Analytic Bethe Ansatz and Baxter equations for long-range psl(2|2) spin chain
aa r X i v : . [ h e p - t h ] J un Analytic Bethe Ansatz and Baxter equationsfor long-range psl (2 | spin chain A.V. Belitsky
Department of Physics, Arizona State UniversityTempe, AZ 85287-1504, USA
Abstract
We study the largest particle-number-preserving sector of the dilatation operator in maximallysupersymmetric gauge theory. After exploring one-loop Bethe Ansatze for the underlying spinchain with psl (2 |
2) symmetry for simple root systems related to several Kac-Dynkin diagrams, weuse the analytic Bethe Anzats to construct eigenvalues of transfer matrices with finite-dimensionalatypical representations in the auxiliary space. We derive closed Baxter equations for eigenvaluesof nested Baxter operators. We extend these considerations for a non-distinguished root systemwith FBBF grading to all orders of perturbation theory in ’t Hooft coupling. We construct gener-ating functions for all transfer matrices with auxiliary space determined by Young supertableaux(1 a ) and ( s ) and find determinant formulas for transfer matrices with auxiliary spaces corre-sponding to skew Young supertableaux. The latter yields fusion relations for transfer matriceswith auxiliary space corresponding to representations labelled by square Young supertableaux.We derive asymptotic Baxter equations which determine spectra of anomalous dimensions ofcomposite Wilson operators in noncompact psl (2 |
2) subsector of N = 4 super-Yang-Mills theory. Introduction
Low-dimensional integrable structures were long known to emerge in Quantum Chromodynamics— the theory of strong interaction. Evolution equations describing logarithmic modification ofscattering amplitudes in kinematical regimes corresponding to different physical phenomena werefound to possess hidden symmetries. These include high-energy and large-momentum transferasymptotics of the theory. The former refers to Regge behavior of cross sections in energyvariable. In leading logarithmic approximation it is governed by reggeon quantum mechanics,i.e., an interacting system with conserved number of particles. Its Hamiltonian was identifiedwith the one of a noncompact Heisenberg magnet with SL (2 , C ) symmetry group [1, 2]. Theregime of scattering amplitudes with large momentum transfer is endowed with operator productexpansion such that evolution equations are equivalent to Callan-Symanzik equations. The latteris in turn a Ward identity for dilatations which is one of the generators of the conformal group,the symmetry of classical Lagrangian of the theory. The dilatation operator acts on the spacespanned by Wilson operators — composite operators built from elementary fields of the theoryand covariant derivatives. At leading order of perturbation theory, the dilatation operator admitsa pair-wise form for a class of quasipartonic operators [3]. In multicolor N c → ∞ limit, onlynearest neighbor interactions survive with long-range effects being suppressed by 1 /N c . It wasrealized that the dilatation operator for aligned-helicity operators coincides with the Hamiltonianof yet another integrable system — spin chain with SL (2 , R ) symmetry group [4, 5, 6]. All four-dimensional gauge theories inherit integrability of one-loop dilatation operator since they allshare the same noncompact sector of operators with covariant derivatives [7, 8]. Supersymmetryenhances the phenomenon to a larger set of operators eventually encompassing all operatorsin maximally supersymmetric gauge theory as was demonstrated in [9, 10, 11, 12, 13, 8]. Thesuper-spin chain Hamiltonian inherits the symmetry group of the classical gauge theory and itsspectrum can be found by means of the nested Bethe Ansatz [14].Beyond leading order of perturbation theory in a generic gauge theory many space-timecharges associated with classical symmetry generators cease to be conserved due to anomalies.This may potentially lead to breaking of integrability in higher loops. The N = 4 super-Yang-Mills theory, on the other hand, is superconformal to all orders and thus its classical symmetrypersists even when quantum effects are taken into account. However, starting from two-looporder the dilatation operator becomes long-ranged, namely more then two partons can be simul-taneously involved in scattering. This invalidates standard procedures to derive Bethe Ansatzequations. Evidence gathered from multiloop perturbative calculations hinted that integrabilityin maximally supersymmetric gauge theory carries on to higher orders and that the spectrum ofcomposite operators is encoded in a long-range super-spin chain model [15, 16, 17]. Bethe Ansatztype equations which depend on the ’t Hooft coupling constant g = g YM √ N c / (2 π ) and which gen-erate known anomalous dimensions in lowest orders of perturbative expansion were conjecturedin Ref. [17]. Integrability of planar N = 4 super Yang-Mills theory then suggests that spectraof anomalous dimensions of composite operators can be computed exactly at finite coupling g and the superconformal nature of the theory implies that this provides an exact solution to it.These Bethe Ansatz equations do not properly incorporate wrapping effect when the range ofthe interaction becomes as long as the spin chain itself and thus the equations are intrinsicallyasymptotic.In this paper we will continue developing an alternative approach to long-range super-spinmagnets based on transfer matrices and Baxter equations [18] initiated in Refs. [19, 20, 21].1ransfer matrices is the main ingredient of this framework. They encode the full set of mutu-ally commuting conserved quantities with Hamiltonian being one of these. Transfer matrices aresupertraces of monodromy matrices in a representation of the symmetry algebra. The Baxter op-erators themselves are certain transfer matrices with a special — spectral parameter-dependent— dimension of representations in the auxiliary space. Thus, they are quantum generalizationof supercharacters and we should expect a transfer matrix (or its eigenvalues) to be a sum ofterms, one per component of corresponding representation in the auxiliary space. The absenceof R − matrices yielding the putative long-range Bethe equations does not prevent us from for-mulating transfer matrices of the model. To accomplish this goal we resort on the analyticBethe Ansatz [22], a techniques which bypasses the microscopic treatment and relies on generalproperties of the macroscopic system like analyticity, unitarity and crossing symmetry.Recently we have addressed Baxter equation for the closed sl (2 |
1) subsector of the N = 4dilatation operator and have shown that it takes the form of a second order finite differenceequation [21] like for the sl (2) long-range magnet [20]. In the present work, we will focus on themaximal particle-number-preserving psl (2 |
2) sector [23] of the maximally supersymmetric gaugetheory studied in a number of papers at one [24], two [25] and all [17] orders, where a similarform of Baxter equations is anticipated [26].The outline of the paper is as follows. In the next section we describe the sl (2 |
2) subsectorof N = 4 super-Yang-Mills theory which arises as a projection of the theory on the light-coneand restricting the particle content to N = 2 hypermultiplet. Then in section 3, we analyze theshort range spin chain describing the spectrum of one-loop dilatation operator. We start withthe distinguished basis and construct transfer matrices with simplest representations in auxiliaryspace which suffice to formulate closed Baxter equations for respective nested Baxter functions.Subsequently we perform a set of Weyl super-reflections with respect to odd roots to obtain Betheequations corresponding to Kac-Dynkin diagram with two isotropic fermionic roots which allowsfor a natural generalization to all orders of perturbation theory. In section 3.3, we constructall transfer matrices with symmetric and antisymmetric atypical representations in the auxiliaryspace and build a determinant representation for transfer matrices labelled by a skew Young su-perdiagram. Subsequently we find finite difference equations for Baxter polynomials. All previousconsiderations are generalized in section 4 to all orders of perturbation within the framework ofasymptotic analytic Bethe Ansatz. Several appendices contain details of projection of su (2 , | psl (2 |
