Canonical Chern-Simons Theory and the Braid Group on a Riemann Surface ∗ Mario Bergeron †, David Eliezer and Gordon Semenoff ‡ Department of Physics 3 University of British Columbia 9 9 Vancouver, British Columbia, Canada V6T 1Z1 1 n February 1, 2008 a J 1 1 1 Abstract v 6 We find an explicit solution of the Schro¨dinger equation for a Chern-Simons theory 3 coupled to charged particles on a Riemann surface, when the coefficient of the Chern- 0 1 Simonstermisarationalnumber(ratherthananinteger) andwherethetotal chargeis 0 zero. We find that the wave functions carry a projective representation of the group of 3 9 large gauge transformations. We also examine the behavior of the wave function under / braiding operations which interchange particle positions. We find that the representa- h t tion of both the braiding operations and large gauge transformations involve unitary - p matriceswhichmixthecomponentsofthewave function. Thesetofwave functionsare e expressed in terms of appropriate Jacobi theta functions. The matrices form a finite h : dimensional representation of a particular (well known to mathematicians) version of v thebraidgroupontheRiemannsurface. Wefindaconstraintwhichrelates thecharges i X of the particles, q, the coefficient of the Chern-Simons term, k and the genus of the ar manifold, g: q2(g−1)/k must be an integer. We discuss a duality between large gauge transformations and braiding operations. ∗Submitted to Physics Letters B †This work is supported in part by a UBC Fellowship and FCAR. ‡This work is supported in part by the Natural Sciences and Engineering Research Council of Canada. 1 It is by now well established that particles confined to a two dimensional space can have fractional statistics. Such particles are called anyons. Interest in them is partially motivated by their physical effects such as their conjectured role in the fractionally quantized Hall effect or high temperature superconductivity, and partially by the fact that the description of anyons uses interesting mathematical structures. Anyons are a generalization of ordinary bosons or fermions where the wave functions of many identical particles, instead of being symmetricorantisymmetric, carryarepresentationofthebraidgrouponthetwodimensional space. This is often described mathematically by coupling the currents of the particles to the gauge field of a Chern-Simons theory, k S = − AdA+ A jµd3x (1) µ 4π Z Z The solutions of this field theory take the form of a set of wave functions which at a given time depend on the particle positions. It is well known that these wave functions carry a non-trivial representation of the braid group, where the representation is specified by the parameter k as well as the number of particles and the topology of the 2 dimensional space. The braid group is an infinite discrete non-Abelian group and has many potentially interesting representations (see, for example [4]). Abelian Chern-Simons theory coupled to classical point particles was solved on the plane by Dunne, Jackiw and Trugenberger [8]. In that case there are no degrees of freedom, the Hilbert space is one-dimensional and the only quantum state is given by a single unimodular complex number. The phase of the complex number depends on the parameters of the model, i.e. the coupling constant k and the histories of time evolution of the positions of the particles in such a way that if we consider a periodic trajectory, z (t), i = 1,...,N, t ∈ [0,1] i and z (1) = z (0) the phase of the wave function changes by the well-known factor i i 1 1 d dt Imln(z (t)−z (t)) i j k dt i<jZ0 X which counts the changes of relative angles of positions of the particles. This can be inter- preted as the wave function carrying a one–dimensional unitary representation of the Nth order braid group on the plane. The statistics parameter is the change in phase of the wave function under interchange of two particle positions. Here it is given by π. k In another interesting paper, Bos and Nair [3] have solved the Schr¨odinger equation for Abelian Chern-Simons theory coupled to particles when the space is a Riemann surface of genus g and when k, the coefficient of the Chern-Simons term, is an integer. In this case the representation of the braid group which is obtained is more interesting. It is generally multi-dimensional and represents the braid group on the Riemann surface which has more (non–Abelian) structure relations than the braid group on the plane. In this Letter we shall show how this solution generalizes to Chern-Simons theory with k a rational number and examine the resulting properties of the braid group representations which we obtain, realizing explicitly the representations found in [4]. In Abelian Chern-Simons theory, unlike the non-Abelian case, there is no reason why k should be quantized but could in principle be any real number. Polychronakos [6] has shown that, even on a torus, k need not be quantized. Furthermore, Witten [2] has argued that 2 the dimension of the Hilbert space of pure Chern-Simons theory on a Riemann surface with genus g is kg when k is an integer. Polykronakos [6] showed that when k is rational, the dimension of the Hilbert space of pure Chern-Simons theory is (k k )g. It is also known that, 1 2 when k is an integer, the representations of the braid group on a Riemann surface obtained by coupling to point particles depends on k and g [4]. We start from Abelian Chern-Simons theory coupled to the current due to a gas of charge particles (1). We shall first demonstrate how to extract the physical degrees of freedom from the non-trivial cohomology classes of the gauge field and the complex structure of the Riemann surface. Later, invariance under large gauge transformations and modular transformations will be used to show that the Hilbert space is finite dimensional. Before quantizing, we rewrite (1) in term of topological quantities on the 2-dimensional Riemann surface, M, of genus g. We decompose A into its exact, coexact and harmonic parts. More precisely, the Hodge decomposition of A, on M, is given by (d and ∗ act on M in this paper) 1 1 2πi g A = d( ∗d∗A)+∗d( ∗dA)+ (γ¯ωl −γ ω¯l) (2) 2′ 2′ k l l l=1 X where 1/2′ is the inverse of the laplacian (2) acting on 0-forms where the prime means that the zero modes are removed. The zero modes of d and ∗d (or 2) acting on a one-form are spanned by the set of Abelian differentials, ωl, on M. We can represent the homology of M in terms of generators a and its conjugate gener- l ators b , l = 1,...,g. The intersection numbers of these generators are given by l ν(a ,a ) = ν(bl,bm) = 0, ν(a ,bm) = −ν(bm,a ) = δm (3) l m l l l where ν(C ,C ) is the signed intersection number (number of right-handed minus number 1 2 of left handed crossings) of the oriented curves C and C . The holomorphic harmonic 1 2 one-forms ωi have the standard normalization [1] ωm = δm, ωm = Ωlm l a bl I l I The matrixΩlm issymmetric andits imaginarypart ispositive definite. This actuallydefines a metric in the space of holomorphic harmonic forms i ωl ∧ω¯m = 2Im(Ωlm) = Glm, G Gmn = δn (4) lm l M Z We will use G and Glm to lower or raise indices when needed and use Einstein summation lm convention over repeated indices. Any linear relation, with integer coefficients, of a and b that satisfy (3) is another valid l l basis for the homology generators. These transformations form a symmetry of the Chern- Simons theory and comprise the modular group, Sp(2g,Z): a a D C → S where S = (5) b b B A ! ! ! 