Loop Groups, Anyons and the Calogero-Sutherland Model

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The idea of using a vertex operator construction to obtain particles with anyon type statistics is not new, see for example [Kl] and more recently [AMOS1, AMOS2, I, H, MS] and references therein. However the vertex operators described in these more recent references are not defined on the Fermion Fock space as limits of implementors of fermion gauge transformations. In other words they do not come from loop group elements. Indeed it is difficult to give a precise meaning to them at all and we do not attempt to do so here. Our vertex operators can be seen to have similar formal properties to those appearing in the papers mentioned, but are well defined in terms of positive energy representations of loop groups in the sense of [PS].

The benefits of our approach are the following. First there is a quantum Hamiltonian acting on the anyon states. This we believe resolves a long standing difficulty in the study of anyons in that it provides a basis for models incorporating interactions. Second we obtain a unifying view of a number of interesting ideas that have emerged in recent times in the physics literature. The most important of these is the connection with the Calogero-Sutherland (CS) model [AMOS1, AMOS2, I, MS] (see also [H, HLV, BHKV, P]). Specifically we find that n-point anyon correlation functions provide useful building blocks for solutions to the CS system. Comparing with the known solutions of the CS system [Fo2] we find that Jack polynomials [St] may be expressed in terms of anyon correlation functions. (Similar relations were previously obtained by different methods in [Fo1].)

From this point of view the anyon Hamiltonian is a second quantized CS Hamiltonian. The final connection we make is with W -algebras, again a connection which has been known from other approaches for some time [AMOS1, AMOS2, I, MS]. In this paper we do not recover the full import of the W -algebra connection in the anyon case. This is a matter we intend to develop more fully elsewhere. However we do construct that part of the W -algebra that we need as an algebra of unbounded operators with a common dense domain. This suffices for our purposes, namely the construction of an anyon Hamiltonian, constructing the CS model solutions as anyon correlation functions, obtaining the link with Jack polynomials, and finding the algebraic relations of the Hamiltonian with the anyon fields.

2 Summary

This paper contains a number of technical sections. In order to make the results accessible we present a summary here. At the same time we take the opportunity to introduce some of our notation. However, the reader will need to take some notation on trust and refer to later sections for the details.

We work on an interval S = [ L/2, L/2] which we will think of as a circle of circumference L − L. We let P be the spectral projections of i ∂ regarded as a self adjoint operator on a dense ± − ∂x domain in L2(S ). We let denote the free fermion Fock space over L2(S ). We choose L F L the usual positive energy condition that the fermion ﬁelds are in a Fock representation of the algebra of the canonical anticommutation relations deﬁned by P−. This means the fermion

2 ﬁelds ψ(f), ψ(g)∗ f, g L2(S ) satisfy { | ∈ L } ∗ Ω, ψ(f) ψ(g)Ω = g, P−f 2 (1) h iF h iL (SL) where Ω is the vacuum or cyclic vector in . We let Q denote the Fermion charge operator F on , and R a unitary charge shift operator on satisfying R−1QR = Q + I (the precise F F choice for R will be explained later).

We will construct regularised anyon ﬁeld operators φν(x) where ν R is a parameter ε ∈ determining the statistics, x S , and ε > 0 is a regularization parameter. For positive ∈ L ε the operator φν(x) is proportional to a unitary operator on which represents a certain ε F U(1) valued loop on SL. These operators are not periodic but obey (the parameter ν0 will be explained below), ν −iπνν0Q ν −iπνν0Q φε (x + L)= e φε (x)e , and in the limit as ε 0 they converge to operator valued distributions φν(x) satisfying ↓ ′ ′ ′ φν (x)φν (y)= e−iπνν sgn(x−y)φν (y)φν (x). (2)

In particular for p Λ∗ = 2π n n Z , the formula ∈ { L | ∈ } L/2 ˆν ipx iπνν0Qx/L ν iπνν0Qx/L φ (p) = lim dx e e φε (x)e (3) ε↓0 Z−L/2 is a well-defined operator on (Proposition 1). Note that we have to insert factors to F ν compensate for the non-periodicity of φε (x) before Fourier transformation. We also find that the statistics parameters ν,ν′ for which Eq. (2) holds cannot be arbitrary but have to be integer multiples of some fixed (arbitrary) number ν0 > 0. (If one is only interested in a single species of anyons one can chose ν = ν .) 0 | | A main focus is on the correlation functions of the anyon fields. These are distributions defined by taking the limit as ε 0 of j ↓ Cν1,...,νN (ν , w , w y ,...,y ) := Ω,Rw1 φν1 (x ) φνN (x )Rw2 Ω (4) ε1,...,εN 0 1 2| 1 N ε1 1 · · · εN N D E for x S , ν /ν Z (for fixed ν ). Using general results for implementors of U(1) loops j ∈ L j 0 ∈ 0 we obtain

Cν1,...,νN (ν , w , w y ,...,y )= δ ε1,...,εN 0 1 2| 1 N w1+w2+(ν1+...+νN )/ν0,0 N N eiπ(w1−w2)ν0(ν1x1+···νN xN )/L b(x x ; ε + ε )νj νk (5) × j − k j k jY=1 k=Yj+1 with π 2π π −i L x − L ε i L x −πε/L π b(x, ε) := e e e = 2ie sin L (x + iε). (6) − − The reason for studying these correlation functions is the connection with the Calogero- Sutherland (CS) Hamiltonian [Su]. This is deﬁned on the set of functions f C2(SN ; C) ∈ L which are zero on (x ,...,x ) SN x = x for some k = j and/or x = L/2 , { 1 N ∈ L | j k 6 j ± } N ∂2 N ( π )2β(β 1) H = + L − , (7) N,β − ∂x2 sin2 π (x x ) j=1 j j,k=1 L j k X jX6=k −

3 2 N 1 and which extends to a self-adjoint operator on L (SL ). We will prove that the eigenfunctions and spectrum of this Hamiltonian can be obtained from anyon correlation functions, namely as ﬁnite linear combinations of functions

ˆν 2π ∗ ˆν 2π ∗ ν ν fν,N (n x) := lim Ω, φ ( nN ) φ ( n1) φε (x1) φε (xN )Ω (8) | ε↓0 L · · · L · · · D E where n N (Theorem 3). We will obtain these results by constructing a self-adjoint j ∈ 0 operator ν,3 which can be regarded as a ‘second quantization’ of the CS Hamiltonian: it H obeys the relations

ν,3 ν ν ν ν φ (x ) φ (x )Ω H 2 φ (x ) φ (x )Ω H ε 1 · · · ε N ≃ N,ν ε 1 · · · ε N where ‘ ’ mean ‘equal in the limit ε 0’ (see Theorem 2 for details). We obtain ν,3 ≃ ↓ H by arguing by analogy with the well known W -algebra associated with fermions. Using analogous formulae we construct the ﬁrst few generators ν,s, s = 1, 2, 3, of an anyon W - H algebra. Understanding the complete anyon W -algebra is a problem we leave for a further investigation.

This main result implies explicit formulas for the eigenvalues and a simple algorithm to construct eigenvectors Ψν,N (n) of HN,ν2 as ﬁnite linear combinations of vectors

η (n)= φˆν( 2π n ) φˆν ( 2π n )Ω, n = (n ,...,n ) NN . (9) ν,N L 1 · · · L N 1 N ∈ 0 These vectors Eq. (9) can be naturally interpreted as N-anyon states with anyon momenta 2π pj = L nj. Using these relations we can compute Ψ (n), ν,3φν (x ) φν(x )Ω ν,N H ε 1 · · · ε N D E in two diﬀerent ways, and in the limit ε 0 we obtain functions (of the variables (x1,...,xN )) 2 N ↓ in L (SL ) which are the promised eigenfunctions of HN,ν2 (Theorem 3). In the last subsection we observe that we recover the known spectrum of the CS Hamil- tonian. Comparing with the known solutions of the CS model [Fo2], we can establish the relationship between the eigenfunctions of Theorem 3 and the Jack polynomials.

3 Preliminaries

The subsequent discussion relies on some standard material which is summarized in this section. We will follow essentially the treatment in [CHu],[PS],[CR].

3.1 Notation

We denote by N and N0 the positive and non-negative integers, respectively. Let 2π Λ∗ = p = n n Z (10) { L | ∈ } 1Since Eq. (7) obviously is a positive symmetric operator, this follows e.g. from Theorem X.23 in Ref. [RS2] (the Friedrich’s extension). Our approach will lead to a particular self-adjoint extension which is related to the standard one [Su] in a simple manner.