2) subsec-tor, superspace realization of these algebras and Serre-Chevalley bases for psl (2 |
2) correspondingto distinguished and symmetric Kac-Dynkin diagrams. Finally we determine numbers of rootsof nested Baxter functions in terms of eigenvalues of Cartan generators. sl (2 | subsector of N = 4 super-Yang-Mills To start with, let us specify the subsector of N = 4 super-Yang-Mills theory which will be thefocus of our current study. Recall that the physical field content of the maximally supersymmetricgauge theory can be accommodated in a single chiral light-cone N = 4 superfield [27, 28, 8],Φ (cid:0) x µ , θ A (cid:1) = ∂ − z A ( x µ ) + θ A ∂ − z ¯ λ A ( x µ ) + i θ A θ B ¯ φ AB ( x µ )+ 13! ε ABCD θ A θ B θ C λ D ( x µ ) − ε ABCD θ A θ B θ C θ D ∂ z ¯ A ( x µ ) . (2.1)2epending on the bosonic four-vector x µ = ( z, x + , x ⊥ ) with its minus component being chirallight-cone coordinate z = x − + ¯ θ A θ A and four Grassmann variables θ A . Truncating this superfieldin one of the superspace coordinates, say θ , one observes the particle content falls into two N = 2superfields [8] Φ( x µ , θ A ) | θ =0 = Φ G ( x µ , θ , θ ) + θ Φ WZ ( x µ , θ , θ ) , (2.2)with one encoding the N = 2 gauge supermultiplet and another N = 2 Wess-Zumino hypermul-tiplet [8], Φ G ( x µ , θ , θ ) = ∂ − z A ( x µ ) + ∂ − z θ j ¯ λ j ( x µ ) + i θ j θ k ¯ φ jk ( x µ ) , (2.3)Φ WZ ( x µ , θ , θ ) = ∂ − z ¯ λ ( x µ ) + iθ j ¯ φ j ( x µ ) + θ θ λ ( x µ ) , (2.4)where the summation runs over the remaining odd directions in superspace, i.e., j, k = 2 , sl (2 |
2) subsector [23] of the full theory as the Ψ WZ component of the N = 4 light-cone superfield projected on the light-cone, i.e., x µ = ( z, , ⊥ ). Forfurther use, it is convenient to introduce new notations for components of the hypermultiplet andodd direction of the N = 2 superspace. Namely, identifying θ ’s as ϑ = θ , ϑ = θ , the gauginofields as ¯ λ = ¯ χ , ψ = λ and the su (2) doublet of scalars as follows ϕ a = ( ¯ φ , ¯ φ ) = ( X, Z ), weget the Wess-Zumino superfield in the formΦ WZ ( Z ) = ∂ − z ¯ χ ( z ) + iϑ a ϕ a ( z ) + 12 ε ab ϑ a ϑ b ψ ( z ) , (2.5)depending on the light-cone superspace variable Z = ( z, ϑ a ). Truncating further in either ϑ or ϑ variable, one gets the closed sl (2 |
1) sector recently discussed in Ref. [21]. Thus in the large- N c limit the sl (2 |
2) sector of the dilatation operator in the maximally supersymmetric Yang-Millstheory is spanned by single trace operators built from the Wess-Zumino superfields, O ( Z , . . . , Z L ) = tr { Φ WZ ( Z ) . . . Φ WZ ( Z L ) } . (2.6)The light-cone superfield Φ WZ ( Z ) defines an infinite-dimensional chiral representation of the sl (2 |
2) algebra with generators realized as differential operators acting on superspace coordinatesas shown in Appendix A. The Wess-Zumino superfield possesses a vanishing conformal spin andinvolves a nonlocal operator ∂ − z ¯ χ (0) as an artefact of the light-cone formalism. To overcome thiscomplication we introduce a regularization by setting the conformal dimension of Φ WZ ( Z ) to ℓ WZ = ǫ and taking the physical limit ǫ → V ǫ arises fromthe expansion of the superfield in Taylor series in even and odd variables around z = ϑ a = 0 andis spanned by polynomials in z and ϑ a , V ǫ = span { , z k +1 , ϑ a z k , ε ab ϑ a ϑ b z k | k ∈ N } . (2.7)The single-trace L − field light-cone operator (2.6) belongs to the L − fold product of these spaces V ⊗ Lǫ . Therefore, the eigenfunctions Ψ ω of the spin chain are classified according to irreduciblecomponents entering this tensor product parameterized by the eigenvalues ω = [ ℓ, t, b, L ] ofgenerators of the psl (2 |
2) Cartan subalgebra, u B (1) automorphism and the length of the operator This regularization affects the form of the generators (A.25) which receive additive addenda: L + → L + ǫ = L + + 2 ǫz , L → L ǫ = L + ǫ , ¯ V a, + → ¯ V a, + ǫ = ¯ V a, + + 2 ǫϑ a and B → B ǫ = B + ǫ . O ω corresponding to superconformal polynomialsΨ ω can be projected out from the light-cone operator (2.6) by means of the sl (2 | − invariantscalar product, O ω = h Ψ ω ( Z , . . . , Z L ) | O ( Z , . . . , Z L ) i≡ Z L Y k =1 [ DZ k ] ǫ Ψ ω ( Z , . . . , Z L ) O ( Z , . . . , Z L ) , (2.8)with the integration measure [29] Z [ DZ ] ǫ = 1Γ(2 ǫ + 1) Z | z |≤ d zπ Z Y a =1 (cid:0) d ¯ ϑ a dϑ a (cid:1) (1 − ¯ zz + ¯ ϑ a ϑ a ) ǫ . (2.9)One can easily convince oneself that the polynomials z k ϑ a . . . , forming the representation space V ǫ , are orthogonal with respect to the sl (2 | − scalar product, i.e., h z k ϑ a . . . | z n ϑ b . . . i ∼ δ kn δ ba . . . .The nonlocal operator ∂ − z ¯ χ (0) associated with the lowest component in the Taylor expansionof the superfield Φ WZ ( Z ) defines an invariant one-dimensional subspace V nonloc = { } . LocalWilson operators belong to the quotient V loc = V ǫ / V nonloc . With the chosen normalization of themeasure (2.9), one finds that h z k | z n i ∼ δ kn / Γ(2 ǫ + k ) such that the vector belonging to V nonloc possesses zero norm and is orthogonal to all states { z k +1 , ϑ z k , ϑ z k , ϑ ϑ z k | k ∈ N } ↔ { ( D + ) k ¯ χ, ( D + ) k X, ( D + ) k Z, ( D + ) k ψ | k ∈ N } , (2.10)belonging to V loc . These will be identified with excitations propagating on the super-spin chainwhich we will turn to next. The eigenvalue problem for the spectrum of one-loop anomalous dimensions of superconformaloperators (2.6) can be reformulated in terms of a short-range psl (2 |
2) quantum super-spin chainwith the one-loop dilatation operator being identified with the nearest-neighbor Hamiltonian ofthe magnet. The spin chain is integrable and it can be diagonalized be means of the nestedBethe Ansatz [14]. It was observed some time ago [30, 31] that for a spin chain model based ona given symmetry (super-)algebra, the nested Bethe Ansatz equations are determined by simpleroot systems of the algebra, generally,( − A pp / u ( p )0 ,k − i w p u ( p )0 ,k + i w p ! L = N r Y q =1 n q Y j =1 u ( p )0 ,k − u ( q )0 ,j + i A pq u ( p )0 ,k − u ( q )0 ,j − i A pq , (3.1)where A pq = ( α p | α q ) is the Cartan matrix and w p are the Kac-Dynkin labels, w p = ( α p | µ )determined by a weight vector µ of a representation of the algebra acting on the spin chain sites.Here N r = rank( G ) is the rank of the algebra G . For projective algebras the upper limit in theproduct is N r = rank( G ) + 1, which is 3 for our psl (2 |
2) sector. As it becomes obvious from the See Appendix B for details. B F F B F B F
F B B F
SW SW
Figure 1: A subset of Kac-Dynkin diagrams for psl (2 | SW generates the diagram standing to its right.above equation, there exists several sets of equivalent Bethe Ansatz equations reflecting the factthat there are several choices of simple root systems { α p | p = 1 , . . ., N r } for a superalgebra, seeFig. 1. These simple root systems are related by reflections with respect to odd simple roots α with vanishing bilinear form ( α | α ) = 0 which form the Weyl supergroup. For Bethe equationsthis corresponds to a particle-hole transformations.To diagonalize the short-range psl (2 |
2) magnet one can use the nested Algebraic Bethe Ansatz[30, 32] following Refs. [33, 34, 35] and construct transfer matrices by a fusion procedure [36].However, the lack of a systematic procedure to construct long-range integrable spin chains corre-sponding to gauge theories, binds one has to resort to techniques which bypass the microscopictreatment and rely on general properties of macroscopic systems. The method of analytic BetheAnsatz, which is a generalization of the inverse scattering method, was developed to determinethe spectrum of transfer matrices for closed chains [22] and serves the purpose. In this approach,one uses general properties such as analyticity, unitarity, crossing symmetry, etc., to completelydetermine eigenvalues of transfer matrices.
