Chern-Simons, WZW models, Twistors, and. Multigluon Scattering Amplitudes

Transcription

1 Chern-Simons, WZW models, Twistors, and Multigluon Scattering Amplitudes V. P. NAIR Physics Department City College of the CUNY New York, NY Notes for lectures at BUSSTEPP 2005 How to read these notes These notes are meant to give you a flavor of the material I shall discuss and are, therefore, very brief. What will be covered in my lectures is contained in sections which have been marked with an asterisk. There are a number of other sections which have been added for a better sense of completion. These contain material which may be useful background or needed results or they provide a brief opening to interesting literature for those of you who might want to pursue related issues.

2 Contents 1 Resumé of canonical quantization *The canonical structure *Rules of quantization The Chern-Simons theory in 2+1 dimensions *Hamiltonian quantization of the CS theory *Quantization of k *The Wess-Zumino-Witten (WZW) action 14 4 *The winding number: a brief aside 18 5 Some uses of CS theory The CS-WZW connection CS theory and knots Integrability conditions Some uses of WZW theory WZW theory as a conformal theory Nonabelian bosonization *The Dirac determinant in two dimensions The gauge-invariant measure in two dimensions *Current correlators of WZW on CP *Twistors, supertwistors The basic idea of twistors An explicit example Conformal transformations Supertwistors Lines in twistor space *Yang-Mills amplitudes and twistors Why twistors are useful The MHV amplitudes Generalization to other helicities *Twistor string theory 53 2

3 10 *Landau levels and Yang-Mills amplitudes The general formula for amplitudes A field theory on CP

4 1 Resumé of canonical quantization 1.1 *The canonical structure We begin by recalling some elements of canonical quantization. Even though this can be done generally for any dynamical system, we shall consider fields as the fundamental dynamical variables and discuss how to obtain a quantum theory of fields. We will consider bosonic fields for the most part. The fields will be denoted by ϕ r (x), r = 1, 2,..., N. The index r or part of it may be a spacetime index for vector and tensor fields; it can also be an internal index labeling the number of independent fields. The Lagrangian L is a scalar function of ϕ r (x) and its spacetime derivatives. The action in a spacetime volume Σ can be written as S = d 4 x L(ϕ r, µ ϕ r ) (1) Σ The spacetime region will be taken to be of the form V [t f, t i ], where V is a spatial region. The equations of motion are given by the variational principle, which says that the classical trajectory ϕ r ( x, t), which connects specified initial and final field configurations ϕ r ( x, t i ) and ϕ r ( x, t f ) at times t i and t f, is an extremum of the action. In other words, we can vary the action with respect to ϕ( x, t) for t i < t < t f and set δs to zero to obtain the equations of motion. Explicitly δl = L δϕ r + ϕ r [ L = ϕ r x µ L ( µ ϕ r ) µδϕ r L ( µ ϕ r ) ] δϕ r + x µ ( L ) ( µ ϕ r ) δϕ r (2) When we integrate the variation of L over the spacetime region Σ to obtain δs, the second term in (2), being a total divergence, becomes a surface integral over Σ. Since we fix the initial and final field configurations ϕ r ( x, t i ), ϕ r ( x, t f ), δϕ r = 0 at t i, t f. Further, we assume that either δϕ r L ( i ϕ r) or vanishes at the spatial boundary V. Generally, we are interested in the limit of large spatial volumes; this condition is physically quite reasonable in this case; alternatively, we could require periodic boundary conditions for the spatial directions. Either way the surface integral is zero and δs = Σ d 4 x [ L ϕ r 4 x µ L ( µ ϕ r ) ] δϕ r (3)

5 The extremization condition δs = 0 now identifies the equations of motion, since δϕ r is arbitrary, as L ϕ r x µ L ( µ ϕ r ) = 0 (4) For the purpose of quantization, we need to consider more general variations of fields, with δϕ r not zero at t i, t f. In this case, the total divergence term in (2) integrates out as Θ(t f ) Θ(t i ), where Θ(t) = V d 3 x L ( 0 ϕ r ) δϕ r (5) This quantity Θ is called the canonical one-form. In the variation of the action when using the variational principle, we specify the initial and final values of the field configurations. Since there is then a unique classical trajectory, we may say that the initial and final values label the classical trajectories. The set of all classical trajectories is defined to be the phase space of the theory. For theories for the equations of motion are of the second order in time-derivatives, we can specify the classical trajectories by the initial data for the equations of motion rather than initial and final values for the field. Since our equations are second order in time-derivatives, the initial data are clearly ϕ r ( x, t) and 0 ϕ r ( x, t), at some starting time t. It will be more convenient for the formalism to use π r ( x, t) = L ( 0 ϕ r ) rather than 0 ϕ r. The phase space for a set of scalar fields is thus equivalent to the set {π r ( x), ϕ r ( x)} (for all x) which is used to label the classical trajectories. The phase space for a field theory is obviously infinite-dimensional. π r is called the canonical momentum conjugate to ϕ r. The canonical one-form Θ can be written as Θ = d 3 x π r δϕ r (7) V (The name is due to the fact that this is a differential one-form on the phase space.) We will denote the phase space variables (coordinates on the phase space) by ξ i ( x) for a general dynamical system. The canonical one-form Θ is identified from the surface term in the variation of the action and has the general form Θ = d 3 x A i (ξ, x) δξ i ( x) (8) (6) 5