4 and 2π 1 Λ∗ = k = (n + ) n Z . (11) 0 { L 2 | ∈ } 2 2 ∗ Our underlying Hilbert space for the fermions we take to be L (SL) ∼= ℓ (Λ0). These are identiﬁed via the Fourier transform deﬁned by

1 L/2 fˆ(k)= dxf(x)e−ikx (12) √ 2π Z−L/2 for k Λ∗. An orthogonal basis of L2(S ) is provided by the functions ∈ 0 L

1 ikx ∗ ek(x)= e , k Λ . (13) √2π ∈ 0 and then we have 2π f = fˆ(k)e . L k Xk 1 ∂ The spectral projection P− corresponding to the negative eigenvalues of i ∂x is deﬁned as (P f)(k)= fˆ(k) for k < 0 and = 0 otherwise. We also use P = I P . − + − − d 3.2 Quasi-free representations of the CAR algebra

∗ 2 Let a(f), a(g) f, g L (SL) be the usual generators of the fermion ﬁeld algebra over 2 { | ∈ } L (SL), satisfying the canonical anticommutation relations (CAR)

∗ ∗ a(f)a(g)+ a(g)a(f) = 0, a(f)a(g) + a(g) a(f)= f, g 2 I. (14) h iL (SL)

In the representation πP− of this algebra determined by the projection P− we write ψ(f) = π (a(f)). If Ω denotes the cyclic (or vacuum) vector in the Fock space on which π acts P− F P− then this representation is speciﬁed by the following conditions,

∗ ψ(P+f)Ω=0= ψ (P−f)Ω. (15)

We also use the notation ψˆ(∗)(k)= ψ(∗)(e ), k Λ∗. (16) k ∈ 0

3.3 Wedge representation of the loop group

2 Each unitary operator U on L (SL), with P±UP∓ Hilbert-Schmidt, deﬁnes an ‘implementer’ Γ(U), on the Fock space satisfying F Γ(U)ψ(f)Γ(U)−1 = ψ(Uf). (17)

Of particular interest is the representation of the smooth loop group = C∞(S ; U(1)) of G L U(1) by implementors of the unitaries U(ϕ) acting on L2(S ). These are deﬁned for ϕ L ∈ G by U(ϕ)f = ϕf, f L2(S ). (18) ∈ L

5 Then Γ gives a projective representation of on . Writing Γ(ϕ) for Γ(U(ϕ)) we may choose G F Γ(ϕ)∗ = Γ(ϕ∗) (19) and we have Γ(ϕ)Γ(ϕ′)= σ(ϕ, ϕ′)Γ(ϕϕ′) (20) where σ(ϕ, ϕ′) is some U(1) valued group two-cocycle on . We will determine this cocycle G next.

The choice of phase of Γ(ϕ) is important for giving an exact formula for σ. For those ϕ = eiα, with α Lie := C∞(S ; R), the map r Γ(eirα) is required to be a one parameter ∈ G L → group such that the generator dΓ(α) of this group satisﬁes Ω, dΓ(α)Ω = 0. Then we have h i [dΓ(α), ψ(g)∗]= ψ(αg)∗ (21) and a standard calculation [CHu],[PS],[CR] gives

[dΓ(α1), dΓ(α2)] = is(α1, α2)I (22) with the Lie algebra two-cocycle 1 L/2 dα (x) dα (x) s(α , α )= dx( 1 α (x) α (x) 2 ). (23) 1 2 4π dx 2 − 1 dx Z−L/2 Hence the Γ(eiα)= eidΓ(α) (24) are Weyl operators satisfying Eq. (3.2) with σ(eiα1 , eiα2 )= e−is(α1,α2)/2.

We will also use dΓ(α) for complex valued α. These are naturally deﬁned by linearity,

dΓ(α + iα )= dΓ(α )+ idΓ(α ) α C∞(S ; R). (25) 1 2 1 2 1,2 ∈ L Then dΓ(α)∗ = dΓ(α∗) (26) (we use the same symbol for Hilbert space adjoints and complex conjugation), and Eqs. ∗∞ (22) and (23) extend to C (SL; C) so that s deﬁnes a complex bilinear form in an obvious way.

Here a technical remark is in order. The operators dΓ(α), α C∞(S ; C) are all un- ∈ L bounded. However, there is a common, dense, domain which is left invariant by all opera- D tors Γ(ϕ), ϕ (this is discussed in more detail in Appendix B). Thus Eqs. (21), (22), (25) ∈ G and similar equations below are all well-deﬁned on . We also note that all vectors in are D D analytic for all the operators dΓ(α), α C∞(S ; C) (see e.g. [CR]). ∈ L It is convenient to decompose loops into their positive, negative and zero Fourier compo- nents, 1 1 α(x)= α+(x)+ α−(x) +α ¯; α±(x)= αˆ(p)eipx, α¯ = αˆ(0) (27) L L ±Xp>0 where L/2 αˆ(p)= dx α(x)e−ipx p Λ∗. (28) ∈ Z−L/2

6 Then dΓ(α)= dΓ(α+)+ dΓ(α−) +αQ ¯ (29) with Q = dΓ(I). Note that

dΓ(α−)Ω = dΓ(α+)∗Ω= QΩ = 0 (30)

(highest weight condition) implying

Ω, dΓ(α )dΓ(α )Ω = Ω, dΓ(α−)dΓ(α+)Ω = is(α−, α+). (31) h 1 2 i 1 2 1 2 D E We also have is(α−, α+) 0 which can be also easily be seen from the explicit formula for s, ≥ p is(α1, α2)= αˆ1( p)ˆα2(p). (32) ∗ 2πL − pX∈Λ Standard arguments now give us (for α real valued)

− + Ω, Γ(eiα)Ω = e−is(α ,α ). (33) D E 2πix/L We also need R = Γ(φ1) which implements the operator U(φ1) where φ1(x)= e (for an explicit construction of Γ(φ1) see e.g. [R]). The phase of this unitary operator will be ﬁxed latter. Notice that R−1dΓ(α)R = dΓ(α) +αI. ¯ (34) (this will be explained in more detail in Appendix A).

General loops in are of the form ϕ = eif with G 2π f(x)= w x + α(x) (35) L with periodic α and integer w = [f(L/2) f( L/2)]/2π (w is the winding number of ϕ). We − − then deﬁne + − Γ(eif ) := eiαQ/¯ 2RweiαQ/¯ 2Γ(ei(α +α )). (36) This ﬁxes the phase for all implementors. With that we get

σ(eif1 , eif2 )= e−iS(f1,f2)/2 (37) where we introduced S(f ,f )= s(α , α ) + (w α¯ α¯ w ). (38) 1 2 1 2 f1 2 − 1 f2 It is worth noting that one can write

L L L L 1 L/2 df (x) df (x) S(f ,f )= f ( )f ( ) f ( )f ( )+ dx( 1 f (x) f (x) 2 ) (39) 1 2 1 2 2 − 2 − 1 − 2 2 2 4π dx 2 − 1 dx Z−L/2 which (up to trivial, but nevertheless important, rescaling of variables) is identical to the antisymmetric two cocycle introduced by Segal [Se]. Notice that our choice of phase for the implementors implies that Ω, Γ(eif )Ω =0 if w = 0. (40) 6 D E

7 We will need the following relation

Γ(eif1 )Γ(eif2 ) Γ(eifN ) = ( e−iS(fj ,fk)/2)Γ(eif1 eif2 eifN ) (41) · · · · · · j

× × Q Q Q We introduce normal ordering × × as follows. For implementors of loops of winding · · · number zero it is deﬁned as

× iα × iS(α−,α+)/2 iα × Γ(e ) × := e Γ(e ) (42)

× iα × with the numerical factor chosen such that Ω, × Γ(e ) × Ω = 1 [cf. Eq. (33)] . We extend h i this to implementors of general loops,

× if × iαQ/¯ 2 w iαQ/¯ 2 × i(α++α−) × × Γ(e ) × := e R e × Γ(e ) × (43) and to products of implementors,

× if1 if2 if × × if1 if2 if × × Γ(e )Γ(e ) Γ(e N ) × := × Γ(e e e N ) × . (44) · · · · · · A straightforward computation then implies the following relations

× if1 ×× if1 × −iS˜(f1,f2)/2 × if1 if1 × × Γ(e ) ×× Γ(e ) × = e × Γ(e )Γ(e ) × (45) with S˜(f ,f )= w α¯ α¯ w + 2S(α−, α+)= S˜(f ,f )∗ (46) 1 2 1 2 − 1 2 1 2 − 2 1 which will be useful in the following. Finally,

m × if × m ∂ × ia1dΓ(α1) iamdΓ(αm) if × × dΓ(α ) dΓ(α )Γ(e ) × := ( i) × e e Γ(e ) × 1 · · · m − ∂a ∂a · · · 1 m aj =0 · · · (47) and × × × × × × × × × × × × × × × AB × = × BA × = × (× A ×)B × = × (× A ×)(× B ×) × (48) extends the definition of normal ordering to arbitrary products of operators dΓ(αj) and Γ(eifk ). We note that by Stone’s theorem [RS1] the differentiations here are well-defined in the strong sense on the dense domain defined in Appendix B. D It is convenient to introduce the operators

ρˆ(p) := dΓ(ǫ ), ǫ (x)= e−ipx, p Λ∗ (49) p p ∈ which allow us to write dΓ(α)= αˆ(p)ˆρ( p). (50) − pX∈Λ Theρ ˆ(p) have a natural interpretation as boson ﬁeld operators and will be further discussed in Appendix A. The subspace (ﬁnite boson vectors) of spanned by vectors of the form Db F η =ρ ˆ( q ) ρˆ( q )RℓΩ, q > 0,n N ,ℓ Z (51) b − 1 · · · − n j ∈ 0 ∈ will be important for us. Note that is dense in (see e.g. [CR]). Db F

8 Appendix A. Relation to quantum ﬁeld theory

In this section we make contact with notation from the more algebraic approach to the results summarized above [K], [KRi]. This notation is close to that commonly used in the physics literature. First the representation πP− of the CAR algebra can be described in terms of the operators ψˆ(∗)(k) Eq. (16) which satisfy the following relations

′ ∗ ′ ∗ L ψˆ(k)ψˆ(k ) + ψˆ(k ) ψˆ(k)= δ ′ I (52) 2π k,k and ψˆ(k)Ω=0= ψˆ∗( k)Ω, k> 0. (53) − We chose the physics notation for the operatorsρ ˆ(p) defined in Eq. (49). These operators satisfy some additional relations easily proved from their definition. For example, Eqs. (21)– (26) imply [ˆρ(p), ψˆ(k)∗]= ψˆ(k p)∗, − L [ˆρ(p), ρˆ(q)] = p δ I, p,q Λ∗, (54) 2π −p,q ∈ andρ ˆ( p) =ρ ˆ(p)∗. Moreover, − ρˆ(p)Ω = 0 p 0 (55) ≥ follows from Eq. (30). If we define the usual Wick ordering for free fermions by

ˆ ′ ˆ ∗ ′ ˆ ∗ ˆ ′ ψ(k )ψ(k) if k = k< 0 : ψ(k) ψ(k ) := − ∗ ′ (56) ( ψˆ(k) ψˆ(k ) otherwise, we can write 2π ρˆ(p)= : ψˆ∗(k p)ψˆ(k) : . (57) L k∈Λ∗ − X0 Since this formally is equivalent toρ ˆ(p) = dx : ψ∗(x)ψ(x) : e−ipx theρ ˆ(p) can be SL interpreted as the Fourier modes of the fermion currents which (formally) are deﬁned as R ρ(x) = : ψ∗(x)ψ(x) :. This motivates our notation for these operators. In particular Q =ρ ˆ(0) is the fermion charge operator.

