore represented projectively by eie[z'zt] and this pro-jective phase is responsible for the fractional spin and statistics of solitons. In addition,Chapter 4. The CP(1) model and Abelian Chern-Simons theory^85Figure 4.18: A negative or left crossing.there are twists corresponding to the exchange of constituents between solitons, the twistn and the winding of A.Putting back the metric in the static energy function (4.4.8) shows that it is confor-mally invariant. The static solutions (4.4.12) can be reduced to a solution inside a smallcircle within a larger space. More than one solution can be considered for a given timeslice. Choosing a proper orientation for the solutions (4.4.12), with the time directiongoing up, corresponds to a soliton of positive charge q, while solutions with the timedirection going down, corresponding to a change of y to —y or 77 to 77*, means that wehave 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 chargeqi, where we use the link as the time direction. We will set Ai = A = constant, so thetopological 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 willsimplify the following calculation. The gauge fields outside the links are then given byu — x)),A" = --(4"^1133/(17)d3u^2 qic""^Y.. x)A dl'^(4.4.19)2We find that the field configurations z outside the links are given by^1^1^gz = ^IAIV( ) exp( iE Ai)V1 +^A^i 50(4.4.20)Chapter 4. The CP(1) model and Abelian Chern-Simons theory^86while the field within one link, Ck, isz exp(E Ai)ziiok sowhere io is an arbitrary reference point and zi given by (4.4.15).The Hopf invariant is then easily computed asN = — ji Aj"d3xik 27r^k+ /(c2, coiokwhere(4.4.2 1)(4.4.22)si =^A jfid'x27ris given by (4.4.16), this is the self-linking number of a loop. It is interesting that the self-linking number is well defined, and an integer, without any regulation as was necessaryin Section 3.2 and 3.5. For i k, I(Ci,Ck) is the Gaussian linking number (3.5.5) of thelinks Ci and Ck.These are not exact static solutions, but are a low energy limit of soliton interactionsas the curvature of the representing links tends to zero.4.5 DiscussionWe have demonstrated how, in the canonical formalism, the presence of the Hopf invariantin the action of the three-dimensional CP(1) model leads to fractional spin and statisticsof quantum solitons.We conclude by discussing the equivalence of the CP(1) and 0(3) non-linear sigmamodels at the quantum level. The canonical transformation constructed in Section 4.4can be used to remove the 0-dependence of the canonical velocity operators, which ap-pear in the quantum Hamiltonian (4.2.39). In the canonically transformed system theChapter 4. The CP(1) model and Abelian Chern-Simons theory^ 87Hamiltonian is now represented by the operator (4.2.39) where the su(2) generators aresa(x) .1 (^2 Y.z(x)aaz(x) zt(x)°"a6zta(x))and the physical state condition now reduces toWphys[eixz, zte^= Tphysk,(4.5.1)(4.5.2)Neither of these equations refer to O. Instead, the wave functionals must be multi-valuedin the sense that if we consider a continuous periodic interpolation it with initial andfinal configurations z, zt the wave function must change phaseillphys[Z, e iON Tphys[Z, 21] (4.5.3)where N is the Hopf invariant of the interpolation. Thus the 0-dependence of the quan-tum mechanical problem has been transferred from the Hamiltonian and physical statecondition to the boundary condition, which the multi-valued wave functionals shouldsatisfy. On multi-soliton states we would interpret these wave functionals as possessingfractional spin and exchange statistics for the solitons.It is straightforward to solve the physical state condition: It implies that the physicalstates 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 as2H = Jd3x {--1 (77(x) x aft,(x)) --Ctia(x) • t' fla(x)}2 (4.5.4)and where we must still take into account the boundary condition (4.5.3). This is theHamiltonian of the 0(3) non-linear sigma model. Also, the normalization integral forwave 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^88The CP(1) model and the 0(3) non-linear sigma model are thus equivalent at the Hamil-tonian level. The additional boundary condition (4.5.3) gives solitons fractional spin andstatistics in both cases.The boundary condition (4.5.3) is trivial when 0 = integer x 27r. The quantummechanical problem is therefore periodic in 0. Physical quantities such as the energyspectrum should also be periodic.Chapter 5Advanced topicsThe 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 modeland the non-Abelian Chern-Simons theory, where topological features play