Let us start with the distinguished Kac-Dynkin (left-most) diagram in Fig. 1 with BBFF gradingencoded in the Cartan matrix A = − − − . (3.2)The nested Bethe equations read, according to Eq. (3.1), − ˜ u (1)0 ,k − i ˜ u (1)0 ,k + i ! L = e Q (1)0 (cid:16) ˜ u (1)0 ,k + i (cid:17)e Q (1)0 (cid:16) ˜ u (1)0 ,k − i (cid:17) e Q (2)0 (cid:16) ˜ u (1)0 ,k − i (cid:17)e Q (2)0 (cid:16) ˜ u (1)0 ,k + i (cid:17) , ˜ u (2)0 ,k + i ˜ u (2)0 ,k − i ! L = e Q (1)0 (cid:16) ˜ u (2)0 ,k − i (cid:17)e Q (1)0 (cid:16) ˜ u (2)0 ,k + i (cid:17) e Q (3)0 (cid:16) ˜ u (2)0 ,k + i (cid:17)e Q (3)0 (cid:16) ˜ u (2)0 ,k − i (cid:17) , (3.3) − e Q (3)0 (cid:16) ˜ u (3)0 ,k − i (cid:17)e Q (3)0 (cid:16) ˜ u (3)0 ,k + i (cid:17) e Q (2)0 (cid:16) ˜ u (3)0 ,k + i (cid:17)e Q (2)0 (cid:16) ˜ u (3)0 ,k − i (cid:17) . They are written in terms of the Baxter polynomials e Q ( p )0 ( u ) = ˜ n p Y k =1 (cid:16) u − ˜ u ( p )0 ,k (cid:17) , (3.4)5arameterized by three sets of Bethe roots ˜ u ( p )0 ,k . Here and below in this section all symbols carrya subscript 0 indicating zeroth order of perturbation theory for the corresponding quantities ingauge theory. These equations yield one-loop anomalous dimensions for Wilson operators in psl (2 |
2) sector of N = 4 super-Yang-Mills theory.In this section we briefly discuss the construction of eigenvalues for transfer matrices in thedistinguished basis echoing considerations of Ref. [37] adopted to psl (2 |
2) algebra. Transfer ma-trices are supertraces of monodromy matrices with certain representation of symmetry algebra inthe auxiliary space. Here we will present only the ones with low-dimensional representations inthe auxiliary space which are sufficient to derive closed Baxter equations for nested Baxter poly-nomials. A full-fledge considerations will be done below for a simple root system correspondingto the symmetric Kac-Dynkin diagram in FBBF grading with two isotropic fermionic roots, see(right-most graph) Fig. 1. Being supertraces over a representation of the algebra, the eigenvalueformulas are expected to be given by a sum of terms, one for each component of the represen-tation. This idea is at the crux of the approach suggested for spin chain based on classical Liealgebras in Ref. [38, 39] and generalized for superalgebras in Ref. [37].We derive transfer matrices with auxiliary space labelled by a particular Young supertableau.The Young superdiagrams are different from classical ones in that there is no limitation onthe number of rows [40]. Thus, we introduce elementary Young supertableaux depending on aspectral parameter u u = (cid:0) u + i (cid:1) L e Q (1)0 (cid:0) u + i (cid:1)e Q (1)0 (cid:0) u − i (cid:1) , (3.5)2 u = (cid:0) u − i (cid:1) L e Q (1)0 (cid:0) u − i (cid:1)e Q (1)0 (cid:0) u − i (cid:1) e Q (2)0 ( u ) e Q (2)0 ( u − i ) , u = (cid:0) u − i (cid:1) L e Q (3)0 (cid:0) u − i (cid:1)e Q (3)0 (cid:0) u − i (cid:1) e Q (2)0 ( u ) e Q (2)0 ( u − i ) , u = (cid:0) u − i (cid:1) L e Q (3)0 (cid:0) u + i (cid:1)e Q (3)0 (cid:0) u − i (cid:1) , parameterized in terms of three Baxter polynomials e Q ( k )0 entering the nested Bethe Ansatz equa-tions (3.3). Each box is labelled by an index with grading ¯1 = ¯2 = 0 and ¯3 = ¯4 = 1 in accordwith the distinguished Kac-Dynkin diagram in Fig. 1.We introduce notations for eigenvalues of transfer matrices t , (1 a ) ( u ) = t , [ a ] ( u ) , t , ( s ) ( u ) = t { s } ( u ) , (3.6)with totally antisymmetric (1 a ) and symmetric ( s ) atypical representations in the auxiliary space.For the lowest dimensional representations, the eigenvalues of transfer matrices can we written6n terms of the elementary boxes as follows t , [1] ( u ) = t { } ( u ) = 1 u + 2 u − u − u , (3.7) t , [2] ( u ) = 12 u − i u + i − u − i u + i − u − i u + i − u − i u + i − u − i u + i + 34 u − i u + i + 33 u − i u + i + 44 u − i u + i ,t { } ( u ) = 1 2 u + i u − i − u + i u − i − u + i u − i − u + i u − i − u + i u − i + 3 4 u + i u − i + 1 1 u + i u − i + 2 2 u + i u − i . The right-hand sides of these expressions are free from pole at positions of Bethe roots as can beeasily proved making use of the nested Bethe Ansatz equations (3.3). When written explicitly interms of Baxter polynomials, the conjugate transfer matrices, i.e., with antichiral representationsin the auxiliary space, can be obtained from these by merely dressing transfer matrices with abar and changing the signs in front of imaginary units. The generating function of all transfermatrices will be given below in Sect. 3.3 though for the symmmetric Kac-Dynkin diagram.Using these transfer matrices and their conjugate one may derive closed equations obeyedby the Baxter polynomials. First, solving the transfer matrix t , [1] ( u ) with respect to e Q (1)0 andsubstituting it into ¯ t ( u ), one finds the Baxter equation for e Q (2)0 t , [1] (cid:0) u + i (cid:1) e Q (2)0 (cid:0) u − i (cid:1) − ¯ t , [1] (cid:0) u − i (cid:1) e Q (2)0 (cid:0) u + i (cid:1) = 0 . (3.8)It is a first order finite-difference equation. Analogously, eliminating the polynomial e Q (3)0 fromthe transfer matrix t , [1] by substituting it twice into t , [2] and shifting its argument in middle ofthe way, one finds (cid:2) t , [2] ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)(cid:3) e Q (1)0 ( u ) + ( u + i ) L t , [1] (cid:0) u + i (cid:1) e Q (1)0 ( u − i ) (3.9)+ ( u − i ) L t , [1] (cid:0) u − i (cid:1) e Q (2)0 (cid:0) u + i (cid:1)e Q (2)0 (cid:0) u − i (cid:1) e Q (1)0 ( u − i ) = 0 . From this one can find a closed equation for the Baxter polynomial e Q (1)0 by merely eliminatingthe ratio of the polynomials e Q (2)0 making use of Eq. (3.8). An alternative, symmetric form of theBaxter equation can be found by deriving first an equation analogous to (3.9) but with conjugatetransfer matrices and eliminating the ratio of e Q (2)0 ’s from them. This yields (cid:2) t , [2] ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)(cid:3) (cid:2) ¯ t , [2] ( u ) − ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1)(cid:3) e Q (1)0 ( u ) (3.10)+ ( u + i ) L t , [1] (cid:0) u − i (cid:1) (cid:2) ¯ t , [2] ( u ) − ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1)(cid:3) e Q (1)0 ( u + i )+ ( u − i ) L ¯ t , [1] (cid:0) u + i (cid:1) (cid:2) t , [2] ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)(cid:3) e Q (1)0 ( u − i ) = 0 . Similarly, solving for e Q (1)0 from t , [1] and eliminating it from t , [2] we find an equation t , [2] ( u ) e Q (3)0 ( u ) + u L t , [1] (cid:0) u − i (cid:1) e Q (3)0 ( u + i ) + u L t , [1] (cid:0) u − i (cid:1) e Q (2)0 (cid:0) u + i (cid:1)e Q (2)0 (cid:0) u − i (cid:1) e Q (3)0 ( u − i ) = 0 , (3.11) Notice that the transfer matrix t , [1] ( u ) with defining fundamental representation (1) in the auxiliary spaceis given by the supertrace str[ L L ( u ) . . . L ( u )] of the product of Lax operators L ( u ) [33, 34, 8] and its eigenvaluesin nested Bethe Ansatz given by Eq. (3.7). e Q (3)0 upon the elimination of the polynomials e Q (2)0 , t , [2] ( u )¯ t , [2] ( u ) e Q (3)0 ( u ) + u L ¯ t , [2] ( u ) t , [1] (cid:0) u − i (cid:1) e Q (3)0 ( u + i ) + u L t , [2] ( u )¯ t , [1] (cid:0) u + i (cid:1) e Q (3)0 ( u − i ) = 0 . (3.12)So far we have derived Baxter equations in terms of transfer matrices with antisymmetricrepresentation in the auxiliary space. The same considerations can be performed with symmetrictransfer matrices. From symmetric matrices we find an analogue to Eq. (3.9) for e Q (1)0 , t { } ( u ) e Q (1)0 ( u ) − ( u + i ) L t , [1] (cid:0) u − i (cid:1) e Q (1)0 ( u + i ) − ( u − i ) L t , [1] (cid:0) u − i (cid:1) e Q (2)0 (cid:0) u + i (cid:1)e Q (2)0 (cid:0) u − i (cid:1) e Q (1)0 ( u − i ) = 0 . (3.13)Eliminating e Q (2)0 , we get yet another Baxter equation t { } ( u )¯ t { } ( u ) e Q (1)0 ( u ) − ( u + i ) L ¯ t { } ( u ) t , [1] (cid:0) u − i (cid:1) e Q (1)0 ( u + i ) (3.14) − ( u − i ) L t { } ( u )¯ t , [1] (cid:0) u + i (cid:1) e Q (1)0 ( u − i ) = 0 , cf. Eq. (3.10). Finally, solving the system of transfer matrices t , [1] and t { } for e Q (1)0 , we get h t { } ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)i e Q (3)0 ( u ) − u L t , [1] (cid:0) u − i (cid:1) e Q (3)0 ( u + i ) (3.15) − u L t , [1] (cid:0) u − i (cid:1) e Q (2)0 (cid:0) u + i (cid:1)e Q (2)0 (cid:0) u − i (cid:1) e Q (3)0 ( u − i ) = 0 , which being solved for e Q (2)0 together with its conjugate gives h t { } ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)i h ¯ t { } ( u ) − ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1)i e Q (3)0 ( u ) (3.16) − u L t , [1] (cid:0) u − i (cid:1) h ¯ t { } ( u ) − ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1)i e Q (3)0 ( u + i ) − u L ¯ t , [1] (cid:0) u + i (cid:1) h t { } ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)i e Q (3)0 ( u − i ) = 0 . The similarity of Baxter equations for Baxter polynomials in terms of symmetric and antisym-metric transfer matrices implies that there a consistency relations between them¯ t , [2] ( u ) t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1) = t , [2] ( u )¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1) , (3.17)¯ t { } ( u ) t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1) = t { } ( u )¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1) . (3.18)The validity of these equations can be explicitly verified using Eqs. (3.7). The Bethe equations (3.3) in the distinguished basis are not particularly convenient for general-ization beyond leading order of perturbation theory in maximally supersymmetric gauge theorysince the corresponding pseudovacuum state is not protected from quantum corrections in ’tHooft coupling constant. Therefore, it is necessary to transform Bethe and Baxter equations to8he basis with protected pseudovacuum state (see Appendix C). For the underlying superalge-bra this reflects non-uniqueness in the choice of the simple root system. The inequivalent rootsystems are related by Weyl group of super-reflections SW ( G ) with respect to odd roots of thesuperalgebra [41, 42, 43], as discussed in Appendix B. In terms of Bethe Ansatz equations thisis known as the particle-hole transformation [44, 45, 33, 34, 46, 47, 17, 24].Let us perform a chain of these transformations on the distinguished Kac-Dynkin diagramyielding non-distinguished one with FBBF grading, Fig. 1, the central node of which will be the sl (2) subalgebra corresponding to scalar Wilson operators with covariant derivatives. At firststep, we reflect the diagram with respect to the odd root α which translates into a dualitytransformation with respect to the fermionic Bethe root ˜ u (2)0 ,k . At first, one rewrites the Betheequation for ˜ u (2)0 ,k as zeros of the polynomial P ( u ) at the positions of these roots,0 = P (˜ u (2)0 ,k ) (3.19)= (cid:16) ˜ u (2)0 ,k + i (cid:17) L e Q (1)0 (cid:16) ˜ u (2)0 ,k + i (cid:17) e Q (3)0 (cid:16) ˜ u (2)0 ,k − i (cid:17) − (cid:16) ˜ u (2)0 ,k − i (cid:17) L e Q (1)0 (cid:16) ˜ u (2)0 ,k − i (cid:17) e Q (3)0 (cid:16) ˜ u (2)0 ,k + i (cid:17) . Obviously, one can extract the polynomial e Q (2)0 ( u ) from P ( u ) with remaining roots encoded intoyet another polynomial Q (2)0 ( u ), such that P ( u ) = Λ e Q (2)0 ( u ) Q (2)0 ( u ) , (3.20)where Λ = i ( L − ˜ n + ˜ n ). The number n of the dual roots u (2)0 ,k of the polynomial Q (2)0 ( u )is related to the ones of the other Baxter functions as n = L + ˜ n − ˜ n + ˜ n −
1. Using Eq.(3.20), one eliminates the polynomial e Q (2)0 ( u ) from the Bethe equations (3.3) and deduces Betheequations corresponding to the Cartan matrix A = − − , (3.21)with the (middle) Kac-Dynkin diagram in Fig. 1. They read1 = Q (2)0 (cid:16) ˜ u (1)0 ,k + i (cid:17) Q (2)0 (cid:16) ˜ u (1)0 ,k − i (cid:17) , u (2)0 ,k − i u (2)0 ,k + i ! L = e Q (1)0 (cid:16) u (2)0 ,k + i (cid:17)e Q (1)0 (cid:16) u (2)0 ,k − i (cid:17) e Q (3)0 (cid:16) u (2)0 ,k − i (cid:17)e Q (3)0 (cid:16) u (2)0 ,k + i (cid:17) , (3.22)1 = Q (2)0 (cid:16) ˜ u (3)0 ,k − i (cid:17) Q (2)0 (cid:16) ˜ u (3)0 ,k + i (cid:17) . At the next step, we reflect the root system respect to the odd root α of the middle Kac-Dynkin diagram in Fig. 1. To dualize the corresponding fermionic Bethe roots ˜ u (1)0 ,k , we introduceyet another polynomial P ( u ) that vanishes, according to the Bethe Ansatz equations (3.22), at u = ˜ u (1)0 ,k , 0 = P (˜ u (1)0 ,k ) = Q (2)0 (cid:16) ˜ u (1)0 ,k + i (cid:17) − Q (2)0 (cid:16) ˜ u (1)0 ,k − i (cid:17) . (3.23)9he Bethe roots ˜ u (1)0 ,k do not exhaust all zeros of the polynomial P ( u ) and, therefore, for arbitrary u it can be rewritten as a product of two polynomials P ( u ) = Λ e Q (1)0 ( u ) Q (1)0 ( u ) , (3.24)with the second one being the dual Baxter polynomial or order n in new Bethe roots u (1)0 ,k . Theproportionality factor Λ and the power n are related to the numbers of “parent” Bethe roots asfollows Λ = in and n = n − ˜ n −
1, respectively. Again, eliminating the polynomial e Q (1)0 ( u ),one gets Bethe equations for the symmetric Kac-Dynkin diagram with two isotropic fermionicroots in (right-most) Fig. 1, and the Cartan matrix A = − − − − , (3.25)which read 1 = Q (2)0 (cid:16) u (1)0 ,k − i (cid:17) Q (2)0 (cid:16) u (1)0 ,k + i (cid:17) , − u (2)0 ,k − i u (2)0 ,k + i ! L = Q (1)0 (cid:16) u (2)0 ,k − i (cid:17) Q (1)0 (cid:16) u (2)0 ,k + i (cid:17) Q (2)0 ( u (2)0 ,k + i ) Q (2)0 ( u (2)0 ,k − i ) Q (3)0 (cid:16) u (2)0 ,k − i (cid:17) Q (3)0 (cid:16) u (2)0 ,k + i (cid:17) , (3.26)1 = Q (2)0 (cid:16) u (3)0 ,k − i (cid:17) Q (2)0 (cid:16) u (3)0 ,k + i (cid:17) , where for conformity of notations we renamed e Q (3)0 ( u ) = Q (3)0 ( u ) and ˜ u (3)0 ,k = u (3)0 ,k . In what follows,we will dub for brevity corresponding basis symmetric. Let us now construct eigenvalues of transfer matrices in symmetric basis labelled a skew Youngsupertableaux [48, 49]. Analogously to (3.5) we identify the elementary Young supertableauxwith a product of ratios of Baxter polynomials1 u = (cid:0) u + i (cid:1) L Q (1)0 (cid:0) u − i (cid:1) Q (1)0 (cid:0) u + i (cid:1) , (3.27)2 u = (cid:0) u + i (cid:1) L Q (1)0 (cid:0) u − i (cid:1) Q (1)0 (cid:0) u + i (cid:1) Q (2)0 ( u + i ) Q (2)0 ( u ) , u = (cid:0) u − i (cid:1) L Q (3)0 (cid:0) u + i (cid:1) Q (3)0 (cid:0) u − i (cid:1) Q (2)0 ( u − i ) Q (2)0 ( u ) , u = (cid:0) u − i (cid:1) L Q (3)0 (cid:0) u + i (cid:1) Q (3)0 (cid:0) u − i (cid:1) , α and labelled by the spectral parameter uα u = Y ( α, u ) , (3.28)we can write generating functions [38, 39, 50, 37] for eigenvalues of transfer matrices in antisym-metric (1 a ) representation h Y (4 , u )e − i∂ u i − h Y (3 , u )e − i∂ u ih Y (2 , u )e − i∂ u ih Y (1 , u )e − i∂ u i − (3.29)= ∞ X a =0 t , [ a ] (cid:18) u − i a − (cid:19) e − ia∂ u , where the powers p α = 1 − α of factors in the left-hand side reflect the grading of the Kac-Dynkindiagram; for symmetric ( s ) representation one finds h − Y (1 , u )e − i∂ u ih − Y (2 , u )e − i∂ u i − h − Y (3 , u )e − i∂ u i − h − Y (4 , u )e − i∂ u i (3.30)= ∞ X s =0 t { s } (cid:18) u − i s − (cid:19) e − is∂ u . Bethe Ansatz equations (3.26) imply that these transfer matrices are pole-free.One can define transfer matrices with auxiliary space labelled by a skew Young superdiagram Y ( m / n ) [51, 38]. Y ( m / n ) is obtained by removing a Young superdiagram Y ( n ) determined bythe partitioning n = { n , n , . . . } with the usual ordering condition on its elements n ≥ n ≥ . . . from a larger superdiagram Y ( m ) with m = { m , m , . . . } and m ≥ m ≥ . . . such that m ≻ n .One enumerates all boxes of the Young superdiagram Y ( m ) starting with the top left one witha pair of integers ( j, k ), j and k enumerating rows and columns, respectively. On the skewsuperdiagram we define a set of admissible skew Young supertableaux Y α ( m / n ) by assigning aflavor α ( j, k ) index to each