It follows from Eq. (17) and the deﬁnition of R that 2π Rψˆ(k)R−1 = ψ(k + ). (58) L From (58) and (54) we deduce the important relation:

−1 R ρˆ(p)R =ρ ˆ(p)+ δp,0I (59) equivalent to Eq. (34). This implies R−wQRw = Q + wI for arbitrary integers w. Notice that as RwΩ is in the eigenspace of Q with eigenvalue w we have Ω,RwΩ = δ . More h i w,0 generally, (Q wI)Γ(eif )Ω = 0, which implies Eq. (40). −

9 Appendix B. Domains for unbounded operators

In this paper we are dealing with an algebra of unbounded operators. For many of the subsequent calculations to make mathematical sense it is essential to understand how the domain on which they all act is obtained. The technical results we need are all contained in [CR, GL].

As mentioned above, implementers dΓ(α) of loops α C∞(S ; C) are unbounded. How- ∈ L ever they have a common invariant dense domain which we now describe. For vectors D η = ψˆ∗(k ) ψˆ∗(k )ψˆ( ℓ ) ψˆ( ℓ )Ω, k ,ℓ > 0,n,m N (60) f 1 · · · n − 1 · · · − m i j ∈ 0 we set ηf if n + m λ Pληf := ≤ , λ N, (61) ( 0 otherwise ∈ and this deﬁnes a family of projection operators on such that s lim P = I; see e.g. F − λ→∞ λ [CR]. Thus := F P F = F for some positive integer λ (62) D0 { ∈ F| λ } is a dense subspace in . The space consists of analytic vectors for the operators dΓ(α), F D0 α C∞(S ; C). This follows from ∈ L dΓ(α)P = P dΓ(α)P , dΓ(α)P C (λ + 2) λ λ+2 λ || λ|| ≤ α where is the operator norm and C a constant depending only on α, see [CR]. It || · · · || α follows that dΓ(α), α C∞(S , R), is essentially self-adjoint on . ∈ L D0 We extend to a space which is also invariant under all implementers of the loop group D0 and deﬁne as the linear span of vectors Γ(ϕ)F , F and ϕ . We summarize the D ∈ D0 ∈ G properties of the domain : D

(i) is a common, dense, invariant set of analytic vectors for all operators dΓ(α), α ∞D ∈ C (SL, C), (ii) is invariant under all operators Γ(ϕ), ϕ , D ∈ G (iii) contains . D Db

(The properties (i) and (ii) follow from the corresponding properties of and the following D0 relation, Γ(e−if )dΓ(α)Γ(eif )= dΓ(α)+ S(f, α)I (63) which is easily proved using Eqs. (20), (24), (37) and (25). Property (iii) follows trivially from the deﬁnitions.)

× if × We ﬁnally justify a formula which we will need below. We observe that formally, × Γ(e ) × Eq. (43) equals + − eiαQ/¯ 2RweiαQ/¯ 2eidΓ(α )eidΓ(α ).

i(A +A−) iA iA− [A ,A−]/2 ± (using e + = e + e e + for A± = dΓ(α ) and Eq. (22) ﬀ, this would account for Eq. (42)). This formula is problematic since the operators eidΓ(α) are only deﬁned for real-valued functions α. However,

−ℓ ± n ′ ′ ℓ′ Ω,R ρˆ(k ) ρˆ(k )dΓ(α ) ρˆ( k ) ρˆ( k ′ )R Ω m · · · 1 − 1 · · · − m D E 10 is always zero for n > max(m,m′) (this is easily proved by using Eq. (30) after applying repeatedly Eq. (22)). Thus ∞ n ± i eidΓ(α ) := dΓ(α±)n n! nX=0 − can be deﬁned as a sesquilinear form on . Since eidΓ(α )RℓΩ= RℓΩ for all ℓ Z, it follows Db ∈ that ∞ in Γ(eif )RℓΩ= eiα¯(w/2+ℓ) dΓ(α+)nRw+ℓΩ (64) n! nX=0 where the r.h.s of this equation is well-deﬁned as an element in the dual of . Db

4 Vertex Operators

4.1 Boson-fermion correspondence

As a motivation and to introduce notation, we ﬁrst recall how the boson-fermion correspondence can be derived from the results summarized in the last Section [PS, CHu]. In Ref. [Se] a so-called ‘blip’ function was introduced which equals, up to the sign, ei(x−y)2π/L λ − , 0 <λ< 1. 1 λei(x−y)2π/L − which is the exponential of a smoothed out step function. Writing it as eify,ε with λ = e−2πε/L one gets 2π f (x)= (x y)+ α+ (x)+ α− (x) (65) y,ε L − y,ε y,ε with ∞ 1 α± (x)= i log(1 e2π(±i(x−y)−ε)/L)= i e±2iπn(x−y)/Le−2πεn/L. (66) y,ε ± − ∓ n nX=1 Note that the winding number of f equals 1. Since f (x) for ε 0 converges to πsgn(x y) y,ε y,ε ↓ − we will also use the following suggestive notation, 1 sgn(x y; ε) := f (x). (67) − π y,ε Later we will also need the function δy,ε(x)= ∂xfy,ε(x)/2π i.e. 1 δ (x)= + δ+ (x)+ δ− (x) (68) y,ε L y,ε y,ε with 1 δ± (x)= e±2πi(x−y)n/Le−2πεn/L. (69) y,ε L n>X0 These functions have the following important properties which we summarize as

Lemma 1: − + + S(αy,ε, αy′,ε′ ) = αy′,ε+ε′ (y) ′ ′ S(fy,ε,fy′,ε′ ) = πsgn(y y ; ε + ε ) (70) ∓ ± ± − S(δ , α ′ ′ ) = δ ′ ′ (y) y,ε y ,ε − y ,ε+ε

11 (The proof of these relations is a straightforward calculation which we skip.)

× × ν iνfy,ε −ν ∗ Then for ε > 0 and integer ν the operators φε (y) := × Γ(e ) ×= φε (y) are well- deﬁned, and from Lemma 1 and Eqs. (20) and (37) we conclude

ν ν′ ′ −iπνν′sgn(y−y′;ε+ε′) ν′ ′ ν φε (y)φε′ (y )= e φε′ (y )φε (y). (71)

For odd integers ν,ν′ and in the limit ε, ε′ 0 these formally become anticommutator re- ±1 ↓ lations. This suggests that the φε (y) in this limit are fermion operators. Indeed one can prove L/2 ˆ∗ 1 1 iky ∗ ψ (k) = lim dy φε(y)e , k Λ0 (72) ε↓0 √ ∈ 2πL Z−L/2 in the sense of strong convergence on a dense domain (see e.g. [CHu, PS]). This is the central result of the boson-fermion correspondence. We note that this relation also ﬁxes the phase of the unitary operator R.

4.2 Construction of anyons

To construct anyons we have to extend the relations Eq. (71) to non-integer ν,ν′. However, the functions eiνfy,ε(x) are not periodic and thus Γ(eiνfy,ε ) does not exist. To circumvent this problem, we note that S(f ,f ) Eq. (38) is invariant under changesα ¯ α¯ λ and w w /λ 1 2 i → i i → i with an arbitrary scaling parameter λ. We use this to construct a function f˜y,ε(x) which has the following properties,

˜ (i) eiνfy,ε(x) is periodic for all ν, (ii) S(f˜y,ε, f˜y,ε)= S(fy,ε,fy,ε).

Since the functions νf˜y,ε(x) have winding numbers different from zero, the first requirement can only be fulfilled for ν values which are an integer multiple of some fixed number ν0 > 0. Then ˜ 2π 2πν0 + − fy,ε(x)= x y + αy,ε(x)+ αy,ε(x) (73) Lν0 − L has the desired properties. Thus the operators

ν × iµν0f˜y,ε × −ν ∗ φ (y) := × Γ(e ) × = φ (y) , ν := ν µ, µ Z (74) ε ε 0 ∈ are well-deﬁned for ε > 0, and they obey the exchange relations Eq. (71) but now for all ′ ν,ν which are integer multiples of ν0. Thus the theory of loop groups provides a simple and ν rigorous construction of regularised anyon ﬁeld operators φε (x).

Remark: To be precise, one should denote the anyon operators deﬁned in Eq. (74) as ν0,µ φε (y). Then Eq. (71) would read

′ 2 ′ ′ ′ ′ ν0,µ ν0,µ ′ −iπν µµ sgn(y−y ;ε+ε ) ν0,µ ′ ν0,µ ′ φ (y)φ ′ (y )= e 0 φ ′ (y )φ (y) µ,µ Z. ε ε ε ε ∈

Making the ν0-dependence manifest would allow us to obtain slightly more general results. However, it would also lead to a proliferation of indices which is a price we are not willing to

12 pay.

