Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States

We generalize the results of [Comm. Math. Phys. 299 (2010), 825-866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the six-vertex model acting in the so-called Matsubara direction. We establish an equivalence between a scaling limit of the partition function...

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Дата:	2011
Автор:	Boos, H.
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Опубліковано:	Інститут математики НАН України 2011
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/146787
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Цитувати:	Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States / H. Boos // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ.

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spelling	irk-123456789-1467872019-02-12T01:24:26Z Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States Boos, H. We generalize the results of [Comm. Math. Phys. 299 (2010), 825-866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the six-vertex model acting in the so-called Matsubara direction. We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particle-hole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand. As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators. In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987). Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations. On the side of the lattice model we use the excited-state TBA approach. We consider in detail the case of the descendant at level 8. 2011 Article Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States / H. Boos // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 82B20; 82B21; 82B23; 81T40; 81Q80 DOI:10.3842/SIGMA.2011.007 http://dspace.nbuv.gov.ua/handle/123456789/146787 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We generalize the results of [Comm. Math. Phys. 299 (2010), 825-866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the six-vertex model acting in the so-called Matsubara direction. We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particle-hole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand. As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators. In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987). Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations. On the side of the lattice model we use the excited-state TBA approach. We consider in detail the case of the descendant at level 8.
format	Article
author	Boos, H.
spellingShingle	Boos, H. Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Boos, H.
author_sort	Boos, H.
title	Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States
title_short	Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States
title_full	Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States
title_fullStr	Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States
title_full_unstemmed	Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States
title_sort	fermionic basis in conformal field theory and thermodynamic bethe ansatz for excited states
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/146787
citation_txt	Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States / H. Boos // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 20 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT boosh fermionicbasisinconformalfieldtheoryandthermodynamicbetheansatzforexcitedstates
first_indexed	2025-07-11T00:34:56Z
last_indexed	2025-07-11T00:34:56Z
_version_	1837308693679439872
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 007, 36 pages Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States? Hermann BOOS †‡ † Fachbereich C – Physik, Bergische Universität Wuppertal, 42097 Wuppertal, Germany E-mail: boos@physik.uni-wuppertal.de ‡ Skobeltsyn Institute of Nuclear Physics, Moscow State University, 119991 Moscow, Russia Received October 07, 2010, in final form January 05, 2011; Published online January 13, 2011 doi:10.3842/SIGMA.2011.007 Abstract. We generalize the results of [Comm. Math. Phys. 299 (2010), 825–866] (hidden Grassmann structure IV) to the case of excited states of the transfer matrix of the six- vertex model acting in the so-called Matsubara direction. We establish an equivalence between a scaling limit of the partition function of the six-vertex model on a cylinder with quasi-local operators inserted and special boundary conditions, corresponding to particle- hole excitations, on the one hand, and certain three-point correlation functions of conformal field theory (CFT) on the other hand. As in hidden Grassmann structure IV, the fermionic basis developed in previous papers and its conformal limit are used for a description of the quasi-local operators. In paper IV we claimed that in the conformal limit the fermionic creation operators generate a basis equivalent to the basis of the descendant states in the conformal field theory modulo integrals of motion suggested by A. Zamolodchikov (1987). Here we argue that, in order to completely determine the transformation between the above fermionic basis and the basis of descendants in the CFT, we need to involve excitations. On the side of the lattice model we use the excited-state TBA approach. We consider in detail the case of the descendant at level 8. Key words: integrable models; six vertex model; XXZ spin chain; fermionic basis, thermo- dynamic Bethe ansatz; excited states; conformal field theory; Virasoro algebra 2010 Mathematics Subject Classification: 82B20; 82B21; 82B23; 81T40; 81Q80 1 Introduction Much progress was made in the understanding of the connection between the one-dimensional XXZ spin chain and two-dimensional quantum field theories (QFT). Many different aspects of this connection were studied in the literature. In the present paper we will touch only some particular aspect related to the hidden fermionic structure of the XXZ model and the corresponding continuum model – conformal field theory (CFT), [1, 2, 3, 4]. The continuum model can be studied through the scaling limit of the six-vertex model com- pactified on an infinite cylinder of radius R. The corresponding direction around the cylinder is sometimes called Matsubara direction. It is well known that the scaling limit of the homoge- neous critical six-vertex model is related to CFT, while introducing inhomogeneities in a special way leads to the sine-Gordon model (sG) which has a mass gap, [5]. An important step forward was done by Bazhanov, Lukyanov and Zamolodchikov in papers [6, 7, 8]. They obtained an integrable structure of CFT by constructing a monodromy matrix with the quantum space re- lated to the chiral bosonic field and the Heisenberg algebra. On the other hand this monodromy ?This paper is a contribution to the Special Issue “Relationship of Orthogonal Polynomials and Spe- cial Functions with Quantum Groups and Integrable Systems”. The full collection is available at http://www.emis.de/journals/SIGMA/OPSF.html mailto:boos@physik.uni-wuppertal.de http://dx.doi.org/10.3842/SIGMA.2011.007 http://www.emis.de/journals/SIGMA/OPSF.html 2 H. Boos matrix satisfies the Yang–Baxter equation with the R-matrix of the six-vertex model. The corre- sponding transfer matrix fulfills Baxter’s TQ-relation [9] and generates the integrals of motion. These integrals of motion are special combinations of the Virasoro generators. Originally they were introduced by Alexander Zamolodchikov in [10]. Zamolodchikov observed that the inte- grals of motion generate that part of the Virasoro algebra that survives under the integrable Φ1,3-perturbation of the CFT. In order to state the full equivalence of the six-vertex model in the scaling limit and the CFT one needs to compare all possible correlation functions. This problem is far from being solved completely. We believe that it can be helpful to use a hidden fermionic structure of the spin-1 2 XXZ chain, [1, 2]. The key idea is to consider a fermionic basis generated by means of certain creation operators t∗, b∗, c∗. These operators act on a space of quasi-local oper- ators W(α). Any operator in this space can be represented in the form q2αS(0)O with some operator O which acts on a chain segment of arbitrary finite length. The operator q2αS(0) was called “primary field” because it fulfills some properties similar to the properties of certain primary field in CFT. The fermionic basis is constructed inductively. Its completeness was shown in [11]. An important theorem proved by Jimbo, Miwa and Smirnov in the paper [3] allows one to reduce any correlation function expressed through the fermionic basis to determi- nants. An interesting feature of the above construction is that it is algebraic in the sense that the fermionic operators are constructed by means of the representation theory of the quantum group Uq(ŝl2). They are independent of any physical data like the magnetic field, the temper- ature, the boundary conditions etc. For example, the temperature can be incorporated via the Suzuki–Trotter formalism [12] by taking inhomogeneity parameters in the Matsubara direction in a special way and then performing the so-called Trotter limit, when the number of sites in the Matsubara direction n → ∞. One can even keep the number n finite and consider the case of arbitrary inhomogeneity parameters in this direction. The fermionic basis will not be affected [3]. There are only two transcendental functions ρ and ω which “absorb” the whole physical information and appear in the determinants in analogy with the situation with free fermions. In [3] these functions were represented in terms of the differential of the second kind in the theory of deformed Abelian integrals for finite n. These functions can be also obtained within the TBA approach, [13] which makes it possible to take the Trotter limit n → ∞. In the recent papers [14, 15] the fermionic basis was used in order to study the one-point func- tions of the sine-Gordon model on a cylinder. In this case the transcendental functions must be modified. Coming back to the similarity with the CFT, let us emphasize that the construction of the fermionic basis bears some similarity with the construction of the descendants in CFT via the action of the Virasoro generators on the primary fields. In the paper [4] we tried to make this similarity more explicit. Namely, we related the main building blocks of CFT – the three-point correlators to the scaling limit of a special partition functions of the six-vertex model constructed with the help of the fermionic operators. More concretely, we believe that the following con- jecture is true. There exist scaling limits τ ∗, β∗, γ∗ of the operators