an importantrole. These models have been at the forefront of a lot of research in theoretical physicsin the last few years. Since a non-Abelian group has more intricate structures than anAbelian group, these models have more elaborate solutions and unfortunately are moredifficult to solve.5.1 The WZW modelSome physical problems are better described by a group-valued field. The CP(1) fielddiscussed 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 Gin a two-dimensional space M is given by1.84 tr(ai,U-18"U)d2xHowever, there is another term that exist in two dimensions that should be considered,this is the Wess-Zumino integral (2.3.9)1247r2 Jc4tr(11-1dUU-idUU-ldU) (5.1.1)89Chapter 5. Advanced topics^ 90where .A-4 is a three-dimensional space that has M as boundary. We showed in section2.3 that this integral is multi-valued under large gauge transformations. We can built aconsistent 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-Wittenmodel (or WZW for short),Swzw =-- —8A7r2^tr(011-104U)d2x -1tr((U-idU)3)^(5.1.2)-12crThe parameter A is dimensionless and is a priory arbitrary, but we will show that weobtain a critical theory when this parameter takes a specific value.The equation of motion for the WZW action can be calculated by performing aninfinitesimal variation of the field variable U. We must insure that SU is such thatU SU still belongs to the group. The best way to achieve this is by doing a right gaugetransformation 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 gaugetransformation U SVU, in which case SUU =- (SV -1) =- iTaSva 0(8v2). Undera right transformation, the variation of the WZW action is given by6Swzwu = A2  1  (9,2(v g^^k ^ ),7-18,0SU^47r^A2\F---.4while for a left variation of U we findSSwzw A2 1^k eguSU^=- 4r \r___:g a„w g(g"' + A2 „v==q)(9UU-1)^(5.1.4)When A2 k something special happens. The term -1(V---,4(e' -^becomes a2^A2 Nr_tiprojection operatorwith the properties1^1p/211^-2 (g4v ^NI-L-.4€4u) (5.1.5)131-LagoPl3v Pga,813"13 =^gcoP4aPvie = goPagPfiv = 0(5.1.3)Chapter 5. Advanced topics^ 91In an orthonormal basis, where gm, = eybriab with the( 0 1 )corresponding to the local coordinates x± -1-ktv 21 0local metric given by 71 ab± X), we find the relationP"'AaB, = PabAaBb = A_B+^where L = ea Lti a (5.1.6)Let us define the currents J+ = —k u -1 a +u and J_ =2irmotion (5.1.3) becomessswzwu = a_j+= 0SU-La uu-i. The equation of2ir(5.1.7)Furthermore, we can use the identitya_(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)uwhich shows that the equation of motion (5.1.7) found from a right transformation isequivalent to the equation found from a left variation6.5ivzw = a+L= 0^ (5.1.8)SUFor A2 = — k, the equation of motion (5.1.4) takes the form (5.1.8), which is equivalentto (5.1.7), but with the change 4= tö+uu-1 and J_ = tru-ia_u instead. The onlydifference 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 bothconserved. For A2 ±k this would not be possible and only one type of current couldbe conserved, representing a very different theory. In fact, under renormalization theparameter 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 foundU = V(x)U01/1/(x)^ (5.1.9)Chapter 5. Advanced topics^ 92where 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 beviewed as the non-Abelian version of non-interfering right and left moving plane wavesolutions of electromagnetism.For an arbitrary field U, the WZW action is invariant under the transformationsU V(x+)UW(x-). In fact, for the classical solution (5.1.9), the WZW action isidentically zero. It can be shown that the generators of these right and left symmetriesare the currents <I± itself. At the quantum level, these symmetries leads to the Gauss'laws constraint on the ground state< 4,0 141410 >=< *01J_ >= oTo find the structure of the states of the WZW model, we need to know the com-mutation relations of the currents, as operators, among each other. This can even becarried out at the classical level by using Poisson bracket, after properly defining thesymplectic structure of the WZW model [88]. It is found that the right and left sectorscommute with each other (as expected) and each sector has the same algebra. With thedecomposition 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 Kac-Moody algebra, compared to a Lie algebra, is the presence of the Schwinger term2/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 correspondsto the algebra of loop group [69] since V(x) corresponds to a map 511 G (for x repre-senting a circle).The study of the algebra (5.1.10) shows that we can separate J as J T +Tt whereChapter 5. Advanced