We note that this deﬁnition and Eq. (43) imply that the anyon ﬁelds are not periodic but the operators

× + − × ˇν iπνν0Qy/L ν iπνν0Qy/L ν/ν0 iνdΓ(αy,ε+αy,ε) φε (y) := e φε (y)e = R × e × (75) are. This suggests that the Fourier modes φˆν(p) of the anyons fields as defined in Eq. (3) are well-defined operators. In fact:

Proposition 1: The φˆν(p) deﬁned in Eq. (3) are operators with as common, dense, Db invariant domain. Especially, φˆν (0)RℓΩ= Rℓ+ν/ν0 Ω ℓ Z. (76) ∀ ∈

The proof of this is given in Appendix C. It implies that all vectors ην,N (n) Eq. (9) are in . This is important due to the following result also proven in Appendix C: Db

Proposition 2: For η , ∈ Db ν ν ν Fη (x1,...,xN ) := lim η, φε (x1) φε (xN )Ω (77) ε↓0 h · · · i exists and has the form 2 2 F ν(x ,...,x )= e−iπν (x1+...+xN )N/L∆ν (x ,...,x ) (ν e−2πix1/L,...,e−2πixN /L) (78) η 1 N N 1 N Pη | where ν2 N N ν2 ∆N (x) := lim b(xj xk; ε) (79) ε↓0  −  jY=1 k=Yj+1 with b given in Eq. (6) and (ν z) a symmetric polynomial.2 Especially, F ν(x) L2(SN ). Pη | η ∈ L

Proposition 2 follows from the following explicit formula derived in Appendix C: for ηb Eq. (51), n N 2 2 ν −iπν (x1+...+xN )N/L −iqj xk ν Fη (x1, ,xN )= δℓ,Nν/ν0 e νe ∆N (x1,...,xN ). (80) b · · · ! jY=1 kX=1 ν2 We note that ∆N (x1,...,xN ) equals, up to a constant, to the well-known ground state wave function of the Sutherland model (see e.g. [Su]). This will be further explored in Section 6.

Using Eqs. (9) and (3) we now obtain L/2 L/2 n ip1x1 ipN xN ˇν ˇν ην,N ( ) = lim dx1e dxN e φε1 (x1) φεN (xN )Ω ε1,...,εN ↓0 · · · · · · Z−L/2 Z−L/2 L/2 L/2 iP1x1 iPN xN ν ν = lim dx1e dxN e φεN (x1) φεN (xN )Ω (81) ε1,...,εN ↓0 · · · · · · Z−L/2 Z−L/2 2i.e. a polynomial which is invariant under permutations of the arguments; see [McD] .

13 2π with pj = L nj and 2π P = P (n)= n + ν2(N j + 1 ) j = 1, 2,...N. (82) j j,ν,N L j − 2 (To derive this formula we used repeatedly eicQRwΩ= eicwRwΩ for c R and w Z.) These ∈ ∈ Pj can be interpreted as anyon momenta, and they will play an important role in Section 6. It is interesting to note how the momentum shifts ν2 appear in our formalism: they are ∝ due to the factors e−πνν0Qx/L in Eq. (3) which are necessary to make the anyon operators periodic.

We ﬁnally formulate a highest weight condition for the Fourier modes of the anyon ﬁeld operators which is analogous to Eq. (53) and will also play an important role in Section 6.

Proposition 3: The vector ην,N (n) Eq. (9) is non-zero only if the following conditions are fulﬁlled,

n + n + . . . + n 0 (83) 1 2 N ≥ N n + 2j−1−ℓn 0 for ℓ = 1, 2,...N. (84) ℓ j ≥ j=Xℓ+1

Again we defer the proof to Appendix C.

4.3 Anyon correlation functions

The results of the last two subsections enable us to complete one of our main aims namely to compute all anyon correlations functions. First eqs. (42), (44), and (70) imply

ν1 ν ν1,···,ν × ν1 ν × φ (x ) φ N (x )= N (x ,...,x ) × φ (x ) φ N (x ) × (85) ε1 1 · · · εN N Jε1,···,εN 1 N ε1 1 · · · εN N where ν1,···,νN (x ,...,x )= b(x x ; ε + ε )νj νk (86) Jε1,···,εN 1 N j − k j k j

w1 × ν1 ν × w2 iπ(w1−w2)ν0(ν1x1+...+ν x )/L Ω,R × φ (x ) φ N (x ) × R Ω = δ e N N . ε1 1 · · · εN N w1+w2+(ν1+...+νN )/ν0,0 D E (87)

Now using equations (33), (40) we obtain Eqs. (4)–(5). Our main interest is in the functions (8) which can be written as

ν fν,N (n x)= F n (x). (88) | ην,N ( ) By a simple computation,

L/2 L/2 −iP1y1 −iPN yN fν,N (n x) = lim lim dy1e dyN e | ε↓0 ε1,...,εN ↓0 · · · Z−L/2 Z−L/2

14 Ω, φ−ν(y ) φ−ν(y )φν(x ) φν(x )Ω × εN N · · · ε1 1 ε 1 · · · ε N 2 2 D L/2 L/2 E −iπν (x1+...+xN )N/L ν −ip1y1 −ipN yN = e ∆N (x1,...,xN ) lim dy1e dyN e ε↓0 · · · Z−L/2 Z−L/2 ν2 −ν2 ˇb(yj yj′ ; εj + εj′ ) ˇb(yj xℓ; 2ε) × ′ − − j>jY Yj,ℓ 2π 2π 2π ˇ − L ε i L x where pj = L nj and b(x, ε) := 1 e e . Comparing with Eq. (78) we see that − L/2 2π L/2 2π −i n1y1 −i nN yN η (n)(z1,...,zN ) = lim dy1e L dyN e L P ν,N ε↓0 · · · Z−L/2 Z−L/2 2 2 2π 2π ν 2π 2π −ν − ε i (yj −y ′ ) − ε i yj 1 e L e L j 1 e L e L zℓ . × ′ − − j>jY Yj,ℓ We now can expand the integrand in a Taylor series in the exponentials and then perform the yj-integrations. The ﬁnal result is

N j−1 N 2 2 N ′ ν ν µ ′ mjℓ η (n)(z1,...,zN )= L − ( 1) jj ( zℓ) (89) ν,N ′ P ′ µjj ! mjℓ ! − − X jY=1 jY=1 ℓY=1 ν2 where ± are the binomial coeﬃcients as usual and ′ here means summation over all n ! µ ′ ,m N such that P jj jℓ ∈ 0 j−1 N N µjj′ µj′j + mjℓ = nj for j = 1, 2 ...N . (90) ′ − ′ jX=1 j =Xj+1 Xℓ=1

4.4 The braid group

The braid group will not play a role in our deliberations however we mention one observation for completeness. We deﬁne operators on the N-anyon subspace as follows. On a vector

φν (x ) φν (x )Ω ε 1 · · · ε N deﬁne, for i 1, 2, ..., N 1 , σ to be the operator which interchanges the ith and (i + 1)th ∈ { − } i arguments and multiplies by the phase:

2 e−iπν sgn(xi−xi+1;ε)/2.

An easy calculation reveals that the braid relations hold:

σ σ = σ σ , i j > 1, i j j i | − | 2 σi = 1 σ σ σ = σ σ σ , i j = 1. j i j i j i | − | So we have a braid group action on each N-anyon subspace.

15 Appendix C. Proofs

C.1 Proof of Proposition 1: According to Eqs. (3) we have to compute

L/2 ( ) := dy eipyφˇν(y)η , · ε b Z−L/2 (we used (75)) for η as in Eq. (51), and show that this has a well-deﬁned strong limit ε 0 b ↓ which is in . We note that Eq. (63) implies for all q Λ∗ Db ∈ φν(x)ˆρ(q)= ρˆ(q) νe−iqx−|q|εI φν(x) (91) ε − ε h i −iqy−|q|ε (we used Eqs. (49), (74) and S(f˜y,ε,ǫq)= e ), and similarly for φˇ. We thus obtain

φˇν(y)η = ρˆ( q ) νeiq1(y+iε)I ρˆ( q ) νeiqn(y+iε)I ε b − 1 − · · · − n − h ∞i (iνh )m i dΓ(α+ )mRℓ+ν/ν0 Ω × m! y,ε mX=0 where we used Eqs. (75) and (64). Now

∞ 1 2π dΓ(α+ )= i ρˆ( 2π j)e−i L j(y−iε), y,ε − j − L jX=1 thus L/2 ( )= dy eipy ρˆ( q ) νeiq1(y+iε)I ρˆ( q ) νeiqn(y+iε)I · − 1 − · · · − n − Z−L/2 ∞ h ∞ i h i νmj 2π −i L j(y−iε)mj 2π mj ℓ+ν/ν0 m e ρˆ( L j) R Ω. × mj!j j − m1,mX2,...=0 jY=1 We see that only terms with

2π ∞ jm = p + δ q + + δ q , δ = 0, 1,m = 0, 1, 2,... (92) L j 1 1 · · · n n i j jX=1 are non-zero after the integration, and this is only a ﬁnite number of terms. Notice that the ε dependence arises only in the scalars multiplying these ﬁnitely many vectors. It is now obvious that the limit φˆν(p)η = lim ( ) exists in norm, and we obtain b ε↓0 · νmj ˆν ′ δ1+...+δn 1−δ1 1−δn ′ 2π mj ℓ+ν/ν0 φ (p)ηb = L ( ν) ρˆ( q1) ρˆ( qn) j m ρˆ( L j) R Ω (93) − − · · · − mj!j j − X Y ′ ′ where means that the sum is over all δi and mj obeying the condition Eq. (92), and j indicates that the product over j is also constrained by (92). This is manifestly a vector in P Q . Db Especially for p = 0 and n = 0 we get Eq. (76). ✷