t∗, b∗, c∗ and scaling limits ρsc and ωsc of the functions ρ and ω as well. The operator τ ∗ generates Zamolodchikov’s integrals of motion mentioned above, while the asymptotic expansion of the function ρsc with respect to its spectral parameter generates their vacuum expectation values. Another conjecture was that the asymptotic expansion of ωsc with respect to the spectral parameters describes the expectation values of descendants for CFT with central charge c = 1− 6ν2/(1− ν) where ν is related to the deformation parameter q = eπiν . Equivalently, one takes the six-vertex model on a cylinder for some special boundary conditions and considers the scaling limit of the cor- responding normalized partition function with inserted quasi-local operator q2αS(0)O. On the other hand one computes a normalized three-point function in the CFT on an infinite cylinder with Virasoro descendants of some primary field φα of conformal dimension ∆α inserted at the Fermionic Basis in CFT and TBA for Excited States 3 origin and two primary fields φ± inserted at +∞ and −∞ respectively. The conjecture states that for any quasi-local operator one can find a corresponding descendant in such a way that the above partition function and the CFT three-point function are equal. In order to state this equality we used the fermionic basis for the quasi-local operators mentioned above. In [4] we established such a correspondence between the fermionic basis and the Virasoro generators only up to the level 6. It corresponds to bilinear combinations of the “Fourier- modes” β∗j and γ∗j with odd index j. For level 8 we have 5 Virasoro descendants l4−2, l−4l 2 −2, l2−4, l−6l−2, l−8. As for the fermionic basis, we also find five linearly independent combinations of the creation operators β∗j and γ∗j . One of them is quartic: β∗1β ∗ 3γ ∗ 3γ ∗ 1. Unfortunately, we could not uniquely fix the transformation matrix between the fermionic basis and the basis generated by the Virasoro descendants. We thought that it could be done if we would insert the simplest excitation corresponding to L−1φ± at ±∞ instead of the primary fields φ± themselves. The difference between the “global” Virasoro generators Ln and the “local” ones ln was explained in [4]. It is also discussed in the next section. The original motivation of the present paper was to fill this gap. It turned out, however, that it was not enough to consider only the simplest excitation. The above uncertainty still remained in this case. One has to take at least the next excitation, namely, that one corresponding to the descendants of the second level L2 −1φ± and L−2φ± in order to fix the unknown elements of the transformation matrix. All these elements are certain rational functions of the central charge c and the conformal dimension of the primary field ∆α.1 They do not depend on the conformal dimensions ∆± of the primary fields φ±. They are also independent of the choice of the excitation at ±∞. This is rather strong condition. We are still unable to prove it for arbitrary excitation. We think that it is interesting to consider excitations also independently of the above concrete problem. Therefore we consider them in a more general setting and come to the solution of the level 8 problem in the very end. The paper is organized as follows. In Section 2 we remind the reader about the main results of the paper [4]. Section 3 is devoted to the TBA approach for the excited states. We derive the equation for the auxiliary function Θ and define the function ωsc in Section 4. In Section 5 we discuss the relation to certain CFT three-point functions. We discuss the solution of the above level 8 problem in Section 6. In Appendix A we show several leading terms of the asymptotic expansion of the function Ψ defined in Section 3 and discuss its relation to the integrals of motion. In Appendices B and C several leading terms for the asymptotic expansions of the functions F̄ , x±, Θ are explicitly shown. 2 Reminder of basic results of [4] As was mentioned in the Introduction, in the paper [4] some specific connection between the conformal field theory (CFT) with the central charge c = 1− 6ν2/(1− ν) and the XXZ model with the deformation parameter q = eπiν was established2. More precisely, the following relation was found to be valid with the left hand side containing the CFT data and the right hand side containing the lattice model data 〈∆−\|Pα ( {l−k} ) φα(0)\|∆+〉〈∆−\|φα(0)\|∆+〉 = lim n→∞, a→0, na=2πR Zκ,s { q2αS(0)O } . (2.1) The left hand side means a normalized three-point function of the CFT defined on a cylinder of radius R parameterized by a complex variable z = x+iy with spacial coordinate x: −∞ < x <∞ 1Look at the formulae (12.4) of [4] or (2.18) and (6.9)–(6.11). 2Usually we take ν in the region 1 2 < ν < 1 called in [7] a “semi-classical domain”. The region 0 < ν ≤ 1 2 demands more accurate treatment. 4 H. Boos and coordinate in the Matsubara direction y: −πR < y < πR. The equivalence of the points x±πiR is implied. At the origin z = 0 some descendant field is inserted which is given by some polynomial Pα({l−k}) of Virasoro generators l−k, k > 0 acting on the primary field φα(z) with conformal dimension ∆α = ν2α(α− 2) 4(1− ν) . (2.2) We called these Virasoro generators “local” in a sense that they are defined in vicinity of z = 0 with the corresponding energy-momentum tensor T (z) = ∞∑ n=−∞ lnz −n−2. The bra- and ket-states \|∆+〉 and 〈∆−\| are to be defined through two primary fields φ± with conformal dimensions ∆± being inserted at x→ ±∞ in such a way that Ln\|∆+〉 = δn,0∆+\|∆+〉, n ≥ 0 when x =∞ and 〈∆−\|Ln = δn,0∆−〈∆−\|, n ≤ 0 at x = −∞. We called the Virasoro gene- rators Ln “global”. They correspond to the expansion obtained via the conformal transformation z → e−z/R T (z) = 1 R2 ( ∞∑ n=−∞ Lne nz R − c 24 ) . In [10] Alexander Zamolodchikov introduced the local integrals of motion which act on local operators as (i2n−1O)(w) = ∫ Cw dz 2πi h2n(z)O(w) (n ≥ 1), where the densities h2n(z) are certain descendants of the identity operator I. An important property is that 〈∆−\|i2n−1 ( O(z) ) \|∆+〉 = (I+ 2n−1 − I − 2n−1)〈∆−\|O(z)\|∆+〉, (2.3) where I±2n−1 denote the vacuum eigenvalues of the local integrals of motion on the Verma module with conformal dimension ∆±. The Verma module is spanned by the elements i2k1−1 · · · i2kp−1l−2l1 · · · l−2lq(φα(0)). (2.4) In case when ∆+ = ∆− the space is spanned by the even Virasoro generators {l−2n}n≥1. In order to describe the right hand side of (2.1) we need the fermionic basis constructed in [1, 2] via certain creation operators. These creation operators called t∗, b∗, c∗ together with the annihilation operators called b, c act in the space3 W(α) = ∞⊕ s=−∞ Wα−s,s, where Wα−s,s is the subspace of quasi-local operators of the spin s with the shifted α-parameter. They all are defined as formal power series of ζ2 − 1 and have the block structure t∗(ζ) : Wα−s,s → Wα−s,s, 3The problem of constructing the annihilation operator corresponding to the creation operator t∗ was discussed in the paper [13] but was not solved completely. Fermionic Basis in CFT and TBA for Excited States 5 b∗(ζ), c(ζ) : Wα−s+1,s−1 → Wα−s,s, c∗(ζ),b(ζ) : Wα−s−1,s+1 → Wα−s,s. The operator t∗(ζ) plays the role of a generating function of the commuting integrals of motion. In a sense it is bosonic. It commutes with all fermionic operators b(ζ), c(ζ) and b∗(ζ), c∗(ζ) which obey canonical anti-commutation relations[ c(ξ), c∗(ζ) ] + = ψ(ξ/ζ, α), [ b(ξ),b∗(ζ) ] + = −ψ(ζ/ξ, α) (2.5) with ψ(ζ, α) = 1 2 ζα ζ2 + 1 ζ2 − 1 . The annihilation operators b and c “kill” the lattice “primary field” q2αS(0) b(ζ) ( q2αS(0) ) = 0, c(ζ) ( q2αS(0) ) = 0, S(k) = 1 2 k∑ j=−∞ σ3 j . The space of states is generated via the multiple action of the creation operators t∗(ζ), b∗(ζ), c∗(ζ) on the “primary field” q2αS(0). In this way one can obtain the fermionic basis. The completeness of this basis was proved in the paper [11]. In the right hand side of equation (2.1) we take the scaling limit of a normalized partition function of the six-vertex model on a cylinder with insertion of a quasi-local operator q2αS(0)O Zκ,s { q2αS(0)O } = TrSTrM ( Y (−s) M TS,M q2κS b∗∞,s−1 · · ·b∗∞,0 ( q2αS(0)O )) TrSTrM ( Y (−s) M TS,M q2κS b∗∞,s−1 · · ·b∗∞,0 ( q2αS(0) )) , (2.6) where the operators b∗∞,j are defined through4 a singular part when ζ → 0 and b∗reg is a regular one ζ−αb∗(ζ)(X) = s−1∑ j=0 ζ−2jb∗∞,j(X) + ζ−αb∗reg(ζ)(X), X ∈Wα−s+1,s−1 for some operator X of the spin s − 1, s > 0 where ζ−αb∗reg(ζ)(X) vanishes at zero. The monodromy matrix TS,M is defined via the universal R-matrix of Uq(ŝl2) on the tensor product of two evaluation representations where the first one HS corresponds to the infinite lattice direction and the second one HM corresponds to the Matsubara direction TS,M = y ∞∏ j=−∞ Tj,M, Tj,M ≡ Tj,M(1), Tj,M(ζ) = x n∏ m=1 Lj,m(ζ) with the standard L-operator of the six vertex model Lj,m(ζ) = q− 1 2 σ3 jσ 3 m − ζ2q 1 2 σ3 jσ 3 m − ζ ( q − q−1 ) (σ+ j σ − m + σ−j σ + m). 4Actually, later we discuss the Fateev–Dotsenko condition (2.13) fulfilled for the case κ′ = κ. In this case the parameters α and s are constrained. We will be mostly interested in the case when 0 < α < 2. Then we need s < 0 and another definition of the functional (2.6) is necessary. It can be done through the replacement of operators b∗∞,j by c∗∞,j . This choice was taken in [15] with identification of the notation there c∗screen,−j = c∗∞,j . 