6 where A i could depend on ξ. (For the scalar field ξ i = (π r, ϕ r ) and A i = (π r, 0).) Given Θ, we define Ω ij ( x, x δ ) = δξ i ( x) A j( x δ ) δξ j ( x ) A i( x) = I A J J A I (9) where in the last line, we have introduced the composite indices I = (i, x) and J = (j, x ) and I = δ/δξ i ( x) to avoid clutter in the notation. Ω is called the symplectic structure or the canonical two-form. (It can be considered as a differential form on the phase space.) Just as the metric tensor defines the basic geometric structure for any spacetime, Ω defines the basic geometric structure of the phase space. The inverse of Ω is defined by (Ω 1 ) IJ Ω JK = δk I which expands out as d 3 x (Ω 1 ) ij ( x, x )Ω jk ( x, x ) = δk i δ(3) (x x ) (10) V As will be clear from the following discussion, it is important to have an invertible Ω IJ. If Ω is not invertible, the Lagrangian is said to be singular. There are many interesting cases, e.g., theories with gauge symmetries, where it is difficult to define an invertible Ω in terms of the obvious field variables. One has to define a nonsingular Ω in such cases, by suitable elimination of redundant degrees of freedom. If F and G are two functions on the phase space, their Poisson bracket is another function on the phase space defined by {F, G} = (Ω 1 ) IJ I F J G = d 3 x d 3 x (Ω 1 ) ij ( x, x δf δg ) δξ i ( x) δξ j ( x ) (11) As willb clear from one of the poblems, the notion of the Poisson bracket arises naturally in the composition of canonical transformations; this is, in fact, why they are important in the canonical framewok. For the ξ I s themselves, we find {ξ I, ξ J } = (Ω 1 ) IJ or {ξ i ( x), ξ j ( x )} = (Ω 1 ) ij ( x, x ) (12) Notice also that if Θ has the simple form (7), the Poisson bracket of two functions F (π, ϕ), G(π, ϕ) of the phase space variables becomes [ δf {F, G} = d 3 δg x δf ] δg (13) δϕ r δπ r δπ r δϕ r V 6

7 A function G on the phase space may be thought of as generating an infinitesimal canonical transformation via the rule δf = {F, G} = (Ω 1 ) IJ J G I F (14) The fact that this transformation is canonical is shown later. G is called the generator of the transformation. Notice that for the simple case of ξ i = (π r, ϕ r ), this is equivalent to δϕ r ( x) = δg δπ r ( x), δπ r( x) = δg δϕ r ( x) (15) These equations show why Poisson brackets are important. The change of any variable, so long as it is canonical, is given by the Poisson bracket of the variable with the generating function for the transformation. We now find the generators of some important canonical transformations for the simple case of a set of scalar fields whose time-evolution is of the second order in time-derivatives. Generators of various transformations Transformation Change of ϕ r ( x) ϕ r ϕ r + a r ( x) π r π r Change of π r ( x) ϕ r ϕ r π r π r + a r ( x) Space translations x i x i + a i a i are constants δϕ r = a i i ϕ r δπ r = a i i π r Generator G = V d3 x a r ( x)π r ( x) G = V d3 x a r ( x)ϕ r ( x) G = V d3 x a i i ϕ r π r = a i P i P i = V d3 x i ϕ r π r P i is the momentum. Time translations H = V d3 x (π r 0 ϕ r L) Lorentz transformations δx µ = ω µν x ν M µν = V d3 x (x µ T ν0 x ν T µ0 ) 7

8 The fact that the generator of time translations is the Hamiltonian H(π, ϕ) can be checked from the equations of motion 0 ϕ r = δh δπ r, 0 π r = δh δϕ r (16) The easiest way to check this is to use to write the action as S = d 4 x π r 0 ϕ r dt H (17) and then use the variational principle to write the equations of motion. The equations of motion so obtained are seen to be (16), showing the consistency of our definition of the generator of time translations. T µν, which occurs in definition of the Lorentz generator, is the energy-momentum tensor. Problem 1. Since δξ I = (Ω 1 ) IJ J F for a canonical transfomation generated by F, we can write V F = (Ω 1 ) IJ J F δ δξ I (18) as an operator generating canonical transformations. Show that the commutator [V F, V G ] = V W, where W = {F, G}. 1.2 *Rules of quantization As with any quantum mechanical system, the states are represented by vectors (actually rays) in a Hilbert space H. The scalar product ϕ α = Ψ α [ϕ] is the wave function of the state α in a ϕ-diagonal representation; it is the probability amplitude for finding the field configuration ϕ( x) in the state α. Observables are represented by linear hermitian operators on H. Fields are in general linear operators on H, not necessarily always hermitian or observable. We have the operator φ r ( x, t) corresponding to ϕ r ( x, t) and the operator π r ( x, t) corresponding to the canonical momentum. The change of any operator F under any infinitesimal unitary transformation of the Hilbert space is given by i δf = F G GF = [F, G] (19) where G is the generator of the transformation; it is a hermitian operator. If we were to start directly with the quantum theory, we can regard this as 8

9 the basic postulate. The fact that observables are linear hermitian operators follow from this because observations or measurements correspond to infinitesimal unitary transformations of the Hilbert space. However, in starting from a classical theory and quantizing it, we need a rule relating the operator structure to the classical phase space structure. The basic rule is that, in passing to the quantum theory, canonical transformations should be represented as unitary transformations on the Hilbert space. The generator of the unitary transformation is obtained by replacing the fields in the classical canonical generator by the corresponding operators. (This replacement rule has ambiguities of ordering of operators; e.g., classically, π r ϕ r and ϕ r π r are the same, but the corresponding quantum versions π r φ r and φ r π r are not the same, since φ r and π r do not necessarily commute. The correct ordering for the quantum theory can sometimes be understood on grounds of desirable symmetries. There is no general rule.) Comparing the rule (14) for the change of a function under a canonical transformation with the rule (19) for the change of an operator under a unitary transformation, we see that i[f, G] should behave as the Poisson bracket {F, G} in going to the classical limit. Therefore the commutator algebra of the operators, apart from ordering problems mentioned above, will be isomorphic to the Poisson bracket algebra of the corresponding classical functions. The finite version of (19) is The transformation law for states is given by F = e ig F e ig (20) α = e ig α (21) Equations (20,21) say that classical canonical transformations are realized as unitary transformations in the quantum theory. Many useful results follow from (19) to (21). From the generators of changes in ϕ r and π r, given in the table, we find, using (19), [ φr ( x, t), φ s ( x, t) ] = 0 [ πr ( x, t), π s ( x, t) ] = 0 [ φr ( x, t), π s ( x, t) ] = i δ rs δ (3) (x x ) (22) These give us the basic commutation rules to be imposed on the field operators of the theory; we have used scalar fields to illustrate the calculation. (More generally, we would have [ξ i ( x, t), ξ j ( x, t)] = i(ω 1 ) ij ( x, x ).) 9