box of the diagram Y ( m / n ) and distributing them according to thefollowing ordering conditions: α ( j, k ) < α ( j, k + 1) and α ( j, k ) < α ( j + 1 , k ) for any two adjacentboxes, with weaker conditions when these indices have coincident gradings, namely, for • bosonic grading ¯ α = 0: α ( j, k ) ≤ α ( j, k + 1) , α ( j, k ) < α ( j + 1 , k ) ; (3.31) • fermionic grading ¯ α = 1: α ( j, k ) < α ( j, k + 1) , α ( j, k ) ≤ α ( j + 1 , k ) . (3.32)Obviously these flavor indices can take four different values, i.e., 1 ≤ α ≤ psl (2 | m = { m , m , . . . , m M } can be equivalentlyrepresented as Y ( m ) = ( s a , s a , . . . , s a ℓ ℓ ) with a + a + · · · + a ℓ = M in case there are coincident m k ’s, i.e., s = m = · · · = m a , s = m a +1 = · · · = m a , ... A transposed Young superdiagramis then obtained by reflection across the main diagonal of horizontal and vertical rows. It can be11 ),1( n )2,1( L )1,1( + n a L ),1( m a )1,2( ),1( n )1,2( + n a L ),2( m a M )1,3( LL )1,3( + n a L ),3( m a )1,( M L ),( M mM a Figure 2: Skew Young supertableau Y α ( m / n ).written as Y ( ˜ m ) = (( a + · · · + a ℓ ) s ℓ , ( a + · · · + a ℓ − ) s ℓ − − s ℓ , . . . ) where ˜ m = { ˜ m , ˜ m , . . . } suchthat ˜ m = M is the hight of the first column of Y ( m ), etc.For the auxiliary space determined by a skew Young supertableau as in Fig. 2, the transfermatrix can be constructed from the elementary boxes (3.27), t ,Y ( m / n ) ( u ) = X Y α Y α ( j,k ) ∈ Y α p α ( j,k ) Y (cid:0) α ( j, k ) , u + i ( ˜ m − m + 2 j − k ) (cid:1) , (3.33)where ˜ m = M . These transfer matrices are functionally dependent. They satisfy a number offunctional relations known as fusion relations, namely, they admit a determinant representation[38, 39, 52, 37] in terms of (anti-)symmetric transfer matrices t , [ a ] and t { s } , t ,Y ( m / n ) ( u ) = det ≤ j,k ≤ m t , [ ˜ m j − ˜ n k − j + k ] (cid:0) u + i ( m − ˜ m + ˜ m j + ˜ n k − j − k + 1) (cid:1) = det ≤ j,k ≤ ˜ m t { m k − n j + j − k } (cid:0) u + i ( m − ˜ m − m k − n j + j + k − (cid:1) , (3.34)with t , [ a< ( u ) = t { s< } ( u ) = 0. For instance for a skew Young superdiagram Y ( m ′ / n ′ ) with m ′ = { , , } and n ′ = { , } , it yields t ,Y ( m ′ / n ′ ) ( u ) = det t { } (cid:0) u − i (cid:1) t { } ( u − i ) t { } (cid:0) u − i (cid:1) t { } ( u ) t { } (cid:0) u + i (cid:1) t { } ( u − i ) (3.35)= det (cid:18) t , [1] (cid:0) u + i (cid:1) t , [4] ( u )1 t , [3] (cid:0) u − i (cid:1) (cid:19) . Equations (3.34) are a generalization of classical formulas for characters on representation deter-mined by corresponding Young supertableaux [40].Using the determinant representation for transfer matrices t ,Y ( m / n ) ( u ), one can immediatelyfind bilinear fusion relations among them [53, 54, 50]. Making use of the Desnanot-Jacobi de-terminant identity, one immediately finds relations for transfer matrices with auxiliary space12orresponding to rectangular Young superdiagrams Y ( m / ∅ ) = ( s a ), t , ( s a ) = t { s } , [ a ] : t { s } , [ a ] (cid:0) u + i (cid:1) t { s } , [ a ] (cid:0) u − i (cid:1) = t { s +1 } , [ a ] ( u ) t { s − } , [ a ] ( u ) + t { s } , [ a +1] ( u ) t { s } , [ a − ( u ) . (3.36)This is a Hirota bilinear difference equations derived in Ref. [37] and recently discussed in [55].Similarly to the distinguished basis, it turns out however that out of the entire tower oftransfer matrices, we will need just the lowest-dimensional ones t , [1] ( u ) = t { } ( u ) = − u + 2 u + 3 u − u (3.37)= (cid:0) u − i (cid:1) L Q (3)0 (cid:0) u + i (cid:1) Q (3)0 (cid:0) u − i (cid:1) Q (2)0 ( u − i ) Q (2)0 ( u ) − ! + (cid:0) u + i (cid:1) L Q (1)0 (cid:0) u − i (cid:1) Q (1)0 (cid:0) u + i (cid:1) Q (2)0 ( u + i ) Q (2)0 ( u ) − ! , and t , [2] ( u ) = − u − i u + i − u − i u + i + 14 u − i u + i + 23 u − i u + i − u − i u + i − u − i u + i + 11 u − i u + i + 44 u − i u + i = u L ( u − i ) L Q (3)0 ( u + i ) Q (3)0 ( u − i ) − Q (2)0 (cid:0) u − i (cid:1) Q (2)0 (cid:0) u − i (cid:1) ! − u L Q (1)0 ( u − i ) Q (1)0 ( u ) Q (3)0 ( u + i ) Q (3)0 ( u ) (cid:16) Q (2)0 (cid:0) u + i (cid:1) − Q (2)0 (cid:0) u − i (cid:1)(cid:17) Q (2)0 (cid:0) u + i (cid:1) Q (2)0 (cid:0) u − i (cid:1) (3.38)+ u L ( u + i ) L Q (1)0 ( u − i ) Q (1)0 ( u + i ) − Q (2)0 (cid:0) u + i (cid:1) Q (2)0 (cid:0) u + i (cid:1) ! , or t { } ( u ) = − u + i u − i − u + i u − i + 1 4 u + i u − i + 2 3 u + i u − i − u + i u − i − u + i u − i + 2 2 u + i u − i + 3 3 u + i u − i = u L ( u − i ) L Q (3)0 ( u + i ) Q (3)0 ( u − i ) Q (2)0 (cid:0) u − i (cid:1) Q (2)0 (cid:0) u + i (cid:1) Q (2)0 (cid:0) u − i (cid:1) Q (2)0 (cid:0) u − i (cid:1) − ! + ( u − i ) L ( u + i ) L Q (3)0 ( u ) Q (3)0 ( u − i ) Q (1)0 ( u ) Q (1)0 ( u + i ) Q (2)0 (cid:0) u − i (cid:1) Q (2)0 (cid:0) u − i (cid:1) − ! Q (2)0 (cid:0) u + i (cid:1) Q (2)0 (cid:0) u + i (cid:1) − ! + u L ( u + i ) L Q (1)0 ( u − i ) Q (1)0 ( u + i ) Q (2)0 (cid:0) u + i (cid:1) Q (2)0 (cid:0) u − i (cid:1) Q (2)0 (cid:0) u + i (cid:1) Q (2)0 (cid:0) u + i (cid:1) − ! , (3.39)and their conjugate, for the derivation of Baxter equations for the polynomials Q ( k )0 . The transfer matrices (3.37), (3.38) and (3.39) and their conjugate can be used to find closedequations for nested Baxter polynomials analogously to the distinguished basis as we discussedin Sect. 3.1. 13irst, solving conjugate transfer matrices for the polynomial Q (2)0 , one immediately finds thefollowing relations involving both Q (1)0 and Q (3)0 , Q (1)0 (cid:0) u + i (cid:1) Q (3)0 (cid:0) u − i (cid:1) t , [1] ( u ) = Q (1)0 (cid:0) u − i (cid:1) Q (3)0 (cid:0) u + i (cid:1) ¯ t , [1] ( u ) , (3.40) Q (1)0 ( u + i ) Q (3)0 ( u − i ) t , [2] ( u ) = Q (1)0 ( u − i ) Q (3)0 ( u + i )¯ t , [2] ( u ) , (3.41) Q (1)0 ( u + i ) Q (3)0 ( u − i ) t { } ( u ) = Q (1)0 ( u − i ) Q (3)0 ( u + i )¯ t { } ( u ) . (3.42)They can be further generalized for arbitrary length of Young supertableaux as shown in Eq.(4.22) for their all-order analogues. The similarity of the last two equations is a consequence offunctional relations between these transfer matrices t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1) ¯ t , [2] ( u ) = ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1) t , [2] ( u ) , (3.43) t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1) ¯ t { } ( u ) = ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1) t { } ( u ) . (3.44)Now, finding Q (2)0 from ¯ t , [1] and eliminating it from t , [2] , we get t , [2] ( u ) + u L ¯ t , [1] (cid:0) u + i (cid:1) Q (1)0 ( u − i ) Q (1)0 ( u + i ) Q (3)0 ( u + i ) Q (3)0 ( u ) + u L ¯ t , [1] (cid:0) u − i (cid:1) Q (1)0 ( u − i ) Q (1)0 ( u ) Q (3)0 ( u + i ) Q (3)0 ( u − i ) = 0 . (3.45)Here is when the relations (3.40) becomes important for derivation of an autonomous finite-difference equations. Using Eqs. (3.40), one can eliminate either Q (1)0 or Q (3)0 from (3.45) and itsconjugate, and obtain a form of Baxter equations for these polynomials, t , [2] ( u )¯ t , [2] ( u ) Q (1)0 ( u ) + u L t , [2] ( u )¯ t , [1] (cid:0) u − i (cid:1) Q (1)0 ( u + i ) (3.46)+ u L ¯ t , [2] ( u ) t , [1] (cid:0) u + i (cid:1) Q (1)0 ( u − i ) = 0 ,t , [2] ( u )¯ t , [2] ( u ) Q (3)0 ( u ) + u L ¯ t , [2] ( u ) t , [1] (cid:0) u − i (cid:1) Q (3)0 ( u + i ) (3.47)+ u L t , [2] ( u )¯ t , [1] (cid:0) u + i (cid:1) Q (3)0 ( u − i ) = 0 . Since the Baxter function Q (3)0 was not affected by the series of particle-hole transformation, itsBaxter equation in symmetric and distinguished (3.12) bases coincide.Similar Baxter equations can be derived using the symmetric transfer matrix t { } . Solvingthe transfer matrices t , [1] and t { } with respect to Q (2)0 , one finds t , [1] (cid:0) u − i (cid:1) t , [1] (cid:0) u + i (cid:1) − t { } ( u ) + u L t , [1] (cid:0) u + i (cid:1) Q (1)0 ( u − i ) Q (1)0 ( u ) + u L t , [1] (cid:0) u − i (cid:1) Q (3)0 ( u + i ) Q (3)0 ( u ) = 0 . (3.48)Deriving a similar