6 H. Boos The “screening operator” Y (−s) M carries spin −s. As was discussed in [4], the functional (2.6) does not depend on the concrete choice of the screening operator under rather mild conditions. Due to common wisdom, in case of an infinite lattice, one can change boundary conditions and, instead of taking the traces in the right hand side of (2.6), insert two one-dimensional projectors \|κ〉〈κ\| and \|κ + α − s, s〉〈κ + α − s, s\| at the boundary, where \|κ〉 is eigenvector of the transfer matrix TM(ζ, κ) = Trj ( Tj,Mq κσ3 j ) with maximal eigenvalue T (ζ, κ) in the zero spin sector and where the eigenvector \|κ+ α− s, s〉 corresponds to the maximal eigenvalue T (ζ, κ+ α− s, s) of the transfer matrix TM(ζ, κ+ α− s) in the sector with spin s. The twist parameter κ plays the role of the magnetic field. So, we can perform the following substitution Zκ,s { q2αS(0)O } → 〈κ+ α− s, s\|TS,M q2κS b∗∞,s−1 · · ·b∗∞,0 ( q2αS(0)O ) \|κ〉〈κ+ α− s, s\|TS,M q2κS b∗∞,s−1 · · ·b∗∞,0 ( q2αS(0) ) \|κ〉 , which does not affect the answer for the case of infinite lattice if 〈κ\|Y (−s) M \|κ+ α− s, s〉 6= 0. The theorem proved by Jimbo, Miwa and Smirnov [3] claims5 that Zκ,s { t∗(ζ)(X) } = 2ρ(ζ\|κ, κ+ α, s)Zκ,s{X}, (2.7) Zκ,s { b∗(ζ)(X) } = 1 2πi ∮ Γ ω(ζ, ξ\|κ, α, s)Zκ,s { c(ξ)(X) }dξ2 ξ2 , Zκ,s { c∗(ζ)(X) } = − 1 2πi ∮ Γ ω(ξ, ζ\|κ, α, s)Zκ,s { b(ξ)(X) }dξ2 ξ2 , where the contour Γ goes around all the singularities of the integrand except ξ2 = ζ2. The direct consequence of the above theorem and the anti-commutation relations (2.5) is the determinant formula Zκ,s { t∗(ζ0 1 ) · · · t∗(ζ0 p )b∗(ζ+ 1 ) · · ·b∗(ζ+ r )c∗(ζ−r ) · · · c∗(ζ−1 ) ( q2αS(0) )} (2.8) = p∏ i=1 2ρ(ζ0 i \|κ, κ+ α, s)× det ( ω(ζ+ i , ζ − j \|κ, α, s) ) i,j=1,...,r . The functions ρ and ω are completely defined by the Matsubara data. The function ρ is the ratio of two eigenvalues of the transfer matrix ρ(ζ\|κ+ α− s, s) = T (ζ, κ+ α− s, s) T (ζ, κ) . (2.9) We will come to the definition of the function ω in Section 4 in more general case of presence of the excited states. The scaling limit in the Matsubara direction means n→∞, a→ 0, na = 2πR, (2.10) where the radius of the cylinder R is fixed. Simultaneously one should rescale the spectral parameter ζ = λāν , ā = Ca (2.11) 5Actually, in [3] the statement was proved for the case s = 0 but as was discussed in [4], this statement can be proved for s 6= 0 also. Fermionic Basis in CFT and TBA for Excited States 7 with some fine-tuning constant C. One of the most important points of [4] was to define the scaling limits of ρ and ω ρsc(λ\|κ, κ′) = lim scaling ρ(λāν \|κ, α, s), (2.12) ωsc(λ, µ\|κ, κ′, α) = 1 4 lim scaling ω(λāν , µāν \|κ, α, s) where κ′ is defined through an analogue of the Dotsenko–Fateev condition [16] κ′ = κ+ α+ 2 1− ν ν s. (2.13) The continuum limit can be taken in both directions of the cylinder. The first conjecture proposed in [4] was that the creation operators are well-defined in the scaling limit for the space direction when ja = x is finite τ ∗(λ) = lim a→0 1 2 t∗(λāν), β∗(λ) = lim a→0 1 2 b∗(λāν), γ∗(λ) = lim a→0 1 2 c∗(λāν) and for the “primary field” Φα(0) = lim a→0 q2αS(0). Asymptotic expansions at λ→∞ look log (τ ∗(λ)) ' ∞∑ j=1 τ ∗2j−1λ − 2j−1 ν , (2.14) 1√ τ ∗(λ) β∗(λ) ' ∞∑ j=1 β∗2j−1λ − 2j−1 ν , 1√ τ ∗(λ) γ∗(λ) ' ∞∑ j=1 γ∗2j−1λ − 2j−1 ν . The next conjecture based on the bosonisation arguments was that the scaling limit of the space Wα−s,s belongs to the direct product of two Verma modules Scaling limit (Wα−s,s) ⊂ Vα+2 1−ν ν s ⊗ V−α and the operators τ ∗(λ), β∗(λ), γ∗(λ) act non-trivially only on the first chirality component and do not touch the second one τ ∗2j−1 : Vα+2 1−ν ν s → Vα+2 1−ν ν s, β∗2j−1 : Vα+2 1−ν ν (s−1) → Vα+2 1−ν ν s, γ∗2j−1 : Vα+2 1−ν ν (s+1) → Vα+2 1−ν ν s. Using the results of the paper [7], we get the asymptotic expansion for λ→ +∞ log ρsc(λ\|κ, κ′) ' ∞∑ j=1 λ− 2j−1 ν Cj ( I+ 2j−1 − I − 2j−1 ) . (2.15) Here the integrals of motion I±2j−1 are the same as in (2.3). They correspond to the “right” and the “left” vacuum and depend on κ and κ′ respectively: I+ 2j−1 = I2j−1(κ), I−2j−1 = I2j−1(κ′). 8 H. Boos We can identify τ ∗2j−1 = Cji2j−1. For the function ωsc we have ωsc(λ, µ\|κ, κ′, α) ' √ ρsc(λ\|κ, κ′) √ ρsc(µ\|κ, κ′) ∞∑ i,j=1 λ− 2i−1 ν µ− 2j−1 ν ωi,j(κ, κ ′, α) (2.16) when λ2, µ2 → +∞. The scaling limit of (2.8) is proportional to Zκ,κ ′ R { τ ∗(λ0 1) · · · τ ∗(λ0 p)β ∗(λ+ 1 ) · · ·β∗(λ+ r )γ∗(λ−r ) · · ·γ∗(λ−1 ) ( Φα(0) )} (2.17) = p∏ i=1 ρsc(λ0 i \|κ, κ′)× det ( ωsc(λ+ i , λ − j \|κ, κ ′, α) ) i,j=1,...,r . If we substitute the expansion (2.14) into the left hand side of (2.17) and the expansions (2.15), (2.16) into the right hand side of (2.17) we can compare the coefficients standing with powers of the spectral parameters and express the functional Zκ,κ ′ R of any monomial constructed from the modes τ ∗2j−1, β∗2j−1, γ∗2j−1 through the integrals of motion I2n−1, coefficients Cn and ωi,j . In [4] we argued that the eigenvalue T (ζ, κ+ α− s, s) in the scaling limit (2.10) becomes equal to T (ζ, κ′). This means that, if we choose α and the spin s in such a way that κ′ = κ, then using (2.9) and (2.12), we get ρsc(λ\|κ, κ′) = 1. This is an important technical point. In this case we were able to apply the Wiener–Hopf technique and obtain the coefficients ωi,j as power series in κ−1 where κ→∞. On the other hand, one can evaluate the left hand side of (2.1) using the operator product expansion (OPE). In order to compare with the result obtained by the above lattice method, one needs to identify the parameters κ, κ′, α with the CFT data. In fact, we already identified α by taking the formula (2.2) for the dimension ∆α of the primary field φα. The next step was to take ∆+ = ∆κ+1 and ∆− = ∆−κ′+1. The most important conjecture of [4] was that for the CFT with central charge c = 1− 6 ν2 1−ν it is always possible to find one-to-one correspondence between a polynomial Pα({l−k}) in the left hand side of (2.1) and certain combinations of β∗2j−1 and γ∗2j−1. It is convenient to introduce β∗2m−1 = D2m−1(α)βCFT∗ 2m−1, γ∗2m−1 = D2m−1(2− α)γCFT∗ 2m−1, where D2m−1(α) = 1√ iν Γ(ν)− 2m−1 ν (1− ν) 2m−1 2 1 (m− 1)! Γ ( α 2 + 1 2ν (2m− 1) ) Γ ( α 2 + (1−ν) 2ν (2m− 1) ) together with even and odd bilinear combinations φeven 2m−1,2n−1 = (m+ n− 1) 1 2 ( βCFT∗ 2m−1γ CFT∗ 2n−1 + βCFT∗ 2n−1 γ CFT∗ 2m−1 ) , (2.18) φodd 2m−1,2n−1 = d−1 α (m+ n− 1) 1 2 ( βCFT∗ 2n−1 γ CFT∗ 2m−1 − βCFT∗ 2m−1γ CFT∗ 2n−1 ) , where dα = ν(ν − 2) ν − 1 (α− 1) = 1 6 √ (25− c)(24∆α + 1− c). (2.19) Fermionic Basis in CFT and TBA for Excited States 9 The Verma module has a basis consisting of the vectors (2.4). Conjecturally the same space is spanned by: i2k1−1 · · · i2kp−1φ even 2m1−1,2n1−1 · · ·φeven 2mr−1,2nr−1φ odd 2m̄1−1,2n̄1−1φ odd 2m̄r̄−1,2n̄r̄−1 ( φα ) . (2.20) Since [ l0, τ ∗ 2j−1 ] = (2j − 1)τ ∗2j−1, [ l0,β ∗ 2i−1γ ∗ 2j−1 ] = (2i+ 2j − 2)β∗2i−1γ ∗ 2j−1, the descendants of the form (2.4) and those created by the fermions of the form (2.20) must be finite linear combinations of each other if the corresponding levels coincide q∑ j=1 2lj = r∑ k=1 2(mk + nk − 1) + r̄∑ k̄=1 2(m̄k̄ + n̄k̄ − 1). As was discussed above, one can choose the parameters α and s in such a way that κ′ = κ and ∆+ = ∆−. With this choice we factor out the integrals of motion. The quotient space of the Verma module modulo the action of the integrals of motion is spanned by the vectors of the form l−2l1 · · · l−2lq(φα(0)). All coefficients of the polynomial Pα({l−k}) are independent of κ. We were able to identify the above basis vectors up to the level 6. The system of equations is overdetermined but nevertheless it has a solution: φeven 1,1 ∼= l−2, φeven 1,3 ∼= l2−2 + 2c− 32 9 l−4, φodd 1,3 ∼= 2 3 l−4, (2.21) φeven 1,5 ∼= l3−2 + c+ 2− 20∆ + 2c∆ 3(∆ + 2) l−4l−2 + −5600 + 428c− 6c2 + 2352∆− 300c∆ + 12c2∆ + 896∆2 − 32c∆2 60(∆ + 2) l−6, φodd 1,5 ∼= 2∆ ∆ + 2 l−4l−2 + 56− 52∆− 2c+ 4c∆ 5(∆ + 2) l−6, φeven 3,3 ∼= l3−2 + 6 + 3c− 76∆ + 4c∆ 6(∆ + 2) l−2l−4 + −6544 + 498c− 5c2 + 2152∆− 314c∆ + 10c2∆− 448∆2 + 16c∆2 60(∆ + 2) l−6, where for simplicity we took ∆ ≡ ∆α. We hope it will not cause some confusion by mixing this ∆ with the anisotropy parameter of the XXZ model. In the above formula (2.21) we imply only a weak equivalence “∼=” between the left hand side and the right hand side. The sign ∼= means that the left and right hand sides being substituted into the corresponding expectation value give the same result by acting on Φα(0) and φα(0) respectively. In other words A ∼= B if and only if Zκ,κR ( A(Φα(0)) ) = 〈∆−\|B ( φα(0) ) \|∆+〉〈∆−\|φα(0)\|∆+〉 with the functional Zκ,κ ′ R defined in (2.17). In the next sections we will generalize the scheme of [4] to the case of excited states. 10 H. Boos 3 TBA for excited states In fact, the Jimbo, Miwa, Smirnov theorem (2.7) is valid not only for eigenvectors \|κ〉, \|κ+α−s, s〉 associated with maximal eigenvalues in the corresponding sectors, but also for eigenvectors corresponding to excited states. Here we consider only special excited states, namely, the so- called particle-hole excitations [17]. Some motivation for this is as follows. In [7] Bazhanov, Lukyanov and Zamolodchikov formulated several assumptions about the analytical properties of the eigenvalues of the Q-operator with respect to the square of the spectral parameter ζ2, in particular, that its zeroes in the complex ζ2-plane are either real or occur in complex conjugated pairs, that an infinite number of zeroes are real and positive and that there may be only a finite number of complex or real negative zeroes. An important assumption is that the real zeroes accumulate towards +∞ in the variable ζ2. Further, in [18] the authors proposed a conjecture that for the asymptotic analysis, when the parameter κ becomes large, only real positive zeroes corresponding either to the vacuum or to the excited states are important. The question why real negative zeroes or complex zeroes are not important for the asymptotic analysis seems hard. Usually one uses experience coming from the numerical study and also the arguments based on the analysis of a small vicinity of the free-fermion point ν = 1/2. Counting arguments play important role here as well. But we do not know any rigorous proof of this statement. Any further discussion of this subtle question would lead us beyond the scope of this paper. Let us start with the case of a lattice with an even finite number of sites n in the Mat- subara direction. The Bethe ansatz equations (BAE) are usually deduced from the Baxter’s TQ-relation [9] for the eigenvalues T (ζ, κ, s), Q(ζ, κ, s) of the transfer matrix TM(ζ, κ) and the Q-operator QM(ζ, κ) defined by the formula (3.3) of [4] T (ζ, κ, s)Q(ζ, κ, s) = d(ζ)Q(qζ, κ, s) + a(ζ)Q ( q−1ζ, κ, s ) , Q(ζ, κ, s) = ζ−κ+sA(ζ, κ, s) with polynomial dependence of the functions T and A on the spectral parameter ζ in every spin sector s, 0 ≤ s ≤ n 2 and a(ζ) = ( 1− qζ2 )n , d(ζ) = ( 1− q−1ζ2 )n . In the TBA approach one introduces auxiliary function a(ζ, κ, s) = d(ζ)Q(qζ, κ, s) a(ζ)Q(q−1ζ, κ, s) which satisfies the BAE a(ξj , κ, s) = −1, j = 1, . . . , n 2 − s (3.1) with n 2 − s zeros ξj of Q(ζ, κ, s) called Bethe roots. The BAE in the logarithmic form look log a(ξj , κ, s) = πimj , (3.2) where mj are pairwise non-coinciding odd integers. The ground state corresponds to s = 0 and the choice mj = 2j − 1. The following picture describes this situation · · · ◦ ◦ ◦ ◦ • • • • • · · · -5 -3 -1 1 3 5 where the black circles correspond to “particles” and have positive odd coordinates mj while the white circles correspond to “holes” and have negative coordinates. The particle-hole excitations can be got when some of “particles” are moved to positions with negative coordinates. Let us denote I(+,k) an ordered subset of positive coordinates I (+,k) 1 < · · · < I (+,k) k ≤ n which Fermionic Basis in CFT and TBA for Excited States 11 correspond to the positions of created holes and I(−,k) correspond