Form factors of descendant operators: $A^{(1)}_{L-1}$ affine Toda theory

F orm factor s of descendan t o p erato rs: A (1) L − 1 affine T o da theor y Oleg Alekseev and Mic h ael Lashk evic h L a ndau Institute for The or etic al Physics, 142 432 Cherno golovka of Mosc ow R e gion, R ussia Abstract In the framew ork of the free field representation w e obtain exact form factors of lo cal operators in the t w o-d imensional affine T oda theories of the A (1) L − 1 series. The construction g eneralizes Lukyano v’s w ell-know n construction to the cas e of descendant operators. Besides, we prop ose a free field rep- resen tation w ith a countable num b er of generators for the ‘stripp ed ’ form factors, which generalizes the recen t prop osal for t h e sine/sinh-Gordon mo del. As a chec k of the construct ion w e compare num bers of the op erators defined by these form factors in level subspaces of t h e chiral sectors with the correspond ing numbers in the Lagrangian formalism. W e argue t hat the construction pro vides a correct counting for op erators with b oth chiral ities. At last we study the prop erties of the op erators with resp ect to th e W eyl group . W e sho w that for generic v alues of parameters th ere exist W eyl inv arian t analytic families of t h e bases in the level subspaces. 1. In tro duction The b o otstr ap appro ach to form factor s in tw o-dimensional integrable q uantu m field theory ma kes it po ssible to calc ula te the form fa c tors exactly by solving a set of difference equa tions for ana lytic functions called form factor axioms [1–3]. Any solution to these equations pr ovides a lo ca l op erator in the theory . The main c o njecture of the appro a ch is that vice versa the set of form factors of any lo cal op era tor is a solution to the b o otstrap equa tio ns. Though a very general approach to solving the for m factor axioms was propo sed b y Smirnov [3], the pr oblem of identification of the op erato r s defined by the b o otstra p form factors to the fields defined in the usual Lagra ngian forma lism is not solved in full gener ality . Moreov er, the in tegral form of Smirnov’s solution and many other prop osals make it difficult to study them. Here we co nsider the t w o-dimensional affine T o da mo dels of the A (1) L − 1 series [4]. These mo dels are mo dels of an ( L − 1 )-comp onent real scalar field ϕ ( x ) with an ex p onential in teraction p otential. In the particula r ca se L = 2, i. e. of the sinh-Gordon mo del, different appro a ches for the fo rm factor s were developed [3, 5–9]. F or gener ic v alue s of L B a bujian and Karowski [10] prop ose d ea rlier a g eneral form of the solution to the form factor axioms in this case in an integral fo r m. W e prop ose a s olution in terms of finite sums base d on Luky anov’s free field formalism for form fa ctors [11]. Lukyano v [12] found the solutions to the b o otstr ap equations that corr esp o nd to the exp onential o p er a tors e i αϕ ( x ) and completely identified them. F ollowing the guidelines of [9] w e find a repr esentation for form factor s of the so calle d descendant op era tors, i. e. the op erators of the for m ( ∂ k 1 µ 1 ϕ i 1 ) . . . ( ∂ k r µ r ϕ i r )e i αϕ . These op erators may b e consider e d as elements of the F o ck spac es gener ated by mo des of the field ϕ ( x ) from the exp onential op erator in the ra dial quantization picture. Though up to now we a re unable to identify these descendant o per ators with particular solutions to the b o otstrap equations, w e can identify some spaces o f s olutions with the F o ck s paces over given exp onential op erato rs. Besides, in the case o f the so called chiral descendants we are able to find bo o tstrap co un terparts of the level s ubspaces of the F o ck spaces. An imp ortant fea ture of the a ffine T o da mo dels is the existence of the so calle d reflection r e la tions betw een the op erators in the theo ry [1 3–16]. These r elations c o nnect op era tors with different v alues o f α related b y the action of the W eyl group. W e prov e the ex istence of these reflection rela tions for our solutions. Moreover, we show that there are analy tic in α families o f W eyl inv ariant ba ses in the F o ck spaces. W e hop e this pro of to b e a step tow a rds solutio n of the identification pro blem. The paper is or g anized as follo ws. In Sec. 2 w e describ e the mo del, fix the notation and recall the main results of Lukyanov’s free field representation. In Sec. 3 w e intro duce an auxiliary commutativ e algebra that allows us to ge ne r alize the free field represe n tation to the descendant op era to rs. W e also cite several 1 simple physical conseque nc e s of the constr uction and carr y out the counting of the descenda n t op erato r s defined b y the bo o tstrap for m factors. Sec t. 4 is dedicated to an a lternative free field construction, which is an imp ortant ing r edient in the pro o f of the reflection relations in Secs. 5 , 6 . In Sec. 5 we use some recurrent re lations to prov e the explicit form of the reflection relations of the expo nential op erator s, while in Sec. 6 they are used to pr ov e the existence of reflection rela tions for the descenda n t op era to rs. The explicit reflection relatio ns for the level 1 descendant op erators are given in Sec. 7. 2. Preliminaries Let h b e the ( L − 1 )-dimensional Cartan subalgebr a o f the s imple Lie algebra A L − 1 and h ∗ be its dual, the brack et h· , ·i be either the K illing fo r m r estricted to h or its dual o n h ∗ . Let α i ∈ h ∗ , i = 1 , . . . , L − 1 be the simple ro ots of the algebr a , h α i , α j i = 2 δ ij − δ i,j +1 − δ i,j − 1 . Let α 0 = − P L − 1 i =1 α i be the a ffine ro ot. Let ρ b e the half sum of the p ositive ro ots. In ter ms of the s imple r o ots ρ = P L − 1 i =1 i ( L − i ) 2 α i and h ρ, α i i = 1 , i > 0. Let H s , s = 1 , . . . , L b e the set of weigh ts o f the fir st fundamental representation π 1 of the algebra , h α i , H s i = δ is − δ i,s − 1 , so that α i = H i − H i +1 . Let ϕ ( x ) ∈ h R be the real ( L − 1)-comp onent field with the action S [ ϕ ] = Z d 2 x h ∂ µ ϕ, ∂ µ ϕ i 8 π − µ 2 L − 1 X i =0 e bα i ϕ ! . (2.1) Let us stress that the sum in the r. h. s. c ontains the summation ov er all simple ro ots of the affine Lie algebra A (1) L − 1 , including the affine ro ot α 0 . The theory is called the affine T o da field the ory asso ciated with the algebra A (1) L − 1 . Below it will b e co n venien t to use the letter s ω , Q , p defined as follows: ω = e 2 π i /L , Q = b + b − 1 , b = r p 1 − p . (2.2) The parameter p is alwa ys thoug ht to b e ir rational. W e shall also use the light-cone v aria bles and deriv atives z = x 1 − x 0 , ¯ z = x 1 + x 0 , ∂ = ∂ ∂ z , ¯ ∂ = ∂ ∂ ¯ z . W e adjusted the definition of the light-cone co or dina tes to fit the usua l conformal field theor y notation system. The sp ectrum o f the mo del consis ts of L − 1 par ticles o f masses [4] M k = [ k ] M 1 , k = 1 , . . . , L − 1 , (2.3) where we us e d the q -num b er type notation: [ k ] = ω k/ 2 − ω − k/ 2 ω 1 / 2 − ω − 1 / 2 = sin π k L sin π L . (2.4) The mass o f the lig htest particle M 1 is prop ortiona l to µ 1 / 2(1 − b 2 ) . The exact rela tio n betw een M 1 and µ is k nown ex plicitly [17]. The spa ce o f lo ca l op er a tors of the mo del consis ts of the exp onential op erator s V a ( x ) = e Q ( a + ρ ) ϕ ( x ) (2.5) and their descendants, i. e. linear combinations of the fields 1 ( α i 1 ∂ l 1 ϕ ) · · · ( α i r ∂ l r ϕ )( α j 1 ¯ ∂ ¯ l 1 ϕ ) · · · ( α j s ¯ ∂ ¯ l s ϕ )e Q ( a + ρ ) ϕ ( x ) (2.6) 1 W e ignore the p ossible factor ϕ m , since it can b e obtained as the m -fold deriv ative in the v ariable a . 