C.2 Proof of Proposition 2: We compute ( ) := η , φν (y ) φν (y )Ω with η Eq. (51). · h b ε 1 · · · ε N i b We obtain ( )= Ω,R−ℓρˆ(q ) ρˆ(q )φν (y ) φν(y )Ω , · n · · · 1 ε 1 · · · ε N D E 16 and by a simple computation,

( )= νe−iq1(y1−iε) + νe−iq1(yN −iε) Ω,R−ℓρˆ(q ) ρˆ(q )φν (y ) + φν(y )Ω · · · · n · · · 2 ε 1 · · · ε N n D E = . . . = νe−iqj (y1−iε) + νe−iqj (yN −iε) Ω,R−ℓφν (y ) φν(y )Ω · · · ε 1 · · · ε N jY=1 D E n N 2 2 −iqj (yk−iε) −iπν (y1+...+yN )N/L ν = νe δℓ,Nν/ν0 e b(yj yk; 2ε) ! − jY=1 kX=1 j

L/2 L/2 2πin1y1/L 2πinN yN /L ην,N (n) = lim dy1e dyN e ( ) ε1,...,εm↓0 · · · · · · Z−L/2 Z−L/2 where ( ) equals · · · ∞ N ν2 1 2πi(yj −yℓ)/L−2π(εj +εℓ) 2π −2πik(yk′ −iεk′ )/L N 1 e exp ν ρˆ( L k) e R Ω. − k − " ′ #! j<ℓY kX=1 kX=1 (Note that the limit is in the strong sense.) Expanding the latter in powers of eiyj /L shows that this is a sum of terms proportional to

eiqjℓ(yj −yℓ+iεj −iεℓ) e−iqk(yk−iεk) j<ℓY Yk ∗ where qjℓ and qk are in Λ and non-negative. Thus ην,N (n) can be non-zero only if

2π ℓ−1 N n q + q q =0 for ℓ = 1, 2,...N L ℓ − jℓ ℓj − ℓ jX=1 j=Xℓ+1 for at least one set of non-negative numbers q ,q Λ∗. Adding these conditions we get jℓ k ∈ N n N q = 0 which implies Eq. (83). Moreover, if these conditions hold then ℓ=1 ℓ − ℓ=1 ℓ P P 2π N qjℓ = nℓ + qℓℓ′ ( 0) j<ℓ L ′ − ≥ ∀ ℓ =Xℓ+1 with ‘( 0)’ terms which always are non-negative. By induction we obtain from this ≥ k N 2π j′−1−ℓ 2π qjℓ = nℓ + 2 nj′ + qj′ℓ′ ( 0) j<ℓ L ′  L ′ ′  − ≥ ∀ j X=ℓ+1 ℓ =Xj +1   which should be positive. Setting k = N this implies Eq. (84). ✷

17 5 W -charges

5.1 Motivation

There are self-adjoint operators W s on obeying F [W s, ψˆ∗(k)] = ks−1ψˆ∗(k) k Λ∗, W sΩ=0 (s N). (94) ∀ ∈ 0 ∈ If we introduce an operator valued distribution ψ∗(x) such that

∗ ∗ ∗ ψˆ (k)= ψ (ek)= dx ek(x)ψ (x), ZSL the commutator relations in Eq. (94) are (formally3) equivalent to

∂s−1 [W s, ψ∗(x)] = is−1 ψ∗(x). (95) ∂xs−1 These operators W s can be represented in terms of the operatorsρ ˆ(p) Eq. (49),

W 1 =ρ ˆ(0) 2 π × × W = × ρˆ(p)ˆρ( p) × L ∗ − pX∈Λ 2 2 3 4π × × π × × W = 2 ρˆ(p1)ˆρ(p2)ˆρ( p1 p2) 2 ρˆ(0) (96) 3L ∗ − − −3L p1,pX2∈Λ etc.

These formulas are known in the physics literature (see e.g. [B]). We shall construct operators ν which obey similar relations with the anyon field operators φε (x). To explain our method, we will first present a construction of operators W s obeying Eq. (94) for all s N. We then ∈ show how to partly extend this to anyons. The extension is essentially trivial for s = 1, 2. The first non-trivial case is s = 3. We propose a natural generalization of W 3 and show that it corresponds to a ‘second quantization’ of the CS Hamiltonian Eq. (7), as described in the Introduction.

To simplify our notation, we set ν0 = ν in the rest of the paper.

5.2 W -charges for fermions

We deﬁne ν ν × iνdΓ(f˜y+a,ε−f˜y,ε) × (y; a) := N (a) × e × I , (97) Wε − with functions f˜y,ε given by equations (73), (66) and the normalization constant i N ν(a)= . (98) 2 ν2 π π 2Lν cos ( L a) tan( L a) In this Section we are mainly interested in the fermion case where ν = 1, but in our discussion on anyons later we will need these formulas for general non-zero ν R. ∈ 3our results below will actually give a precise mathematical meaning to this

18 We claim that

L/2 ∞ s−1 s−2 ν ν ( ia) ν ν,s (a) := lim dy ε (y; a)= − W (99) W ε↓0 −L/2 W (s 1)! Z sX=1 − deﬁnes an operator valued generating function for operators W ν,s, s N. To be more precise: ∈

Lemma 2: For all a R and non-zero ν R, the operators ν (a) Eqs. (97)–(99) are ∈ ∈ W well-defined on and leave invariant. Especially, Db Db ν (a)Ω = 0. (100) W Moreover, Eq. (99) defines a family of operators W ν,s, s N, which have as a common, ∈ Db dense invariant domain of definition.

The proof of this result is in Appendix D. We now show how to compute these operators W ν,s explicitly. We deﬁne

1 (1 ν) δ˜ (x) := ∂ f˜ (x)= δ (x)+ − (101) y,ε −2π y y,ε y,ε L

∂ 4 where ∂y = ∂y , and (1 ν) ρ˜ (y) := dΓ(δ˜ )= ρ (y)+ − Q. (102) ε y,ε ε L With that we obtain ∞ as dΓ(f˜ f˜ )= 2π ∂s−1ρ˜ (y), y+a,ε − y,ε − s! y ε kX=1 and one can expand ν(y; a) Eq. (99) in a formal power series in a. A straightforward Wε computation then gives

L/2 ν,1 × × W = dy × ρ˜ε(y) × −L/2 ε↓0 Z L/2 ν,2 × 2 × W = π dy × ρ˜ε(y) × −L/2 ε↓0 Z 2 L/2 2 ν,3 4π × 3 × π 2 ν,1 × × W = dy ρ˜ε(y) + 2 2 (2 3ν )W (103) 3 −L/2 ε↓0 3L ν − Z etc.

(this list can be easily extended with the help of a symbolic programming language like MAPLE). Note that for ν = 1, these are identical to the operators in Eq. (96), W 1,s = W s for s = 1, 2, 3. Later we will also need the following formulas which are obtained by simple computations from the deﬁnitions above,

W ν,1 = (2 ν)Q − π W ν,2 = W 2 + (1 ν)(1 3ν)Q2 L − − 4 L/2 −ipx Note thatρ ˆ(p) = lim ↓ dxρ (x)e , which motivates our notation. ε 0 −L/2 ε R 19 π 4 π 2 W ν,3 = W 3 + 4 (1 ν)QW 2 + (1 ν)2(4 ν)Q3 L 3 L − − − 2 π 2 (1 ν)(1 ν 3ν2)Q. (104) 3 L − − − − etc.

The result described in the last subsection can now be stated as follows.

Theorem 1: The operators W 1,s obey the relations Eq. (94) i.e. W 1,s = W s for all s N. ∈

Proof: We recall Eq. (100) for ν = 1. Here we will show that

[ 1(a), ψˆ∗(k)] = e−ikaψˆ∗(k) (105) W These two relations prove the result, as can be seen by an expansion in a formal power series in a and using Eq. (99).

To prove Eq. (105) we use the boson-fermion correspondence Eq. (72). We thus compute 1 1 ifx,ε the commutator of ′ (y) with φ (x)=Γ(e ). With Eqs. (45), (46) and (70) we obtain Wε ε

1 1 × i[fx,ε+f ′ −f ′ ] × [ ′ (y; a), φ (x)] = ( ) × Γ(e y+a,ε y,ε ) × Wε ε · · · with

π 1 sin L (y + a x + iε˜) i π ( ) := N (a) π − c.c. = cot L (y x + iε˜) c.c. · · · sin L (y x + iε˜) − ! 2L − − − whereε ˜ = ε + ε′ and c.c. means the same term complex conjugated. We now use that i 1 cot π (y x iε˜)= + δ± (y) (106) ± 2L L − ± 2L x,ε˜ which is easily seen by expanding the l.h.s as a Taylor series in e±i(y−x)2π/Le−ε2π/L. Thus ( )= δ (y) independent of a (!), and we obtain · · · x,ε˜

1 1 × i[fx,ε+f ′ −f ′ ] × [ ′ (y; a), φ (x)] = δ ′ (y) × Γ(e y+a,ε y,ε ) × . Wε ε x,ε+ε Using Eqs. (99) and (72) we thus obtain for the l.h.s. of Eq. (105),