to negative coordinates of the moved particles6 −I(−,k) k < · · · < −I(−,k) 1 : · · · ◦ ◦ ◦ · · · · ◦ • ◦ · · · · ◦ • ◦ · · · · · · ◦ ◦ ◦ • • • · · · • ◦ • · · · • ◦ • · · · −I(−,k) k · · · · −I(−,k) 1 · · · · · − 5− 3− 1 1 3 5 · · · · ·I(+,k) 1 · · · · I(+,k) k We will denote the corresponding vector and co-vector \|κ; I(+,k), I(−,k)〉 and 〈κ; I(+,k), I(−,k)\|. Let ξ−r , r = 1, . . . , k be the Bethe roots corresponding to the particles moved into the positions with negative coordinates −I(−,k) r and ξ+ r correspond to the holes with positive positions I (+,k) r . Following the papers [5, 7, 19, 20], we can rewrite the BAE (3.1) in form of the non-linear integral equation log a(ζ, κ, s) = −2πiν(κ− s) + log ( d(ζ) a(ζ) ) − ∫ γ(s,k) K(ζ/ξ) log (1 + a(ξ, κ, s)) dξ2 ξ2 , (3.3) where the contour γ(s,k) goes around all the Bethe roots ξj including the moved ones ξjr = ξ−r in the clockwise direction and the kernel K(ζ, α) = 1 2πi ∆ζψ(ζ, α), K(ζ) = K(ζ, 0), ∆ζf(ζ) = f(qζ)− f(q−1ζ). For the case of the ground state s = 0, k = 0 we have γ(0,0) = γ with the contour γ used in Section 3 of [4]. Let us for simplicity stay with the case s = 0. One can rewrite (3.3) replacing the contour γ(0,k) by γ and taking into account the contribution of the residues from the moved Bethe roots ξ−r and holes ξ+ r . There are also other equations coming from (3.2). Altogether we have the following set of equations log a(ζ, κ) = −2πiνκ+ log ( d(ζ) a(ζ) ) + k∑ r=1 ( g(ζ/ξ+ r )− g(ζ/ξ−r ) ) − ∫ γ K(ζ/ξ) log (1 + a(ξ, κ)) dξ2 ξ2 , log a(ξ±r , κ) = ∓πiI(±,k) r , r = 1, . . . , k, (3.4) where g(ζ) = log 1−q2ζ2 1−q−2ζ2 . Now we can consider the scaling limit and generalize the analysis of Section 10 of [4]. Let us start with a small remark. So far, we considered only particles which “jump” to the left. In principle, for the case of finite n we should also consider particles which “jump” to the right. But in the scaling limit when n→∞ it is implied that the right tail of the Bethe ansatz phases in the ground state is infinite. So, the jumps to the right are irrelevant in the scaling limit. Therefore we will not discuss them here. Let us introduce the functions T sc(λ, κ) = lim n→∞, a→0, 2πR=na T (λāν , κ), Qsc(λ, κ) = lim n→∞, a→0, 2πR=na āνκQ(λāν , κ) and taking into account that for 1/2 < ν < 1 the ratio a(ζ)/d(ζ)→ 1 in the scaling limit with ζ related to λ as in (2.11), we get the scaling limit of the auxiliary function a as asc(λ, κ) = Qsc(λq, κ) Qsc(λq−1, κ) . 6The case of s > 0 can be treated similarly to the case s = 0. The only difference is that for the finite n case one has less Bethe roots. 12 H. Boos We would like to study the asymptotic behavior of asc for λ2 → ∞ and κ → ∞ in such a way that the variable t = c(ν)−1 λ 2 κ2ν is kept fixed with c(ν) = Γ(ν)−2eδ ( ν 2R )2ν , δ = −ν log ν − (1− ν) log(1− ν). (3.5) Then the function F (t, κ) := log asc(λ, κ) satisfies the equation F (t, κ)− ∫ ∞ 1 K(t/u)F (u, κ) du u = −2πiνκ+ k∑ r=1 ( g(t/t+r )− g(t/t−r ) ) (3.6) − ∫ eiε∞ 1 K(t/u) log ( 1 + eF (u,κ) )du u + ∫ e−iε∞ 1 K(t/u) log ( 1 + e−F (u,κ) )du u , where ε is a small positive number and t±r = c(ν)−1 ξ±r 2 (κā)2ν . With a slight abuse of notation we will write K(t) = 1 2πi · 1 2 ( tq2 + 1 tq2 − 1 − tq−2 + 1 tq−2 − 1 ) , g(t) = log tq2 − 1 tq−2 − 1 . Introducing the resolvent R(t, u) R(t, u)− ∫ ∞ 1 dv v R(t, v)K(v/u) = K(t/u) (t, u > 1) and G(t;u, v) = G(t, u)−G(t, v), G(t, u) = ((I +R)g)(t, u) = g(t/u) + ∫ ∞ 1 R(t, v)g(v/u) dv v , (3.7) one can rewrite the equation (3.6) as follows F (t, κ) = κF0(t) + k∑ r=1 G(t; t+r , t − r ) − ∫ eiε∞ 1 R(t, u) log ( 1 + eF (u,κ) )du u + ∫ e−iε∞ 1 R(t, u) log ( 1 + e−F (u,κ) )du u , (3.8) where t±r depend on κ and can be defined from the equations F (t±r , κ) = ∓πiI(±,k) r , r = 1, . . . , k (3.9) Fermionic Basis in CFT and TBA for Excited States 13 and F0 is the same as in [4]. It satisfies the integral equation ((I −K)F0)(t) = −2πiν, (3.10) where as in (3.7), the contraction is defined on the interval [1,∞) Kf(t) = ∫ ∞ 1 K(t/u)f(u) du u . It follows from the WKB technique [7] that the asymptotic behavior at t→∞ F0(t) = const · t 1 2ν +O ( t− 1 2ν ) . Then the equation (3.10) can be uniquely solved by the Wiener–Hopf factorization technique F0(t) = ∫ R− i 2ν −i0 dl tilS(l) −if l(l + i 2ν ) (t > 1), where f = 1 2 √ 2(1− ν) (3.11) and the function S(k) = Γ(1 + (1− ν)ik)Γ(1/2 + iνk) Γ(1 + ik) √ 2π(1− ν) eiδk with δ defined in (3.5) satisfies factorization condition 1− K̂(k) = S(k)−1S(−k)−1, K̂(k) = sinh(2ν − 1)πk sinhπk , where K̂ stands for the Mellin transform of the kernel K K̂(k) = ∫ ∞ 0 K(t)t−ik dt t , K(t) = 1 2π ∫ ∞ −∞ K̂(k)tikdk. Below we also use the asymptotic expansion of S(k) at k →∞ S(k) ' 1 + ∞∑ j=1 Sj(ik)−j . For example, the two leading terms look S1 = (1 + ν)(2ν − 1) 24(1− ν)ν , S2 = (1 + ν)2(2ν − 1)2 1152(1− ν)2ν2 . (3.12) The same method leads to R(t, u) = ∫ ∞ −∞ dl 2π ∫ ∞ −∞ dm 2π tiluimS(l)S(m)K̂(m) −i l +m− i0 = K(t/u) + ∫ ∞ −∞ dl 2π ∫ ∞ −∞ dm 2π tiluimS(l)S(m)K̂(l)K̂(m) −i l +m− i0 . (3.13) 14 H. Boos One can solve the equation (3.8) iteratively using the asymptotic expansion F (t, κ) = ∞∑ n=0 κ−n+1Fn(t). An important difference of this expansion with the formula (10.3) of [4] is that here also even degrees of κ may appear. Since − 1 2πi u ∂ ∂u g(t/u) = K(t/u), we can get the following equation by differentiating (3.7) − 1 2πi u ∂ ∂u G(t/u) = ((I +R)K)(t, u) = R(t, u) and then using (3.13) G(t, u) = −2πi ∫ ∞ −∞ dl 2π ∫ ∞ −∞ dm 2π tiluimS(l)K̂(l)S(m) −i l +m− i0 −i m+ i0 , where we used the regularization m+ i0 in such a way that G(t, u)→ 0 for u→∞ but in fact, this does not matter because G(t, u) enters into the equation (3.8) only through the difference G(t, u)−G(t, v). One introduces the function Ψ(l, κ) which has an asymptotic expansion Ψ(l, κ) ' ∞∑ n=0 κ−n+1Ψn(l), Ψ0(l) = − if l(l + i 2ν ) (3.14) related to the function F (t, κ) via F (t, κ) = κF0(t) + ∫ ∞ −∞ dl tilS(l)K̂(l)(Ψ(l, κ)− κΨ0(l)). (3.15) It is convenient to introduce p := fκ, where f is defined in (3.11) and consider the asymptotic expansion with respect to p instead of κ. Using (3.15), one can rewrite (3.6) in the following equivalent form Ψ(p)(l, p)− pΨ(p) 0 (l) = k∑ r=1 Ĝ(l; t+r , t − r )− i p {∫ −i∞+ε 0 dx 2π R̂(l, eix/p) log ( 1 + eF (p)(eix/p,p) ) + ∫ i∞+ε 0 dx 2π R̂(l, e−ix/p) log ( 1 + e−F (p)(e−ix/p,p) )} , (3.16) where ε is a small positive number and by definition F (p)(t, p) := F (t, κ), Ψ(p)(l, p) := Ψ(l, κ), Ψ (p) 0 (l) := 1 f Ψ0(l) = − i l(l + i 2ν ) and then R̂(l, eix/p) = resh [ e−hx/p l + h S(h) ] , Ĝ(l; y, z) = −resh [ e−hy/p − e−hz/p h(l + h) S(h) ] , Fermionic Basis in CFT and TBA for Excited States 15 where resh is the coefficient at h−1 in the expansion at h =∞. Taking F (p)(eix/p, p) = −2π ( x− F̄ (x, p) ) , we obtain the Taylor series at x = 0 F̄ (x, p) = x+ resh [ e−hx/pS(h)iΨ(p)(h, p) ] . (3.17) As the asymptotic series with respect to p at p→∞, it starts with p−1. The 2k parameters t±r participating in (3.9) and (3.16) are functions of κ or equivalently of p. We take them in the following form t±r (κ) = e ix±r (p) p , x±r (p) = ∞∑ j=0 x±r,jp −j . (3.18) Then we get from (3.16) Ψ(p)(l, p) = k∑ r=1 Ĝ(l;x+ r (p), x−r (p))− ip l(l + i 2ν ) +H(l, p), (3.19) where H(l, p) = −2i p ∫ ∞ 0 dx 2π { resh ( e−hx/p l + h S(h) ) × ∞∑ n=0 1 n! F̄ (x, p)n ( − ∂ ∂x )n} even log(1 + e−2πx). (3.20) Here {f(x, ∂∂x)}even = 1 2(f(x, ∂∂x) + f(−x,− ∂ ∂x)). The parameters x±r (p) are to be determined from the condition F̄ (x±r (p), p) = x±r (p)∓ i 2 I(±,k) r (3.21) which is O(p−1) if x±r,0 = ± i 2 I(±,k) r . (3.22) So, we come to the iterative scheme which allows us to compute F (p)(t, p) to any order of p−1. One can see that the following expansions are consistent with the above equations: F̄ (x, p) = ∞∑ j=0 F̄j(x)p−j−1, (3.23) H(l, p) = ∞∑ j=0 Hj(l)p −j−1, (3.24) Ĝ(l; y, z) = ∞∑ j=0 Ĝj(l; y, z)p −j−1 (3.25) and in agreement with (3.14) Ψ(p)(l, p) = ∞∑ j=0 Ψ (p) j (l)p−j+1. (3.26) 16 H. Boos Let us explain the very first iteration step. We first compute H0 in the expansion (3.24) using (3.20), the formula∫ ∞ 0 dx 2π xm ( − ∂ ∂x )n log(1 + e−2πx) = m!(1− 2−m−1+n) ζ(m− n+ 2) (2π)m−n+2 and the fact that only the term with n = 0 in the sum at the right hand side of (3.20) contributes because F̄ is of order p−1. So, we easily come to the result H0(l) = − i 24 . Then we substitute it into (3.19) and calculate the leading order of the function Ĝ taking into account the condition (3.22). As a result we obtain a few leading orders of Ψ(p): Ψ(p)(l, p) = − ip l(l + i 2ν ) + ( k∑ r=1 i 2 (I(+,k) r + I(−,k) r )− i 24 ) p−1 +O ( p−2 ) . This we substitute into the formula (3.17) and get F̄0 from (3.23) F̄0(x) = − ( i 4ν + iS1 2 ) x2 + S1 ( k∑ r=1 i 2 ( I(+,k) r + I(−,k) r ) − i 24 ) , where S1 is given by (3.12). Then we take the equation (3.21) up to the order p−1 and easily solve it x±s,1 = ( i 4ν + iS1 2 )( I (±,k) s 2 )2 + S1 ( k∑ r=1 i 2 (I(+,k) r + I(−,k) r )− i 24 ) . One can do the second iteration by repeating this procedure. In Appendices A and B we show the result of such a calculation for few next orders with respect to p−1. 