2 for any in tegers r , s ≥ 0 and l 1 , . . . , ¯ l s > 0. The pair of num b ers ( l , ¯ l ), defined as l = r X p =1 l p , ¯ l = s X p =1 ¯ l p , (2.7) is called the level of a desc endant op erator, while the num b ers l and ¯ l se parately will b e referred to as chiral levels. The difference S = l − ¯ l is the Lorentz spin of the o per ator, while the sum D = Q 2 h a + ρ, a + ρ i + l + ¯ l is the scaling dimensio n in the ultraviolet re g ion. The descendant o p er ators with ¯ l = 0 ar e ca lled chiral, while those with l = 0 are called antic hiral. In the ra dial quantization pictur e op era tors a t some point, e. g . x = 0, are put in one-to-one corres p ondence to vectors of some auxiliary vector spa ce. Namely , the field ϕ ( x ) ca n b e expa nded in a kind of La urent series in the vic inity of the p oint x = 0 : 2 α i ϕ ( x ) = Q i − i P i log z ¯ z + X n 6 =0 a in i n z − n + X n 6 =0 ¯ a in i n ¯ z − n , i = 1 , . . . , L − 1 , (2.8) where the op erators Q i , P i , a in , ¯ a in form a Heisenber g a lgebra with the c o mm utation r elations [ P i , Q j ] = − i h α i , α j i , [ a im , a j n ] = m h α i , α j i δ m + n, 0 , [ ¯ a im , ¯ a j n ] = m h α i , α j i δ m + n, 0 . (2.9) The op erator V a (0) corres po nds in this pictur e to the vector | a i rad , such that a in | a i rad = ¯ a in | a i rad = 0 ( n > 0) , P i | a i rad = Q h α i , a + ρ i| a i rad . (2.10) The radial v a cuum vector | v a c i rad = | − ρ i rad corres p onds to the unit op e rator so that | a i rad = e i Q ( a + ρ ) Q | v ac i rad . Up to so me c -num ber facto r s the op erato rs (2.6) corr esp ond to the vectors a i 1 , − l 1 · · · a i r , − l r ¯ a j 1 , − ¯ l 1 · · · ¯ a j s , − ¯ l s | a i rad . (2.11) These vectors span a F o ck mo dule with the highes t weight vector | a i rad , which can b e written as a tensor pro duct F a ⊗ ¯ F a of tw o chiral co mponents. The mo dule F a is the F ock mo dule s pa nned on the vectors (2.11) with ¯ l = 0, while the mo dule ¯ F a is spanned on thos e with l = 0. In tur n, each of the mo dules can be s plit into a sum of the level subspaces . F or example, F a ≃ L ∞ l =0 F a,l , where F a,l is spa nned by the level l vectors. The dimensio ns of the s ubs pa ces F a,l are given by the well-kno wn generating function: ∞ X l =0 q l dim F a,l = ∞ Y m =1 1 (1 − q m ) L − 1 . (2.12) Besides, ther e is a natural isomorphism T : F a → ¯ F a due to the (time-reversal) map a in ↔ ¯ a in . This isomorphism preserves the level: T : F a,l → ¯ F a,l . F or an y vector v ∈ F a ⊗ ¯ F a we define a n op erato r Φ a [ v ]( x ) as the oper ator corr esp onding to the state v | a i rad . The reflection symmetry conjecture [13 –16] declare s the following prop erty . Let W b e the W eyl gro up of the A L − 1 simple Lie alge bra. Then for a gener ic v alue of the parameter a and any element w ∈ W there exists a map R a ( w ) : F ⊗ F → F ⊗ F , such that Φ wa [ v ]( x ) = Φ a [ R a ( w ) v ]( x ) . (2.13) Since this prop erty originates from the conformal field theory , it pre s erves the level of the op era tors: R a ( w ) F al ⊗ ¯ F a ¯ l = F wa, l ⊗ ¯ F wa, ¯ l . Besides, it factorizes in to chiral comp onents: R a ( w ) = r a ( w ) ⊗ T r a ( w ) T − 1 , r a ( w ) : F a → F wa . (2.14) 2 Surely , this expansion only holds f or very small vicinit y of zero, | z | , | z ′ | ≪ M − 1 1 , where the field can be considered as a massless free b oson field. 3 W e shall see b elow, that ther e is ano ther bijection betw een F a and ¯ F a ′ , which is natural fr om the po in t of vie w of the expr essions fo r for m factor s. Let w ∗ ∈ W be the element defined by the relation w ∗ α i = − α L − i ⇔ w ∗ H s = H L +1 − s . (2.15) The W eyl gr oup is known to b e genera ted by the reflections w i such that w i a = a − h a, α i i α i , i = 1 , . . . , L − 1 . In terms of these g enerators the element w ∗ is given by w ∗ = w 1 ( w 2 w 1 )( w 3 w 2 w 1 ) · · · ( w L − 1 w L − 2 · · · w 1 ) . Since w 2 ∗ = 1 , we ca n define a n automo r phism o f the W eyl group e w = w ∗ ww ∗ , (2.16) so that f w i = w L − i . F or generic v a lues of a there exists a bijection T ∗ a : F w ∗ a → ¯ F a , pr eserving the lev el, T ∗ a F w ∗ a,l = ¯ F a,l , such that the factorization prop erty of the reflection op erator reads R a ( w ) = r a ( w ) ⊗ T ∗ wa r w ∗ a ( e w ) T ∗− 1 a . (2.17) Such corresp ondence, seeming utterly artificial from the La grangia n po in t o f view, turns out to be a symmetry of the expressio ns for form factors. Consider the ex po ne ntial fields. Since dim F 0 = 1, we have G − 1 a V a ( x ) = G − 1 wa V wa ( x ) , ∀ w ∈ W , (2.18) where G a = h V a ( x ) i is the v a cuum ex p ectatio n v alue of the o per ator V a ( x ), which is known ex actly [16]. Now we recall Lukyanov’s r e pr esentation [12 ] for form factor s of exp onential o pe r ators in the mo d- els (2.1 ). W e do no t need the deta iled descriptio n in terms of the auxiliar y free field, and we only formulate the result. Let us introduce the vertex op erator s Λ s ( θ ), s = 1 , . . . , L , with the tw o -p o int tr ace functions h h Λ s ( θ ′ )Λ s ( θ ) i i = R ( θ − θ ′ ) , h h Λ s ′ ( θ ′ )Λ s ( θ ) i i = R ( θ − θ ′ ) F  θ − θ ′ + sign( s − s ′ ) i π L  , (2.19) where log R ( θ ) = − 4 Z dt t sh( L − 1) t s h pt s h(1 − p ) t sh 2 Lt ch L ( π − i θ ) t π and F  θ ± i π L  = sh  θ 2 ± i π p L  sh  θ 2 ± i π (1 − p ) L  sh θ 2 sh  θ 2 ± i π L  . (2.20) Below we need generic multipoint trace functions and the normal pro duct : · · · :. W e define b oth by the equations Λ s N ( θ N ) · · · Λ s 1 ( θ 1 ) = :Λ s N ( θ N ) · · · Λ s 1 ( θ 1 ): Y 1 ≤ m 0 \ L Z > 0 . (3.14) 3 In fact, the ov erall sign in the exponent of ω can b e assumed arbitrari ly . Change of this s ign only redefines k i → L − k i . 7 It is s traightforw ard to chec k that, according to (3.9), the element ι n pro duces a common fac to r in all terms in (3.7) r elated to the sums (3 .6 ), resulting in the identit y f ι n g a ( θ 1 , . . . , θ N ) k 1 ...k N = N X m =1 [ k m n ] [ n ] e nθ m ! f g a ( θ 1 , . . . , θ N ) k 1 ...k N , n ∈ Z \ L Z . (3.15 ) for any g ∈ A ⊗ ¯ A . In the factor in parentheses, one can reco g nize the eigenv alue of the (appropria tely normalized) spin n in tegral of motion I n . It means that V ι n g a ( x ) = [ V g a ( x ) , I n ] . (3.16) This result is consistent with [1 8, 19 ]. Note also that the set of the integrals of motion I n is the natural deformation of the set of the integrals of mo tion in the mas sless (confo r mal) limit, and the commutators (3.16) add the v alue n to the spin of the o pe r ator and the v a lue | n | to the c hiral (antic hiral) level of the des c e ndant op erator for n > 0 ( n < 0 ). It means that if we could identify some o per ator V g a with a desc e nda n t op era tor in the Lagra ngian formulation, we would hav e a la r ge subspace of F a ⊗ ¯ F a that consists of the o per ators generated fr o m V g a by the integrals of motion ide ntified. 