L/2 L/2 1 ikx × i[fx,ε+f ′ −f ′ ] × lim dx e lim dy δx,ε+ε′ (y) × Γ(e y+a,ε y,ε ) × ε↓0 √ ε′↓0 2πL Z−L/2 Z−L/2 L/2 1 ikx × ifx+a,ε × = lim dx e × Γ(e ) × ε↓0 √ 2πL Z−L/2

ifx,ε 1 in the sense of strong convergence on a dense domain. Recalling Γ(e )= φε(x) and using Eq. (72) again we obtain the r.h.s. of Eq. (105). ✷

We ﬁnally discuss a technical point which will be important in the next Section: Our proof above shows that [ 1(a), φ1(x)] φ1(x + a) W ε ≃ ε

20 where ‘ ’ means equality after smearing with appropriate test functions, and taking the ≃ strong limit ε 0 on an appropriate dense domain. It will be useful to characterize ‘ ’ more ↓ ≃ explicitly as follows. Using Eq. (74) we deﬁne

L/2 × ˜ ˜ ˜ × ˜ν iν[fx,ε+fy+a,ε′ −fy,ε′ ] φε (x; a) := lim dy δx,ε+ε′ (y) × Γ(e ) × (107) ε′↓0 Z−L/2 Then φ˜ν (x; a) φν(x + a). We now deﬁne ε ≃ ε s−1 s−1 ∂ε ν ∂ ˜ν s−1 φε (x) := s−1 φε (x; a) (108) ∂εx ∂x a=0

for s = 1, 2,..., which we regard as ε-deformed diﬀerentiations. We specify the relation between these and the ordinary diﬀerentiations in the following

Lemma 3: s−1 s−1 ∂ε ν ∂ ν × s,ν ν × × × s−1 φε (x)= s−1 φε (x)+ ε cε (x)φε (x) (109) ∂εx ∂x where cs,ν(x) is a well-deﬁned operator-valued distribution for ε 0. Especially,5 ε ↓ 1,ν 2,ν cε (x)= cε (x) = 0,

2 3,ν (iν) i(p1+p2)x 1 −ε(|p1+p2|) −ε|p1|−ε|p2| cε (x)= 2 ρˆ(p1)ˆρ(p2) e e e . (110) L ∗ ε − p1,pX2∈Λ (The proof is a straightforward computation which we skip.)

5.3 W -charges for anyons

The considerations of the preceding section may be extended to cover the case of anyons i.e. ν an arbitrary non-zero real number. Using an argument similar to that in the proof of Theorem 1, we compute

ν ν × iν[f˜x,ε+f˜ ′ −f˜ ′ ] × [ ′ (y; a), φ (x)] = ( ) × Γ(e y+a,ε y,ε ) × Wε ε · · · with ( ) equal to · · · ν2 sin π (y + a x + iε˜) N ν (a) L − c.c.  sin π (y x + iε˜) −  L − ! 2  2  = N ν(a) cosν ( π a) 1 + tanh( π a) cot π (y x + iε˜) ν + c.c. L L L − δ (y) 1 (ν2 1)a∂ δ (y)+ (a2) (111) x,ε˜ − 2 − y x,ε˜ O whereε ˜ = ε+ε′ (in the last line we Taylor expanded in a and used cot2(z)= 1 d cot(z)/dz − − and Eq. (106)). Integrating this in y, performing a partial integrations, and using Eq. (107) we thus obtain

ν ν ν 2 × ν × 3 [ (a), φ (x)] = φ˜ (x; a)+ iπν(ν 1)a × [˜ρ (x + a) ρ˜ (x)]φ˜ (x; a) × + (a ). W ε ε − ε − ε ε O 5 s,ν We will only need this for s = 1, 2, 3 and thus do not specify the cε (x) for s> 3.

21 Comparing now equal powers of a on both sides of the last equation using Eqs (107)–(110) we see that the generalization of Theorem 1 to anyons holds true only for s = 1, 2,

∂s−1 [W ν,s, φν (x)] = ν2−sis−1 φν (x) s = 1, 2 (112) ε ∂xs−1 ε but s> 2 we get correction terms, e.g.

2 2 ν,3 ν i ∂ε ν 2 × ′ ν × × × [W , φε (x)] = 2 φε (x) + 2πi(ν 1) ρ˜ε(x) φε (x) (113) ν ∂εx − ′ whereρ ˜ε(x) := ∂xρ˜ε(x). We deﬁne 1 ν,1 := W ν,1, ν,2 := W ν,2 (114) H ν H which according to Eq. (112) are the anyon W -charges for s = 1, 2.

In the following we only consider the ﬁrst non-trivial case s = 3. To proceed, it is crucial to observe that correction term in Eq. (113) can be partly canceled using the following operator,

L/2 × + − + − × = πi dy × [ρε (y) ρε (y)]∂y[ρε (y)+ ρε (y)] × C − −L/2 − ε↓0 Z 2π × × = × pρˆ(p)ˆρ( p) × (115) − L − p>X0 where ± ± ρy,ε := dΓ(δy,ε). (116) This operator obeys the remarkable relations,

ν ν × ′ ν × × ν × φ (x)+ φ (x) = 2πiν × ρ˜ (x) φ (x) × +2 × φ (x) × . (117) C ε ε C ε ε C ε The proof of this, which we now outline, is by a computation similar to the one leading to Eq. (113). We consider the operator

+ − + − × −iadΓ(iδ −iδ )+ibdΓ(∂y δ +∂yδ ) × (y; a, b) :=× e y,ε y,ε y,ε y,ε × (118) Vε and observe that L/2 ∂ ∂ = π lim dx ε(y; a, b) . (119) C − ε↓0 −L/2 ∂a ∂bV Z a=b=0

Using Eqs. (45), (46) and (70) one then computes

ν ν ′ (y; a, b)φ (x)+ φ (x) ′ (y; a, b) Vε ε ε Vε which by a Taylor expansion in a and b and integrating in y gives Eq. (117). (For details see Appendix D, Proof of Lemma 4.)

We also note that Eq. (55) implies

RℓΩ = 0 ℓ Z. (120) C ∀ ∈ Thus the operator ν,3 = νW ν,3 + (1 ν2) (121) H − C

22 obeys the relation

2 ν,3 ν 2 ∂ε ν 2 × ν × ν × × [ , φε (x)] = i 2 φε (x) + 2(1 ν )( φε (x) φε (x) ). H ∂εx − C − C Again there are correction terms, however, in contrast to the one in Eq. (113) it vanishes when applied to vectors RwΩ! We obtain

∂2 [ ν,3, φν (x)]RwΩ= i2 φν (x)RwΩ (122) H ε ∂x2 ε

× s,ν ν × w (we used Lemma 3 and × cε (x)φε (x) × R Ω = 0). This seems to be the best we can do to generalize the relation Eq. (95) for s = 3 to the anyon case.

To fully appreciate this operator ν,3 one has to extend the computation above to a H product of multiple anyon operators. We thus obtain our main result:

Theorem 2. The operator ν,3 obeys the following relations, H ν,3 ν ν w ν ν w [ , φε (x1) φε (xN )]R Ω= HN,ν2,εφε (x1) φε (xN )R Ω (123) H 1 · · · N 1 · · · N for all integer w, where

N 2 N π 2 2 2 ∂ ( L ) ν (ν 1) HN,ν2,ε = + − + CN,ν2,ε(x) (124) − ∂x2 sin2 π (x x isgn(k ℓ)(ε + ε )) k=1 k k,ℓ=1 L k ℓ k ℓ X kX6=ℓ − − −

6 is a regularised version of the CS Hamiltonian Eq. (7), i.e. the function CN,ν2,ε(x) is non- singular and vanishes uniformly as ε 0 for all j = 1, 2....N. j ↓

The proof of this Theorem is a straightforward but tedious extension of the computation leading to Eq. (122) (which is the special case N = 1), and the interested reader can ﬁnd it in Appendix D.

Appendix D: Proofs

Proof of Lemma 2

The argument here is very similar to the Proof of Proposition 1 and thus we can be brief.

For ηb given by equation (51) we obtain

n × iν(f˜y+a,ε−f˜y,ε) × iqj (y+a)−|qj |ε iqj y−|qj|ε × Γ(e ) × η = ρˆ( q ) νe I + νe I b − j − jY=1 h i ∞ ∞ m mj ν j 2π 2π 2 −ij L (y+a−iε) −ij L (y−iε) 2π mj −2πiν aℓ/L ℓ m e e ρˆ( L j) e R Ω. × mj!j j − − m1,mX2,...=0 jY=1 6The interested reader can ﬁnd the deﬁnition of this function Eq. (130) below.