4 The function ωsc Also we need to generalize the expressions (11.5), (11.6) of [4] for the function ωsc to the case of excited states. Still we take the condition κ = κ′ for which we can choose the excited state for the spin 0 sector and for the sector with spin s in such a way that ρ(λ\|κ, κ′) = 1. The reasoning here is quite similar to that one described in Section 4 of [4]. Actually, one can start with the similar expression to (11.1) of [4] but with a generalized dressed resolvent Rdress ωsc(λ, µ\|κ, κ, α) = ( fleft ?k fright + fleft ?k Rdress ?k fright ) (λ, µ) + ω0(λ, µ\|α), (4.1) where7 fleft(λ, µ, α) = 1 2πi δ−λ ψ0(λ/µ, α), fright(λ, µ, α) = δ−µ ψ0(λ/µ, α), δ−λ f(λ) = f(qλ)− f(λ), ω0(λ, µ\|α) = δ−λ δ − µ ∆−1 λ ψ0(λ/µ, α), ψ0(λ, α) = λα λ2 − 1 , ∆λf(λ) = f(qλ)− f(q−1λ) 7In comparison with (11.1) of [4] we take instead of the function ψ defined in (2.5) the function ψ0. The result does not change if we also change the kernel Kα → Kα,0 as it is done in (4.2). Fermionic Basis in CFT and TBA for Excited States 17 with ∆−1 λ ψ0(λ, α) = −V P ∫ ∞ 0 ψ0(µ, α) 2ν ( 1 + (λ/µ) 1 ν ) dµ2 2πiµ2 , where the principal value is taken for the pole µ2 = 1. The contraction ?k means f ?k g = ∫ γ̃(0,k) f(λ)g(λ)dm(λ), dm(λ) = dλ2 λ2(1 + asc(λ, κ)) with the contour γ̃(0,k) which corresponds to γ(0,k) from the equation (3.3) but taken for variab- le λ2 instead of ζ2 = λ2ā2ν . We remind the reader that the contour γ(0,k) was taken around all the Bethe roots in the clockwise direction in case of the excited state with k particles and k holes for the zero spin sector. The resolvent Rdress fulfills the integral equation Rdress −Rdress ?k Kα,0 = Kα,0, Kα,0(λ) = ∆λψ0(λ, α). (4.2) Applying a similar trick which we used to derive (3.4), namely, deforming the contour γ̃(0,k) → γ̃(0,0) and taking into account additional terms coming from the residues corresponding to particles and holes, we can obtain Rdress(t, u)−R(t, u, α) = 2πi k∑ r=1 ( R(t, t+r , α)Rdress(t + r , u) F ′(t+r , κ) − R(t, t−r , α)Rdress(t − r , u) F ′(t−r , κ) ) − ∫ eiε∞ 1 R(t, v;α)Rdress(v, u) 1 + e−F (v,κ) dv v − ∫ e−iε∞ 1 R(t, v;α)Rdress(v, u) 1 + eF (v,κ) dv v , (4.3) F ′(t, κ) := t ∂ ∂t F (t, κ), where as in [4], we introduced the “undressed” resolvent R(t, u, α) which satisfies the equation R(t, u, α)− ∫ ∞ 1 dv v R(v, u, α)K0(t/v, α) = K0(t/u, α) (4.4) with the kernel K0(t, α) = 1 2πi ( (tq2)α/2 tq2 − 1 − (tq−2)α/2 tq−2 − 1 ) corresponding to the above kernel Kα,0. The solution of (4.4) again can be got by the Wiener– Hopf factorization technique R(t, u, α) = K0(t/u, α) + ∫ ∞ −∞ dl 2π ∫ ∞ −∞ dm 2π tiluimS(l, α)S(m, 2− α)K̂(l, α)K̂(m, 2− α) −i l +m− i0 with the Mellin-transform K̂(k, α) = sinhπ ( (2ν − 1)k − iα 2 ) sinhπ ( k + iα 2 ) corresponding to the kernel K0(t, α) and S(k, α) = Γ ( 1 + (1− ν)ik − α 2 ) Γ ( 1 2 + iνk ) Γ ( 1 + ik − α 2 )√ 2π(1− ν)(1−α)/2 eiδk, 18 H. Boos 1− K̂(k, α) = S(k, α)−1S(−k, 2− α)−1. We assume that 0 < α < 2. Now we take the ansatz for Rdress(t, u) Rdress(t, u) = K0(t/u, α) (4.5) + ∫ ∞ −∞ dl 2π ∫ ∞ −∞ dm 2π tiluimS(l, α)S(m, 2− α)K̂(l, α)K̂(m, 2− α)Θ(l,m\|p, α) with the asymptotic expansion8 at p→∞ Θ(l,m\|p, α) ' ∞∑ n=0 Θn(l,m\|α)p−n, Θ0(l,m\|α) = − i l +m (4.6) and substitute it into the equation (4.3). As a result we get the equation which allows us to calculate all Θn by iterations Θ(l,m\|p, α)−Θ0(l,m\|α) = 1 p resl′resm′ [ S(l′, 2− α)S(m′, α)Θ(m′,m\|p, α)/(l + l′) × ( −i k∑ r=1 e−(l′+m′)x+ r (p)/p/(F̄ ′(x+ r (p), p)− 1) + i k∑ r=1 e−(l′+m′)x−r (p)/p/(F̄ ′(x−r (p), p)− 1) + 2 ∞∑ n=0 1 n! ∫ ∞ 0 dx { e−(l′+m′)x/pF̄ (x, p) n ( − ∂ ∂x )n} odd 1 1 + e2πx )] , (4.7) where the odd part {f(x, ∂∂x)}odd = 1 2(f(x, ∂∂x)− f(−x,− ∂ ∂x)) and F̄ ′(x, p) := ∂ ∂x F̄ (x, p). Performing iterations implies that enough many orders in the expansion of F̄ (x, p) and x±(p) with respect to p−1 were obtained by means of the iteration scheme described in the previous section. For a few leading terms in the expansion for Θ(l,m\|p, α) we get Θ(l,m\|p, α) = − i l +m + ( 1 24ν − 1 2ν k∑ r=1 ( I(+,k) r + I(−,k) r )) × ( −iν(l +m)− 1 2 + ∆α ) p−2 +O ( p−3 ) , where ∆α is given by (2.2). Some other terms of this expansion will be shown in Appendix C for the case k = 1. One can derive relation like (11.5) of [4] using the form (4.1), integral equations (4.2), (4.3) for the dressed resolvent Rdress and the ansatz (4.5) ωsc(λ, µ\|κ, κ, α) (4.8) 8Here Θn(l,m\|α) are different from those introduced in [4] since we should take into account the contribution of terms with odd degrees n also. Fermionic Basis in CFT and TBA for Excited States 19 ' 1 2πi ∫ ∞ −∞ dl ∫ ∞ −∞ dm S̃(l, α)S̃(m, 2− α)Θ(l + i0,m\|p, α) ( eδ+πiνλ2 κ2νc(ν) )il ( eδ+πiνµ2 κ2νc(ν) )im , where we returned to the variables λ, µ and S̃(k, α) = Γ ( −ik + α 2 ) Γ ( 1 2 + iνk ) Γ ( −i(1− ν)k + α 2 )√ 2π(1− ν)(1−α)/2 . The asymptotic expansion at λ, µ → ∞ can be obtained by computing the residues of the functions S̃(l, α) and S̃(m, 2− α) ωsc(λ, µ\|κ, κ, α) ' ∞∑ r,s=1 1 r + s− 1 D2r−1(α)D2s−1(2− α)λ− 2r−1 ν µ− 2s−1 ν Ω2r−1,2s−1(p, α),(4.9) where D2n−1(α) = 1√ iν Γ(ν)− 2n−1 ν (1− ν) 2n−1 2 1 (n− 1)! Γ ( α 2 + 1 2ν (2n− 1) ) Γ ( α 2 + (1−ν) 2ν (2n− 1) ) (4.10) and Ω2r−1,2s−1(p, α) = −Θ ( i(2r − 1) 2ν , i(2s− 1) 2ν ∣∣∣p, α)(r + s− 1 ν )(√ 2pν R )2r+2s−2 . (4.11) The relations (4.8)–(4.11) look the same as (11.5)–(11.7) of [4]. However there is an important difference. It was pointed out in [4] that the expansion coefficients Θn ( i(2r−1) 2ν , i(2s−1) 2ν ∣∣∣p, α) satisfy the so-called vanishing property i.e. for given r and s they vanish starting from 2n = r+s. It is equivalent to the fact that the function Θ is proportional to a polynomial with respect to p. This is true only for the case of ground state k = 0 and also for the case k = 1 with I(+,1) = I(−,1) = 1. For both cases all the coefficients Θn with odd n vanish. We will see that in both cases the space of the CFT-descendants is one-dimensional. 5 Relation to the CFT Here we would like to study generalization of the relation (2.1) between the lattice six vertex model and the CFT to the case of excited states. As discussed in Section 2, for the case of the ground state we inserted at ±∞-points of the cylinder the two primary fields φ± with dimensions ∆±. We also identified ∆+ = ∆κ+1, ∆− = ∆−κ′+1 and ∆+ = ∆− for the case κ = κ′. The corresponding states were denoted \|∆+〉 and 〈∆−\| respectively. For the six vertex model we introduced in Section 3 the states \|κ; I(+,k), I(−,k)〉 marked by the two ordered sets I(+,k) and I(−,k) of k odd, positive, non-coinciding numbers. In case of the spin s sector we denote such a state \|κ, s; I(+,k), I(−,k)〉. In the scaling limit we can identify these states with descendants of the primary fields φ± at level N = 1 2 k∑ r=1 (I (+,k) r + I (−,k) r ) \|κ; I(+,k), I(−,k)〉 →scal \|∆+; I(+,k), I(−,k)〉 = ∑ n1≥···≥nm≥1 n1+···+nm=N A(I(+,k),I(−,k)) n1,...,nm L−n1 · · ·L−nm \|∆+〉, 〈κ+ α− s, s; I(+,k), I(−,k)\| →scal 〈∆−; I(+,k), I(−,k)\| = ∑ n1≥···≥nm≥1 n1+···+nm=N Ā(I(+,k),I(−,k)) n1,...,nm 〈∆−\|Lnm · · ·Ln1 . (5.1) 20 H. Boos We normalize 〈∆−; I(+,k), I(−,k)\|φα(0)\|∆+; I(+,k), I(−,k)〉 = 1 (5.2) but we do not demand orthogonality of the states with different sets. In other words, the “scalar product” 〈∆−; I(+,k), I(−,k)\|φα(0)\|∆+; I ′(+,k), I ′(−,k)〉 is not necessarily zero. So, instead of (2.1) we take ∑ n1≥···≥nm≥1 n1+···+nm=N ∑ n′1≥···≥n ′ m′ ≥1 n′1+···+n′ m′ =N A(I(+,k),I(−,k)) n1,...,nm Ā (I(+,k),I(−,k)) n′1,...,n ′ m′ × 〈∆−\|Ln′ m′ · · ·Ln′1Pα ( {l−j} ) φα(0)L−n1 · · ·L−nm \|∆+〉 (5.3) = lim n→∞, a→0, na=2πR 〈κ+ α− s, s; I(+,k), I(−,k)\|TS,Mq 2κSb∗∞,s−1 · · ·b∗∞,0 ( q2αS(0)O ) \|κ; I(+,k), I(−,k)〉〈κ+ α− s, s; I(+,k), I(−,k)\|TS,Mq2κSb∗∞,s−1 · · ·b∗∞,0 ( q2αS(0) ) \|κ; I(+,k), I(−,k)〉 . In this relation the polynomial Pα ( {l−j} ) does not depend on κ and the choice of the excitation i.e. it is independent of k and of the both sets I(±,k). As was pointed out above, the coefficients of this polynomial are rational functions of the conformal charge c and the conformal dimension ∆α of the primary field φα only. If the operator O = βCFT∗ 2i1−1 · · ·βCFT∗ 2in−1γ CFT∗ 2jn−1 · · ·γCFT∗ 2j1−1 then the level of descendants participating in Pα ( {l−j} ) is M = 2 n∑ l=1 (il + jl − 1). On the other hand, the coefficients An1,...,nm , Ān1,...,nm are independent of the choice of the operator O. So, our strategy is to take for a given level M all linear independent operators O represented in terms of the fermionic basis modulo integrals of motion and as many different excitations as necessary in order to fix the corresponding polynomials Pα ( {l−j} ) and the coefficients9 An1,...,nm , Ān1,...,nm . All other relations for this level partially determine further coefficients A, Ā and the rest of the equations fulfills automatically. In [4] we were able to fix the polynomials Pα ( {l−j} ) up to M = 6 using only the ground state data when k = 0. It means that for M ≤ 6 the relation (5.3) fulfills automatically for any excitation i.e. for any k and any two sets I(±,k). We checked this for the case k = 1 with I (+,1) 1 = I (−,1) 1 = 1 and I (+,1) 1 = 1, I (−,1) 1 = 3, I (+,1) 1 = 3, I (−,1) 1 = 1. Starting with M = 8 the situation changes. We do not have enough equations in order to fix the polynomials Pα ( {l−j} ) if we restrict ourselves with the case of the ground state. We need additional equations involving excitations. In the next section we will consider in detail the case M = 8. Before we go further let us make one remark. It is interesting to note that the above fermionic basis operators are marked in exactly the same way as the particle-hole excitations, namely, by two ordered sets of n odd, positive integers I(+,n) = {2i1 − 1 < · · · < 2in − 1} and I(−,n) = {2j1 − 1 < · · · < 2jn − 1}. So, we can denote such an operator OI(+,n),I(−,n) = βCFT∗ 2i1−1 · · ·βCFT∗ 2in−1γ CFT∗ 2jn−1 · · ·γCFT∗ 2j1−1. (5.4) In case we take this operator in the right hand side of (5.3) we will use the following shorthand notation r.h.s. of (5.3) := 〈I(+,k), I(−,k)\|OI(+,n),I(−,n) \|I(+,k), I(−,k)〉. Now let us describe in general how we compute both sides of the relation (5.3). We start with the right hand side of (5.3) determined by the lattice data. We pointed out the fact that 9In fact, not all coefficients A, Ā can be fixed but rather some of their products. We checked for one particular case that one can fix them completely taking Ān1,...,nm = An1,...,nm but a’priori it is not quite clear to us why it should be so. Fermionic Basis in CFT and TBA for Excited States 21 the theorem by Jimbo, Miwa, Smirnov works for the case of excited states also with the same fermionic operators. Only two transcendental functions ρ and ω are sensible to the changes that happen in the Matsubara direction. It means that we still can apply the Wick theorem and after taking the scaling limit come to the same determinant formula (2.17) but with a new functional determined by the right hand side of (5.3) instead of Zκ,κ ′ R . With our choice of parameters α, s, κ, κ′ the function ρsc is still 1 and the function ωsc is now determined through the asymptotic expansion (4.9). So, we come to 〈I(+,k), I(−,k)\|OI(+,n),I(−,n) \|I(+,k), I(−,k)〉 = det ( Ω2ir−1,2jr′−1(p, α) ir + jr′ − 1 ) r,r′=1,...,n , (5.5) where the function Ω is defined by (4.11). Now let us proceed to the left hand side of (5.3) which involves the CFT data. For simplicity let us put the radius of the cylinder R = 1. We need to calculate 〈∆−\|Ln′ m′ · · ·Ln′1Pα ( {l−j} ) φα(0)L−n1 · · ·L−nm \|∆+〉 with n1 ≥ · · · ≥ nm ≥ 1, n′1 ≥ · · · ≥ n′m′ ≥ 1 and N = m∑ j=1 nj = m′∑ j=1 n′j or picking out some monomial with respect to the local Virasoro generators l−j , we need the following value (n′1, . . . , n ′ m′ ; a1, . . . , ad;n1, . . . , nm) := 〈∆−\|Ln′ m′ · · ·Ln′1 l−2ad · · · l−2a1φα(0)L−n1 · · ·L−nm \|∆+〉 (5.6) with d positive integers 1 ≤ a1 ≤ · · · ≤ ad. In order to compute it we follow the scheme described in Section 6 of [4]. First we define the function W (z1, . . . , zK ;w) := 〈∆−\|T (zK) · · ·T (z1)φα(w)\|∆+〉, K = d+m+m′, where T (z) is the energy-momentum tensor as a function of the point z on the cylinder with the OPEs T (z)T (w) = − c 12 χ′′′(z − w)− 2T (w)χ′(z − w) + T ′(w)χ(z − w) +O(1), T (z)φα(w) = −∆αφα(w)χ′(z − w) + φ′α(w)χ(z − w) +O(1) and χ(z) = 1 2 coth (z 2 ) = ∞∑ n=0 B2n (2n)! z2n−1, (5.7) where B2n are Bernulli numbers. We also need the expansion: χ(z) = ±1 2 ± ∞∑ j=1 e∓jz, <(z)→ ±∞. (5.8) As was discussed above we can use two different expansions for the energy-momentum tensor as well 22 H. Boos • the “local” expansion in vicinity of z = 0 T (z) = ∞∑ n=−∞ lnz −n−2, • the “global” expansion when <(z)→ ±∞ T (z) = ∞∑ n=−∞ Lne nz − c 24 . (5.9) The action of the local Virasoro generators ln on a local field O(w) is defined through the contour integral (lnO)(w) = ∫ Cw dz 2πi (z − w)n+1T (z)O(w), where Cw encircles the point w anticlockwise. The conformal Ward–Takahashi identity allows to determine the function W (z1, . . . , zK) re- cursively: W (z1, . . . , zK ;w) = − c 12 K∑ j=2 χ′′′(z1 − zj)W (z2, . . . ĵ . . . , zK ;w) (5.10) + { K∑ j=2 ( −2χ′(z1 − zj) + (χ(z1 − zj)− χ(z1 − w)) ∂ ∂zj ) −∆αχ ′(z1 − w) + (∆+ −∆−)χ(z1 − w) + 1 2 (∆+ + ∆−)− c 24 } W (z2, . . . , zK ;w). Of course, the term containing the difference ∆+ −∆− drops for the case ∆+ = ∆− which is only interesting for us here. In order to calculate the above object (n′1, . . . , n ′ m′ ; a1, . . . , ad;n1, . . . , nm) given by (5.6), we proceed in several steps: step 1: take the recursion (5.10) and expand χ(z1−· · · ), χ′(z1−· · · ), χ′′(z1−· · · ), χ′′′(z1−· · · ) using the expansion (5.8) for <(z1) → −∞ and then having in mind the expansion (5.9) take there the coefficient at en ′ 1z1 , step 2: repeat this procedure consequently for the variables z2, z3, . . . , zm′ taking every time the coefficients at en ′ 2z2 , en ′ 3z3 , . . . , en ′ m′zm′ , step 3: similarly we proceed with the next m variables zm′+1, . . . , zm′+m taking the expansion (5.8) for <(zm′+1)→∞, . . . ,<(zm′+m)→∞, further using the recursion (5.10) and picking up coefficients at e−n1zm′+1 , . . . , e−nmzm′+m , step 4: now one can easily compute the limit w → 0 and then apply (5.10) with respect to the variable zm′+m+1, take the local expansion (5.7) of χ(zm′+m+1), χ′(zm′+m+1), χ′′(zm′+m+1), . . . and calculate the contour integral ∫ C0 dzm′+m+1z −2a1+1 m′+m+1 · · · with the expression obtained in this way, step 5: repeat the step 4 with respect to the residual variables zm′+m+2, . . . , zK every time calculating the contour integrals ∫ C0 dzm′+m+2z −2a2+1 m′+m+2 · · · etc. up to the last integral∫ C0 dzKz −2ad+1 K · · · . Fermionic Basis in CFT and TBA for Excited States 23 In this way we can obtain, for example, for the case d = 0, N = 2 (2;∅; 2) = c 2 + 4 (∆2 −∆ + ∆+), (5.11) (1, 1;∅; 2) = (2;∅; 1, 1) = 2(∆3 −∆ + 3∆+), (1, 1;∅; 1, 1) = ∆(∆− 1)(∆2 −∆ + 2) + 4∆+(2∆2 − 2∆ + 1) + 8∆2 +, where again we used the shorthand notation ∆ ≡ ∆α. For d = 1, a1 = 1, N = 2 we get (2; 1; 2) = −1 3 ∆(∆2 − 61∆ + 42) + c ( −1 6 ∆2 + 17 8 ∆ + 1 ) − c2 48 + ∆+ ( 4∆2 + 35 3 ∆ + c 3 + 8 ) + 4∆2 +, (1, 1; 1; 2) = (2; 1; 1, 1) = − 1 12 ∆(∆ + 1) ( 2∆2 + c∆− 86∆ + 72− 7c ) + ∆+ ( (2∆ + 3) ( ∆2 + 9 2 ∆ + 4 ) − c 4 ) + 6∆2 +, (1, 1; 1; 1, 1) = − 1 24 ∆(∆− 1) ( 2∆3 + (c− 50)∆2 − (c+ 44)∆ + 2c− 96 ) (5.12) + ∆+ 6 ( 6∆4 + 32∆3 − 2(c− 71)∆2 + 2(c+ 29)∆ + 48− c ) + ∆2 + 3 ( 24∆2 + 70∆ + 60− c ) + 8∆3 +. And for d = 1, a1 = 2, N = 2: (2; 2; 2) = ∆ ( ∆2 60 + 359 60 ∆ + 1921c 480 − 3 ) + 1921∆∆+ 60 , (1, 1; 2; 2) = ∆(∆ + 1) ( ∆2 120 + 179 120 ∆ + c− 1 ) + ∆ ( 12∆ + 1441 40 ) ∆+, (5.13) (1, 1; 2; 1, 1) = ∆2(∆− 1) ( ∆(∆− 1) 240 + 241 120 ) + ∆ ( 121 30 ∆2 + 599 30 ∆ + 1921 60 ) ∆+ + 481∆∆2 + 30 . 6 The level M = 8 Here we show how the procedure generally described in the previous section works for the case of level M = 8. As was pointed out in Introduction, there are 5 monomials of the local Virasoro generators which generate linear independent descendant states modulo integrals of motion. Let us arrange them as a vector Vl =  l4−2 l−4l 2 −2 l2−4 l−6l−2 l−8  . 24 H. Boos As for the fermionic operators (5.4), we also have 5 possibilities for M = 8 which we also take as a vector Vfermi =  φeven 1,7 φodd 1,7 φeven 3,5 φodd 3,5 βCFT∗ 1 βCFT∗ 3 γCFT∗ 3 γCFT∗ 1  , where we used the even and odd combinations (2.18). We would like to determine the 5 by 5 transformation matrix U Vfermi ∼= UVl, (6.1) where the weak equivalence “∼=” should be understood in the same way as in the formula (2.21) and all the matrix elements of U depend only on the conformal dimension ∆ ≡ ∆α and central charge c. In [4] we were not able to uniquely fix this matrix from the consideration of the ground state case N = 0. If we substitute the both sides of (6.1) into (2.1) where the matrix elements of U in every row correspond to the coefficients in the polynomial Pα ( {l−k} ) and use (5.5) for k = 0 then we get 〈∆−\|(UVl)φα(0)\|∆+〉〈∆−\|φα(0)\|∆+〉 = V (0), (6.2) where V (0) :=  (Ω (0) 1,7 + Ω (0) 7,1)/2 (Ω (0) 7,1 − Ω (0) 1,7)/(2dα) (Ω (0) 3,5 + Ω (0) 5,3)/2 (Ω (0) 5,3 − Ω (0) 3,5)/(2dα) Ω (0) 1,1Ω (0) 3,3/3− Ω (0) 1,3Ω (0) 3,1/4  (6.3) and Ω (0) j,j′ denote the functions Ωj,j′(p, α) given by (4.11) which are calculated for the ground state case N = 0, k = 0 as was explained in Section 11 of [4]. From the vanishing property follows that all five equations in (6.2) are polynomials with respect to p2 of degree 4. In all 5 cases one of the equations is fulfilled automatically. It means that every equation leaves an one-parametric freedom. In other words, in every row of U one matrix element is left undetermined. Therefore in order to fix the matrix U completely we need to involve excitations. Here we restrict ourselves with the case k = 1 and N ≤ 2 only. Let us remind the reader that the descendant level for the excitations N = 1 2 k∑ r=1 (I (+,k) r + I (−,k) r ). So, for the case N = 1 we have only one possibility I(+,1) = I(−,1) = {1}. As in the ground state case N = 0, the descendant space is one-dimensional because there is only one vector L−1\|∆+〉 here. When we wrote the paper [4] we thought that considering this excitation would help us to fix the above mentioned freedom. As appeared it is not the case because it does not produce any additional constraints on the transformation matrix U . We leave the checking of this fact as an exercise for the reader. So, we have to consider the case N = 2. There are two possibilities I(+,1) = {1}, I(−,1) = {3} and I(+,1) = {3}, I(−,1) = {1} that correspond to the two-dimensional descendant space spanned by two vectors L−2\|∆+〉 and L2 −1\|∆+〉. Now let us consider the states (5.1) \|∆+; {1}, {3}〉 = AL−2\|∆+〉+BL2 −1\|∆+〉, (6.4) Fermionic Basis in CFT and TBA for Excited States 25 \|∆+; {3}, {1}〉 = CL−2\|∆+〉+DL2 −1\|∆+〉, 〈∆−; {1}, {3}\| = Ā〈∆−\|L2 + B̄〈∆−\|L2 1, 〈∆−; {3}, {1}\| = C̄〈∆−\|L2 + D̄〈∆−\|L2 1, where we use simpler notation for the coefficients A ({1},{3}) 2 ≡ A, A ({1},{3}) 1,1 ≡ B, A ({3},{1}) 2 ≡ C, A ({3},{1}) 1,1 ≡ D and similar for Ā, B̄, C̄, D̄. As appeared we can take them Ā = A, B̄ = B, C̄ = C, D̄ = D. Now we should satisfy the normalization condition (5.2) 〈∆−; {1}, {3}\|φα(0)\|∆+; {1}, {3}〉 = 〈∆−; {3}, {1}\|φα(0)\|∆+; {3}, {1}〉 = 1. Substituting the above formulae (6.4) here, using the notation (5.6) and the fact that (1, 1;∅; 2)= (2;∅; 1, 1), we get A2(2;∅; 2) +B2(1, 1;∅; 1, 1) + 2AB(1, 1;∅; 2) = C2(2;∅; 2) +D2(1, 1;∅; 1, 1) + 2CD(1, 1;∅; 2) = 1, (6.5) where (2;∅; 2), (1, 1;∅; 1, 1), (1, 1;∅; 2) are given by (5.11). Now we can use the first row of (2.21), the formulae (5.5) and (6.5) in order to obtain(2;∅; 2) (1, 1;∅; 1, 1) (1, 1;∅; 2) (2; 1; 2) (1, 1; 1; 1, 1) (1, 1; 1; 2) (2; 2; 2) (1, 1; 2; 1, 1) (1, 1; 2; 2)  A2 B2 2AB  =  1 Ω (1,3) 1,1 3 4 ( Ω (1,3) 3,1 − Ω (1,3) 1,3 ) /dα  . (6.6) Here we denoted Ω (1,3) j,j′ the function Ωj,j′(p, α) given by the formula (4.11) where Θ(ij/(2ν), ij′/(2ν)\|p, α) is determined by the equation (4.7) with k = 1 and I(+,1) = {1}, I(−,1) = {3} while dα is taken from (2.19). The matrix elements of the second and the third row in the left hand side of (6.6) are given by (5.12) and (5.13) respectively. The matrix equation (6.6) can be solved with respect to A and B by inverting the 3 by 3 matrix. These equations are overdetermined. The first two of them give A2, B2 and we can check that the last equation which gives 2AB is fulfilled automatically. In a similar way one can get the coefficients C and D from the equation which can be obtained from (6.6) by changing A→ C, B → D and Ω (1,3) j,j′ → Ω (3,1) j,j′ where Ω (3,1) j,j′ is defined through the equations (4.11) and (4.7) with I(+,1) = {3}, I(−,1) = {1}. The result for few leading orders with respect to p looks as follows A = 1 4 (pν)−1 + c− 22 192 (pν)−2 − ( ∆(∆− 1) 16 + c2 + 148c− 860 18432 ) (pν)−3 +O ( p−4 ) , B = 1 8 (pν)−2 − c+ 14 384 (pν)−3 +O ( p−4 ) . (6.7) The coefficients C and D can be got from A and B respectively through the substitution of p by −p. As in the N = 0 case described above, one can substitute (6.1) into (5.3) and use (5.5) in order to get A2〈∆−\|L2(UVl)φα(0)L−2\|∆+〉+B2〈∆−\|L2 1(UVl)φα(0)L2 −1\|∆+〉 26 H. Boos + 2AB〈∆−\|L2 1(UVl)φα(0)L−2\|∆+〉 = V (1,3), (6.8) where V (1,3) can be obtained from V (0) given by (6.3) via the substitution Ω (0) j,j′ → Ω (1,3) j,j′ V (1,3) :=  ( Ω (1,3) 1,7 + Ω (1,3) 7,1 ) /2 (Ω (1,3) 7,1 − Ω (1,3) 1,7 ) /(2dα)( Ω (1,3) 3,5 + Ω (1,3) 5,3 ) /2( Ω (1,3) 5,3 − Ω (1,3) 3,5 ) /(2dα) Ω (1,3) 1,1 Ω (1,3) 3,3 /3− Ω (1,3) 1,3 Ω (1,3) 3,1 /4  . Both sides of the equation (6.8) can be represented as a power series with respect to p−1 starting with p0. Then one can equate coefficients standing at powers p−j and get as many equations as necessary. It turned out that all the equations obtained by taking coefficients at p0 up to p−8 fulfill automatically. The unknown matrix elements of the transformation matrix U are fixed only in the order p−9. All equations which stem from further orders with respect to p−1 should be fulfilled automatically. Unfortunately, so far we could not prove it or even check any further equations because of complexity of calculations in the intermediate stage. We hope to do it in future. We checked that if one substitutes A → C, B → D in the left hand side of the equation (6.8) and Ω (1,3) j,j′ → Ω (3,1) j,j′ in the right hand side of (6.8) then the equation obtained is fulfilled automatically up to p−9. Let us explicitly show the final result using even and odd combinations (2.18): φeven 1,7 ∼= l4−2 + ( 4(c− 4)∆ + 4(c+ 8) ) / ( 3(∆ + 4) ) l−4l 2 −2 − ( −43c2 + 924c− 16340− (21c2 + 1102c− 18535)∆ − (11c2 − 198c+ 775)∆2 + 40(c− 25)∆3 ) / ( 45(∆ + 4)(∆ + 11) ) l2−4 − ( 2(−c2 − 1540c+ 17264)− 2(43c2 − 298c+ 2652)∆ − 12(c2 − 31c+ 574)∆2 + 32(c− 28)∆3 ) / ( 15(∆ + 4)(∆ + 11) ) l−6l−2 − ( −45c3 + 1637c2 − 137176c+ 2033360− 20(2c3 + 261c2 − 7623c+ 67411)∆ − 4(10c3 − 439c2 + 7142c− 5825)∆2 + 96(2c2 − 81c+ 705)∆3 ) / ( 105(∆ + 4)(∆ + 11) ) l−8, (6.9) φodd 1,7 ∼= ( 4∆ ) /(∆ + 4)l−4l 2 −2 + ( 20(c− 28) + 24(c+ 