3.2. F actorization pr op erty Let h, h ′ ∈ A . Co nsider the Λ → ∞ asy mptotics of the function f h ¯ h ′ a ( θ 1 , . . . , θ M , θ M +1 + Λ , . . . , θ N + Λ) k 1 ...k M k M +1 ...k N . It is easy to chec k that F ( θ ± Λ) , R ( θ ± Λ) → 1 in this limit. Let s i = ( s (1) i , . . . , s ( k i ) i ) b e sets of integers 1 ≤ s (1) i < · · · < s ( k i ) i ≤ L . F rom (3.9) we immediately g et ( a s 1 ( x 1 ) . . . a s M ( x M ) a s M +1 ( x M +1 e Λ ) . . . a s N ( x N e Λ ) , h ) = ( a s M +1 ( x M +1 e Λ ) . . . a s N ( x N e Λ ) , h ) , ( a s 1 ( x 1 ) . . . a s M ( x M ) a s M +1 ( x M +1 e − Λ ) . . . a s N ( x N e − Λ ) , h ) = ( a s 1 ( x 1 ) . . . a s M ( x M ) , h ) for h ∈ A . Finally , we obtain the following asymptotic factorization pr op ert y [20]: f h ¯ h ′ a ( θ 1 , . . . , θ M , θ M +1 + Λ , . . . , θ N + Λ) k 1 ...k M k M +1 ...k N = f h a ( θ M +1 + Λ , . . . , θ N + Λ) k M +1 ...k N f ¯ h ′ a ( θ 1 , . . . , θ M ) k 1 ...k M as Λ → + ∞ . (3.17) This pro p er ty mea ns that it is alwa ys p oss ible to extrac t chiral parts of desc e ndant s using this limit. Roughly spe aking, the high velo city right moving pa r ticles only k now ab out the chiral part of a lo cal op erator, while the high velocity left moving particles only know ab out its antic hir al part. F rom the factorization prop erty together with (3.11) w e immediately obtain that if the reflection prop erty holds, it should pos sess the factor ized form (2.1 7). In other words, if it is p oss ible define an action of R a ( w ) on the a lgebra A ⊗ ¯ A such tha t V R a ( w ) h ¯ h ′ a ( x ) = V h ¯ h ′ a ( x ), we necessarily hav e V h ¯ h ′ a ( x ) = V ( r a ( w ) h )( r w ∗ a ( e w ) h ′ ) a ( x ) , (3.18) where r a ( w ) is the restriction of R a ( w ) on A ≃ A ⊗ ¯ A 0 . The p o ssibility to define such a ction w ill b e prov en in Sec. 6. Another co nsequence concerns the identification of op era tors. F o r any h ∈ A n and h ′ ∈ A ¯ n define the op erator V h ¯ h ′ a ( x ) = M n + ¯ n 1 V h ¯ h ′ a ( x ). T he o per ators V h a and V ¯ h ′ a are level ( n, 0) and level (0 , ¯ n ) descendants corres p onding ly . Mor eov e r , they must b e linear combinations of the vectors o f the form (2.6) with µ - independent co efficients. Hence, their form factors are prop or tional to G a M n 1 ∝ M Q 2 h ρ + a,ρ + a i + n 1 and G a M ¯ n 1 ∝ M Q 2 h ρ + a,ρ + a i + ¯ n 1 . It follo ws fro m (3.17) and (3 .8 ) that the leading term in the op erator V h ¯ h ′ a is prop ortiona l to G a M n + ¯ n 1 ∝ M Q 2 h ρ + a,ρ + a i + n + ¯ n 1 . It mea ns that the op era tor V h ¯ h ′ a is a nonzero le vel ( n, ¯ n ) descendant plus some op erato rs of lesser dimensions. 8 3.3. Desc endants c ounting W e wan t to prov e that, for generic v alues of a , the op er a tors V g a with differen t v alues of g differ. F or this purp ose we fir st prove this fact for a particular asymptotics in a . Then we apply the deformation argument. Let a ( τ ) = Lτ 2 π i ρ, a ( τ ) H s = Lτ 2 π i  L + 1 2 − s  . W e shall consider the limit τ → + ∞ . Evidently , 2 π i Lτ a ( τ ) H 12 ...k = k ( N − k ) 2 , 2 π i Lτ a ( τ ) H s 1 s 2 ...s k < k ( N − k ) 2 for s k > k . Therefore e − τ k ( L − k ) 2 T k ( z ) | ˆ a = a ( τ ) = Λ 12 ...k ( z ) + O (e − τ ) as τ → + ∞ . (3 .1 9) Then the functions J g N ,a hav e the following a symptotics: e − τ 2 P N i =1 k i ( L − k i ) J g N ,a ( x 1 , . . . , x N ) k 1 ...k N    τ →∞ = P g ( X 1 | . . . | X L − 1 ) , X k = { x i | k i = k } . (3.20) The functions P g are p olyno mials defined by the following relations: P g 1 g 2 = P g 1 P g 2 , P C 1 g 1 + C 2 g 2 = C 1 P g 1 + C 2 P g 2 ( ∀ g 1 , g 2 ∈ A , C 1 , C 2 ∈ C ) . (3.21a) P α i c − n ( X 1 | . . . | X L − 1 ) = ω i − 1 2 S n ( X i ) + L − 1 X k = i +1 ω i − k 2 (1 − ω ) S n ( X k ) , (3.21b) P α L − i ¯ c − n ( X 1 | . . . | X L − 1 ) = − ω − i − 1 2 S − n ( X i ) + L − 1 X k = i +1 ω k − i − 1 2 (1 − ω ) S − n ( X k ) , (3.21c) Here S n ( x 1 , . . . , x N ) = N X i =1 x n i . The pro ducts o f p olyno mia ls S n for n > 0 of a given power for m a basis of the symmetric p olynomials of the resp ective p ower for a la rge eno ugh num ber of v a riables. First, consider the case g ∈ A . Since we ar e interested in the whole colle c tio ns of form factors rather than the fo rm factor s for particula r num b ers of particles, we may consider the functions z in = S n ( X i ), i = 1 , . . . , L − 1 as indep enden t v aria bles. The equation (3.21b) defines a map from the algebra A to the algebra o f p olynomia ls in the v ariables z in . This map is invertible. Indeed, Eq. (3.21b) makes it p ossible to expr ess any monomial z in in ter ms of P α i c − n and the monomia ls z j n , j > i . Applying it recursively we can expr ess any monomial z in in ter ms of a linea r combination of the po lynomials P α j c − n with j ≥ i . Hence, z in = P g in , where g in = P L − 1 j = i A j α j c − n with some uniquely defined co efficients A j . This defines a ma p from the a lgebra o f p olynomia ls in the v a riables z in to the algebra A . This pro v es that differen t elements g 1 6 = g 2 ∈ A pro duce different po lynomials P g 1 6 = P g 2 of the v ariables z kn . Since the form factors ar e analytic functions of the v ariable a , these element s pro duce different collections of form factors f g 1 a 6 = f g 2 a . Now supp ose that g = P h i ¯ h ′ i , where { h i } , { h ′ i } ⊂ A l are sets o f linearly indep endent elemen ts. Suppo se that f g a = 0 . Then due to the fac to rization pr o pe r ty the c o m bination X f h i a ( θ 1 , . . . , θ M ) f ¯ h ′ i a ( θ ′ 1 , . . . , θ ′ N ) = 0 . This contradicts to the linear independenc e of the form factors f h i for generic a . This prov es the Theorem 1 F or generic values of a t he line ar m ap g 7→ f g a fr om A ⊗ ¯ A int o the sp ac e of c ol le ctions of functions is an inje ction, i. e. it is invertible as a map onto its image. As an immedia te cons e q uence we have the 9 Prop ositio n 1 F or generic values of a the dimension of the sp ac e of t he op er ators V g a with g ∈ A l ⊗ ¯ A ¯ l is e qual to the dimension of the c orr esp onding s u bsp ac e of the F o ck sp ac e dim( F l ⊗ F ¯ l ) = dim F l · dim F ¯ l . The dimensions of the sp ac es of t he op er ators, V g a with g ∈ A l or g ∈ ¯ A l ar e e qual to that of the su bsp ac e, dim F l . F or chiral (antic hiral) op era to rs Pr op osition 1 mea ns that the conjecture that the chiral (antic hiral) descendants ar e the op erators V g a ( x ) with g ∈ A ( g ∈ ¯ A ) is consistent with the op era tor counting from the bo sonic picture (2.6), (2.7). 