10 The generator of time-translations is the Hamiltonian and we get from (19) i F = [F, H] (23) t This is the quantum equation of motion, or the Heisenberg equation of motion. Using the canonical commutation rules, one can also work out the commutator algebra of various operators of interest. For example, using expressions of the generators given in the table and replacing the fields and their canonical momenta by operators, we get the operators P µ, M µν, which give the quantum action of the Poincaré transformations on any quantity as in (19). In particular, using the canonical commutation rules, one can check that these operators obey the Poincaré algebra commutation relations. Problem 2. Our general formulation of quantization in terms of Ω can be applied in many unusual situations. A simple example is the light-cone quantization of a scalar field. We introduce light-cone coordinates, corresponding to a light-cone in the z-direction as u = 1 (t + z) 2 v = 1 2 (t z) (24) Instead of considering evolution of the fields in time t, we can consider evolution in one of the the light-cone coordinates, say, u. The analog of space is given by the other light-cone coordinate v and the two coordinates x T = x, y transverse to the light-cone. They correspond to equal-u hypersurfaces. A scalar field can be written as ϕ(u, v, x, y) and the action is S = du dv d 2 x T [ u ϕ v ϕ 1 2 ( T ϕ) 2 U(ϕ) ] (25) This is first order in the u-derivatives, the analog of the time-derivatives. The phase space is thus given by field configurations ϕ(v, x T ). Show that the Poisson brackets are given by {ϕ(u, v, x T ), ϕ(u, v, x T )} = 1 4 ɛ(v v ) δ (2) (x T x T ) (26) and that the Hamiltonian for u-evolution is given by [ ] 1 H = dv d 2 x T 2 ( T ϕ) 2 + U(ϕ) (27) 10

11 Here ɛ(v v ) is the signature function, equal to 1 for v v > 0 and equal to 1 for v v < 0. 2 The Chern-Simons theory in 2+1 dimensions 2.1 *Hamiltonian quantization of the CS theory The Chern-Simons (CS) theory is a gauge theory in two space (and one time) dimensions [1]. The action is given by S = k d 3 x ɛ µνα Tr [A µ ν A α + 23 ] 4π A µa ν A α (28) Σ [t i,t f ] Here A µ is the Lie-algebra-valued gauge potential, A µ = it a A a µ. t a are hermitian matrices forming a basis of the Lie algebra in the fundamental representation of the gauge group. We shall consider the gauge group to be SU(N) in what follows, and normalize the t a as Tr(t a t b ) = 1 2 δab. k is a constant whose precise value we do not need to specify at this stage. The spatial manifold can be taken to be some Riemann surface Σ; we will concentrate on the simplest case, the sphere S 2, and we shall be using complex coordinates. The equations of motion for the theory are F µν = 0 (29) The theory is best analyzed, for our purposes, in the gauge where A 0 is set to zero. In this gauge, the equations of motion (29) tell us that A z = 1 2 (A 1 + ia 2 ) and A z = 1 2 (A 1 ia 2 ) are independent of time, but must satisfy the constraint F zz z A z z A z + [A z, A z ] = 0 (30) This constraint is just the Gauss law of the CS gauge theory. (From now on, we shall use A, Ā for A z, A z, for simplicity.) In the A 0 = 0 gauge, the action becomes S = ik 2π dtdµ Σ Tr(Ā 0A A 0 Ā) (31) From the boundary term in the variation of the action, we can identify the canonical one-form as Θ = ik Tr ( ĀδA 2π AδĀ) + δρ[a] (32) Σ 11

12 where ρ[a] is an arbitrary functional of A. The freedom of adding δρ is the freedom of canonical transformations. Θ is defined on A, the space of gauge potentials on Σ. A is the phase space of the theory before reduction by the action of gauge symmetries. The canonical two-form Ω is given by Ω ab ĀA (x, x ) = Ω ab AĀ(x, x ) = ik 2π δab δ (2) (x x ) (33) The fundamental Poisson brackets for Ā, A are obtained by inverting the components of Ω and lead to the basic commutation rules [A a (z), A b (w)] = 0 [Āa (z), Āb (w)] = 0 [A a (z), Āb (w)] = 2π k δab δ (2) (z w) (34) Gauge transformations are given by A g = gag 1 dgg 1 (Dθ) (35) where the second line is for infinitesimal transformations, g 1 + θ. The commutation rules show that the generator of infinitesimal gauge transformations is G a = ik 2π F z z a (36) Classically the reduction of the phase space can be performed by setting F to zero. This is also the equation of motion we found for the component A 0. On the sphere, there will be no degrees of freedom left; on the torus, we will have the single mode due to the zero mode of z, namely, a of equation (90). In the following, we will actually be interested in the quantization of the theory in terms of A, Ā, and then imposing the Gauss law on the states as a condition selecting physical states. The wave functions should depend only on half of the phase space coordinates. Given the complex components, a good choice would be to take the wave functions ψ to depend only on Ā; this would be analogous to the coherent state description of the harmonic oscillator. The commutation rules show that, on such holomorphic wave functions, we can represent A by A a ψ[ā] = 2π k 12 δ ψ[ā] (37) δāa

13 The inner product for such wave functions can be seen to be 1 2 = [dāa, da a ] e K(Ā,Aa) ψ1 ψ 2 (38) where K = k 2π Ā a A a (39) (K is also the Kähler potential on A). The necessity for a factor e K can be seen from the requirement that A and Ā should be adjoints of each other. Equations (37, 38) are in complete analogy with the coherent states for the harmonic oscillator. We have obtained the quantization in terms of the A s; now we have to make a reduction of the Hilbert space by imposing gauge invariance on the states, i.e., by setting the generator Fz z a to zero on the wave functionals. This amounts to ( δ D z k ) δāa 2π zāa ψ[ā] = 0. (40) The solution of this equation can be constructed using the WZW action. Making an ansatz of the form ψ = exp(kw ), we find that this equation becomes D z A z Ā = 0 A a = δw δāa (41) Comparing this with the property (54) (or its conjugate), we find that W = S W ZW (M ). Notice that, although the wave function ψ[ā] is not gauge-invariant by itself, the normalization integral in (38) involves the gauge-invariant quantity exp[ks(m M)] = exp[ks(h)]. For physical wave functions obeying the Gauss law (40), we must restrict the measure of integration in (38) to the gauge-invariant measure dµ from (114). Problem 3. Check the inner product (38) by verifying that A, Ā are indeed adjoints with this inner product. 2.2 *Quantization of k The coefficient k which appears in the Chern-Simons action (28) is called the level number of the CS theory. As in the case of the WZW action, this 13