47)∆ + 8(2c− 11)∆2 ) / ( 15(∆ + 4)(∆ + 11) ) l2−4 + ( −40(c− 28) + 8(13c− 44)∆ + 16(c− 8)∆2 ) / ( 5(∆ + 4)(∆ + 11) ) l−6l−2 − ( 20(c− 28)(c+ 79)− 4(15c2 + 1316c− 20143)∆ − 4(15c2 − 398c+ 4359)∆2 + 32(c− 33)∆3 ) / ( 35(∆ + 4)(∆ + 11) ) l−8, φeven 3,5 ∼= l4−2 + ( 8(2c+ 13) + 12(c− 16)∆ ) / ( 9(∆ + 4) ) l−4l 2 −2 + ( 2(262c2 − 4271c+ 82750) + 3(59c2 + 2338c− 67785)∆ + (79c2 − 1502c− 6965)∆2 + 120(c− 25)∆3 ) / ( 405(∆ + 4)(∆ + 11) ) l2−4 + ( 4(68c2 + 7571c− 86380) + 4(188c2 − 3379c+ 10310)∆ + 24(4c2 − 127c+ 1125)∆2 ) / ( 135(∆ + 4)(∆ + 11) ) l−6l−2 + ( 2(420c3 − 9013c2 + 711929c− 10449400) + 4(70c3 + 9741c2 − 370938c+ 3325745)∆ Fermionic Basis in CFT and TBA for Excited States 27 + 20(14c3 − 629c2 + 9532c− 48805)∆2 + 96(c− 25)(2c− 31)∆3 ) / ( 945(∆ + 4)(∆ + 11) ) l−8, (6.10) φodd 3,5 ∼= ( 4∆ ) / ( 3(∆ + 4) ) l−4l 2 −2 + ( −4(17c− 236)− 24(c− 43)∆ + 8(4c− 127)∆2 ) / ( 135(∆ + 4)(∆ + 11) ) l2−4 + ( 8(17c− 236) + 8(47c− 536)∆ + 48(c− 18)∆2 ) / ( 45(∆ + 4)(∆ + 11) ) l−6l−2 + ( 28(15c2 − 571c+ 7708) + 4(35c2 + 2708c− 66859)∆ + 20(7c2 − 222c+ 767)∆2 + 96(c− 33)∆3 ) / ( 315(∆ + 4)(∆ + 11) ) l−8 and for the fourth-order combination we get βCFT∗ 1 βCFT∗ 3 γCFT∗ 3 γCFT∗ 1 ∼= 1/12 · l4−2 + ( 3c− 54 + 2(c− 22)∆ ) / ( 18(∆ + 4) ) l−4l 2 −2 − ( 43c2 − 1426c+ 11664 + 2(15c2 − 302c− 1069)∆ + 2(c2 + 86c− 2667)∆2 + 16(c− 25)∆3 ) / ( 216(∆ + 4)(∆ + 11) ) l2−4 + ( 215c2 − 8714c+ 74952 + (125c2 − 3856c+ 13208)∆ + 2(5c2 − 73c− 1816)∆2 + 16(c− 28)∆3 ) / ( 180(∆ + 4)(∆ + 11) ) l−6l−2 − ( −25c3 + 1755c2 − 39410c+ 326592− 4(30c2 − 1393c+ 9520)∆ + 8(11c+ 289)∆2 + 192∆3 ) / ( 72(∆ + 4)(∆ + 11) ) l−8. (6.11) The elements of the above matrix U can be easily got from these data. It is interesting to note that the determinant of U has relatively simple factorized form detU = −32(c− 25)(c− 28)(c− 33)(∆− 1) (6.12) × (c+ 2− 2(c+ 11)∆ + 48∆2)(−c2 + 18c+ 175 + 16(c− 25)∆ + 192∆2) 382725(∆ + 4)(∆ + 11) . The numerator of (6.12) is proportional to the following product (ν − 2)2(ν − 3)(ν − 4)(2ν − 3)(3ν − 4)(αν − 1)(αν − 2)(αν + 1− ν)(αν − 1− ν) × (αν + 2− ν)(αν − 2 + ν)(αν − 2− ν)(αν + 1− 2ν)(αν + 2− 2ν)(αν + 2− 3ν). It would be interesting to understand the meaning of the degeneration points like ν = 2, ν = 3, ν = 4, ν = 2/3, ν = 3/4 and α = 1/ν, α = 2/ν, α = −(1 − ν)/ν etc. where the determinant detU = 0. 7 Conclusion In this paper we demonstrated that the method developed in [4] works for excited states as well. As we saw with the example of the descendant level M = 8, it is not possible to completely determine the transformation matrix between the usual basis constructed through the action of Virasoro generators on the primary field and the fermionic basis constructed in [1, 2] without involving excitations. On the other hand the matrix elements of the transformation matrix should not depend on the fact which excitation is taken. This should provide an interesting compatibility condition on the structure of the three-point functions both from the point of view of the CFT and the lattice XXZ model we started with. In this paper we were able to treat only the case of excitations corresponding to the descendant level N ≤ 2. It would be interesting to check the above mentioned compatibility for the case of other excitations with N > 2 and for the higher descendants M > 8 as well. Of course, it is very important to find 28 H. Boos a general proof of the compatibility condition. We think it is also interesting to study singular points of the transformation matrix mentioned in the end of the previous section. This may shed new light on the structure of the Virasoro algebra, Verma modules and singular vectors. One more important generalization of the results obtained here, which is still out of our reach, would be to treat the case of general values α, κ, κ′ and also the case of different excitations inserted at +∞ and −∞ on the cylinder. In both cases the function ρ is not 1 and we would have to generalize the whole Wiener–Hopf factorization technique. We hope to return to these questions in future publications. A The function Φ(p) and integrals of motion Here we show several further orders of the 1/p-expansion of the functions used in Section 3 for the case of excitations with k = 1. For simplicity we use shorter notation m0 ≡ I (+,1) 1 , m1 ≡ I(−,1) 1 . First let us show expressions for several coefficients Ψ (p) j (l) of the expansion (3.26) for the function Ψ(p)(l, p): Ψ (p) 0 (l) = − i l(l + i/(2ν)) , Ψ (p) 1 (l) = 0, Ψ (p) 2 (l) = i 24 (−1 + 12(m0 +m1)), Ψ (p) 3 (l) = − l − i/(2ν) 8 (m2 0 −m2 1), Ψ (p) 4 (l) = − l − i/(2ν) 2880ν(1− ν) ( ilν(1− ν)(7/2 + 60(m3 0 +m3 1)) + 5(2ν2 − 11(1− ν))(m3 0 +m3 1) + 5(1 + ν)(2ν − 1)(m0 +m1)(6(m0 +m1)− 1) + ν2 + 3(1− ν) ) , Ψ (p) 5 (l) = l − i/(2ν) 147456ν2(1− ν)2 (m2 0 −m2 1) ( 384l2ν2(1− ν)2(m2 0 +m2 1) − 16ilν(1− ν) ( 5(2ν2 + 11(1− ν))(m2 0 +m2 1) + 2(1 + ν)(2ν − 1)(12(m0 +m1)− 1) ) − (20ν4 − 220ν3 + 681ν2 − 922ν + 461)(m2 0 +m2 1) − 2(1 + ν)(2ν − 1)(2ν2 + 23(1− ν))(12(m0 +m1)− 1)) ) , Ψ (p) 6 (l) = l − i/(2ν) 8360755200ν3(1− ν)3 { 8640il3(−31 + 252(m5 0 +m5 1))(1− ν)3ν3 + 1440l2(1− ν)2ν2 ( 567(2ν2 + 11(1− ν))(m5 0 +m5 1) + 21(1 + ν)(2ν − 1)(m0 +m1)(7− 10(m2 0 +m2 1 −m0m1) + 120(m3 0 +m3 1)) − 164ν2 − 755(1− ν) ) − 90ilν(1− ν) ( 21(6085− 12170ν + 8769ν2 − 2684ν3 + 244ν4)(m5 0 +m5 1) + (1 + ν)(2ν − 1)(420(34ν2 + 295(1− ν))(m4 0 +m4 1) + 840(14ν2 + 113(1− ν))(m3 0m1 +m0m 3 1) − 2520(2ν2 + 23(1− ν))m2 0m 2 1 + 70(274ν2 − 257(1− ν))(m3 0 +m3 1) + 2520(1 + ν)(2ν − 1)(m0 +m1)(−m0 −m1 + 12m0m1) + 7(158ν2 + 761(1− ν))(m0 +m1))− 96(159− 318ν + 240ν2 − 81ν3 + 8ν4) ) Fermionic Basis in CFT and TBA for Excited States 29 − 21(210403− 631209ν + 813759ν2 − 575503ν3 + 219354ν4 − 36804ν5 + 1288ν6)(m5 0 +m5 1)− (1 + ν)(2ν − 1) ( 420(20281− 40562ν + 25053ν2 − 4772ν3 + 52ν4)(m4 0 +m4 1)− 840(−6407 + 12814ν − 6771ν2 + 364ν3 + 436ν4)m0m1(m2 0 +m2 1)− 2520(2489− 4978ν + 3837ν2 − 1348ν3 + 308ν4)m2 0m 2 1 + 70(−31607 + 63214ν + 16989ν2 − 48596ν3 + 3316ν4)(m3 0 +m3 1) + 12600(1 + ν)(−1 + 2ν)(2ν2 + 35(1− ν))(m0 +m1) × (−m0 −m1 + 12m0m1)− 7(−39599 + 79198ν − 52347ν2 + 12748ν3 + 2452ν4)(m0 +m1) ) + 103680(1− ν)(5− 10ν + 10ν2 − 5ν3 + ν4) } . (A.1) One can check that the Ψ-function in the ground state case is reproduced if one takes m0 = m1 = 0. There is a connection of the function Ψ with the integrals of motion described in [7]. In the ground state case this connection was given by the formula (10.17) of [4]: I2n−1 = −iΨ(p) ( i(2n− 1) 2ν , p ) n(2n− 1)(2ν2)n−1p2n−1. (A.2) In the case of excitations the integrals of motion are given by matrices. Let us consider the first three integrals of motion I1, I3, I5. Their explicit expressions via the Virasoro generators can be found in [6]. Again let us consider an example of the excitations with N = 2 and k = 1 with two pos- sibilities I(+,1) = {1}, I(−,1) = {3} and I(+,1) = {3}, I(−,1) = {1} which correspond to the two-dimensional descendant space spanned by two vectors L−2\|∆+〉 and L2 −1\|∆+〉 or their linear combinations \|∆+; {1}, {3}〉, \|∆+; {3}, {1}〉 given by the formula (6.4). Let us start with the simplest case of the very first integral of motion I1 given by the formula (11) of the paper [6]: I1 = L0 − c 24 . This operator is diagonal for the above two-dimensional space I1 ( L−2\|∆+〉 L2 −1\|∆+〉 ) = ( ∆+ + 2− c 24 )(L−2\|∆+〉 L2 −1\|∆+〉 ) and ∆+ + 2− c 24 = 2(pν)2 + 47 24 . We can easily check that we get exactly the same result if we substitute the expansion (3.26) with the coefficients given by (A.1) for l = i/(2ν) and m0 = 1, m1 = 3 or m0 = 3, m1 = 1 into the formula (A.2) with n = 1. We see that in this case all Ψ (p) j (i/(2ν)) = 0 with j ≥ 3 even without fixing m0 and m1. The next case of I3 is a bit less trivial since I3 is not diagonal any longer. Following [6], we have I3 = 2 ∞∑ n=1 L−nLn + L2 0 − c+ 2 12 L0 + c(5c+ 22) 2880 . 30 H. Boos We see that only the first term here is not diagonal. Using the Virasoro algebra, one can check that I3 ( L−2\|∆+〉 L2 −1\|∆+〉 ) = ( c 6 12∆+ 4 )( L−2\|∆+〉 L2 −1\|∆+〉 ) + ( 8∆+ + (∆+ + 2)2 − (c+ 2)(∆+ + 2) 12 + c(5c+ 22) 2880 )( L−2\|∆+〉 L2 −1\|∆+〉 ) . We can diagonalize the matrix( c 6 12∆+ 4 ) = ( C D A B )−1( λ+ 0 0 λ− )( C D A B ) with the eigenvalues λ± = −6ν2 − 5(1− ν) 2(1− ν) ± 3 √ 64ν2(1− ν)2p2 + (1 + ν)2(2ν − 1)2 2(1− ν) (A.3) and the matrix elements A, B, C, D taken from the definition of the vectors \|∆+; {1}, {3}〉, \|∆+; {3}, {1}〉 given by (6.4). Hence these vectors are eigenvectors of the matrix I3. Several leading terms of 1/p-expansion of A, B where shown in (6.7) while C, D can be got from A, B by changing p→ −p. So, for the eigenvalues of I3 we obtain10 I (±) 3 = λ± + 8∆+ + (∆+ + 2)2 − (c+ 2)(∆+ + 2) 12 + c(5c+ 22) 2880 ' 4(pν)4 + 47 2 (pν)2 ∓ 12pν − 4804ν2 − 5769(1− ν) 960(1− ν) ∓ 3 32 (1 + ν)2(2ν − 1)2 (1− ν)2 (pν)−1 +O ( p−2 ) . (A.4) We can check at least for number of first orders that we get the same 1/p-expansion if, like in the previous case we substitute the expansion (3.26) this time for l = 3i/(2ν) and m0 = 3, m1 = 1 in case of I (+) 3 and m0 = 1, m1 = 3 in case of I (−) 3 into the formula (A.2) with n = 2. In contrast to the case of the first integral of motion I1, the series expansion (A.4) does not terminate because of the square root in the expression (A.3) for the eigenvalues λ±. Also we get both even and odd powers with respect to p in contrast to the case N = 0 corresponding to the ground state. The next case can be treated similarly. First we take I5 from [6] I5 = ∑ n1+n2+n3=0 : Ln1Ln2Ln3 : + ∞∑ n=1 ( c+ 11 6 n2 − 1− c 4 ) L−nLn + 3 2 ∞∑ r=1 L1−2rL2r−1 − c+ 4 8 L2 0 + (c+ 2)(3c+ 20) 576 L0 − c(3c+ 14)(7c+ 68) 290304 with the normal ordering : : for which the Virasoro generators with bigger indices are placed to the right. Then using the Virasoro algebra, one can come to the following formula I5 ( L−2\|∆+〉 L2 −1\|∆+〉 ) = 5 ( 5 6 + c 24 + ∆+ )( c 6 12∆+ 4 )( L−2\|∆+〉 L2 −1\|∆+〉 ) 10We hope the reader will not mix these eigenvalues I (±) j with the integrals of motion I±j mentioned in Section 2 that equal to each other when κ′ = κ and ∆+ = ∆−. Fermionic Basis in CFT and TBA for Excited States 31 + ( ∆3 + + 236− c 8 ∆2 + + ( c2 192 − 227c 288 + 3845 72 ) ∆+ ) + ( ∆3 + − c3 13824 + 1361c2 145152 − 7325c 5184 + 221 36 )( L−2\|∆+〉 L2 −1\|∆+〉 ) . We see that the matrix I5 can be diagonalized by the same similarity transformation as I3 because all the integrals of motion commute. The result for the asymptotic expansion of the two corresponding eigenvalues I±5 looks I (±) 5 ' 8(pν)6 + 235 2 (pν)4 ∓ 120(pν)3 − 4808ν2 − 12503(1− ν) 96(1− ν) (pν)2 ∓ 15(4ν4 + 36ν3 + 21ν2 − 114ν + 57) 16(1− ν)2 (pν) + 822432ν4 + 2338220ν3 − 821605ν2 − 3033230ν + 1516615 96768(1− ν)2 +O ( p−1 ) . Again this result perfectly matches