0 1 with SES⊤ = E and E = . The g ×g matrices A,B,C,D have integer entries. −1 0 ! 3 We can define k 1 ξ = − ∗d∗A , F = ∗dA 2π2′ So we then have (including the A term) 0 2π 1 A = A dt− dξ +∗d( F)+2πi(γ¯ωl −γ ω¯l) (6) 0 k 2′ l l Similarlywecanwritethecurrentj = jµ ∂ intoaone-formj dxµ = j dt+˜j,usingthemetric ∂xµ µ 0 which we assume to be flat in the time direction (g = 1, g = g = 0 and the remaining 00 01 02 components forming the metric on M). We can use again the Hodge decomposition of ∗˜j on M ∗˜j = −dχ+∗dψ +i(j ω¯l −¯j ωl) (7) l l The continuity equation ∂j0d3x+d∗˜j ∧dt = 0 can be used to solve for ψ ∂t 1 ∂j 0 ψ = − 2′ ∂t We shall consider a set of point charges moving on M, with trajectories z (t) and charge i q , where z (t) 6= z (t) for i 6= j. For technical reasons (involving the solution of Gauss’ i i j law) we shall assume here that the total charge is zero, q = 0. We believe that this is i i only a technical restriction and can be generalized. We shall discuss this generalization, the P inclusion of the nonzero chrge sector, in a future publication.1 The current is represented by 1 j (z,t) = q δ(z −z (t)), ˜j(z,t) = q δ(z −z (t)) (z˙ (t)dz¯+z¯˙ (t)dz) (8) 0 i i i i i i 2 i i X X Integrating (8) with the harmonic forms ωl , we find the topological components of the current in (7) jl(t) = q z˙ (t)ωl(z (t)) , ¯jl(t) = q z¯˙ (t)ω¯l(z¯(t)) (9) i i i i i i i i X X To solve for χ, it is best to use complex notation R = ψ +iχ = R(z,z¯) where we find ∗dχ+dψ = ∂ R¯dz +∂ Rdz¯. From (7), (8) and using (9) we find z z¯ 1 ∂ R¯ +¯j ωl(z) = q z¯˙ δ(z −z (t)) z l i i i 2 i X 1Gauss’ Law (see ahead (17)) implies that the (single) additional degree of freedom associated with the nonzero charge should commute with those in the zero charge sector. This (and the additive structure of the Hamiltonian) would imply that the full wavefunction is a product of the zero chargewave function (see aheadequation(36))andanadditionalfactorforthe nonzerocharge,depending onlyonthe singlequantum 2πi variable ξ0 conjugate to F, and fixed by Gauss’ Law to be exp− k Qξ0. Thus our results will be largely anaffected by this inclusion of the zero charge sector. R 4 1 ∂ R+j ω¯l(z¯) = q z˙ δ(z −z (t)) (10) z¯ l i i i 2 i X To solve (10) for R, we will need the prime form E(z,w) = (h(z)h(w))−21 ·Θ 1/2 ( wω|Ω) 1/2 ! z Z 1/2 where h(z) = ∂ Θ (u|Ω)| ·ωl(z). The prime form is antisymetric in the variables ∂ul 1/2 u=0 ! z and w and behaves like z−w when z ≈ w (the h(z) which appear in the denominator are for normalization).2 The theta functions [7] are defined by α Θ (z|Ω) = eiπ(nl+αl)Ωlm(nm+αm)+2πi(nl+αl)(zl+βl) (11) β ! n Xl where α, β ∈ [0,1], have the following property α α Θ (zm +sm +Ωmlt |Ω) = e2πiαlsl−iπtmΩmltl−2πitm(zm+βm)Θ (z|Ω) β l β ! ! for integer–valued vectors sm and t . For a non-integer constant c l Θ α (zm +cΩmlt |Ω) = e−iπc2tmΩmltl−2πictm(zm+βm)Θ α−ct (z|Ω) β l β ! ! The solution of (10) is ∂ 1 E(z,z (t)) z R = [− q log( i )]−j (t) (ω¯l −ωl) (12) i l ∂t 2π E(z ,z (t)) i 0 i Zz0 X where we have chosen R such that R(z ,z¯ ) = 0 for an arbitrary point z (We can choose 0 0 0 z = ∞ for genus zero). The important fact about R is that it is a single-valued function. If 0 we move z around any of