14 number must be quantized as an integer. There are many ways to see this. Consider the CS action on a three-dimensional space M 3. Under a gauge transformation A A g = gag 1 dgg 1, we get S(A g ) = S(A) k 4π M 3 ɛ µν Tr(g 1 µ ga ν ) + 2πk Q[g] (42) where Q[g] is the winding number for g, which is an integer, as discussed before. In a functional integral analysis, we can impose that the gauge transformations go to the identity at the boundary of M 3, so that the surface term can be taken to be zero. There can, of course, be transformations for which Q[g] is not zero. We see that e is will be gauge-invariant for all transformations, including those for which Q[g] is not zero, if k is an integer. Problem 4 (Challenge problem). Construct a purely Hamiltonian argument for the quantization of k along the following lines. Consider a twoparameter set of configurations, labeled by parameters σ, λ, A µ = σ µ gg 1 (43) where 0 σ, λ 1 and g(x 1, x 2, λ) is a group-valued function with Q[g] = 1. We can think of σ, λ as two coordinates in the space of potentials A. We have g(x 1, x 2, 0) = g(x 1, x 2, 1) = 1, so that A µ = 0 at λ = 0, 1 and at σ = 0. It is a pure gauge at σ = 1. Thus (43) defines a closed two-surface in the gauge-invariant configuration space. Ω on this surface is like the field of a magnetic monopole. Using this and the Dirac quantization condition for monopoles, show that k must be an integer. 3 *The Wess-Zumino-Witten (WZW) action The Wess-Zumino-Witten (WZW) action gives a class of important field theories in two dimensions. It is intimately related to anomalies in two dimensions and has many applications. It is used for nonabelian bosonization. It defines a conformal field theory in two dimensions; various rational conformal field theories can be obtained as either a WZW model or gauged versions of it [3]. The field variables of the WZW action are invertible matrices M(x), i.e., elements of GL(N, C) or suitable subgroups and cosets of it, so it may be regarded as a particular type of sigma model. We will denote this target space by G. We shall discuss the action in two-dimensional space with 14

15 Euclidean signature; the Minkowski version is briefly discussed later. The action for the WZW theory is given by S W ZW = 1 d 2 x g g ab Tr( a M b M 1 ) + Γ[M] (44) 8π M 2 Γ[M] = i d 3 x ɛ µνα Tr(M 1 µ MM 1 ν MM 1 α M) (45) 12π M 3 Here M 2 denotes the two-dimensional space on which the fields are defined. It can in general be a curved manifold with a metric tensor g ab. (g ab is the inverse metric and g denotes the determinant of g ab as a matrix.) M 2 will be taken as a closed manifold. The model can be defined and used for fields on R 2 as well, by choosing the boundary condition M 1 (or some fixed value independent of directions) as x ; topologically, such fields are equivalent to fields on the closed manifold S 2. The second term in the action, Γ[M], is known as the Wess-Zumino term. It is defined by integration over a three-dimensional space M 3 which has M 2 as its boundary. The integrand does not require metrical factors for the integration; it may be considered as a differential three-form. However, it requires an extension of the fields to the three-space M 3. There can be many spaces M 3 with the same boundary M 2, or equivalently, there can be many different ways to extend the fields to the three-space M 3. The physical results of the theory are independent of how this extension is chosen. To see how this works out, notice that if M and M are two different extensions of the same field into the three-space, we write M = MN, where N = 1 on M 2, the boundary of M 3. By direct computation, we observe that Γ[MN] = Γ[M] + Γ[N] i 4π M 2 d 2 x ɛ ab Tr(M 1 a M b NN 1 ) (46) The last term vanishes for N = 1 on M 2 = M 3. Since N = 1 on the boundary of M 3, N is equivalent to a map from a closed three-space to G. In general, there are homotopically distinct classes of such maps. For example, if we take M 2 = S 2 (or R 2 with the boundary condition indicated), M 3 is a ball in three dimensions. With the prescribed behavior for N, it is equivalent to a map from the three-sphere S 3 to G. The homotopy classes of such maps are given by Π 3 [G]. (The set of all continuous maps S 3 G falls into distinct disjoint classes with all maps within a class being continuously deformable to each other. These are the homotopy classes.) If G contains any compact nonabelian Lie group, Π 3 [G] is nonzero. In particular, Π 3 [G] = Z for all simple nonabelian Lie groups, 15

16 except for SO(4), in which case it is Z Z. The winding number of the map N(x) : S 3 G is given by Q[N] = 1 24π 2 d 3 x ɛ µνα Tr(N 1 µ NN 1 ν NN 1 α N) (47) S 3 Q[N] is an integer for any N(x). (Some of the salient properties of Q are discussed below.) Thus, for Γ[N], we have two cases to discuss. Γ[N] is zero for N close to identity, to linear order in NN 1 ; hence, by successive transformations, Γ[M] is independent of the extension to M 3 for all N connected to identity, i.e., for N belonging to a homotopically trivial element. On the other hand, if N is homotopically nontrivial, the integral Γ[N] gives 2πi times the winding number of the map N(x) : S 3 G. Since Q[N] is an integer, exp( k Γ[M]) is independent of how the extension into the three-space is made, if k is an integer. Thus, by using the action S = k S W ZW (48) where k is an integer, we can construct a field theory on the two-space M 2. Since the theory can be defined by using exp( S) = exp( ks W ZW ) to construct the functional integral, this will be well-defined, not requiring more than the specification of field configurations on M 2 itself. The action (48) defines the WZW theory; k is referred to as the level number of this theory. (Even though we presented the arguments for quantization of the coefficient of the action for M 2 = S 2, similar arguments and results hold more generally.) For the simplest case of M 2 = R 2 with the appropriate boundary conditions on the field M(x), we can write the WZW action using complex coordinates as S W ZW = 1 Tr( z M z M 1 ) + Γ[M] (49) 2π M 2 For the first term in this expression, we have used complex coordinates z = x 1 ix 2, z = x 1 + ix 2. This action obeys a very useful identity known as the Polyakov-Wiegmann identity. Using (46), one can easily verify that S W ZW [M h] = S W ZW [M] + S W ZW [h] 1 Tr(M 1 z M z h h 1 ) π M 2 (50) Notice the chiral splitting; we have only the antiholomorphic derivative of M and the holomorphic derivative of h. This shows that the equations of motion are given by z (M 1 z M) = M 1 z ( z M M 1 )M = 0 (51) 16