equation for conjugate transfer matrices and solving the resulting system with14espect to either Q (1)0 or Q (3)0 , one deduces two Baxter equations h t { } ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)i h ¯ t { } ( u ) − ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1)i Q (1)0 ( u ) (3.49) − u L ¯ t , [1] (cid:0) u − i (cid:1) h t { } ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)i Q (1)0 ( u + i ) − u L t , [1] (cid:0) u + i (cid:1) h ¯ t { } ( u ) − ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1)i Q (1)0 ( u − i ) = 0 , h t { } ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)i h ¯ t { } ( u ) − ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1)i Q (3)0 ( u ) (3.50) − u L t , [1] (cid:0) u − i (cid:1) h ¯ t { } ( u ) − ¯ t , [1] (cid:0) u + i (cid:1) ¯ t , [1] (cid:0) u − i (cid:1)i Q (3)0 ( u + i ) − u L ¯ t , [1] (cid:0) u + i (cid:1) h t { } ( u ) − t , [1] (cid:0) u + i (cid:1) t , [1] (cid:0) u − i (cid:1)i Q (3)0 ( u − i ) = 0 . Again, the equations for Q (3)0 in the distinguished and symmetric bases coincide by the sametoken as before. Transfer matrices are polynomials in spectral parameter with coefficients deter-mined by local conserved charges of the spin chain. The above equations provide quantizationconditions on these charges and determine the Baxter polynomials Q (1 , . This data is used inturn to find the momentum-carrying roots of Q (2)0 via Eq. (3.37). The latter determines one-loop anomalous dimensions of superconformal harmonics in the decomposition of the compositelight-cone operators (2.6). Now we turn to multiloop generalization of Bethe Ansatz equations which describe the spectrumof anomalous dimensions of Wilson operators in sl (2 |
2) sector of N = 4 super-Yang-Mills the-ory to all-orders in ’t Hooft coupling g = g YM √ N c / (2 π ). As we pointed out in Sect. 3.2, thedeformation one-loop equations beyond leading order of perturbation theory is achieved in thesymmetric basis (3.26). There are several important changes which occur when the long-rangeeffects enter the game. The long-rage spin chain is written in terms of the renormalized spectralparameter [15] x [ u ] = (cid:16) u + p u − g (cid:17) . (4.1)The scattering matrix of momentum-carrying excitations in long-range Bethe equation in thesymmetric basis acquire an additional phase factor due to renormalization of the superconformalspin and also a nontrivial scattering phase absorbing deviations from strong-coupling calculations.Then the conjectured form of all-order psl (2 |
2) asymptotic Bethe equations read [17]1 = n Y j =1 x (1) k − x (2)+ j x (1) k − x (2) − j , (4.2) x (2)+ k x (2) − k ! L = n Y j = k =1 x (2) − k − x (2)+ j x (2)+ k − x (2) − j (cid:16) − g x (2)+ k x (2) − j (cid:17)(cid:16) − g x (2) − k x (2)+ j (cid:17) e iθ “ x (2)+ k ,x (2) j ” e iθ “ x (2) − k ,x (2) j ” n Y m =1 x (2)+ k − x (1) m x (2) − k − x (1) m n Y n =1 x (2)+ k − x (3) n x (2) − k − x (3) n , n Y j =1 x (3) k − x (2)+ j x (3) k − x (2) − j . θ ( x j , x k ) [56] will be irrelevant for the present study sincethe construction is purely algebraic and relies on perturbative analyticity. As in previous discus-sion we will assume the approach based on conjectured form of nested Bethe Ansatz equationsand subsequent use of the analytic Bethe Ansatz to find transfer matrices. The cancellation ofthe pole will be done only in perturbative, asymptotic sense, thus neglecting poles generated atfinite coupling. To start with, we define three Baxter polynomials with zeros determined by three sets of Betheroots Q ( p ) ( u ) = n p Y k =1 (cid:16) u − u ( p ) k ( g ) (cid:17) . (4.3)The roots u ( p ) k ( g ) depend on the ’t Hooft coupling and admit an infinite perturbative expansion u ( p ) k ( g ) = u ( p )0 ,k + g u ( p )1 ,k + . . . , (4.4)with the lowest order term being the one-loop Bethe roots u ( p )0 ,k obeying the Bethe Ansatz equa-tions (3.26). Following our one-loop considerations in preceding sections, the eigenvalues oftransfer matrices will be built in terms of elements which parameterize components of a Youngsupertableaux associated with an auxiliary space. The single-box Young supertableaux, depend-ing on the spectral parameter u and labelled by the flavor index α = 1 , . . ., u = ( x + ) L e ∆ + ( x + )+ ∆ − ( x + ) b Q (1) (cid:0) u − i (cid:1)b Q (1) (cid:0) u + i (cid:1) , (4.5)2 u = ( x + ) L e ∆ + ( x + ) Q (2) ( u + i ) Q (2) ( u ) b Q (1) (cid:0) u − i (cid:1)b Q (1) (cid:0) u + i (cid:1) , u = ( x − ) L e ∆ − ( x − ) Q (2) ( u − i ) Q (2) ( u ) b Q (3) (cid:0) u + i (cid:1)b Q (3) (cid:0) u − i (cid:1) , u = ( x − ) L e ∆ + ( x − )+ ∆ − ( x − ) b Q (3) (cid:0) u + i (cid:1)b Q (3) (cid:0) u − i (cid:1) . They depend on the dressing factor∆ ± ( x ) = σ (2) ± ( x ) − Θ( x ) , (4.6)which is composed of a “trivial” one [20] σ ( p ) η ( x ) = Z − dtπ ln Q ( p ) (cid:0) η i − gt (cid:1) √ − t − p u − g u + gt ! , (4.7)roughly accounting for renormalization of the superconformal spin in higher orders of pertur-bation theory, as explained in Ref. [19], and a “nontrivial” scattering factor corresponding to16 ( x j , x k ) in long-range Bethe Ansatz equations admitting an integral form in terms of the nestedBaxter polynomials [21]Θ( x ) = g Z − dt √ − t ln Q (2) (cid:0) − i − gt (cid:1) Q (2) (cid:0) i − gt (cid:1) − Z − ds √ − s s − t × Z C [ i,i ∞ ] dκ πi ( πκ ) ln (cid:18) g xx [ κ + gs ] (cid:19) (cid:18) − g xx [ κ − gs ] (cid:19) , (4.8)cf. Ref. [57]. Here we also used a notation for a product of the nested Baxter functions and the“trivial” dressing which emerges in all formulas b Q ( p ) ( u ) = e σ ( p )0 ( x [ u ]) Q ( p ) ( u ) , (4.9)for p = 1 ,
3. For vanishing gauge coupling g = 0, Eqs. (4.5) reduce to (3.27).Introducing a symbolic notation for the elementary Young supertableaux (4.5) α u = Y ( α, u ) , (4.10)one can write generating functions for transfer matrices with antisymmetric (1 a ) h Y (4 , u )e − i∂ u i − h Y (3 , u )e − i∂ u ih Y (2 , u )e − i∂ u ih Y (1 , u )e − i∂ u i − (4.11)= ∞ X a =0 t [ a ] (cid:0) x [1 − a ] (cid:1) e − ia∂ u , and symmetric ( s ) finite-dimensional chiral representations, h − Y (1 , u )e − i∂ u ih − Y (2 , u )e − i∂ u i − h − Y (3 , u )e − i∂ u i − h − Y (4 , u )e − i∂ u i (4.12)= ∞ X s =0 t { s } (cid:0) x [1 − s ] (cid:1) e − is∂ u . In order to write transfer matrices in a concise fashion, we used u [ ± n ] ≡ u ± i n , x [ ± n ] ≡ x (cid:0) u [ ± n ] (cid:1) . (4.13)The transfer matrices t [ s ] and t { s } are free from poles upon the use of Bethe Ansatz equations.Notice that in our treatment we assume perturbative analyticity in the spectral parameter u ,ignoring dynamical pole generated at finite coupling constant. The incorporation of these intothe analysis — a problem which was recently addressed in a related context in Ref. [58] — goesbeyond the scope of this paper.The eigenvalues of transfer matrices with auxiliary space determined by a skew Young su-pertableau in Fig. 2 are build from the elementary boxes (4.5) using the same algorithm as spelledout in Sect. 3.3 and takes the same functional form as Eq. (3.33), t Y ( m / n ) ( x ) = X Y α Y α ( j,k ) ∈ Y α p α ( j,k ) Y (cid:0) α ( j, k ) , u [ ˜ m − m +2 j − k ] (cid:1) . (4.14)17hese transfer matrices admit a determinant representation in terms of (anti-)symmetric transfermatrices t [ a ] and t { s } , t Y ( m / n ) ( x ) = det ≤ j,k ≤ m t [ ˜ m j − ˜ n k − j + k ] (cid:0) x [ m − ˜ m + ˜ m j +˜ n k − j − k +1] (cid:1) = det ≤ j,k ≤ ˜ m t { m k − n j + j − k } (cid:0) x [ m − ˜ m − m k − n j + j + k − (cid:1) , (4.15)with boundary conditions t [ a< = t { s< } = 0. For auxiliary space labelled by a rectangular Youngsupertableaux, the all-order transfer matrices obey Hirota-type equations t { s } [ a ] ( x + ) t { s } [ a ] ( x − ) = t { s +1 } [ a ] ( x ) t { s − } [ a ] ( x ) + t { s } [ a +1] ( x ) t { s } [ a − ( x ) . (4.16)These can be extended to arbitrary Young supertableaux upon proper choice of boundary condi-tions as was recently discussed for short-range super-spin chains in Ref. [55] generalizing earlierconsiderations for classical Lie algebras [59]. Finally, let us derive a set of closed equations for the polynomials (4.3). Again it suffices toconsider the lowest-dimensional transfer matrices only. Using the generating functions (4.11)and (4.12), we obtain explicit form of transfer matrices in defining fundamental representation t [1] ( x ) = ( x − ) L b Q (3) (cid:0) u + i (cid:1)b Q (3) (cid:0) u − i (cid:1) (cid:18) e ∆ − ( x − ) Q (2) ( u − i ) Q (2) ( u ) − e ∆ + ( x − )+ ∆ − ( x − ) (cid:19) + ( x + ) L b Q (1) (cid:0) u − i (cid:1)b Q (1) (cid:0) u + i (cid:1) (cid:18) e ∆ + ( x + ) Q (2) ( u + i ) Q (2) ( u ) − e ∆ + ( x + )+ ∆ − ( x + ) (cid:19) , (4.17)cf. Refs. [17, 58], and the rest deferred to Appendix D. Removing excitation associated witheither first Q (1) = 1 or last Q (3) = 1 node of the symmetric Kac-Dynkin diagram in Fig. 1 orboth, we reduce to all-order transfer matrices in either sl (2 |