4. The s tri pp ed b osonization T o prov e the r e fle c tion prop erty for the descendant o pe r ators we shall need a free field r epresentation of the functions J g N ,a , which a re o btained from the functions f g a by stripping o ut the R facto r s. W e shall call it stripp ed b o sonization. This b os onization differs fro m that desc r ibe d in [1 1 , 1 2] in that, firs t, the Heisenberg algebra is gener ated by a countable set o f ele ments rather than a contin uous o ne a nd, second, the functions J g N ,a for all g ∈ A ⊗ ¯ A are expre s sed in terms of matrix elements rather than tra ces. The price paid for these adv antages is that the residue of the kinematic p ole is not a c -n umber, but a new vertex op erator. W e sha ll se e b elow tha t this new vertex op erator will b e a n imp ortant ingredient of our pro of. Consider the Heisenberg algebra with the generato r s d ( s ) n , s = 1 , . . . , L , n ∈ Z , n 6 = 0 and the commutation relations [ d ( s ) m , d ( s ) n ] = 0 , [ d ( s ′ ) m , d ( s ) n ] = mδ m + n, 0 A sign( s ′ − s ) n ( s ′ 6 = s ) (4.1) with A ± n = ( ω ∓ pn − ω ∓ n )(1 − ω ± pn ) . (4.2) Note that A − n = A + − n = ω n A + n . (4.3) Let ˆ a be a n additional central e lemen t. Define the v acuums | 1 i a and a h 1 | by the relations d ( s ) n | 1 i a = 0 , ˆ a | 1 i a = a | 1 i a , a h 1 | d ( s ) − n = 0 , a h 1 | ˆ a = a h 1 | a, a h 1 | 1 i a = 1 ( n > 0) (4.4) and let : · · · : b e the corr esp o nding norma l ordering op eration. W e sha ll als o wr ite h . . . i a ≡ a h 1 | . . . | 1 i a . The F o ck space generated by the op erator s d ( s ) n , n > 0, from the v acuum a h 1 | will b e denoted a s D R a , while that gener ated by d ( s ) − n , n > 0, fro m the v acuum | 1 i a will b e denoted as D L a . They admit a natural grading D R a = L ∞ n =0 D R a,n , D L a = L ∞ n =0 D L a,n so that D R a,m d ( s ) n ⊆ D R a,m + n , d ( s ) n D L a,m ⊆ D L a,m − n . Let us intro duce the vertex op erators λ s ( z ) = exp X n 6 =0 d ( s ) n n z − n . (4.5) Note that the ex p onential in the r. h. s . do es no t need any normal ordering due to c o mm utativity of all elements d ( s ) n with a g iven s . It is ea s y to chec k that λ s ( z ′ ) λ s ( z ) = : λ s ( z ′ ) λ s ( z ): , λ s ′ ( z ′ ) λ s ( z ) = λ s ( z ) λ s ′ ( z ′ ) = f  z z ′  : λ s ′ ( z ′ ) λ s ( z ): , s ′ > s, z z ′ 6 = 1 , ω . (4.6) Here f ( z ) = F  log z − i π L  = ( z − ω p )( z − ω 1 − p ) ( z − 1)( z − ω ) , f ( z ) = f ( ω / z ) . (4.7) Let λ s 1 ...s k ( z ) = : k Y m =1 λ s m  z ω k +1 − 2 m 2  : , 1 ≤ s 1 < · · · < s k ≤ L . (4.8) 10 Note that the vertex op erator λ 12 ...L ( z ) is no t equal to one and plays an impo rtant role below. Its impo rtant prop erty is λ 12 ...L ( z ) λ s ( x ) = L − 1 Y m =1 f  z x ω L +1 − 2 m 2  : λ 12 ...L ( z ) λ s ( x ): . (4.9) It is neces sary to stress tha t the c o e fficien t in the r. h. s . is s indep endent. Now define the stripp ed W alg ebra cur r ents t k ( z ) = X 1 ≤ s 1 < ··· 0 . Then ˜ J h ¯ h ′ N ,a ( x 1 , . . . , x N ) k 1 ,...,k N = J π − 1 LR ◦ π RL ( h ¯ h ′ ) N ,a ( x 1 , . . . , x N ) k 1 ,...,k N . (4.17 ) More explicitly , ta ke the pro duct π R ( h ) π L ( h ′ ) and push these tw o factor s through each other. W e g e t a combination of the form π R ( h ) π L ( h ′ ) = X i π L ( h ′ i ) π R ( h i ) . Then ˜ J h ¯ h ′ N ,a ( x 1 , . . . , x N ) k 1 ,...,k N = X i J h i ¯ h ′ i N ,a ( x 1 , . . . , x N ) k 1 ,...,k N . 11 The most imp ortant feature of this expressio n is that the function ˜ J h ¯ h ′ with h ∈ A l , h ′ ∈ A ¯ l is expre s sed in terms o f the functions J h i ¯ h ′ i with h i ∈ A l i , h ′ i ∈ A ¯ l i such that l i ≤ l , ¯ l i ≤ ¯ l and vice versa. It mea ns, in particula r , that the fa ctorization pr o pe r t y (3.17) holds as well for the for m fac to rs co rresp onding to the ˜ J functions. These ‘physical’ vectors form the ‘ph ysical’ subspaces in the spaces D R a and D L a : D R, phys a,n = n a h h |    h ∈ A n o , D R, phys a = ∞ M n =0 D R, phys a,n , D L, phys a,n = n | h i a    h ∈ A n o , D L, phys a = ∞ M n =0 D L, phys a,n . (4.18) Evidently , dim D R, phys a,n = dim D L, phys a,n = dim F n . (4.19) It is conv enie nt to find a definition of the subspaces D R, phys , D L, phys as kernels of some op er ators. Int ro duce a s e t o f op era tors D n ( n 6 = 0) s uc h that [ D n , π R ( h )] = [ D n , π L ( h )] = 0 . (4.20) They a re g iven b y D n = L X s =1 ω − L +1 − 2 s 2 n d ( s ) n . (4.21) These e lemen ts commute with each other: [ D m , D n ] = 0 . (4.22) Note that λ 1 ...L ( z ) = exp X n 6 =0 D n z − n n . (4.23) Let a h U | , | V i a be some states from the F o ck mo dules ov er a h 1 | , | 1 i a . It can b e shown tha t a h U | D − n = 0 ∀ n > 0 ⇔ ∃ h ∈ A : a h U | = a h 1 | π R ( h ) , D n | V i a = 0 ∀ n > 0 ⇔ ∃ h ∈ A : | V i a = π L ( h ) | 1 i a . (4.24) Therefore, the ‘ph ysical’ subspaces can b e also defined as D R, phys a,n = n a h v | ∈ D R a    a h v | D − m = 0 ∀ m > 0 o , D L, phys a,n = n | v i a ∈ D L a    D m | v i a = 0 ∀ m > 0 o , (4.25) 5. Recurren t relations and reflection prop erty for exp onent ial op erators Our first step in the pro of of the r eflection prop erty is to prov e it for the form factors of the ex po nen tial op erators V a ( x ). Since the express ions (2.2 5), (4.1 1) are not explicitly inv ariant under the transforma tio ns of the W ey l group, we need a nother representation for the J N ,a functions. It turns out that such a W eyl inv ar iant r epresentation can be found in the for m of a recurrent relation in the num b er of par ticles N starting for m the explicitly in v aria nt expression for N = 0. In this section we set I = (1 , . . . , N ), X = ( x 1 , . . . , x N ). Besides, we us e the notation ˆ I n = I \ { n } , ˆ X n = X \ { x n } . It follows from Eq. (5.5) be low that any function J N ,a ( . . . ) k 1 ...k N can b e expressed in terms of the function J P k i ,a ( . . . ) 1 ... 1 . Hence, it is sufficient to obtain a r ecurrent r elation for an y subset of the J functions that contains the functions with k 1 = · · · = k N = 1. W e choose the subset that co nsists of the functions with an arbitrary k 1 and with fixed k 2 = · · · = k N = 1. Namely , co ns ider the function J k,N +1 ,a ( z ; X ) = N Y n =1 k − 1 Y m =1 f − 1  z x i ω k +1 − 2 m 2  h t k ( z ) t 1 ( x 1 ) . . . t 1 ( x N ) i a , (5.1) 12 which will b e cons idered as an ana lytic function of the v ariable z , while the o ther v ar iables X will be considered as parameter s. The prefactor in this expr ession cancels r edundant p oles. Indeed, co nsider the pr o duct of tw o vertex op erators t k ( z ) t 1 ( x ). It po ssesses four simple (see Appe ndix A) p oles at the p oints z = xω ± k +1 2 , xω ± k − 1 2 , (5.2) ( k − 2) double p oles at the p oints z = xω k +1 − 2 m 2 , m = 2 , . . . , k − 1 , (5.3) and 2 ( k − 1) simple zeros at the p oints z = xω ± ( p − k +1 − 2 m 2 ) m = 1 , . . . , k − 1 . (5.4) Two more simple ze ros hav e no fixed p o sition and dep e nd on a matrix element. The pro duct k − 1 Y m =1 f − 1  z x ω k +1 − 2 m 2  = k − 1 Y m =1 ( z − xω k +1 − 2 m 2 )( z − xω − k +1 − 2 m 2 ) ( z − xω p − k +1 − 2 m 2 )( z − xω − p + k +1 − 2 m 2 ) just cancels all fixe d zeros and all poles except thos e at z = xω ± ( k +1) / 2 . Thus, the o nly p oles of the pro duct k − 1 Y m =1 f − 1  z x ω k +1 − 2 m 2  t k ( z ) t 1 ( x ) are lo cated at z = xω ± ( k +1) / 2 , where the residues are pr op ortional to t k +1 ( x ): Res z = xω ± k +1 2 k − 1 Y m =1 f − 1  z x ω k +1 − 2 m 2  t k ( z ) t 1 ( x ) = ± xω ± k +1 2 κ p t k +1 ( xω ± k 2 ) , (5.5) where κ p = ( ω 1 − p − 1)( ω p − 1) ω − 1 = 2i sin π p L sin π (1 − p ) L sin π L . (5.6) It means that the prefactor in (5.1) es sentially reduces the num b er of p oles of the resulting function. The case k = L − 1 is a s pecia l one since bo th p oles coincide: xω ± L/ 2 = − x . F r om the physical p oint of view, it corresp onds to a kinema tica l ra ther tha n a dyna mical p ole . T he res idue is g iven by Res z = − x L − 2 Y m =1 f − 1  − z x ω − m  t L − 1 ( z ) t 1 ( x ) = xκ p  t L ( − xω 1 / 2 ) − t L ( − xω − 1 / 2 )  . (5.7) It is imp ortant that t L ( z ) canno t b e extracted from the op er ator pro ducts of t k for k < L . In s pite of its apparent triviality , it is a new s ubs ta nce and this fact will b e esse ntially used in the next section. The prefac to r in (5.1 ) makes the analytic structure of the function J k,N +1 ,a ( z ; X ) to b e simple. Its only dynamic p oles are lo cated at z = x i ω ± ( k +1) / 2 . The equatio n (5.5) makes it p ossible to calculate residues at these p o les: Res z = x n ω ± k +1 2 J k,N +1 ,a ( z ; X ) = ± x n ω ± k +1 2 R ± N ,n ( X ) J k +1 ,N , a ( x n ω ± k 2 ; ˆ X n ) , (5.8) where R ± N ,n ( X ) = κ p Y m ∈ ˆ I n f  x m x n  ± 1  . (5.9) F or the sp ecial case k = L − 1 we hav e Res z = − x n J L − 1 ,N +1 ,a ( z ; X ) = x n  R − N ,n ( X ) − R + N ,n ( X )  J 1 ,N − 1 ,a ( ˆ X n ) , (5.10) 13 since J L,N , a ( x ; ˆ X n ) = J 1 ,N − 1 ,a ( ˆ X n ) due to o (4.9). F o rtunately , there is no need to c o nsider this ca se separately while deriving the recur r ent relations. It will be taken in to acc ount implicitly b y the cyclic prop erty (5 .22) b elow. One c an sepa rate the p ole co n tributions from the r egular pa rt: J k,N +1 ,a ( z ; X ) = J ( ∞ ) k,N +1 ,a ( z ; X ) + N X n =1 x n ω k +1 2 z − x n ω k +1 2 R + N ,n ( X ) J k +1 ,N , a ( x n ω k 2 ; ˆ X n ) − N X n =1 x n ω − k +1 2 z − x n ω − k +1 2 R − N ,n ( X ) J k +1 ,N , a ( x n ω − k 2 ; ˆ X n ) , (5.11 ) where the function J ( ∞ ) k,N +1 ,a ( z ; X ) is r egular in the v ar iable z everywhere except the points z = 0 and z = ∞ . Note, that the sum o ver the po les is of the or de r O ( z − 1 ) as z → ∞ . Th us , the a symptotic behavior of the function J k,N +1 ,a ( z ; X ) as a function of z is governed b y J ( ∞ ) k,N +1 ,a ( z ; X ): J k,N +1 ,a ( z ; X ) − J ( ∞ ) k,N +1 ,a ( z ; X ) = O ( z − 1 ) as z → ∞ . (5.12) W e may use another expansio n J k,N +1 ,a ( z ; X ) = J (0) k,N +1 ,a ( z ; X ) − N X n =1 x − 1 n ω − k +1 2 z − 1 − x − 1 n ω − k +1 2 R + N ,n ( X ) J k +1 ,N , a ( x n ω k 2 ; ˆ X n ) + N X n =1 x − 1 n ω k +1 2 z − 1 − x − 1 n ω k +1 2 R − N ,n ( X ) J k +1 ,N , a ( x n ω − k 2 ; ˆ X n ) , (5.13) where the function J (0) k,N +1 ,a ( z ; X ) is a gain r egular ev erywhere exce pt z = 0 , ∞ . It is eviden t that the behavior of the function J k,N +1 ,a ( z , X ) in the vicinity of the po int z = 0 is gov erned by J (0) k,N +1 ,a ( z ; X ): J k,N +1 ,a ( z ; X ) − J (0) k,N +1 ,a ( z ; X ) = O ( z ) a s z → 0 . (5.14) W e shall use the notatio n D k,N , a ( X ) = N X n =1 R + N ,n ( X ) J k +1 ,N , a ( x n ω k 2 ; ˆ X n ) − N X n =1 R − N ,n ( X ) J k +1 ,N , a ( x n ω − k 2 ; ˆ X n ) . ( 5.15) F rom the definitions (5.1 1), (5.1 3) it is easy to derive that J (0) k,N +1 ,a ( z ; X ) − J ( ∞ ) k,N +1 ,a ( z ; X ) = D N ,a ( X ) . (5.16) This relation shows that the bo th functions are nea r ly the same except the zer o mo de in z . I t means that it is sufficient establish the singular pa rts of J (0) and J ( ∞ ) in the vicinit y of z = 0 and z = ∞ corres p onding ly and the zero mo de o f o ne o f them. What ha v e been said above p ertains equally to form fac to rs of ex p one ntial a nd descendan t oper- ators. Now we w an t to restrict ourselves to the exponential op era tors. In order to fix the functions J ( ∞ ) k,N +1 ,a ( z ; X ) and J (0) k,N +1 ,a ( z ; X ) we hav e to calcula te the asymptotics of the function J k,N +1 ,a ( z ; X ) as z → 0 and z → ∞ . Since f (0) = f ( ∞ ) = 1, it is finite in these limits a nd we hav e J (0) k,N +1 ,a ( z ; X ) = J ( ∞ ) k,N +1 ,a ( z ; X ) = K k,a J 1 ,N ,a ( X ) , (5.17) where K k,a = J k, 1 ,a (0) = J k, 1 ,a ( ∞ ) = X 1 ≤ s 1 < ··· i + 1, h w i a, H s 1 ...s m ...s k i = h a, H s 1 ...,s m − 1 ,...s k i , if s m = i + 1, s m − 1 < i , h w i a, H s 1 ...s k i = h a, H s 1 ...s k i otherwise . Thu s, any simple reflection acts as a p ermutation of the s et {h a, H s 1 ...s k i} of functions of the v aria ble a . Since the sum in (5.18 ) runs ov er the whole set, the function K k,a is inv a riant under simple W eyl reflections. As a n ex a mple o f an application of the recurrent relations (5.20)—(5.22) w e pr ov e the eq ua tion o f motion fo r qua n tum fields in App endix B. 6. Reflection prop e rt y for de scendan t op erators The pro of o f the re fle c tio n prop er ty rep eats in the main features that for the sine/sinh-Gor don mo del [9]. The idea o f the pro o f stems from the conjecture of [21], developed further in [22], that a ll form factor s can b e o btained from those o f the pr imary o per ators a s co efficients o f larg e θ expansions . Theorem 4 F or generic values of a ther e exists a r epr esentation of the Weyl gr oup r a on the algebr a A such that for any h, h ′ ∈ A the e quation holds ˜ J h ¯ h ′ N ,a ( x 1 , . . . , x N ) k 1 ...k N = ˜ J ( r a ( w ) h )( r w ∗ a ( e w ) h ′ ) N ,w a ( x 1 , . . . , x N ) k 1 ...k N . (6.1) The firs t step of the pro of is to prov e that the whole F o ck space of the Heisenberg alg ebra (4.1) can be s panned on vectors cre a ted by pro ducts o f the op er ators t k ( x ), 1 ≤ k ≤ L . More prec is ely , conside r the expansion a h 1 | t k 1 ( ξ − 1 1 z ) · · · t k K ( ξ − 1 K z ) = ∞ X n =0 z − n a h n ; k 1 ξ 1 , . . . , k K ξ K | . (6.2) F or shortness, we shall write Ξ = ( k 1 , ξ 1 , . . . , k K , ξ K ). W e wan t to prov e that for gener ic v alues of a and large eno ug h v alues of K o ne can choo se a se t Ξ ( i ) , i = 1 , . . . , dim D R n , s uc h that the vectors a h n ; Ξ ( i ) | form a basis in the space D R n . Firs t, let us prove it in the limit a = a ( τ ), τ → + ∞ a lready used in Subsection 3.3. In this limit e − τ k ( L − k ) 2 t k ( z ) | ˆ a = a ( τ ) = λ 12 ...k ( z ) + O (e − τ ) as τ → ∞ , (6.3) which is the full a nalog o f (3.19). Therefore e − τ 2 P K i =1 k i ( L − k i ) a ( τ ) h 1 | t k 1 ( ξ − 1 1 z ) · · · t k K ( ξ − 1 K z ) = a ( τ ) h 1 | λ 1 ...k 1 ( ξ − 1 1 z ) · · · λ 1 ...k K ( ξ − 1 K z ) + O (e − τ ) = F ( ξ 2 /ξ 1 , . . . , ξ K /ξ 1 ) a ( τ ) h 1 | : λ 1 ...k 1 ( ξ − 1 1 z ) · · · λ 1 ...k K ( ξ − 1 K z ): + O (e − τ ) , (6.4) where F ( z 2 , . . . , z K ) is a pro duct of functions f of a ppropriate arguments. The particular fo rm of this pro duct is not essential fo r o ur pur p os es. Let us calculate the state in the last line: a ( τ ) h 1 | : λ 1 ...k 1 ( ξ − 1 1 z ) · · · λ 1 ...k K ( ξ − 1 K z ): = a ( τ ) h 1 | exp ∞ X n =1 L X s =1 κ ( s ) n d ( s ) n z − n n , (6.5) where κ ( s ) n = K X i =1 k i ≥ s ω k i +1 − 2 s 2 n ξ n i . (6.6) Consider the expansion a ( τ ) h 1 | : λ 1 ...k 1 ( ξ − 1 1 z ) · · · λ 1 ...k K ( ξ − 1 K z ): = ∞ X n =0 z − n ( − ) h n ; Ξ | . 