17 The WZW action has invariance under infinitesimal left translations of M by a holomorphic function, M (1 + θ(z))m, and right translations of M by an antiholomorphic function, M M(1 + χ( z)). This is easily checked using (50). These transformations have the associated currents J z = k π zm M 1 J z = k π M 1 z M (52) (These currents are for a level k WZW model.) By the equations of motion (51), these currents obey z J z = 0, z J z = 0 (53) The Polyakov-Wiegmann property also gives another result for the WZW action which is very useful. Consider a small variation of the field M given by M + δm = (1 + θ)m, where θ = δm M 1 is infinitesimal. Using the Polyakov-Wiegmann property, we then get δs W ZW = 1 Tr ( z (δmm 1 ) z MM 1) π = 1 Tr(δMM 1 z A z ) = 1 Tr(δMM 1 D z Ā) π π = 1 π Tr(ĀδA z) (54) where A z = z MM 1 and D z is the covariant derivative in the adjoint representation, D z Ā = z Ā + [A z, Ā]. Ā is defined by Notice that this obeys the equation Ā = z M M 1 (55) z A z z Ā + [Ā, A z] = 0 (56) Problem 5. Prove the Polyakov-Wiegmann identity (50) by direct computation. 17

18 4 *The winding number: a brief aside We shall now briefly discuss some properties of the winding number. Consider an arbitrary element of SU(2). The group SU(2) can be thought of as the group of (2 2) unitary matrices of unit determinant. We can parametrize an element of SU(2) as g = φ 0 + iφ i σ i (57) where σ i are the Pauli matrices and the condition g = g 1 implies that φ 0, φ i are real. The condition det g = 1 requires φ i φ 2 i = 1 (58) The group SU(2) is thus topologically a three-sphere. The φ µ ( x), µ = 0, 1, 2, 3, thus give a mapping R 3 S 3. Further we have g 1 as x. For the case of such maps, we can think of space R 3 itself as being a threesphere. Explicitly this can be realized as follows. We define a mapping y 0 = x2 1 x 2 + 1, y i = 2x i x (59) Evidently y0 2 + i y2 i = 1; the y s define a three-sphere. (59) thus gives a description of R 3 as a three-sphere, spatial infinity, which corresponds to x, being mapped to the pole y 0 = 1, y i = 0. Now, given an SU(2)- valued function on S 3, i.e., given g(y) where µ y2 µ = 1, we can use (59) to write it as a function on R 3 with g g, g not dependent on angles as x. In particular, choosing g = 1 requires g(1, 0, 0, 0) = 1. We are thus concerned with maps S 3 S 3, where the first S 3 represents space R 3 via the map (59) and the second S 3 is SU(2). The classes of such maps will be given by Π 3 [S 3 ] = Z. We can analyze these maps by defining a winding number Q. Consider φ µ : S 3 S 3. We want to determine how many times the target sphere S 3 is covered by the map φ µ (x) as we cover the spatial S 3 once. Since the volume of S 3 is 2π 2, we get for the winding number Q[g] = 1 2π 2 {volume traced out by φµ (x)} 1 = 12π 2 d 3 x ɛ µναβ ɛ ijk φ µ i φ ν j φ α k φ β (60) 18

19 This can also be written directly in terms of g(x) as Q[g] = 1 24π 2 d 3 x Tr ( g 1 i g g 1 j g g 1 k g ) ɛ ijk = 1 24π 2 Tr(g 1 dg) 3 (61) Notice that there is again no need for a metric tensor when we extend this expression to a curved space. The winding number is independent of the metric of the spatial manifold. One of the key properties of Q[g] is To see how this arises, notice that Q[g 1 g 2 ] = Q[g 1 ] + Q[g 2 ] (62) (g 1 g 2 ) 1 d(g 1 g 2 ) = g 1 2 (g 1 1 dg 1)g 2 + g 1 2 dg 2 = g 1 2 (A + B)g 2 (63) where A = g1 1 dg 1 and B = dg 2 g2 1. Using this decomposition Q[g 1 g 2 ] = 1 24π 2 Tr(A + B)(A + B)(A + B) = 1 [TrA 3 24π 2 + TrB 3 + (3A 2 B + 3B 2 A) ] = Q[g 1 ] + Q[g 2 ] 1 8π 2 Tr( da B + AdB) = Q[g 1 ] + Q[g 2 ] + 1 8π 2 d(trab) (64) = Q[g 1 ] + Q[g 2 ] (65) (The wedge products or antisymmetrized products are left understood in this equation.) We see that the winding numbers add when we take products of the group-valued functions g 1 and g 2. The possible surface contribution in passing from (64) to (65) is zero, because the antisymmetric product of A and B falls off sufficiently fast as x. Another important result which follows from this is Q[gh] = Q[g] for group elements h close to the identity; this is because Q[h] = 0 to linear order in h 1. Thus Q is invariant under small deformations of the map g(x) : S 3 SU(2); it is a topological invariant. An example of a configuration for which Q = 1 is given by g 1 (x) = x2 1 x i 2x i x σ i (66) 19