topics^ 93T is a destruction operator and Tt is a creation operator. The ground state is definedby^>, 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. Forexample, the well known Ising model or the 3-state Potts model [21].5.2 Non-Abelian Chern-Simons theoryThe non-Abelian Chem-Simons action (3.0.7), derived in Chapter 3,S --k 1 tr(AdA —2 A3) + tr(AmMd3x47r JM^3^m (5.2.1)is more complicated to solve than the Abelian version. For simplification, we will workon le and set the current j° = 0 for now. We can separate the time part from the spacepart of (5.2.1), which brings the action into—k I Oiltr(AiA^n3i)d3x +^tr(A0(--k B))d3x47r n3 ^27rwhere B = OijFii. This indicates that the commutation relation are given by[A7(x),Abi(y)J ___2krisaboija(x _ y)^(5.2.2)where we assume the normalization tr(T Tb)^saba We will choose the polarizationdefined by the Al_ variables (with the same notation for x± as in Section 5.1), thus27riA' k SABy working in the Ao 0 gauge, we find the Gauss' law generators6^kGa 27r^6A+a^27r^+They satisfy the commutation relations(5.2.3)(5.2.4)[Ga(x),Gb(y)]^ifabaGa(x)8(x — y)^ (5.2.5)Chapter 5. Advanced topics^ 94which is identical with the algebra of the group G. This allows us to consistently definethe ground state by the Gauss' law constraintGa(x)41° = o^ (5.2.6)It is possible to defined a group-valued field U by the conditionA+(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-zerocurvature. From this, we can build the ground state wave function in terms of the WZWaction (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, that8Swzw = „ R tr(a_(U-10+U)U"SU)d2x = zr R tr(S(U"a+U)U-1,9_U)d2xzr 2^ 2From (5.2.3), (5.2.7) and (5.2.8) this meansA_‘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 onthe 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 particleof 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 ofWilson 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^ 95For a curve C starting at x1 and ending at x2, it can be shown thatw(c )mwo = ti71 (x i )uK.1 (x2)To^(5.2.11)where the indices I, J, K labels the groups element of U. The equation (5.2.11) can bepictured as the creation of a charge at x1 represented by U./1- (xi), which is later destroyedat x2 represented by U.KJ(x2). The contraction of the index K corresponds to the curveC from x1 to x2. In general, to each particle of unit charge corresponds a representationof G. For two charges, we can associate the stateIL0 >. II > IJ >. Is > +la >where Is > and la > are the symmetric and antisymmetric component of IL0 >. Theexchange operator of the two charges and its inverse, after some non-trivial algebra, aregiven byIL_ >=.- EEIL0 >^and^IL+ >= EE-11L0 >^(5.2.12)Where El/ > IJ >. IJ > II > is the permutation operator and the matrix E is given byE = 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 ofSU(N) we have the identity1 c cTbricL -= SILSJK — i—\7 °IJ°KLfrom whichIL_ >,____ exp(i (N —Nk )I> exp( i(N Nk+ 1)71-1),1^a >^(5.2.14)IL+ >,_ exp(—i (N — 1)7 )ls > exp(i (N + 1)7,1 (5.2.15)Nk^Nk )1a >andChapter 5. Advanced topics^ 96The equations (5.2.14) and (5.2.15) forms a two particle representation (not a pure0-statistic representation) of the braid group. This can be generalized (not trivially) toan arbitrary number of particles.Recently a lot of work has been done on quantum groups [23, 57] (an extension ofthe standard Lie group algebra). The algebra of braiding of quantum states of the WZWmodel produces interesting quantum group representations.Chapter 6ConclusionIn Chapter 2, we described a number of advanced mathematical subjects. These subjectsare not very well known in the physics community in general. Once understood, thesemathematical tools give a deeper understanding of what really is a gauge theory, andwhat are the possible topological structures of the gauge field. Hopefully, this new kindof mathematical formalism will be used more often for a good understanding of physicalmodels 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 perfectmodel for a representation of the quantum Hall effect, or other physical models understudy. Instead, we concentrated our efforts into building a consistent model of a Chern-Simons theory, by working out its general solutions and properties. The fundamentalconstraints (3.2.14)e'2irq1(Q+Mg-1)) = 1of Chern-Simons theory imposes a rather interesting set of restrictions among the differentparameters of the model. In principle, these restrictions might be used to check thevalidity of a Chern-Simons model as the representation of a physical system. Recentworks [52] on the