23 Just as in the proof of Proposition 1 in Appendix C this shows that L/2 × iν(f˜y+a,ε−f˜y,ε) ( ) := dy × Γ(e )η · b Z−L/2 is a sum of a finite number of terms. As the ε dependence lies in the coefficients of this finite dimensional subspace the norm limit ε 0 exists and is in . Thus ν(a)η . ↓ Db W b ∈ Db Especially for η = Ω, we obtain ( ) = Ω, which implies Eq. (100). ✷ b ·

D2. Proof of Theorem 2

We write ν ν ν ν × ν × Φε(x):= φε (x1) φε (xN )= ε (x) × Φε(x) × 1 · · · N J where ν ν1,···,νN ε(x)= ε ,···,ε (x1,...,xN ) J J 1 N are deﬁned in Eq. (86). We also use the short-hand notation ν ν × ν × A Φ (x) := (x) × AΦ (x) × . (125) ∗ ε Jε ε We compute L/2 ν ν × iν[f˜ ′ −f˜ ′ ] × ν [ (a), Φε(x)] = lim dy ( ) × Γ(e y+a,ε y,ε ) × Φε(x) W ε′↓0 · · · ∗ Z−L/2 with 2 N π ν sin (y + a xj + iε˜j) ( )= N ν(a) L − c.c.  π  · · · sin L (y xj + iε˜j) ! − jY=1 − 2 N  π ν  ν sin L (y + a xj + iε˜j) = ( )jN (a) π − c.c. ·  sin L (y xj + iε˜j) ! −  Xℓ=1 − ′   whereε ˜j = εj + ε and ν2 N sin π (y + a x + isgn(j k)˜ε ) ( ) = L − ℓ − ℓ . · j sin π (y x + isgn(j k)˜ε ) k=1 L ℓ ℓ ! kY6=j − − Using Eq. (111) we obtain

N L/2 ν ν [ (a), Φε(x)] = lim dy δxj ,ε˜j (y) W ε′↓0 −L/2 jX=1 Z 1 2 2 × iν[f˜ ′ −f˜ ′ ] × ν 1+ (ν 1)a∂y + (a ) ( )j × Γ(e y+a,ε y,ε ) × Φε(x). × 2 − O · ∗ h i By a simple computation we see that this equals N Φ˜ ν (x; ae )+ iπν(ν2 1)a[˜ρ (y + a) ρ˜ (y )] Φ˜ ν (x; ae ) ε j − εj j − εj j ∗ ε j jX=1 N 2 2 π π π ˜ ν 3 + ν (ν 1) cot (ykj + a) cot (ykj) Φε(x; aej)+ (a ). − L L − L O j,k=1 jX6=k

24 where y = y y + isgn(k j)(ε + ε ) and jk j − k − j k ˜ ν ν ˜ν ν Φε(x; aej):= φε (x1) φε (xj; a) φε (xN ) 1 · · · j · · · N ˜ν 2 with φε (x; a) deﬁned in Eq. (107). Collecting the terms proportional to a on both sides of this equation we obtain,

N ν,3 ν 1 ˜ ν 2 ′ ν [W , Φε(x)] = HN,ν2,εΦε(x)+ 2πi(ν 1)˜ρε (yj) Φε(x) (126) ν − j ∗ jX=1 with N 2 N ( π )2ν2(ν2 1) ˜ ∂ε L HN,ν2,ε = + − (127) − ∂ x2 sin2 π (x x isgn(k ℓ)(ε + ε )) k=1 ε k k,ℓ=1 L k ℓ k ℓ X kX6=ℓ − − − 2 2 and ∂ε /∂εxj as characterized in Lemma 3. To proceed, we need to generalize Eq. (117):

Lemma 4: The operator given in Eq. (115) satisﬁes the following relations C N ν ν × ν × ′ ν Φε(x)+Φε(x) = 2 × Φε(x) × + 2πiνρ˜ε (yj) Φε(x). (128) C C C j ∗ jX=1

Thus with ν,3 Eq. (121), H ν,3 ν ˜ ν 2 × ν × ν [ , Φ (x)] = HN,ν2,εΦ (x) + 2(1 ν )(× Φ (x) × Φ (x) ). (129) H ε ε − ε C − ε C Applying this equation to the state RℓΩ, eq. (120) implies that only the second term on the r.h.s. vanishes, and we obtain Eq. (123) where we set

N ν ℓ ν × 3,ν ν × ν ℓ CN,ν2,ε(x)Φ (x)R Ω:= εjφ (x1) × c (xj)φ (xj) × φ (xN )R Ω. (130) ε − ε1 · · · εj εj · · · εN jX=1

It is easy to see that this deﬁnes a function CN,ν2,ε(x) which can be calculated explicitly, however, we only need that this function is non-singular and vanishes uniformly as ε 0 for ↓ all j, which is obvious (see Eqs. (110) and (91)). ✷

D3. Proof of Lemma 4

We consider the operator (y; a, b) Eq. (118) and recall Eq. (119). Using Eqs. (45), (46) and Vε (70) we compute

ν ν × ν × (⋆) := ε′ (y; a, b)Φε(x)+Φε(x) ε′ (y; a, b) = ( )2 × ε′ (y; a, b)Φε(x) × (131) V V · · · V

25 where

N + ν(−a+ib∂y ) δ ′ (y) ( ) = e j=1 xj ,εj +ε + c.c. · · · 2 N N P + − + − = 1 νa [δ ′ (y)+ δ ′ (y)] + iνb∂y[δ ′ (y) δ ′ (y)] − xj ,εj +ε xj ,εj +ε xj ,εj +ε − xj ,εj +ε jX=1 jX=1 2 N iν + 2 − 2 2 2 ab ∂y[δ ′ (y) δ ′ (y) ]+ (a )+ (b ). − 2 xj ,εj +ε − xj ,εj +ε O O jX=1 Moreover, using Eq. (116) we expand

× + − × × + − × (y; a, b)= I + a × [ρ (y) ρ (y)] × +ib∂ × [ρ (y)+ ρ (y)] × Vε ε − ε y ε ε × + − + − × 2 2 +iab × [ρ (y) ρ (y)]∂ [ρ (y)+ ρ (y)] × + (a )+ (b ). ε − ε y ε ε O O Using the relation

L2 σ σ′ σ′ ′ dy δ ′ (y)ρ ′ (y)= δ ′ ρ ′ (x), σ,σ = (132) x,ε+ε ε σσ ε+2ε ± Z−L/2 and L2 ± ± dy ρε′ (y)ρε′ (y) = 0 (133) Z−L/2 we may calculate L/2 ∂ ∂ π lim dx (⋆) , − ε↓0 −L/2 ∂a ∂b Z a=b=0

✷ and obtain Eq. (128).

6 The Calogero-Sutherland Hamiltonian and its eigenfunctions

We are now ready to show how the results of the last Section provide the means to construct eigenfunctions and the corresponding eigenvalues of the CS Hamiltonian Eq. (7).

6.1 Eigenfunctions from anyon correlation functions

We claim that Theorem 2 essentially relates these eigenfunctions of the Sutherland Hamilto- ν,3 nian H 2 Eq. (7), to the eigenvectors of the operator . In fact the key step is just to N,ν H observe the following elementary corollary of Theorem 2.

Proposition 4. Let η . Then ∈ Db ν,3 ν ν ν lim < η, φε (x1) φε (xN )Ω >= HN,ν2 Fη (x1,...,xN ) (134) ε↓0 H · · ·

26 ν ν,3 where Fη is deﬁned in Eq. (77) Especially, if η is an eigenvector of with the eigenvalue ν H then F is an eigenfunction of H 2 with the same eigenvalue . E η N,ν E

The immediate next question is to ask if our method constructs all eigenvectors of (7). We answer this in two steps. We ﬁrst state and prove another consequence of Theorem 2.

Proposition 5. The vectors η (n) deﬁned in Eq. (9) are in , and they obey ν,N Db ∞ ν,3η (n)= (n)η (n)+ γ nη (n + n[e e ]) (135) H ν,N Eν,N ν,N ν,N j − ℓ Xj<ℓ nX=1 with m (n)= P 2, (136) Eν,N j jX=1 Pj deﬁned in Eq. (82), 2π 2 γ := 2ν2(ν2 1) , (137) L − and e1 = (1, 0,..., 0), e2 = (0, 1,..., 0), ..., em = (0, 0,..., 1).

Proof: We use Eqs. (9), (3) and Theorem 2 to write

ν,3η (n) = ( ) + ( ) + ( ) H ν,N · 1 · 2 · 3 with

N L/2 L/2 iP1x1 iPN xN ( )1 = lim dx1e dxN e · ε1,...,εN ↓0 −L/2 · · · −L/2 jX=1 Z Z 2 ∂ ν ν 2 φε1 (x1) φεN (xN )Ω, × −∂xj ! · · ·

L/2 L/2 ip1x1 ipN xN ( )2 = lim dx1e dxN e · ε1,...,εN ↓0 · · · Z−L/2 Z−L/2 ˇν ˇν CN,ν2,ε(x1,...,xN )φ (x1) φ (xN )Ω × ε1 · · · εN and

L/2 L/2 ip1x1 ipN xN ( )3 = γ lim dx1e dxN e · ε1,...,εN ↓0 −L/2 · · · −L/2 Xj<ℓ Z Z (x x ; ε + ε )φˇν (x ) φˇν (x )Ω ×S j − ℓ k ℓ ε1 1 · · · εN N where γ is deﬁned in Eq. (137) and

∞ 1 2πin(r+iε)/L (x; ε)= 2 π = 2 ne S 2sin ( L (r + iε)) − nX=1

27 (the last equality is obtained by a Taylor expansion in e2πi(r+iε)/L). Recalling Eq. (81), a simple computation implies ∞ ( ) = γ nη (n + n[e e ]). · 3 ν,N j − ℓ Xj<ℓ nX=1 Moreover, using Eq. (130) we see that ( ) = 0. The remaining term is easily computed by · 2 partial integrations, N ( ) = P 2η (n). · 1 j ν,N jX=1 This gives Eqs. (135)–(136). ✷

Based on this result we can now present a simple algorithm to construct eigenvectors of the operator ν,3. For that we ﬁnd it convenient to use the following notation. For H µ NN(N−1)/2 we write ∈ 0 N N µ = (µ ) = µ E , µ N (138) jℓ 1≤j<ℓ≤N jℓ jℓ jℓ ∈ 0 jX=1 ℓ=Xj+1 NN(N−1)/2 which deﬁnes a canonical basis (Ejℓ)1≤j<ℓ≤N in 0 . Moreover, we write n ˙ µ := n µ [e e ]. (139) ± ± jℓ j − ℓ Xj<ℓ E.g., n+˙ nE = n + n[e e ]. We also write 0 for the zero element in NN(N−1)/2, i.e. jℓ j − ℓ 0 n ˙ 0 = n. ± Proposition 4 suggests the ansatz