20 which is equivalent to φ µ (x) = y µ. The φ s form a three-sphere; so do the y s. φ µ = y µ gives one covering of the S 3, which is SU(2), when y s cover the S 3 corresponding to space (or equivalently R 3 ). Thus Q = 1 for this configuration. This may also be verified directly from the integral (61). g 1 (x) is a smooth configuration with g 1 1 as x. Now the configuration g(x) = 1 everywhere evidently has Q = 0. Since we have Q = 0 for g = 1 and Q = 1 for g = g 1 (x) and Q is invariant under smooth deformations, it is clear that g 1 (x) cannot be smoothly deformed to the identity everywhere. Further we can consider g(x) = g 1 (x)g 1 (x), which has Q = 2 by (65). Also we have g = g 1 (x), which has Q = 1, which follows from Q[g g] = Q[g ] + Q[g] and Q[g g] = Q[1] = 0. Putting all this together, we see that the classes of maps g(x) : S 3 S 3 have the structure of the additive group of integers Z. Since a general compact Lie group contains an SU(2) subgroup, the formula for the winding number and the nature of the classes of maps are also valid for any compact Lie group G. (There is an exception for SO(4) which is locally SU(2) SU(2) and hence Π 3 [SO(4)] = Z Z.) Problem 6. Consider a map R 3 SU(2) given by U(x) = cos F (r) + iσ i x i r sin F (r) (67) where F (r) is a function of the radial variable r. This can be considered as a map from S 3 to SU(2) only if sin F is zero at r = 0, ; why? Using the formula for the winding number, show that Q[U] = 1 (F (0) F ( )) (68) π 5 Some uses of CS theory 5.1 The CS-WZW connection We have already seen that the wave function for the CS theory involves the WZW action. This is just part of a more general connection between CS theory and the WZW theory which was shown by Witten. Since we considered the sphere with no charges on it, classically all physical fields are gauge equivalent to zero; correspondingly, there is only one quantum state ψ = exp[ks(m )]. If we introduce point sources with charges given by 20

21 matrices in representation {R i }, the Gauss law is modified as ik 2π F a z z(x) = n t a R i δ (2) (x x i ) (69) i Now there are nontrivial solutions and the physical Hilbert space is no longer one-dimensional. In this case, the wave functions of the level k, group G, CS theory are precisely the chiral blocks F I (z 1,, z n ) of a correlator φ R1 (x 1 ) φ Rn (x n ), where φ Ri (x) is an operator corresponding to the representation R i, of a level k, group G, WZW theory. This relation between chiral blocks of the WZW theoy and the CS theory holds for higher genus Riemann surfaces as well, in which case, one can have nontrivial solutions to F z z = 0 even without charges. 5.2 CS theory and knots One of the most impressive and fascinating uses of the CS theory is in knot theory. There are clearly many topologically distinct knots and links possible in three dimensions. The topological classification of knots and links has been a long-standing problem in mathematics. The standard approach has been to associate a polynomial in some formal variable to each knot (or link). A number of different polynomials have been proposed, with calculational rules on how they can be determined for a given knot. It would be nice to have a single polynomial which uniquely specifies a knot and vice versa. This is not the case. Generally, if two knots have different polynomials, then they are distinct; however, it is possible to have two different knots with the same polynomial. The original polynomial, the Alexander polynomial, was proposed around In 1984, Jones obtained an important generalization; the Jones polynomial is a finer description in the sense that two knots which have the same Alexander polynomial can have different Jones polynomials. There are a number of other polynomials known by now, the Alexander-Conway polynomial, two polynomials due to Kauffman, the HOMFLYPT polynomial (which is named after the simultaneous discoverers Hoste, Ocneanu, Millet, Freyd, Lickorish, Yetter, Przytycki, Traczyk), to name a few. We shall first describe briefly the rules for the HOMFLYPT polynomial before coming to the role of the CS theory. We consider oriented links L in S 3 ; equivalently, we can consider links of finite extent in R 3. There is an equivalence relation between links, L 1 L 2 if L 1 can be mapped to L 2 by an orientation-preserving coordinate transformation of S 3. (This is 21

22 the mathematical statement of the notion that we can pull, deform and do unlinking transformations on the link L 1 to bring it to the form L 2, without cutting the strands.) Let Z[l ±1, m ±1 ] denote the set of polynomials in l ±1, m ±1 with integer coefficients. The HOMFLYPT polynomial P L [l, m] is then a mapping from oriented links L in S 3 (or finitely extended links in R 3 ) to Z[l ±1, m ±1 ]. It is defined by the following three rules. 1. P L is defined on equivalence classes of links. 2. The unknot U is a simple unlinked circle in S 3 and P U = l P L+ + l 1 P L + mp L0 = 0. This is known as the skein rule. L ± refer to the overcrossing and undercrossing of link segments projected to two dimensions as shown, L 0 refers to no crossing at all. L+ L Lo One can obtain the polynomial for any link from these rules, in a recursive way, starting from the unknot and building up to complicated knots. The Jones polynomial corresponds to a special choice of l, m as indicated later. In a gauge theory, we are familiar with the Wilson loop defined by [ ( W L = Tr P exp it a A a dx µ )] µ L ds ds [ R 1 = Tr e 1 ɛ ita A a µ dxµ ds ds R ɛ e 0 ita A a µ dxµ ds] ds (70) Here x µ (s), 0 s 1, defines a closed curve in R 3 or S 3. t a are matrices in the Lie algebra of the gauge group. Since they are not necessarily mutually commuting, one needs to specify the ordering of these matrices to define the exponential of the matrix-valued integral by its expansion in a power series. The symbol P indicates that the matrices are ordered along the path, with the matrices at the start, at s = 0, at the right end and the matrices at the end, at s = 1, to the left, as indicated. One can take expectation values of this Wilson operator using any gauge theory action. The important result, 22

23 which was shown by Witten, is that if we take the expectation value of a Wilson operator defined over a curve L using the level k CS action, we get a version of the HOMFLYPT polynomial for the curve L; i.e., W L W U 1 W U dµ e is CS(A) W L [A] = P L [ iq N/2, i(q 1 2 q 1 2 )] (71) where q = exp(iπ/(k + N)). (N = 2 corresponds to the Jones polynomial.) This version of the polynomial has the advantage that it is intrinsically defined in three dimensions; unlike the earlier methods which use the recursive build-up of polynomials using the skein rule which applies to a projected version of the knot or link. It is also easy to generalize this definition to different three-dimensional spaces, not just S 3 or R 3. For these reasons there has been a tremendous amount of work in mathematics along these lines. 5.3 Integrability conditions The basic feature of the CS theory is that the equations of motion express the vanishing of the field strength or curvature. There are many situations where such a condition expresses the set of equations of interest. For example, for instantons in four dimensions, part of the curvature, namely, the self-dual (or antiself-dual) part vanishes. While they occur as a subset of solutions of the Yang-Mills theory, one can ask whether there is an action which gives the instantons as the only solutions, in other words, a theory of instantons in their own right. It is possible to set up a CS theory, with a suitable projection to the self-dual part, which realizes this. There are many other similar situations. The vanishing of some components of the curvature occurs for integrable systems, and one can set up a CS-type description for many integrable models. Another interesting line of development involves supersymmetry. A supersymmetric version of a CS theory will not have zero curvature, since there are matter fields coupling to the gauge field. Nevertheless, the equations of motion will correspond to the vanishing of certain generalized curvatures of field strengths in an appropriate superspace, preserving the notion of integrability. One gets an interesting set of soliton solutions in such cases. A particularly important case is the N = 4 super Yang-Mills theory. In a supersymmetric theory, we can introduce the superspace (x µ, θ Ai, θȧi ) and 23