the expansion of (A.2) for n = 3 with Ψ(p) (5i/(2ν), p) taken for m0 = 3, m1 = 1 in case of I (+) 5 and for m0 = 1, m1 = 3 in case of I (−) 5 . Similar to I (±) 3 , the integral I (±) 5 and all further integrals I (±) 2n−1 are not polynomials with respect to p in contrast to the ground state case. B The functions F̄ (x, p) and x± r (p) In Section 3 we discussed several functions defined within the TBA approach. In the previous appendix we showed few orders of the function Ψ(p)(l, p). We also need the functions F̄ (x, p) and x±r (p). As was described in Section 3, the asymptotic expansions of these functions and the function Ψ(p)(l, p) are calculated order by order with respect to 1/p via the iterative procedure. There we explained how does the very first iteration work. Once the function Ψ(p)(l, p) is found up to some order, the next order of the function F̄ (x, p) can be obtained via the equation (3.17). The coefficients of (3.18) are determined from the equation (3.21). The further iteration steps are straightforward but the answer becomes rather cumbersome already after several iterations. For the reader who wants to check his own calculations we show few orders of the expansions (3.23) and (3.18) in the case k = 1. We do not think it would be instructive to show further orders. As in the previous appendix we use the shorthand notation m0 ≡ I(+,1) 1 , m1 ≡ I(−,1) 1 . F̄0(x) = −i(1 + ν)(2ν − 1) 576ν(1− ν) (1− 12(m0 +m1))− ix2 2ν2 + 11(1− ν) 48ν(1− ν) , F̄1(x) = i (1 + ν)(2ν − 1)(2ν2 + 23(1− ν)) 9216ν2(1− ν)2 − x(1 + ν)2(2ν − 1)2 27648ν2(1− ν)2 (1− 12(m0 +m1)) + x3 4ν4 − 44ν3 + 309ν2 − 530ν + 265 6912ν2(1− ν)2 , F̄2(x) = − i(1 + ν)(2ν − 1) 2388787200ν3(1− ν)3 ( 120(436ν4 + 364ν3 − 6771ν2 + 12814ν − 6407) × (m3 0 +m3 1) + 3600(1 + ν)(2ν − 1)(2ν2 + 23(1− ν))(m0 +m1) × (1− 6(m0 +m1)) + 2452ν4 + 9148ν3 − 50547ν2 + 82798ν − 41399 ) + x(1 + ν)(2ν − 1) 3317760ν3(1− ν)3 (556ν4 − 1676ν3 + 2859ν2 − 2366ν + 1183)(m2 0 −m2 1) 32 H. Boos − ix2(1 + ν)(2ν − 1) 19906560ν3(1− ν)3 (556ν4 − 2036ν3 + 3039ν2 − 2006ν + 1003) × (1− 12(m0 +m1))− ix4 9953280ν3(1− ν)3 (1112ν6 − 2796ν5 − 5154ν4 + 64603ν3 − 154059ν2 + 146109ν − 48703), F̄3(x) = −i (1 + ν)(2ν − 1) 15288238080ν4(1− ν)4 (m2 0 −m2 1) ( 3(7736ν6 − 49644ν5 + 41430ν4 + 233059ν3 − 658107ν2 + 649893ν − 216631)(m2 0 +m2 1) − 4(1 + ν)(2ν − 1)(932ν4 + 788ν3 − 22227ν2 + 42878ν − 21439) × (1− 12(m0 +m1)) ) + x (1 + ν)(2ν − 1) 229323571200ν4(1− ν)4 ( 1080(1496ν6 − 12924ν5 + 42270ν4 − 75161ν3 + 78753ν2 − 49407ν + 16469)(m3 0 +m3 1) − 960(1 + ν)(2ν − 1)(556ν4 − 1676ν3 + 2859ν2 − 2366ν + 1183)(m0 +m1) × (1− 6(m0 +m1)) + 138728ν6 − 926052ν5 + 2802450ν4 − 4743023ν3 + 4847079ν2 − 2970681ν + 990227 ) + ix2 (1 + ν)(2ν − 1) 637009920ν4(1− ν)4 (m2 0 −m2 1) × (4568ν6 − 40572ν5 + 138270ν4 − 239513ν3 + 230049ν2 − 132351ν + 44117) + x3 (1 + ν)2(2ν − 1)2 5733089280ν4(1− ν)4 (2284ν4 − 8084ν3 + 12111ν2 − 8054ν + 4027) × (1− 12(m0 +m1)) + x5 4777574400ν4(1− ν)4 (9136ν8 − 76576ν7 + 196840ν6 + 149096ν5 − 3382325ν4 + 10503740ν3 − 14391926ν2 + 9334868ν − 2333717), F̄4(x) = i(1 + ν)(2ν − 1) 2157476157849600ν5(1− ν)5 ( 6048(4048ν8 + 208064ν7 − 1145688ν6 + 1757040ν5 + 1432457ν4 − 8416452ν3 + 12210950ν2 − 8061828ν + 2015457)(m5 0 +m5 1) + 3809872ν8 − 215990816ν7 + 939983096ν6 − 1142084632ν5 − 1598906179ν4 + 6538494484ν3 − 8737091978ν2 + 5620794700ν − 1405198675 − 196(1 + ν)(2ν − 1)(m0 +m1) ( 240(11528ν6 − 109308ν5 + 124842ν4 + 705307ν3 − 2193591ν2 + 2209125ν − 736375)(m3 0 +m3 1) + 120(27112ν6 − 287566ν5 + 627666ν4 + 187193ν3 − 2262129ν2 + 2602239ν − 867413)m0m1(m2 0 +m2 1) + 10(247064ν6 + 795588ν5 − 9542418ν4 + 12330151ν3 + 6743697ν2 − 15490527ν + 5163509)(m2 0 +m2 1) + 10(575464ν6 + 728508ν5 − 17201838ν4 + 25221881ν3 + 6701007ν2 − 23174337ν + 7724779)m0m1 + 360(2ν2 + 23(1− ν))(676ν4 − 3716ν3 + 12489ν2 − 17546ν + 8773)m0m1(m0 +m1)− 5040(1 + ν)(2ν − 1) × (68ν4 + 92ν3 − 2223ν2 + 4262ν − 2131)(m0 +m1) + 7(35272ν6 − 272436ν5 + 362346ν4 + 559733ν3 − 2128749ν2 + 2218659ν − 739553) )) − x(1 + ν)(2ν − 1) 1284211998720ν5(1− ν)5 (m2 0 −m2 1) ( 8(114704ν8 − 186848ν7 − 1703896ν6 + 8391976ν5 − 17873683ν4 + 22437364ν3 − 17519338ν2 + 8605900ν − 2151475) × (m2 0 +m2 1) + 7(1 + ν)(2ν − 1)(11240ν6 − 137412ν5 + 480594ν4 − 893735ν3 Fermionic Basis in CFT and TBA for Excited States 33 + 965295ν2 − 622113ν + 207371)(1− 12(m0 +m1)) ) − i x2(1 + ν)2(2ν − 1)2 77052719923200ν5(1− ν)5 ( 120(1247080ν6 − 6770580ν5 + 15390258ν4 − 18566683ν3 + 12601659ν2 − 3981981ν + 1327327)(m3 0 +m3 1) + 1680(4568ν6 − 40572ν5 + 138270ν4 − 239513ν3 + 230049ν2 − 132351ν + 44117)(m0 +m1)(1− 6(m0 +m1)) + 7(1201400ν6 − 6364860ν5 + 14007558ν4 − 16171553ν3 + 10301169ν2 − 2658471ν + 886157) ) + x3(1 + ν)(2ν − 1) 321052999680ν5(1− ν)5 (m2 0 −m2 1)(2622064ν8 − 14133568ν7 + 33585976ν6 − 49220560ν5+ 55866331ν4− 52813996ν3+ 43115890ν2− 21866764ν + 5466691) − ix4(1 + ν)(2ν − 1) 3852635996160ν5(1− ν)5 (2622064ν8 − 14517280ν7 + 35135944ν6 − 51550552ν5+ 56686759ν4− 51453364ν3+ 41083762ν2− 20513692ν + 5128423) × (1− 12(m0 +m1))− ix6 4815794995200ν5(1− ν)5 (5244128ν10− 25645072ν9 + 45932736ν8 − 30480936ν7 + 22676526ν6 − 291308781ν5 + 1140456861ν4 − 2039726202ν3 + 1941710088ν2 − 954519025ν + 190903805). For the parameters x±1 (p) ≡ x±(p) = ∞∑ j=0 x±j p −j which determine the Bethe roots via the rela- tion (3.18) we had the initial conditions x+ 0 = im0 2 , x−0 = − im1 2 and then x+ 1 = i(2ν2 + 11(1− ν)) 192ν(1− ν) m2 0 − i(1 + ν)(2ν − 1) 576ν(1− ν) (1− 12(m0 +m1)), x+ 2 = i(20ν4 − 220ν3 + 681ν2 − 922ν + 461)m3 0 55296ν2(1− ν)2 + i(1 + ν)(2ν − 1)(2ν2 + 23(1− ν)) 55296ν2(1− ν)2 (−m0 + 18m2 0 + 12m0m1 − 6m2 1), x+ 3 = i(1288ν6 − 36804ν5 + 219354ν4 − 575503ν3 + 813759ν2 − 631209ν + 210403)m4 0 159252480ν3(1− ν)3 + i(1 + ν)(2ν − 1)(52ν4 − 4772ν3 + 25053ν2 − 40562ν + 20281)m3 0 9953280ν3(1− ν)3 − i(1 + ν)(2ν − 1)(436ν4 + 364ν3 − 6771ν2 + 12814ν − 6407)m1(3m2 0 +m2 1) 19906560ν3(1− ν)3 + i(1 + ν)2(2ν − 1)2(2ν2 + 35(1− ν))(−m0 −m1 + 12m0m1 + 6m2 1) 663552ν3(1− ν)3 + i(1 + ν)(2ν − 1)(3316ν4 − 48596ν3 + 16989ν2 + 63214ν − 31607)m2 0 79626240ν3(1− ν)3 − i(1 + ν)(2ν − 1)(308ν4 − 1348ν3 + 3837ν2 − 4978ν + 2489)m0m 2 1 3317760ν3(1− ν)3 − i(1 + ν)(2ν − 1)(2452ν4 + 12748ν3 − 52347ν2 + 79198ν − 39599) 2388787200ν3(1− ν)3 , 34 H. Boos x+ 4 = i 458647142400ν4(1− ν)4 ( −3 ( 18704ν8 + 1330336ν7 − 16442440ν6 + 75087544ν5 − 183046975ν4 + 267969460ν3 − 244932514ν2 + 133379452ν − 33344863 ) m5 0 − 150(1 + ν)(2ν − 1) ( 3256ν6 + 51156ν5 − 738570ν4 + 2884979ν3 − 5217867ν2 + 4530453ν − 1510151 ) m4 0 + 30(1 + ν)(2ν − 1) ( 53816ν6 − 1086444ν5 + 6642390ν4 − 7848221ν3 − 4235067ν2 + 9791013ν − 3263671 ) m3 0 − 90(1 + ν)(2ν − 1) ( 7736ν6 − 49644ν5 + 41430ν4 + 233059ν3 − 658107ν2 + 649893ν − 216631 ) m1(4m3 0 −m3 1) + 7200(1 + ν)2(2ν − 1)2(2ν2 + 23(1− ν))2(−3m2 0 − 2m0m1 + 36m2 0m1 +m2 1 − 12m3 1) + (1 + ν)(2ν − 1) ( 74632ν6 − 808308ν5 + 3707610ν4 − 8492107ν3 + 10979811ν2 − 8080509ν + 2693503 ) m0 − 60(1 + ν)(2ν − 1) ( 9976ν6 − 100044ν5 + 431430ν4 − 1092901ν3 + 1621773ν2 − 1290387ν + 430129 ) m0m 2 1(3m0 − 2m1) ) . The coefficients x−j can be got from x̃+ j obtained from x+ j by the replacement m0 ↔ m1 as follows: x−2j = −x̃+ 2j , x−2j+1 = x̃+ 2j+1, j = 0, 1, . . . . C The function ωsc In Section 4 we explained how we determine our main object: the function ωsc. The asymptotic expansion for ωsc is related to the function Θ by the formula (4.9). In it’s turn the coefficients Θn of asymptotic expansion (4.6) of the function Θ Θ(l,m\|p, α) ' ∞∑ j=0 Θj(l,m\|α)p−j can be found with help of the TBA data of Section 3 and Appendices A, B via solving the equation (4.7) by iterations. In this appendix we show the result for several leading coefficients again in the case k = 1. With the exception of the coefficient Θ0, all other coefficients Θj are polynomials with respect to l and m: Θ0(l,m\|α) = − i l +m , Θ1(l,m\|α) = 0, Θ2(l,m\|α) = −ν 2α(2− α) + 2i(1− ν)(−i+ 2ν(l +m)) 96ν(1− ν) (1− 12(m0 +m1)), Θ3(l,m\|α) = − m2 0 −m2 1 512ν2(1− ν)2 ( −3ν4α2(2− α)2 − 24iα(2− α)(1− ν)ν2(−i+ ν(l +m)) + 32(1− ν)2(−i+ ν(l +m)) (−i+ 2ν(l +m)) ) , Θ4(l,m\|α) = − 1 184320ν3(1− ν)3 ( −5ν6 ( 5(m3 0 +m3 1)− 42(m0 +m1)2 + 7(m0 +m1) ) × α3(2− α)3 + 10iν4(1− ν) ( −24(−3i+ 2(l +m)ν)(m3 0 +m3 1) + 2m(0,1)(−16i+ 9(l +m)ν)− (−i+ (l +m)ν) ) α2(2− α)2 + 8ν(1− ν) × ( 2ν(1− ν)(7 + 120(m3 0 +m3 1))(−i+ (l +m)ν)2 − 3ν(1− ν) ( 1 + 10m(0,1) ) × (−i+ 2lν)(−i+ 2mν)− 20iν(1 + ν)(2ν − 1)(−i+ (l +m)ν)m(0,1) Fermionic Basis in CFT and TBA for Excited States 35 − iν(−i+ (l +m)ν) ( 4ν2 + 5(1 + 24(m3 0 +m3 1))(1− ν) )) α(2− α) − 16iν4(1− ν)(2− ν) ( 1 + 10m(0,1) ) (l −m)α(1− α)(2− α) + 8i(1− ν)3 × (7 + 120(m3 0 +m3 1))(−i+ (l +m)ν)(−i+ 2(l +m)ν)(−3i+ 2(l +m)ν) + 8(1− ν)2(1 + ν)(2ν − 1) ( 1 + 10m(0,1) )( 2(−i+ (l +m)ν)(−i+ 2(l +m)ν) + (−i+ 2lν)(−i+ 2mν) )) , where m(0,1) = m3 0 + m3 1 + 6(m0 + m1)2 − m0 − m1. Unfortunately, we cannot show all the orders in the expansion of Θ and other functions from the previous Appendices that we needed in order to get the formulae (6.9)–(6.11) because the expressions become very cumbersome with growing order. Again the ground state result can be obtained by taking m0 = m1 = 0. Acknowledgements Our special thanks go to M. Jimbo, T. Miwa and F. Smirnov with whom the work on this paper was started. Also we would like to thank F. Göhmann, A. Klümper and S. Lukyanov for many stimulating discussions. We are grateful to the Volkswagen Foundation for financial support. References [1] Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Hidden Grassmann structure in the XXZ model, Comm. Math. Phys. 272 (2007), 263–281, hep-th/0606280. [2] Boos H., Jimbo M., Miwa T., Smirnov F., Takeyama Y., Hidden Grassmann structure in the XXZ model. II. Creation operators, Comm. Math. Phys. 286 (2009), 875–932, arXiv:0801.1176. 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[20] Klümper A., Free energy and correlation length of quantum chains related to restricted solid-on-solid lattice models, Ann. Physik (8) 1 (1992), 540–553. http://dx.doi.org/10.1103/PhysRevLett.26.1301 http://dx.doi.org/10.1143/PTP.46.401 http://dx.doi.org/10.1143/PTP.48.2187 http://dx.doi.org/10.1143/PTP.48.2187 http://dx.doi.org/10.1103/PhysRevLett.59.259 http://dx.doi.org/10.1103/PhysRevLett.59.259 http://dx.doi.org/10.1016/0550-3213(85)90271-8 http://arxiv.org/abs/hep-th/0307108 http://dx.doi.org/10.1088/0305-4470/24/13/025 http://dx.doi.org/10.1002/andp.19925040707 1 Introduction 2 Reminder of basic results of HGSIV 3 TBA for excited states 4 The function sc 5 Relation to the CFT 6 The level M=8 7 Conclusion A The function (p) and integrals of motion B The functions (x,p) and xr(p) C The function sc References

Fermionic Basis in Conformal Field Theory and Thermodynamic Bethe Ansatz for Excited States

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