1) [21] or sl (2) [20] subsectors of thetheory, respectively. Performing the same steps as earlier in Sect. 3.4, we find that the Baxterequations take the form of second order finite-difference equations with coefficients determinedby either antisymmetric, t [2] ( x )¯ t [2] ( x ) b Q (1) ( u ) + x L e
12 ∆ − ( x )+ 12 ∆ + ( x ) t [2] ( x )¯ t [1] ( x − ) b Q (1) ( u + i ) (4.18)+ x L e
12 ∆ − ( x )+ 12 ∆ + ( x ) ¯ t [2] ( x ) t [1] ( x + ) b Q (1) ( u − i ) = 0 ,t [2] ( x )¯ t [2] ( x ) b Q (3) ( u ) + x L e
12 ∆ − ( x )+ 12 ∆ + ( x ) ¯ t [2] ( x ) t [1] ( x − ) b Q (3) ( u + i ) (4.19)+ x L e
12 ∆ − ( x )+ 12 ∆ + ( x ) t [2] ( x )¯ t [1] ( x + ) b Q (3) ( u − i ) = 0 ,
18r symmetric transfer matrices and deformed by dressing factors (cid:2) t { } ( x ) − t [1] ( x + ) t [1] ( x − ) (cid:3) (cid:2) ¯ t { } ( x ) − ¯ t [1] ( x + )¯ t [1] ( x − ) (cid:3) b Q (1) ( u ) (4.20) − x L e
12 ∆ − ( x )+ 12 ∆ + ( x ) ¯ t [1] ( x − ) (cid:2) t { } ( x ) − t [1] ( x + ) t [1] ( x − ) (cid:3) b Q (1) ( u + i ) − x L e
12 ∆ − ( x )+ 12 ∆ + ( x ) t [1] ( x + ) (cid:2) ¯ t { } ( x ) − ¯ t [1] ( x + )¯ t [1] ( x − ) (cid:3) b Q (1) ( u − i ) = 0 , (cid:2) t { } ( x ) − t [1] ( x + ) t [1] ( x − ) (cid:3) (cid:2) ¯ t { } ( x ) − ¯ t [1] ( x + )¯ t [1] ( x − ) (cid:3) b Q (3) ( u ) (4.21) − x L e
12 ∆ − ( x )+ 12 ∆ + ( x ) t [1] ( x − ) (cid:2) ¯ t { } ( x ) − ¯ t [1] ( x + )¯ t [1] ( x − ) (cid:3) b Q (3) ( u + i ) − x L e
12 ∆ − ( x )+ 12 ∆ + ( x ) ¯ t [1] ( x + ) (cid:2) t { } ( x ) − t [1] ( x + ) t [1] ( x − ) (cid:3) b Q (3) ( u − i ) = 0 . On top of these equations, the product of Baxter polynomials Q (1 , obey consistency conditions b Q (1) (cid:0) u [+ a ] (cid:1) b Q (3) (cid:0) u [ − a ] (cid:1) t [ a ] ( x ) = b Q (1) (cid:0) u [ − a ] (cid:1) b Q (3) (cid:0) u [+ a ] (cid:1) ¯ t [ a ] ( x ) , (4.22) b Q (1) (cid:0) u [+ s ] (cid:1) b Q (3) (cid:0) u [ − s ] (cid:1) t { s } ( x ) = b Q (1) (cid:0) u [ − s ] (cid:1) b Q (3) (cid:0) u [+ s ] (cid:1) ¯ t { s } ( x ) , (4.23)with (anti-)symmetric transfer matrices, in their turn, obeying functional relations identical tothe one for a short-range magnet (3.43). Once the nested Baxter polynomials are determinedfrom Eqs. (4.17) – (4.21), they generate the spectrum of transfer matrices associated with a skewYoung supertableaux via Eq. (4.15). The main focus of the present study was a closed psl (2 |
2) subsector of the dilatation operatorin maximally supersymmetric Yang-Mills theory. The sector is encoded into the N = 2 Wess-Zumino supermultiplet embedded into the N = 4 light-cone superfield and obeys autonomousrenormalization group evolution to all orders in ’t Hooft coupling.Due to non-uniqueness of simple root systems for superalgebras we chose the one which allowsfor a straightforward generalization of Bethe equations to all orders of perturbation theory.We concentrated on a Kac-Dynkin diagram having two isotropic odd nodes which reduces tononcompact sl (2) sector when the number of fermionic excitations vanishes. We used closerelation of transfer matrices to representation theory and analytic Bethe ansatz to construct theirform for auxiliary spaces associated with skew Young supertableaux in terms of Baxter functions.The latter possess determinant representation in terms of transfer matrices with symmetric ( s ) orantisymmetric (1 a ) atypical representations in the auxiliary space. For zero spectral parameter u = 0 these relations reproduce well known supercharacter formulas for supergroups. BetheAnsatz equations ensure that these transfer matrices are pole-free at positions of nested Betheroots. We have used these equations in perturbative, asymptotic sense when the dynamical polesat u = g/ (2 u k ) are not reachable. Proper incorporation of the latter into the formalism and aproof of their cancellations would contribute to resolution of the notorious wrapping problem forthe underlying long-range spin chain, i.e., when the range of interaction is even or higher themthe length of chain itself.We have formulated equivalent closed systems of Baxter equations (4.17), (4.18) and (4.19)or (4.19), (4.20) and (4.21) for nested Baxter functions. Transfer matrices encode a full set ofmutually commuting conserved quantities. The solutions to these sets determine the spectra19f quantized charges and roots of Baxter functions and thus determine spectra of anomalousdimensions of Wilson operators to all orders in gauge coupling constant via the equation γ ( g ) = ig Z − dtπ √ − t (cid:0) ln Q (2) (cid:0) i − gt (cid:1) − ln Q (2) (cid:0) − i − gt (cid:1)(cid:1) ′ . (5.1)Perturbative solutions to these equations in lowest few orders in ’t Hooft coupling were given forthe sl (2 |
1) long-range spin chain in Ref. [21] and applies in a straightforward fashion to the caseat hand.The framework of Baxter equations is advantageous for a number of reasons. While bothBethe Ansatz and Baxter equations produce identical results for models based on representa-tions with highest and/or lowest weight vectors, the Baxter framework applies even when thepseudovacuum state in the Hilbert space of the chain is absent. For noncompact super-spinchains, the number of eigenstates is infinite for a finite length of the spin chain and the analy-sis of spectra in this approach is preferable. Another advantage of Baxter approach within thepresent context of the putative long-range magnet is that it clearly demonstrates the limitation ofthe asymptotic equations, i.e., their invalidity beyond wrapping order. Since the Baxter equationis a polynomial equation in the renormalized spectral parameter x [ u ], by expanding everythingin perturbative series in coupling constant, one gets nonpolynomial terms. However, it is notthis non-polynomiality per se which breaks beyond wrapping order rather it is the fact that bysolving the system of equations stemming from coefficients in front of powers of u , one finds thatit becomes overdetermined and inconsistent. This may serve as a starting point to elaborate oncorrections terms which yield all-order anomalous dimensions even for short spin chains. Anotherproblem which deserves a dedicated study is the origin of the dressing phase. Recall that theeigenvalues of the Baxter operator have a clear physical meaning of the wave functions of themagnet in separated variables [60], a property which should be preserved to all order in ’t Hooftcoupling. The crossing symmetry [61, 62] implemented in terms of Q yields a relations whichshould be understood in physical terms as a certain constraint on the analytical properties of thiswave function. It remains to formulate psu (2 , |
4) Baxter equations for the full theory relyingon the analytic Bethe Ansatz, to start with.This work was supported by the U.S. National Science Foundation under grant no. PHY-0456520.