24 the spinorial derivatives D Ai = θ Ai + i(σµ ) A Ȧ θȧi x µ, DiȦ = θȧi iθ Ai (σ µ ) A Ȧ x µ (72) where i = 1, 2,..., N and σ µ = (1, σ i ). We also have the usual derivative / x µ. We then introduce gauge potentials A Ai, ĀȦi, A µ, which are functions of x µ, θ Ai, θȧi, corresponding to these derivatives. Generally speaking this will give too many degrees of freedom and one has to impose constraints which reduce them to the required number of fields for the chosen value of N. These constraints take the form of the vanishing of certain components of the curvatures. We then solve these in terms of certain unconstrained fields and use the latter to construct the action. This procedure works nicely for N = 1, 2. For N = 4, the constraints are along with a subsidiary condition F AiBj + F BiAj = 0 F ij ȦḂ ḂȦ = 0 (73) F j iaḃ = 0 W ij = 1 2 ɛ ijkl W kl (74) where F AiBj = ɛ AB W ij. These constraints are stringent and lead to the equations of motion via the Bianchi identity [2]. If we are interested in constructing an action with manifest N = 4 supersymmetry, this is bad news. We do not have fields which are off the mass-shell. However, the good news is that this property shows that the second order classical equations of motion of the theory are equivalent to a set of first order equations in an appropriate superspace. This suggests a certain integrability for the N = 4 Yang-Mills theory. Further, one can seek some sort of generalized CS action in superspace as the appropiate action for the theory. We will explore some of these possibilities in what follows. 6 Some uses of WZW theory 6.1 WZW theory as a conformal theory Conformal transformations are spacetime coordinate transformations or diffeomorphisms which preserve the metric up to a scale factor; i.e., the change 24

25 in the metric tensor is given by fg µν where f is a scalar. In this case, ds 2 (1 + f)ds 2. The propagation of light rays is given by ds 2 = 0 and this condition is preserved by conformal transformations; this is one reason why conformal symmetry is important. In a quantum field theory, the behavior of long wave length modes can be obtained by the renormalization group techniques. If we have an infrared fixed point for which the β-function vanishes, then we have a scale-invariant field theory. In a local field theory, scale invariance extends to full conformal invariance. Thus such conformal theories can describe the behavior of physical systems near a second-order phase transition point, giving another reason why these are important. W ZW as an infrared fixed point One has to use algebraic techniques to prove that the WZW theory is a conformal field theory for a general (but integral) value of k. But, for large values of k, an ordinary one-loop calculation can be done to show that the β-function is zero. For this, we consider the general sigma model S(g) = λ 8π M 2 Tr( µ g µ g 1 ) + kγ[g] (75) The coefficient of the first term is now arbitrary; this parameter λ is related to the coupling constant. The coefficient of Γ(g) has to be quantized for topological reasons and so must remain an integer. We also consider the group SU(N) for this calculation. To obtain the β-function, we can do a background field calculation of the effective action. We make the replacement g g e ita ϕ a and expand to quadratic order in ϕ a. g is considered as a background value and ϕ a as the fluctuations. This gives S(gh) S(g) + λ 16π µϕ a µ ϕ a λδ µν + ikɛ µν 4π f abc Tr[g 1 µ g( it a )]ϕ b ν ϕ c + (76) The one-loop correction to the effective action is now obtained as [ ] CA λ S = 8πλ (k2 λ 2 ) I Tr( µ g µ g 1 ) (77) 8π where I is the logarithmically divergent integral d 2 p 1 I = (2π) 2 p 2 (78) 25

26 C A is the Casimir defined by f abc f kbc = δ ak C A. The coefficient of I in (77) is the β-function for λ. Result (77) shows that we do have a fixed point at λ = k. This argument is applicable when λ is large. Problem 6. Obtain the one-loop correction to the action given in (77). T he nature of conformal transformations Making a change x µ x µ + ξ µ (x) in the expression for the metric ds 2, we find [ δ(ds 2 ) = ξ α g µν x α + g ξ α αν x µ + g ξ α ] µα x ν dx µ dx ν (79) Conformal transformations are characterized by ξ α α g µν + g αν µ ξ α + g µα ν ξ α = f g µν (80) This is the conformal Killing equation. For the flat Euclidean metric this simplifies to µ ξ ν + ν ξ µ = f δ µν (81) The contraction of this with δ µν gives, in two dimensions, f = µ ξ µ. The conformal Killing equation becomes µ ξ ν + ν ξ µ ( α ξ α ) δ µν = 0 (82) In components, this works out to z ξ = 0, z ξ = 0, where ξ = ξ 1 + iξ 2. We see that conformal transformations in two dimensions are generated by holomorphic and anti-holomorphic transformations in terms of the complex coordinates z = x 1 ix 2, z = x 1 + ix 2. W hat is a conformal field theory? In a two-dimensional conformal field theory, the basic symmetry algebra of various correlation functions is the conformal algebra. Since conformal transformations are holomorphic transformations, the correlators can be constructed in terms of holomorphic functions which carry a representation of the confomal algebra. More specifically, every correlator of a conformal field heory admits a factorization of the form 0 N φ(z i, z i ) 0 = i ĪJ F Ī ( z 1, z 2,..., z N ) h ĪJ F(z 1, z 2,..., z N ) (83) 26

27 where F I (z 1, z 2,..., z N ) are holomorphic functions. (Technically, they are sections of a line bundle, since there can be branch cuts). They are called the conformal blocks or chiral blocks. The number of chiral blocks, or the range of Ī, J, will depend on the theory and on the particular correlator we are discussing. The coefficients h ĪJ are partially determined by singlevaluedness of the correlator. For the WZW model, the chiral blocks carry an even larger symmetry. As a result of the conservation of the chiral currents, namely, z J z = z J z = 0, the action of J s on a holomorphic F will leave it holomorphic. The J s act as a chiral symmetry algebra on the F s. (This is traced to the holomorphic symmetry of the action, M M + θ(z)m.) The chiral blocks can be constructed as solutions of the Ward-Takahashi identities corresponding to the J s. One can turn this around and pose this as a classification problem. Can we construct all the possible chiral blocks and conformal field theories by classifying all the representations of a chiral algebra? This would be equivalent to classifying all possible second order phase transitions in two dimensions. The WZW theory and various gauged versions of it are sufficient to obtain all the known so-called rational conformal field theories. For the algebraic formulation, algebra of the currents is important. This is most easily done in Minkowski space, in light-cone coordinates. (The components J u, J v correspond to the Minkowski continuations of J z, J z.) The action then takes the form S = k Tr( u gg 1 v gg 1 ) + k Γ W Z (84) 4π Choosing the light-cone coordinate u as time, the algebra of the J v is given by [J v (θ), J v (ϕ)] = ij v (θ ϕ) i k v θ a ϕ a (85) 4π where (θ ϕ) a = f abc θ b ϕ c. The representations of this algebra can be used to classify second order phase transitions in two dimensions. Problem 7 (Challenge problem). Starting with the Minkowksi version of the WZW action (84), show that the canonical two-form is Ω ab (v, v ) = k [ ) ( ) 4π Tr E a (v) v (E b (v)δ(v v ) E b (v) v E a (v)δ(v v ) +[E a (v), E b (v )] v gg 1] (86) where E a is matrix defined by δgg 1 = E a δϕ a, g = e ita ϕ a. Obtain the commutation rules for J v as given in (85). 27

28 6.2 Nonabelian bosonization The WZW action is the key to nonabelian bosonization in two dimensions (or one spatial dimension), which was the context in which Witten first introduced the model. In two dimensions, there is no real difference between fermions and bosons; a fermionic field theory can be written as a bosonic field theory and vice versa. In particular, correlation functions of observables can be calculated using a bosonic description or a fermionic description. Abelian bosonization, applicable to one flavor of fermions, has been known for a long time. In fact, one can explicitly construct the fermion field operator in terms of a scalar field [4]. If we have N flavors of fermions, the free Dirac action for them can be transformed to a bosonic action which is, in fact, a level 1 WZW action for g U(N), ψ i (iγ )ψ i S W ZW ]k=1,g=u(n) (87) This equivalence is established by analyzing the current algebra (85). First, one can show that the same algebra is obeyed by the currents ψγ v ψ, the extra c-number on the right hand side of (85) arising from regularization of products of field operators. (It is the so-called Schwinger term in the commutators of currents.) In terms of representation theory, it is known that the algebra (85), for k = 1 has only one irreducible representation, up to equivalence. This mathematical fact then establishes the equivalence of the fermionic and bosonic descriptions. 6.3 *The Dirac determinant in two dimensions The functional integrals over the fermion fields in two dimensions lead to the determinant of the two-dimensional Dirac operator. For massless fermions, this determinant can be exactly evaluated using the WZW action. The Dirac matrices relevant for two dimensions can be taken as σ i, i = 1, 2, since they obey the relation σ i σ j + σ j σ i = 2 δ ij. Consider a set of massless fermion fields in two dimensions which belong to an irreducible representation R of U(N) and which are coupled to a U(N) gauge field. (Other groups can be treated similarly.) The Lagrangian for these fermions can be written as L = ψ(d 1 + id 2 )ψ + χ(d 1 id 2 )χ (88) where ψ, χ may be taken as the two chiral components of a two-spinor. D i = i + A i are the covariant derivatives. It is convenient to use the 28

29 complex components D z = 1 2 (D 1 + id 2 ) and D z = 1 2 (D 1 id 2 ), since the Lagrangian is naturally split in terms of these. A parametrization f or gauge potentials The evaluation of the determinants det D z and det D z can be done most efficiently using an elegant general parametrization of the complex components of the gauge field A z and A z given by A z = z M M 1 A z = M 1 z M (89) where M is a complex matrix; it has unit determinant if the gauge group is SU(N). For the gauge group U(1), we can use the elementary fact that A i can be written as a gradient plus a curl, A i = i θ + ɛ ij j φ. This leads to the parametrization (89) with M = exp(φ + i θ). The result (89) holds for gauge groups U(N) in general. The proof of this parametrization is based on the invertibility of z, which is Liouville s theorem of complex analysis. If the space is not simply connected, one can have zero modes for z ; there are in general flat potentials a which are not gauge equivalent to zero. One can then generalize (89) to include this degree of freedom. As an example, consider the torus S 1 S 1. This can be described by two real coordinates ξ 1, ξ 2, 0 ξ i 1, with the identification of ξ 1 = 0 and ξ 1 = 1 and likewise for ξ 2. The complex coordinate can be taken as z = ξ 1 + τξ 2, where τ = Re τ + i Im τ is a complex number called the modular parameter of the torus. Gauge fields on a torus can be parametrized as [ ] iπ a A z = M M 1 z M M 1 (90) Im τ There is an ambiguity in the parametrization (89) (and (90)). Notice that M and MV ( z), where V ( z) is purely antiholomorphic, will lead to the same potential A z. On a sphere, or on the complex plane with suitable boundary conditions, there are no nonsingular antiholomoprhic functions (Liouville s theorem), and there is no ambiguity. Any gauge-invariant function of M, M will be an observable. On other spaces where z has zero modes, if we use the parametrization analogous to (89), we have to ensure that only quantities which are independent of this ambiguity are considered as observables. Even on the sphere, such an ambiguity can exist when there are charges since singularities are possible at the locations of charges. Physical quantities defined in terms of the potential will be free of such ambiguities. 29