Ψ= α(µ)ην,N (n+˙ µ). (140) NN(N−1)/2 µ∈ 0X It is important to note that due to Proposition 3, there is actually only a ﬁnite number of non-zero terms in this sum, i.e. Ψ . With Eq. (135) the eigenvalue equation ν,3ψ = Ψ ∈ Db H E implies ∞ [ (n+˙ µ)]α(µ)= γ nα¯ (µ n¯E ) E−Eν,N − jℓ n¯X=0 Xj<ℓ where α(µ) := 0 if ην,N (n+˙ µ) = 0. (141) Setting µ = 0 we get, = (n), (142) E Eν,N and α(0) is arbitrary. Moreover, the other coeﬃcients α(µ) are then uniquely determined by Eq. (145) provided that b (n, µ) := [ (n+˙ µ) (n)] remains always non-zero. A ν,N Eν,N −Eν,N simple computation shows that

2 2π 2 N N j−1 N b (n, µ)= 2 µ n n + (ℓ j)ν2 + µ µ (143) ν,N L  jℓ j − ℓ −  ℓj − jℓ  j=1 ℓ=j+1 h i ℓ=1 ℓ=j+1 X  X X X      28 which is strictly positive for all µ NN(N−1)/2 if ∈ 0 n n n 0 (144) 1 ≥ 2 ≥···≥ N ≥ (the last inequality here is due to Eq. (84)). Note that Eq. (145) then allows to compute the α(µ) recursively, γ ∞ α(µ)= nα¯ (µ n¯Ejℓ). (145) −bν,N (n, µ) − n¯X=0 Xj<ℓ We summarize these calculations and their implication from Proposition 4 in the following

Theorem 3: For n NN satisfying Eq. (144), the equations (138)–(145) and the normal- ∈ ization condition α(0) = 1 (146) determine a unique vector Ψ=Ψ (n) which is an eigenvector of the operator ν,3 ν,N ∈ Db H with the eigenvalue (n). Thus Eν,N ˜ ν ν ψν,N (n x1,...,xN ) := lim Ψν,N (n), φε (x1) φε (xN )Ω (147) | ε↓0 h · · · i 2 N is in L (SL ) and is an eigenvector of the CS Hamiltonian Eq. (7) with the same eigenvalue,

[H 2 (n)]ψ˜ (n x ,...,x ) = 0. (148) N,ν −Eν,N ν,N | 1 N

Remark: We have shown that condition (144) is suﬃcient for the construction of the vectors Ψν,N (n) and we show below that all eigenvectors of the Sutherland model are thereby obtained. Nevertheless, we believe that it would be interesting to explore the signiﬁcance of condition (144) more fully. However, this is beyond the scope of the present paper.

Below we shall compare the eigenfunctions we have obtained with the known ones from the literature [Su, Fo2]. For that it is useful to have the corresponding (but simpler) relation for s = 2, m ν,2η (n)= P η (n) (149) H ν,N j ν,N jX=1 with Pj given in Eq. (82). This relation is a simple consequence of

N L/2 L/2 ν,2 iP1y1 iPN yN ην,N (n)= lim dy1e dyN e H ε1,...,εN ↓0 −L/2 · · · −L/2 jX=1 Z Z ∂ ν ν i φε1 (y1) φεN (yN )Ω × ∂yj ! · · · which is obtained by using Eqs. (9), (3) and (112). By partial integrations we obtain Eq. (149). Finally using this and an argument similar to that proving Theorem 3 we obtain

N ∂ N i Pj ψ˜ν,N (n x1,...,xN ) = 0. (150)  ∂xj −  | jX=1 jX=1   29 6.2 Relation to Jack polynomials

We show below that the eigenfunctions of the CS Hamiltonian which we have obtained are related to the standard ones in the literature [Su] via the following transformation,7

2 ψ (n,µ x) := eiπ(ν +2µ)(x1+...+xN )N/Lψ˜ (n x), µ N . (151) ν,N | ν,N | ∈ 0 Note that the physical interpretation of the phase factor is that it represents a free motion (i.e. plane wave) of the center-of-mass of the system, thus Eq. (151) can be regarded as a trivial change of our wave functions.

We obtain from Eq. (148) (p := ν2 + 2µ)

N 2 iπp(x1+...+xN )N/L π ∂ π HN,ν2 ψν,N = e HN,ν2 2 pN i + N pN ψν,N ,  − L ∂xj L  jX=1   and with Eq. (150), HN,ν2 ψν,N = Eψν,N (152) where

N N 2 2 2 2 π π 2π 1 2 E = Pj 2 pNPj + pN = nj µ + 2 ν (N + 1 2j) . (153) − L L ! L − − jX=1 jX=1 h i We thus reproduce all known eigenvalues of the CS Hamiltonian [Su]. Note that according to Proposition 2, these eigenfunctions have the form

2 ψ (n,µ x ,...,x )= e2πiµ(x1+...+xN )N/L∆ν (x ,...,x ) ν,N | 1 N N 1 N −2πix /L −2πix /L n (e 1 ,...,e N ) (154) ×P ;ν with n := n a symmetric polynomial [McD]. Similarly as in [Fo2] one may use P ;ν PΨν,N ( ) results from [St] to prove that the polynomials n are proportional to the Jack polynomial P ;ν associated with the partition n and the parameter 1/ν2 [St]. (See Appendix E for details.) It is worth noting that due to this, Theorem 3 can be used to formulate an algorithm for explicitly constructing the Jack polynomials in terms of the polynomials given in Eq. (89). It would be interesting make this algorithm more explicit, but this is beyond the scope of the present paper.

6.3 Duality

It is known that the eigenfunctions of the CS Hamiltonians Eq. (7) with couplings β = ν2 and β = 1/ν2 are closely related to each other [St]. In our approach this duality appears as follows.

We note that Eqs. (121) implies

ν,3 = ν2 −1/ν,3 + ν[W ν,3 W −1/ν,3]. (155) H − H − 7Note that the phase factor here is not periodic, thus the wave functions ψ and ψ˜ correspond to diﬀerent self-adjoint extensions of the symmetric operator deﬁned in Eq. (7).

30 Using Eq. (104) we obtain by a straightforward computation π ν[W ν,3 W −1/ν,3]= 4 (ν2 + 1)QW −1/ν,2 + E (Q; ν) (156) − − L 0 with

E (Q; ν)= c 4(4ν4 3ν3 4ν2 9ν 5)Q2 3ν4 + 2ν3 + 5ν2 2ν 3) Q (157) 0 ν,L − − − − − − − 2(ν2+1) π 2 where cν,L = 3ν2 L . By an argument similar to the one leading to Theorem 3 we conclude: Theorem 4: The vectors Ψ (n) characterized in Theorem 3 are eigenvectors of −1/ν,N ∈ Db the operator ν,3 with corresponding eigenvalues H N π N ˜ (n)= ν2 P˜2 4 (ν2 + 1)N P˜ + E˜ (N,ν) (158) Eν,N − j − L j 0 jX=1 jX=1 ˜ where Pj := Pj,−1/ν,N as deﬁned in Eq. (82) and E0(N,ν) Eq. (157). Thus

ν ν lim Ψ−1/ν,N (n), φε (x1) φε (xN )Ω (159) ε↓0 · · · D E is an eigenvector of the CS Hamiltonian H 2 Eq. (7) with the same eigenvalue ˜ (n). N,ν Eν,N

Appendix E. On Jack polynomials

As discussed in Ref. [Fo2] (see also [Su]), all eigenvalues of the CS Hamiltonian HN,ν2 are of the form Eq. (153) with µ a non-positive integer and the n integers such that n n j 1 ≥ 2 ≥ ...n 0. Moreover, the corresponding eigenfunctions are given by8 N ≥ 2 2 −2πiNµ(x1+...+xN )/L (1/ν ) 2πix1/L 2πix1/L ν ψ = e Cn (e ,...,e )∆ (x1,...,xN ) (160)

(α) where Cn is the Jack polynomial [St] associated with the partition n and parameter α [Fo2]. ∗ Note that the complex conjugate ψ of ψ is also an eigenfunction of HN,ν2 with the same eigenvalue, and our eigenfunction Eq. (154) has the same form as ψ∗. Note also that ψ∗ can ′ ′ be written also in the form Eq. (160) with the parameters µ,nj replaced by µ ,nj which are such that n′ µ′ = µ n , n′ 0. (161) j − − N−j N ≥ This follows from the fact that E Eq. (153) is invariant under the transformation µ,nj ′ ′ → µ ,nj. We now derive the precise relation of our solutions to the Jack polynomials. Similarly to [Fo2] we deduce from Eqs. (152)–(154) that n = n (e−2πix1/L,...,e−2πix1/L) obeys the P ;ν P ;ν equation9

N 2 2 ∂ n;ν 2πν π(xk xj) ∂ ∂ n n P2 cot − ;ν = (E E0) ;ν (162) − ∂xj − L L ∂zk − ∂zj ! P − P jX=1 1≤j

31 where 2π 2 N E E = n2 + ν2n (N + 1 2j) . (163) − 0 L j j − jX=1 Moreover, N ∂ n;ν 2π P = i nj n;ν (164) ∂xj − L N P jX=1 jX=1 follows from Eq. (150). Comparing with the diﬀerential equation deﬁning the Jack polyno- −2πix1/L −2πix1/L mials [St], these equations imply that n;ν(e ,...,e ) equals, up to a constant, (1/ν2) P the Jack polynomial Cn (e−2πix1/L,...,e−2πix1/L) associated with the partition n [Fo2].

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