16 Then ( − ) h n ; Ξ | = a ( τ ) h 1 | n X r =1 X n 1 ,...,n r > 0 n 1 + ··· + n r = n C n 1 ...n r r Y j =1 L X s =1 κ ( s ) n j d ( s ) n j (6.7) with some po sitive c o nstants C n 1 ...n r . It means that all p ossible pro ducts of d ( s ) n ent er the r. h. s . F or large enough car dinal num ber s # { i | k i = s } , 1 ≤ s ≤ L , the functions κ ( s ) n ′ , 1 ≤ s ≤ L , 1 ≤ n ′ ≤ n are functiona lly independent a nd ca n b e considered a s indep endent v ariables. Besides, the monomials κ ( s 1 ) n 1 · · · κ ( s r ) n r are line a rly independent. Hence, for an y nonzero s e t of num b ers A s 1 ...s r n 1 ...n r , r = 1 , . . . , n , n 1 , . . . , n r > 0, n 1 + · · · + n r = n , we have X r X s 1 ,...,s r n 1 ,...,n r A s 1 ...s r n 1 ...n r κ ( s 1 ) n 1 · · · κ ( s r ) n r 6 = 0 for so me v alues of the v ar iables κ ( s ) n ′ . Therefore, the vector genera ted by the n umbers A s 1 ...s r n 1 ...n r is not orthogo nal to at least one vector gener ated by the num b ers κ ( s 1 ) n 1 · · · κ ( s r ) n r . It means that there is no vector in D R n orthogo nal to all of the v ectors generated by the pro ducts o f κ ( s j ) n j . It pr ov es that there exists a bas is ( − ) h l ; Ξ ( I ) | in the space D R l . No w the deformatio n argument pr ov es the sa me statement for generic v alues of a . Now we b egin the seco nd step o f the pro of. Let a h l ; I | = a h l ; Ξ ( I ) l | be a basis in the space D R l related to any particular set o f v alues o f the par ameters { Ξ ( I ) l | I = 1 , . . . , dim D R l } . F rom the reflection pro per t y for the exp onential o pe r ators (5.2 6) we immediately conclude tha t for any no nnegative integers l , ¯ l w e hav e a h l ; I | t k 1 ( x 1 ) . . . t k N ( x N ) | ¯ l ; J i a = wa h l ; I | t k 1 ( x 1 ) . . . t k N ( x N ) | ¯ l ; J i wa ∀ w ∈ W . This iden tit y provides a map r a ( w ) : D R l → D R l such that r a ( w )( a h l ; I | ) = wa h l ; I | . Note that the left subscript a at the vectors is essential, s inc e the ele ment of D R l generated by Ξ ( I ) l depe nds on its v alue. Now o ur aim is to prove that this ma p is consis ten t with the restriction (4.24), w hich selects ‘physical’ states generated by (4.12). Let the v ectors a h 1 | π R ( h a,l,µ ) = a h f l ; µ | = P I v µ I ( a ) a h l ; I | form a basis in the subspace D R, phys a,l . Similarly , let π L ( h ′ a, ¯ l,ν ) | 1 i a = | f ¯ l , ν i a = P J ¯ v ν J | ¯ l, J i a . Due to (4.23) we hav e a h 1 | t k 1 ( x 1 ) . . . t k M ( x M ) D n t k M +1 ( x M +1 ) . . . t k N ( x N ) | 1 i a = wa h 1 | t k 1 ( x 1 ) . . . t k M ( x M ) D n t k M +1 ( x M +1 ) . . . t k N ( x N ) | 1 i wa . W e hav e 0 = a h f l ; µ | D − n | l − n ; J i a = X I v µ I ( a ) a h l ; I | D − n | l − n ; J i a = X I v µ I ( a ) wa h l ; I | D − n | l − n ; J i wa . Therefore, X I v µ I ( a ) wa h l ; I | D − n = 0 and ther e exists an ele ment h w wa, l,µ such that wa h 1 | π R ( h w wa, l,µ ) = X I v µ I ( a ) wa h l ; I | . Similarly , there exis ts an element h ′ w wa, ¯ l,ν such that π L ( h ′ w wa, ¯ l,ν ) | 1 i wa = X J ¯ v ν J ( a ) | ¯ l ; J i wa . Finally , we find 17 h π R ( h a,l,µ ) t ( x 1 ) . . . t ( x N ) π L ( h ′ a, ¯ l,ν ) i a = a h f l ; µ | t k 1 ( x 1 ) . . . t k N ( x N ) | f ¯ l , ν i a = X I ,J v µ I ( a ) ¯ v ν J ( a ) a h l ; I | t k 1 ( x 1 ) . . . t k N ( x N ) | ¯ l, J i a = X I ,J v µ I ( a ) ¯ v ν J ( a ) wa h l ; I | t k 1 ( x 1 ) . . . t k N ( x N ) | ¯ l , J i wa = h π R ( h w wa, l,µ ) t k 1 ( x 1 ) . . . t k N ( x N ) π L ( h ′ w wa, ¯ l,ν ) i wa . Thu s, we obtained the map r a ( w ) on the s ubspaces D R, phys a , D L, phys : r a ( w )( a h 1 | π R ( h a,l,µ )) = wa h 1 | π R ( h w wa, l,µ ) , r a ( w )( π L ( h ′ a, ¯ l,ν ) | 1 i a ) = π L ( h ′ w wa, ¯ l,ν ) | 1 i wa . Comparing with the pr op erty (3.11), which holds for the ˜ J functions as well as for the J ones, we ca n define r a ( w ) h a,l,µ = h w wa, l,µ , r w ∗ a ( e w ) h ′ a, ¯ l,ν = h ′ w wa, ¯ l,ν . Again this prov es the factorized form (2.17) of the reflection map. An alternative construction. It is easy to derive the commutation rela tion [ D n , t k ( z )] = A n [ k n ] [ n ] z n t k ( z ) , (6.8) where the co efficients A n = − A + n L X s =2 ω − L +1 − 2 s 2 n = ( ( − ) n ω n/ 2 A + n , n 6∈ L Z , (1 − L ) A + n , n ∈ L Z . are all nonzero fo r ir rational v alues o f p . Besides, the ratio [ k n ] [ n ] = k X s =1 ω k +1 − 2 s 2 n is well defined (and equal to k ) for n ∈ L Z . The co mm utation r elation (6 .8) mea ns that the pro duct (6 .2 ) sa tis fy the rela tions a h 1 | t k 1 ( ξ − 1 1 z ) . . . t k K ( ξ − 1 K z ) D − n = 0 , 1 ≤ n ≤ l , (6.9) sub ject to the equa tions K X m =1 [ k m n ] [ n ] ξ n m = 0 , 1 ≤ n ≤ l, (6.10) are satisfied. Hence, a h n ; k 1 ξ 1 , . . . , k K ξ K | D − n ′ = 0 , 1 ≤ n ≤ l, n ′ ≥ 1 . (6.11) If w e a lso define t ¯ k 1 ( η 1 z ) . . . t ¯ k ¯ K ( η K z ) | 1 i a = ∞ X n =1 z n | n ; ¯ k 1 η 1 , . . . , ¯ k ¯ K η ¯ K i a , (6.12) we have D n ′ | n ; ¯ k 1 η 1 , . . . , ¯ k ¯ K η ¯ K i a = 0 , 1 ≤ n ≤ ¯ l, n ′ ≥ 1 , (6.13) sub ject to the equa tions ¯ K X m =1 [ ¯ k m n ] [ n ] η n m = 0 , 1 ≤ n ≤ ¯ l, (6.14) are satisfied. W e conclude that these vectors pr o duce the W ey l in v aria nt matrix elements, a h n ; k 1 ξ 1 , . . . , k K ξ K | t κ 1 ( x 1 ) . . . t κ M ( x M ) | n ′ ; ¯ k 1 η 1 , . . . , ¯ k ¯ K η ¯ K i a = wa h n ; k 1 ξ 1 , . . . , k K ξ K | t κ 1 ( x 1 ) . . . t κ M ( x M ) | n ′ ; ¯ k 1 η 1 , . . . , ¯ k ¯ K η ¯ K i wa , w ∈ W , (6.15) which are form fac tors of so me descenda n t op erators for 1 ≤ n ≤ l , 1 ≤ n ′ ≤ ¯ n . 18 Theorem 5 F or generic values of the p ar ameter a the ve ctors a h n ; k 1 ξ 1 , . . . , k K ξ K | with the c ondi- tion (6.10) sp an the whole sp ac e D R, phys n for 0 ≤ n ≤ l , while the ve ctors | n ; ¯ k 1 η 1 , . . . , ¯ k ¯ K η ¯ K i a with (6.14) sp an the sp ac e D L, phys n for 0 ≤ n ≤ ¯ l . Indeed, consider the bra-vectors. Sub ject to the condition (6.10 ), the co efficients κ ( s ) n satisfy the equation L X s =1 κ ( s ) n = 0 . It mea ns that the r. h. s. o f Eq . (6.7) only co n tains differences of d ( i +1) n − d ( i ) n as it must be . Besides, this r. h. s. only dep e nds o n L − 1 parameter s for a given n , e. g. κ ( i ) n , i = 1 , . . . , L − 1. Now the same argumentation p ers uades us that these remaining κ s ma y b e considered a s indep endent v aria bles, a nd the same argumentation prov es that there is no vector orthog onal to the se t of mono mials. Then the same deformation argument should b e applied. W e a rrive to the Theorem 6 F or any l ther e exists an analytic in a family of sets { h inv a,l,µ ∈ A l } dim A l µ =1 , which ar e b ases in A l for generic values of a , such that r a ( w ) h inv a,l,µ = h inv wa, l,µ . This alternative deriv a tion has tw o adv antages. First, it prov es the existence of an analy tic in a W eyl inv ariant bas is in the spa ce of the op erator s V g a . Sec ond, it pro vides a prescr iption to get form factors of these basic elemen ts indep en dent ly of the r epr esentation for form factors in use. Moreov er, it is e a sy to se e, that it is not necessary to solv e the equations (6.10) and (6.14) explicitly . As it is shown in App endix C the form factors are ra tionally expres s ed in terms o f an appropria tely chosen set of indep endent v ariables . The App endix pr ovides a c onstructive w ay to o btain the form factor s in terms of the indep enden t v ariables . 