fractional quantum Hall effect, where v is the filling fraction, has beenconcentrated around a variation of Chern-Simons theory1S = _- kif A1dAJIJwhere km is a matrix with integer entries. The indices I labels different 'levels' of97Chapter 6. Conclusion^ 98the quasiparticles. For one level only, this is the standard Chern-Simons theory, it isconjectured that the filling fraction is related to the permutation phase by a = ei", thuswe find the relation v = Alternatively, the Laughlin wave function [3, 4, 70], thatdescribes phenomenologically the fractional quantum Hall effect, is very similar to thewave 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 hasshown how a dynamical field theory coupled to a Chern-Simons action is consistent withthe results found in Chapter 3 about Chern-Simons theory. In particular, we found thatthe soliton field configurations obey a certain type (two-colors) of braid group. It is notclear if this modified CP (1) model might be related to the ordered phase of a quantumHeisenberg 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 statis-tics, will be useful in future work on these recently discovered physical problems, andpossibly new ones, whose dynamics are primarily two dimensional.Appendix ATheta functionsThe theta functions [51, 62] are defined byao) (z p)^E ei7r(711-Faigem(nm-Fam,)+2/ri(nii-cei)(z1-1-#1)ni(A .0.1 )where a, E [0, 1]. They satisfy the following properties( a )P(zm^nml) = e2iriaisi_iirt,ornit,_27ritm(zm+7-1,30 a ) (zin)for integer-valued vectors .sm and t1, while for a non-integer constant cao (zm^= c i 2tm1mt tt2iriCtm(zm+13m)e^a + ct(42)Under the modular transformation (2.4.7), the theta functions are transformed asa (z1D) --*^ao^ (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 newvariables a',^are given by1^ 1a' = Da -^- -2(CDT)d,^,3/ = -Bad- AP - -2 (ABT)dwhere (M)d are the diagonal elements Mdd.99Appendix A. Theta functions^ 100Closely related to the theta functions is the prime formE(z,w) (h(z)h(w)) - i 0 (112 ) (jz co151)1/2a ( 1/2 )7u^(02)1,, 01/2variables z and w and behaves like z — w when z w (the h(z), which appear in thedenominator, are for normalization).This formalism can also be extended to include the sphere, where there are no har-monic 1-forms at all (the space of cohomology generators has dimension zero) by properlydefining the prime form. We use stereographic projection to map the sphere onto thecomplex 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 equationôR = 45(2)(z — zo)^ (A.0.2)on a Riemann surface of genus g. Since on the Riemann sphere1 ^1^18(2)(z — zo) = az( z — zo) = —7r .9. (az log(z — zo))the solution to the differential equation (A.0.2) is given byR = 1—raz(log(E(z, zo)))^(A.0.3)Explicit formulas can be worked out for the special case of genus one. In this case itis customary to choose co(z) = 1 and 12 = T, a complex number with Im(r) > 0. Thetorus 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 withz = u +T and z = TV is identified with z =1+ TV. Alternatively, we can view the toruswhere h(z) = • col(z). The prime form is antisymmetric in theAppendix A. Theta functions^ 101as the complex plane where z is identified, with z m nr with m and n integer. Usingthe 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)(ziT) = E(-1)nynqin2 f(q) H (1 - yqk+1)(1 - y-Y+1)( 1/2) (*) = Eyn-F5e+112/2 =^f(q) (1 + yqk+1)(1 Y-lqk)0^ k=0( 1/2 ) 0.0^(zir) = iE(-1)nyn+lq(n+1)2/2 = iyiqkf(q) 11(1 — yqk+1)(1 — y-igk)1/2^n^ k=0where f(q) = 111°,11(1 — qk) = En(-1)nqin(3n+1) and ii(r) = q* f (q) is the Dedekindfunction. We also find that h(z) = —271-773(r) is independent of z. Thus the genus oneprime form isE(z, w) —± ( ,„1^fi (1 - ye)(1 y-le)27r \a^`Y k=1^(1 - q92where now y = e27r1(z-").The solution (A.0.3) of (A.0.2), with y = e21ri(z-z0), takes the form0^1^1^'_R(z,z0) = —2iy—ay {const. + log(y-i — y--) + E log(1 — yqk) + log(1 y-iqk)}k=1= —2z. { 1  y^1 y-1-^00 Yqk^y-1 qk2 1 — y + 2 1 — y-1 + E^k +^1 1 _ yq^1 _ y-1 qkk=When z 7.-_, zo + m + nr, we can expand R as1 ^1 R(z, zo) = + Regular in z(z — zo — m — nr)0 k=01/2( 0k=0Appendix A. Theta functions^ 102Thus, we find that adi = 6(2)(z — zo — m — nr) for z around zo + m + nr. In general, forthe whole torus we finda.,R(z, zo) = E 6(2)(z — zo + m + nr) :.-=-_- (52) s(z — zo)as claimed.Bibliography[1] I. Affleck, Nucl. Phys. B265, 409 (1986)[2] 0. Alvarez, Comm. of Math. Phys. 100, 279 (1985)[3] D. Arovas, J.R. Schrieffer and F. Wilczek, Phys. Rev. Lett. 53, 722 (1984)[4] J.E. Avron and R. Seiler, Phys. Rev. Lett, 54, 259 (1985)[5] M.F. 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