the homology cycles, R returns to its original value. In fact, this is also true for windings of z , an important relation since it is only a reference point. So 0 ∂ 1 E(z,z (t)) i z χ = [− q Imlog( i )]+ [(j (t)+¯j (t)) (ω¯l −ωl)] (13) i l l ∂t 2π E(z ,z (t)) 2 i 0 i Zz0 X Now we are ready to solve the Chern-Simons theory on M. By putting (6) and (7) back into (1) we find 1 k S = (ξPF˙ −ξ˙PF)d3x+iπk (γlγ¯˙ −γ˙lγ¯)dt+ A (j − F)d3x 2 l l 0 0 2π Z Z Z 2This formalism can also be extended to include the sphere, where there are no harmonic 1-forms at all (the space of cohomology generators has dimension zero) by properly defining the prime form. We use stereographicprojectionto mapthe sphereinto the complexplane anduse E(z,w)=z−w asthe definition of the prime form. 5 2π ∂j − ( ξ 0 +FPχ)d3x+2πi (j γ¯l −¯jlγ )dt (14) l l k ∂t Z Z where P is a projection operator onto the space orthogonal to the zero mode. From this we obtain the equal-time commutation relations of the quantum theory δ [ξ(z),F(w)] = −iPδ(z −w) or F(z) = iP (15) δξ(z) and 1 1 ∂ 1 ∂ [γ ,γ¯ ] = − G or γ¯ = G = (16) l m 2πk lm l 2πk lm∂γ 2πk∂γl m The projection operator in (15) changes the delta function to δ(z −w)−1/Area(M). The functional derivative must also be defined using this projection operator. With this holo- morphic polarization [3] it is convenient to use the following measure in γ space (Ψ |Ψ ) = e−2πkγmGmlγ¯lΨ∗(γ¯)Ψ (γ)|G|−1 dγmdγ¯m 1 2 1 2 Z m Y where |G| = det(G ). With this measure, we find that γ† = γ¯ as it should be. mn A is a Lagrange multiplier which enforces the Gauss’ law constraint 0 2π δ 2π F(z)− j (z) = iP − j (z) ≈ 0 (17) 0 0 k δξ(z) k Notethat since thetotalchargeis zero, the totalmagnetic fluxshould bezero, asis necessary on a compact space when the gauge field A is a globally defined 1-form. Under a modular transformation, the basis γl, γ¯l will be transformed accordingly. This will not change the choice of polarization, since the modular transformations do not mix γ and γ¯. From (14), (15) and (16), we find that the hamiltonian, in the A = 0 gauge, can be 0 separated into two commuting parts H = − A∧∗ ˜j = H +H 0 T M Z where 2π ∂j δ H = ( ξ 0 +iχP )d2x (18) 0 k ∂t δξ M Z (note that d2x = −idz ∧dz¯) while the additional part that takes care of the topology is 2 1 ∂ H = i(2π¯j γl − jl ) (19) T l k ∂γl To solve the Schr¨odinger equation, we will use the fact that the hamiltonian separates, thus writing the wave function as Ψ(ξ,γ,t) = Ψ (ξ,t)Ψ (γ,t) 0 T 6 with the Gauss law constraint (17) δ 2π (iP − j )Ψ (ξ,t) = 0 0 0 δξ k which is solved by 2πi Ψ (ξ,t) = exp[− ξ(z)j (z,t)d2x]Ψ (t) (20) 0 0 c k M Z The first Schr¨odinger equation is ∂Ψ (ξ,t) 2π ∂j δ i 0 = H Ψ (ξ,t) = ξ 0 +iχP d2x Ψ (ξ,t) 0 0 0 ∂t " M k ∂t δξ! # Z which has the solution [2] 2πi t Ψ (t) = exp − χ(z,t′)j (z,t′)d2xdt′ (21) c 0 k (cid:20) Z0 ZM (cid:21) For a system of point charges, the use of (13) allows us to write (20) and (21) as 2πi i t ˙ Ψ (ξ,t) = exp − q ξ(z (t))+ q q dtθ (t)+Φ(t) (22) 0 i i i j ij  k 2k  i ij Z0 X X   where π t t′ t t′ Φ(t) = j (t′)dt′ ¯jl(t′′)dt′′ − ¯j (t′)dt′ jl(t′′)dt′′ l l k "Z0 Z0 Z0 Z0 # π t t t t + ¯j (t′)dt′ ¯jl(t′)dt′ − j (t′)dt′ jl(t′)dt′ (23) l l 2k (cid:20)Z0 Z0 Z0 Z0 (cid:21) and E(z (t),z (t)) i j θ (t) = Imlog ij "E(zi(t),z0)E(z0,zj(t))# zi(0) zj(t) zj(0) zi(t) +Im ωl (ω +ω¯ )+ ωl (ω +ω¯ ) (24) l l l l "Zz0 Zzj(0) Zz0 Zzi(0) # is a multi-valued function defined using the prime form. We will need the phase (23) for the topological part of the wave function. The function θ (t) is the angle function for particle i ij and j. For i = j, we choose a framing z (t) = z (t)+ǫf (t) which lead to the replacement of i j i E(z (t),z (t)) by f (t). So the wave function (22), with the angle function (24), accurately i i i forms an Abelian representation of the braid group [2, 8]. Now, the topological part of the hamiltonian is used to find the part of the wave function affected by the currents going around the non-trivial loops of M. The Schr¨odinger equation for (19) is ∂Ψ (γ,t) 1 ∂ i T = H Ψ (γ,t) = i 2π¯j γl − jl Ψ (γ,t) ∂t T T l k ∂γl! T which has the solution t 2π t t′ Ψ (γ,t) = exp 2πγl ¯j (t′)dt′ − j (t′)dt′ ¯jl(t′′)dt′′ Ψ˜ (γ,t) (25) T l l T " Z0 k Z0 Z0 # 7 Note that with the phase (23), the double integral above will turn into tj (t′)dt′· t¯jl(t′)dt′, 0 l 0 a topological expression. R R The remaining equation for Ψ˜ (γ,t) T ˜ ˜ ∂Ψ (γ,t) 1 ∂Ψ(γ,t) T = − jl (26) ∂t k ∂γl is easily solved in the form 1 t Ψ˜ (γl,t) = Ψ˜ (γl − jl(t′)dt′) (27) T T k Z0 The wave function (27) is not arbitrary, but must satisfy the invariance of the action (1) under large gauge transformations, when there is no current around. So let us set jµ = 0 for a while and find the condition on Ψ˜ . T In general, the large U(1) gauge transformations are given by the set of single-valued gauge functions, with sm and t integer-valued vectors, m U (z) = exp(2πi(t ηm(z)−smη˜ (z)) s,t m m where z z ηm(z) = i (Ω¯mlω −Ωmlω¯ ) , η˜ (z) = −i (ω −ω¯ ) l l m m m Zz0 Zz0 If we change the endpoint of integration by z → z+a ul+bmv with u,v integer and a,b l m defined in (3), we find ηm → ηm+um, η˜ → η˜ +v and U → U e2πi(tmum−smvm) = U . m m m s,t s,t s,t The transformation of the gauge field (2) under U is given by s,t γm → γm +sm +Ωmlt , γ¯m → γ¯m +sm +Ω¯mlt (28) l l The classical operator that produces the transformation (28) ∂ ∂ c (γ,γ¯) = exp (sm +Ωmlt ) +(sm +Ω¯mlt ) s,t " l ∂γm l ∂γ¯m# must be transformed into the proper quantum operator acting on the wave function Ψ˜ . T By using the commutation (16) to replace ∂ by −2πkγ we find the operators C which ∂γ¯m m s,t implement the large gauge transformations [5] Cs,t(γ) = exp −2πk(sm +Ω¯mltl)γm −πk(sm +Ω¯mltl)Gmn(sn +Ωnltl) e(sm+Ωmltl)∂γ∂m (29) h i The quantum operators C do not commute among themselves for non-integer k. From s,t now on we will set k = k1 for integer k and k . Now, in contrast with their classical k2 1 2 counterparts, the operators C satisfy the clock algebra s,t C C = e−2πik(sm1 tm2−sm2 tm1)C C (30) s1,t1 s2,t2 s2,t2 s1,t1 Their action on the wave function is C (γ)Ψ˜ (γm) = exp −2πk(sm +Ω¯mlt )γ s,t T l m h 8 −πk(sm +Ω¯mlt )G (sn +Ωnlt ) Ψ˜ (γm +sm +Ωmlt ) (31) l mn l T l i OntheotherhandC commutes witheverything andmust berepresented onlybyphases k2s,k2t φ . This implies, using (31), s,t Ψ˜ (γm +k (sm +Ωmlt )) = exp −iφ +2πk (sm +Ω¯mlt )γ T 2 l s,t 1 l m h +πk k (sm +Ω¯mlt )G (sn +Ωnlt ) Ψ˜ (γm) (32) 1 2 l mn l T i The only functions that are doubly (semi-)periodic are combinations of the theta functions (11). After some algebra, we find that the set of functions α α+k1p+k2r Ψ (γ|Ω) = eπkγmγmΘ k1k2 (k γ|k k Ω) (33) p,r β β 1 1 2 ! ! where p = 1,2,...,k and r = 1,2,...,k with α, β ∈ [0,1] solve the above algebraic 2 1 conditions (32). Their inner product is given by (Ψ |Ψ ) = e−2πkγmGmlγ¯lΨ (γ)Ψ (γ)|G|−1 dγmdγ¯m (34) p1,r1 p2,r2 p1,r1 p2,r2 ZP m Y = |G|−12δ δ p1,p2 r1,r2 The integrand is completely invariant under the translation (28), thus we restrict the inte- gration to one of the plaquettes P (γm = um +Ωmlv with u,v ∈ [0,1]), the phase space of l the γ’s. Under a large gauge transformation Cs,tΨp,r αβ (γ) = e2πikpmsm+iπksmtm+2kπ2i(αmsm−βmtm)Ψp+t,r αβ (γ) ! ! α = [Cs,t]p,p′Ψp′,r β (γ) (35) p′ ! X The matrix [Cs,t]p,p′ forms a (k2)g dimensional representation of the algebra (30) of large gauge transformations. The parameters α and β appear as free parameters, but in fact they may be fixed such that we obtain a modular invariant wave function. The modular transformation (5) on our set of functions (33) is Ψ α (γ|Ω) → |CΩ+D|−21e−iπφΨ α′ (γ′|Ω′) p,r β p,r β′ ! ! where γ′ = (CΩ+D)−1⊤γ, Ω′ = (AΩ + B)(CΩ + D)−1 and φ is a phase that will not concern us here (and G′ = [(CΩ + D)−1] G [(CΩ¯ + D)−1] ). Most important are the lm lr rs sm new variables k k k k α′ = Dα−Cβ − 1 2(CD⊤) β′ = −Bα+Aβ − 1 2(AB⊤) d d 2 2 9 where (M) mean [M] , the diagonal elements. d dd A set of modular invariant wave functions [4, 5, 6] can exist only when k k is even, 1 2 where we set α = β = 0 (and also φ = 0). In the case of odd k k , we can set α, β to either 1 2 0 or 1, which amount to the addition of a spin structure on the wave functions. This will 2 increase the number of functions by 4g which will now transform non trivially under modular transformations. Considering a set of point charges leads to the set of wave functions α t 2πi Ψ (ξ,γ,t|Ω) = exp πkγmγ +2πγm (¯j −j )dt′ − q ξ(z (t)) p,r β ! " m Z0 m m k i i i X i π t t + q q (θ (t)−θ (0))+ (j −¯j )dt′ · (jm −¯jm)dt′ i j ij ij m m 2k 2k  ij Z0 Z0 X  α+k1p+k2r t ·Θ k1k2 (k γm −k jmdt′|k k Ω) (36) 1 2 1 2 β ! Z0 The wave function depends on charge positions through the integrals over the topological components of the current jm,¯jm, and through the function θ (t) − θ (0). Consider for a ij ij moment motions of the particles which are closed curves, and are homologically trivial. We focus first on the integrals over jm,¯jm. If, for example a single particle moves in a circle, we find that the integral of these topological currents vanishes, we conclude that these currents contribute nothing additional to the phase of the wave function under these kinds of motions. The function θ (t) − θ (0) must be treated differently here, because it has ij ij singularities when particles coincide, and thus, while motions that encircle no other particles may be easily integrated to get zero, this is not true when other particles are enclosed by one of the particle paths, and the result is nonzero in this case, in fact it is 2π. Nevertheless, this function is still independent of the particular shape of the particle path. Actually the definition of θ in term of the prime form E(z,w) is just the generalization to an arbitrary ij Riemann surface of the well known angle function on the plane, that is as the angle of the line joining the particle i and j compare to a fixed axis of reference, determined by z here. 0 Thus, we may conclude that, under permutations of particles of charge q, the wave functions defined here acquire the phase σ = eiπq2. k For homologically nontrivial motions of a single particle on M , the current integral tjl(t′)dt′ will in general change as tjl(t′)dt′ −→ tjl(t′)dt′ + sl + Ωlmt , where sl and 0 0 0 m t are integer-valued vectors whose entries denote the number of windings of the particle Rm R R around each homological cycle. However, now, for multi-particle non-braiding paths, there is no contribution comming from θ . Thus, the wave functions become ij 2πi 2πi i Ψ (t) = exp − r sm − ((α−k α ) sm −(β −k β )mt )− J p,r m 2 0 m 2 0 m k k 2k (cid:20) 1 (cid:21) ·Ψp,r+t(0) = [Bs,t]r,r′Ψp,r′(0) (37) r′ X with q zi(0)ωl = αl +Ωlmβ and where J = q2(f (t)−f (0)) is a self-linking term. i i z0 0 0m i i i i P R P 10