7. The s i mplest exampl e : l e v el (1 , 0) descendan ts Here we study the form factors of the level (1 , 0) op erators. Fir st, we find the rec ursion relations for these form factors and, second, we find the W eyl inv ariant combinations by means of the fir s t approach from the previous section. No te that finding recur sion relations is not nec e ssary for obtaining the form factors explicitly , but re cursion re lations often turn o ut to b e a useful to ol to prov e theorems. As we have a lready ment ioned, the whole reas oning (5.8 )–(5.16) is v alid for the J functions related to a rbitrary op erators . Co ns ider the function J α i c − 1 k,N +1 ,a ( z ; X ) = N Y n =1 k − 1 Y m =1 f − 1  z x n ω k +1 − 2 m 2  a h 1 | π R ( α i c − 1 ) t k ( z ) t 1 ( x 1 ) . . . t 1 ( x N ) | 1 i a . (7.1) It admits the e xpansions (5.11) and (5.1 3) with appropria te functions J (0) and J ( ∞ ) . Therefore, to obtain the recurrent r elations for this function it is sufficient to find the as ymptotics as z → 0 or z → ∞ . Rewrite the e xpression (7 .1) in the for m J α i c − 1 k,N +1 ,a ( z , X ) N Y n =1 k − 1 Y m =1 f  z x n ω k +1 − 2 m 2  = a h 1 | [ π R ( α i c − 1 ) , t k ( z )] t 1 ( x 1 ) . . . t 1 ( x N ) | 1 i a + a h 1 | t k ( z ) π R ( α i c − 1 ) t 1 ( x 1 ) . . . t 1 ( x N ) | 1 i a . (7.2) F rom (4.13 a) we hav e [ π R ( α i c − 1 ) , t k ( z )] = z X 1 ≤ s 1 <... 1 in the for m d ( s ) 1 = d (1) 1 − s − 1 X i =1 ( d ( i ) 1 − d ( i +1) 1 ) = d (1) 1 − A + 1 s − 1 X i =1 ω L +1 − 2 i 2 R ( i ) 1 . After substituting it int o (7.6), (7.7) the co efficient at the element d (1) 1 cancels out due to (7.8). 20 Let us s ing le o ut one so lution: c ( L − 1) 1 = · · · = c ( L − 1) L − 1 = 0 , c ( L − 1) L = 1 . (7.9) This solution corr e spo nds to the int egral of motion a h C ( L − 1) | = a h ι 1 | . (7.10) Now, let C ( σ ) , σ = 1 , . . . , L − 2 b e any ba sis in the subspace o f the s olutions to the equation (7.8) with c L = 0 . Then a h C ( σ ) | = a h h ( σ ) 1 ,a | , wher e h ( σ ) 1 ,a = − L − 1 X i =1 α i c − 1 L − 1 X k =1 c ( σ ) k [ k ] K k,a k X m =1 ω m − i − k +1 2 X 1 ≤ s 1 < ··· i ω h a,H s 1 ...s k i , σ = 1 , . . . , L − 2 . (7.11) In particular, h ( L − 1) 1 ,a = ι 1 assuming (7.9 ). If w e only allow a -indep enden t solutions C ( σ ) , the co efficients in the linear c o m bination (7.7) are W eyl inv ariant. Hence, by the co nstruction a h h ( σ ) 1 ,a | t k 1 ( x 1 ) . . . t k N ( x N ) | 1 i a = wa h h ( σ ) 1 ,w a | t k 1 ( x 1 ) . . . t k N ( x N ) | 1 i wa ∀ w ∈ W for σ = 1 , . . . , L − 1. 8. Discussion In this pa p er we extended the ma in res ults of [9 ] to the case of the A (1) L − 1 affine T o da mo dels. W e constr uct spaces of solutions to the form factor b o o tstrap equations, which, as we argue, can b e bijectiv ely mapp ed onto the F o ck spaces o f descendant op erato rs over the e xpo nent ial o per ators V a ( x ) for gener ic v alues of a . W e prop ose a constructio n to find W eyl inv ariant families of bases in these spa ces base d o n high r apidity asymptotic expansions o f the form factors of exp onential op erators . In principle, it is p ossible, at least for the low est levels, to obtain W eyl inv ar iant families of base s in the F o ck s paces of descendant op erators in the Lagrangia n formalism [24]. How ev er, the iden tification of b oth t ypes of bases cannot b e unique without some a dditional informa tion. Probably , we could fix iden tification at some sp ecial res onant po in ts, but it has not b een do ne up to now. Thus, the field iden tification problem o f the bo otstrap form factor progr am r emains unsolved. Recently , in the remark able pap er s [25, 26 ] it was shown using the sc a ling limit from a lattice mo del, that the spa ces of descenda n t o pe r ators, a t least in the case of the sine-Gordon mo del, can b e created by use of so me fermionic op e rators acting in the s pace of lo cal op erator s of the theory . In particular , it turned out to b e p ossible to c alculate ex actly all the exp ectation v alues of descendant op erator s in the theory [26]. It would b e utterly unnatural, if such fermionic op erator s w ould no t induce an action on the algebra A ⊗ A in o ur construction. Hence, revealing such fermions in a co nstruction for form factors would be an imp or ta n t step toward the field identification, if not its complete solution. Ac knowledgmen ts The a utho r s are gr ateful to M. Bershtein, B. F e ig in, Y a . P ug ai and F. Smirnov for stimulating discussions. The work was supp orted, in part, b y RFBR under the grants 08– 01–00 720 and 09– 02–12 446, by RFBR and CNRS under the g rant 09–02 –9310 6 , b y the Progr am for Supp ort of Leading Scie n tific Scho ols under the gra nt 3472 .2008.2 and by the F ederal Progr am “Scientific and Scientific-Pedagogical Personnel of Innov ational Russia” under the state c o nt ract No. P13 39. The v is it of M. L. to LP THE, Universit ´ e Paris 6 in August–Septem ber of 2009 , where a part of the work was done, w as supp orted by CNRS in the framework of the LIA “Physique Th´ eor ique et Mati` er e Condens´ ee” (ENS–La ndau progr am). App endix A. Simple p ol e s of t k ( z ) t 1 ( x ) Here we prove that the p oles (5.2) a re simple. The pro duct t k ( z ) t 1 ( x ) co uld p o ssess do uble p oles at the po in ts z = xω ± ( k − 1) / 2 due to tw o p oles of the function f ( z ). F o r example, for the p ole z = xω − ( k − 1) / 2 21 the double p ole s appear in the t w o type s of terms. Let us o btain the ter ms o f the firs t type. Let σ 1 , σ 2 be tw o in tegers s uch that 2 ≤ σ 1 < σ 2 ≤ L . Then the e xpression (4.1 0) for t k contains terms with s k − 1 = σ 1 − 1 and s k = σ 1 , while that for t 1 contains a term with s = σ 2 . Hence, the pro duct t k ( z ) t 1 ( x ) contains terms of the for m : λ ∗ λ σ 1 − 1  z ω 3 − k 2  λ σ 1  z ω 1 − k 2  λ σ 2 ( x ): f  z x ω 3 − k 2  f  z x ω 1 − k 2  , where λ ∗ means the pro duct of all other λ s and f s. The pr o duct of the tw o f functions produces a double p ole at the po int z = xω − k +1 − 2 i 2 . In the vicinity of this p oint it b ehaves as : λ ∗ λ σ 1 − 1 ( ω x ) λ σ 1 ( x ) λ σ 2 ( x ): − (1 − ω 1 − p ) 2 (1 − ω p ) 2 (1 − ω ) 2  z x ω 1 − k 2 − 1  2 + O   z x ω 1 − k 2 − 1  − 1  . (A.1) The second type o f ter ms co nsists o f those with s i − 1 = σ 1 − 1, s i = σ 2 , s = σ 1 : : λ ∗ λ σ 1 − 1  z ω 3 − k 2  λ σ 2  z ω 1 − k 2  λ σ 1 ( x ): f  z x ω 3 − k 2  f  x z ω − 1 − k 2  . It p ossess es a double p o le at the sa me po in t, wher e it b ehav es just as minus the expres sion (A.1). It is easy to se e that the factors denoted by λ ∗ are the same for b oth expr essions in this limit if a ll other s j coincide. Hence, bo th double p ole contributions cancel each other. The same reasoning is v alid for the po le at z = xω ( k − 1) / 2 . App endix B. Equation of motion In this app endix we prov e that for m factors a re c onsistent with the equation of motion α i ∂ ¯ ∂ ϕ = π µb 2 (2 e bα i ϕ − e bα i − 1 ϕ − e bα i +1 ϕ ) . The deriv a tives of a field pro duce multiplication of its form facto rs by the comp onents of the momentu m according to the us ual rule P µ ↔ i∂ µ . Int ro duce the notation S k n ( z ; X ) = sin π kn L sin π n L z n + S n ( X ) . Let z = e θ , x n = e θ n . Then the co mponents of the mo mentum a re given by P z ( θ, θ 1 , . . . , θ N ) k, 1 ,... , 1 = − m 2 S k 1 ( z ; X ) P ¯ z ( θ, θ 1 , . . . , θ N ) k, 1 ,... , 1 = m 2 S k − 1 ( z ; X ) . Let a = L − 1 X i =1 a i α i , ν i = pα i − ρ. Then we hav e h v ac | α i ∂ ¯ ∂ ϕ | θ , θ 1 , . . . , θ N i k, 1 ,... , 1 = m 2 4 Q S k 1 ( z ; X ) S k − 1 ( z ; X ) d da i f a ( θ, θ 1 , . . . , θ N ) k 1 ... 1     a = − ρ and h v ac | e bα i ϕ | θ, θ 1 , . . . , θ N i k 1 ... 1 = ω ∓ ( k + N ) h v ac | e bα i ± 1 ϕ | θ, θ 1 , . . . , θ N i k 1 ... 1 = G ν i f ν i ( θ, θ 1 , . . . , θ N ) k 1 ... 1 . All v alues G ν i are evidently equal. Let J ′ k,N +1 ,i ( z ; X ) = d da i J k,N +1 ,a ( z ; X )     a = − ρ . 22 The equation of mo tion can be rewritten as S k 1 ( z ; X ) S k − 1 ( z ; X ) J ′ k,N +1 ,i ( z ; X ) = A sin 2 π ( k + N ) L J k,N +1 ,ν i ( z ; X ) , (B.1) where A = 8 π µG ν i (1 − p ) m 2 . According to [23] it reads A = π L sin π L sin π p L sin π (1 − p ) L . (B.2) It is easy to chec k Eq. (B.1) with (B.2) for N = 0 b y using (5.21 ), (5.24). No w we prov e it fo r N > 0 by induction. It is more co n venien t to use induction in the v ariable k + N ra ther than just N . Supp ose that the equation (B.1) is v alid for some v alue M = k + N for ar bitrary k = 1 , . . . , L − 1. T aking der iv atives o f bo th sides o f the recurrent relation (5.20) we get J ′ k,N +1 ,i ( z ; X ) = N X n =1 x n ω k +1 2 z − x n ω k +1 2 R + N ,n ( X ) J ′ k +1 ,N , i ( x n ω k 2 ; ˆ X n ) − N X j =1 x n ω − k +1 2 z − x n ω − k +1 2 R − N ,n ( X ) J ′ k +1 ,N , i ( x n ω − k 2 ; ˆ X n ) . Multiplying it by S k 1 ( z , X ) S k − 1 ( z , X ) and using the ident ity (5.19) we get S k 1 ( z , X ) S k − 1 ( z , X ) J ′ k,N +1 ( z ; X ) = N X n =1 [ k ] x n ω k +1 2 R + N ,n ( X ) S k +1 − 1 ( x n ω k 2 ; ˆ X n ) J ′ k +1 ,N ( x n ω k 2 ; ˆ X n ) + N X n =1 x n ω k +1 2 z − x n ω k +1 2 R + N ,n ( X ) S k +1 1 ( x n ω k 2 ; ˆ X n ) S k +1 − 1 ( x n ω k 2 ; ˆ X n ) J ′ k +1 ,N ( x n ω k 2 ; ˆ X n ) − N X n =1 [ k ] x n ω − k +1 2 R − N ,n ( X ) S k +1 − 1 ( x n ω − k 2 ; ˆ X n ) J ′ k +1 ,N ( x n ω − k 2 ; ˆ X n ) − N X n =1 x n ω − k +1 2 z − x n ω − k +1 2 R − N ,n ( X ) S k +1 1 ( x n ω − k 2 ; ˆ X n ) S k +1 − 1 ( x n ω − k 2 ; ˆ X n ) J ′ k +1 ,N ( x n ω − k 2 ; ˆ X n ) . (B.3) Due to the induction hypothesis we hav e S k +1 1 ( x n ω − k 2 ; ˆ X n ) S k +1 − 1 ( x n ω − k 2 ; ˆ X n ) J ′ k +1 ,N , i ( x n ω − k 2 ; ˆ X n ) = A sin 2 π ( k + N ) L J k +1 ,N , ν i ( x n ω − k 2 ; ˆ X n ) . Hence, the sum of the fir st and the thir d terms in the r ight ha nd side o f Eq. (B.3) ar e equa l to A sin 2 π ( k + N ) L N X n =1 [ k ] x n ω k +1 2 S k +1 1 ( x n ω k 2 ; ˆ X n ) R + N ,n ( X ) J k +1 ,N , ν i ( x n ω k 2 ; ˆ X n ) − N X n =1 [ k ] x n ω − k +1 2 S k +1 1 ( x n ω − k 2 ; ˆ X n ) R − N ,n ( X ) J k +1 ,N , ν i ( x n ω − k 2 ; ˆ X n ) ! = − A sin 2 π ( k + N ) L ( J k,N +1 ,ν i ( − [ k ] − 1 S 1 ( X ); X ) − K k,ν i J 1 ,N ,ν i ( X )) , while the tw o remaining terms rea ds A sin 2 π ( k + N ) L N X n =1 x n ω k +1 2 z − x n ω k +1 2 R + N ,n ( X ) J k +1 ,N , ν i ( x n ω k 2 ; ˆ X n ) 23 − N X n =1 x n ω − k +1 2 z − x n ω − k +1 2 R − N ,n ( X ) J k +1 ,N , ν i ( x n ω − k 2 ; ˆ X n ) ! = A sin 2 π ( k + N ) L ( J k,N +1 ,ν i ( z ; X ) − K k,ν i J 1 ,N ,ν i ( X )) . Gathering these terms we get S k 1 ( z ; X ) S k − 1 ( z ; X ) J ′ k,N +1 ,i ( z ; X ) = A sin 2 π ( k + N ) L ( J k,N +1 ,ν i ( z ; X ) − J k,N +1 ,ν i ( − [ k ] − 1 S 1 ( X ); X )) . It is nearly what we need. T o prove the equation o f motion for k + N = M + 1 it r emains to prov e that J k,N +1 ,ν i ( − [ k ] − 1 S 1 ( X ); X ) = 0 . (B.4) Consider firs t the case k = 1. Then the function J 1 ,N +1 ,ν i ( x N +1 ; x 1 , . . . , x N ) is symmetric with r esp ect to a ll of the v a riables x 1 , . . . , x N +1 . Hence, any of thes e v ariables can b e chosen for z . It means that J 1 ,N +1 ,ν i ( − S 1 ( X ); X ) = J 1 ,N +1 ,ν i ( − S 1 ( x N +1 , ˆ X j ); x N +1 , ˆ X j ) . The le ft hand side is x N -indep enden t, while the right hand side is x j -indep enden t. It means that the function J 1 ,N +1 ,ν i ( − S 1 ( X ); X ) is constant in its all N v aria bles. Therefore, it is s ufficien t to pr ov e that it is zero e. g. fo r x N → ∞ . Let us use the recurrent relation J 1 ,N +1 ,ν i ( − S 1 ( X ); X ) = K 1 ,ν i J 1 ,N ,ν i ( X ) + N X n =1 x n ω S 2 1 ( x n ω 1 2 ; ˆ X n ) R + N ,n ( X ) J 2 ,N ,ν i ( x n ω 1 2 ; ˆ X n ) − N X n =1 x n ω − 1 S 2 1 ( x n ω − 1 2 ; ˆ X n ) R − N ,j ( X ) J 2 ,N ,ν i ( x n ω − 1 2 ; ˆ X n ) . Since the le ft hand side is a co nstant, we may ca lc ulate it in the limit x N → ∞ . In this limit the o nly nonv a nis hing ter ms in the sums in the right hand side are those with n = N . T aking into account that R + N ,n ( X ) = R − N ,n ( X ) = − 2i sin π p L sin π (1 − p ) L sin π L + O ( x − 1 N ) as x N → ∞ , we o btain J 1 ,N +1 ,ν i ( − S 1 ( X ); X ) → ( K 1 ,ν i ) 2 + 4 sin π L sin π p L sin π ( p − 1) L sin 2 π L K 2 ,ν i ! J 1 ,N − 1 ,ν i ( ˆ X N ) = 0 . Hence, J 1 ,M ,ν i ( − S 1 ( X ); X ) = 0. Now, it is straightforward to chec k (B.4) by fusing t 1 ’s into t k . That is why we use d inductio n in the v ariable k + N rather than N . App endix C. Solutions to Eqs. (6 . 10), (6.14 ) and form factors As the equations (6.10 ) and (6 .1 4) hav e the sa me for m, we restrict the co nsideration to the bra -vectors. Since t k ( z ) for k = 2 , . . . , L − 1 ca n b e o btained by the fusion o f t 1 ( z ) accor ding to (5.5 ), it is sufficient to cons ider t 1 ( z ) and t L ( z ) without loss of genera lit y . Besides, since the differences t L ( z ω ) − t L ( z ) also app ear in such fusion a ccording to (5.7 ), it is conv enien t to consider the ‘symmetrized’ version of t L ( z ): t sym L ( z ) = L − 1 X m =0 t L ( z ω m ) , (C.1) which is an o per ator v alued function of z L . Similarly to (6.2) consider the expansio n a h 1 | t 1 ( ξ − 1 1 z ) . . . t 1 ( ξ − 1 r z ) t sym L ( ζ − 1 /L 1 z ) . . . t sym L ( ζ − 1 /L s z ) = ∞ X n =0 z − n a h n ; ξ 1 , . . . , ξ r ; ζ 1 , . . . , ζ s | . (C.2) 24 The equations (6 .10) reduce in this case to r X m =1 ξ n m = 0 , n 6∈ L Z , r X m =1 ξ n m = − L s X m =1 ζ n/L m = Σ n/L , n ∈ L Z , (C.3) for 1 ≤ n ≤ l . Here we intro duced the v ariables Σ ν , ν = 1 , . . . , λ = ⌊ l /L ⌋ , which will be us eful below. Due to the Newton–Gir ard identities, for 1 ≤ n ≤ l the quantit ies σ 1 n v anish, if n 6∈ L Z , while σ 1 L , σ 1 2 L , . . . , σ 1 λL are in one-to- one co rresp ondence with Σ 1 , . . . , Σ λ . The qua n tities σ L n for n = 1 , . . . , λ are also in one-to-one corres po ndence with Σ 1 , . . . , Σ λ and, hence , with σ 1 L , σ 1 2 L , . . . , σ 1 λL . The for m factor s a h n ; ξ 1 , . . . , ξ r ; ζ 1 , . . . , ζ s | t k 1 ( x 1 ) . . . t k N ( x N ) | h ′ i are rational symmetric functions of the v a riables ξ 1 , . . . , ξ r and the v a riables ζ 1 , . . . , ζ s , that is they are ratios of symmetric p olynomials. Therefore, if we learn how to calculate elementary symmetric po lynomials σ 1 n = σ n ( ξ 1 , . . . , ξ r ), n = 1 , . . . , r , and σ L n = σ n ( ζ 1 , . . . , ζ s ), n = 1 , . . . , λ on s olution of the equations (C.3), we will b e able to ca lculate the for m factors. W e hav e r + s v a riables and l eq uations, that is r + s − l independent v ariables. Let r 0 = r − l + λ and s 0 = s − λ so that r 0 + s 0 = r + s − l . T ake ξ 1 , . . . , ξ r 0 and ζ 1 , . . . , ζ s 0 for indep endent v ariables. Then the v aria ble s ξ 1 , . . . , ξ r and ζ 1 , . . . , ζ s are so lutions to the equations ξ r m + λ X n =1 ( − ) Ln σ 1 Ln ξ r − Ln m + r X n = l +1 ( − ) n σ 1 n ξ r − n m = 0 , (C.4) ζ s m + s X n =1 ( − ) n σ L n ζ r − n m = 0 . (C.5) Consider the equations (C.4) for m = 1 , . . . , r 0 as a system o f r 0 linear inho mogeneous equations for r 0 v ariables σ 1 L , σ 1 2 L , . . . , σ 1 λL , σ 1 l +1 , σ 1 l +1 , . . . , σ 1 r . 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