A Superconformal algebra and superspace realization
The N -extended superconformal algebra su (2 , |N ) contain 15 even charges P µ , M µν , D and K µ and 4 N odd charges Q αA , ¯ Q ˙ αA , S Aα , ¯ S ˙ αA which are two-component Weyl spinors carryingan su ( N ) index A = 1 , , . . . , N . There are additional bosonic chiral charge R and, in case ofextended N > su ( N ) charges T AB satisfying the commutation relations[ T AB , T C D ] = δ BC T AD − δ DA T C D . The nontrivial commutation relations, on top of conventional20osonic su (2 ,
2) relations, read { Q αA , ¯ Q B ˙ β } = 2 δ BA ¯ σ µα ˙ β P µ , [ Q αA , M µν ] = σ µν αβ Q βA , [ Q αA , D ] = i Q αA , [ Q αA , K µ ] = ¯ σ µα ˙ β ¯ S ˙ βA , [ S Aα , P µ ] = ¯ σ µα ˙ β ¯ Q ˙ βA , [ S Aα , D ] = − i S Aα , [ S Aα , M µν ] = σ µναβ S Aβ , [ Q αA , R ] = Q αA , [ S Aα , R ] = − S Aα , [ S Aα , ¯ S ˙ βB ] = 2 δ AB ¯ σ µα ˙ β K µ , [ T AB , Q αC ] = t BDAC Q αD , [ T AB , S Cα ] = − t BCAD S Dα , { S Aα , Q βB } = 2 iδ AB δ βα D + δ AB σ µν αβ M µν + (cid:0) N − (cid:1) δ AB δ βα R − T BA δ βα , (A.1)and their complex conjugate. Here t ACBD = δ AD δ CB − N δ AB δ CD . The remaining (anti-)commutatorsvanish. Here and below we use conventions for Clifford algebra from Ref. [63].The extended superspace with coordinates X = ( x µ , θ αA , ¯ θ ˙ αA ) admits a coset manifold pa-rameterization g ( X ) ≡ e ix µ P µ + iθ αA Q αA + i ¯ θ ˙ αA ¯ Q ˙ αA , (A.2)with the multiplication law g ( X ) g ( X ) = g ( X ) , (A.3)and transformed coordinates being X = (cid:16) x µ + x µ − iθ αA ¯ σ µα ˙ β ¯ θ ˙ β A + iθ αA ¯ σ µα ˙ β ¯ θ ˙ β A , θ αA + θ αA , ¯ θ αA + ¯ θ αA (cid:17) . (A.4)In this parametrization, a superfield in this superspace is defined asΦ( X ) = g ( X )Φ(0) g − ( X ) . (A.5)Using conventional technique of induced representations one easily computes a representationof generators in the superspace. The center elements are[ M µν , Φ(0)] = − Σ µν Φ(0) , [ D , Φ(0)] = − id Φ(0) , [ R , Φ(0)] = r Φ(0) , [ T AB , Φ(0)] = t BA Φ(0) , (A.6)all the rest vanish . The representation of generators as differential operators acting on thecoordinates X of the superfield Φ, [ G , Φ ( X )] ≡ GΦ ( X ) , (A.7)is for su (2 ,
2) bosonic iP µ = ∂ µ , (A.8) iM µν = x µ ∂ ν − x ν ∂ µ − i Σ µν − i θ βA σ µν βα ∂ θ αA − i ¯ θ ˙ βA ¯ σ µν ˙ β ˙ α ∂ ¯ θ ˙ αA ,iD = d + x µ ∂ µ + θ αA ∂ θ αA + ¯ θ ˙ αA ∂ ¯ θ ˙ αA ,iK µ = (cid:0) x µ x ν − x g µν (cid:1) ∂ ν + 2¯ θ ˙ αA σ µ ˙ αβ θ Bβ ¯ θ ˙ βB σ ν ˙ βα θ Aα ∂ ν + 2 dx µ − ix ν Σ µν + iε µνρσ ¯ θ ˙ αA σ ν ˙ αβ θ Aβ Σ ρσ + ir (cid:0) N − (cid:1) ¯ θ ˙ αA σ µ ˙ αβ θ Aβ − i ¯ θ ˙ αA σ µ ˙ αβ θ Bβ t AB + ( x ν θ αA ¯ σ µα ˙ β σ ν ˙ βγ − i ¯ θ ˙ αB σ µ ˙ αβ θ Aβ θ γB ) ∂ θ γA + ( x ν ¯ θ ˙ αA σ µ ˙ αβ ¯ σ νβ ˙ γ + 2 i ¯ θ ˙ αA σ µ ˙ αβ θ Bβ ¯ θ ˙ γB ) ∂ ¯ θ ˙ γA , We assumed here that the superfield Φ is a tensor with respect to SU ( N ) group. The superfields of gaugetheories are SU ( N ) scalars thus t BA = 0. iQ αA = ∂ θ αA + i ¯ σ µ α ˙ β ¯ θ ˙ βA ∂ µ , (A.9) i ¯ Q ˙ αA = ∂ ¯ θ ˙ αA + iσ µ ˙ αβ θ Aβ ∂ µ ,iS Aα = − (cid:0) d + r (cid:0) N − (cid:1)(cid:1) θ Aα + σ µν αβ θ Aβ Σ µν + 4 t AB θ Bα − ( x ν ¯ σ ν α ˙ β σ µ ˙ βγ θ Aγ − iθ Bα ¯ θ ˙ αB σ µ ˙ αβ θ Aβ ) ∂ µ + 4 θ Bα θ βA ∂ θ βB − (cid:0) θ Bα ¯ θ ˙ βB − ix µ ¯ σ µα ˙ β (cid:1) ∂ ¯ θ ˙ βA ,i ¯ S ˙ αA = − (cid:0) d − r (cid:0) N − (cid:1)(cid:1) ¯ θ ˙ αA + ¯ σ µν ˙ α ˙ β ¯ θ ˙ βA Σ µν − t BA ¯ θ ˙ αB − ( x ν σ ν ˙ αβ ¯ σ µβ ˙ γ ¯ θ ˙ γA + 2 i ¯ θ ˙ αB ¯ θ ˙ βA σ µ ˙ βα θ Bα ) ∂ µ + 4¯ θ ˙ αB ¯ θ ˙ βA ∂ ¯ θ ˙ βB − (cid:0) θ ˙ αB θ βB − ix µ σ µ ˙ αβ (cid:1) ∂ θ βA , and finally su ( N ) and u (1) generators iT BA = it AB + i (cid:0) θ αA ∂ θ αB − ¯ θ ˙ αB ∂ ¯ θ ˙ αA (cid:1) − i N δ AB (cid:0) θ αC ∂ θ αC − ¯ θ ˙ αC ∂ ¯ θ ˙ αC (cid:1) , (A.10) iR = ir − iθ αA ∂ θ αA + i ¯ θ ˙ αA ∂ ¯ θ ˙ αA , cf. Ref. [64]. A.1 Light-cone reduction of superspace
In this paper we are interested in a closed sl (2 |
2) subsector of the dilatation operator, whichacts on the light-cone composite operators (2.6) built from N = 2 Wess-Zumino superfield of thetruncated maximally supersymmetric gauge theory. The sl (2 |
2) subalgebra of the superconformalalgebra is spanned by the following generators P + , M − + , D , K − , Q + αA , ¯ Q ˙ αA + , S A − α , ¯ S ˙ α − A , R , T AB , M . (A.11)The vector indices of generators are contracted with the light-cone vectors n µ and n ∗ µ obeyingthe following conditions n = n ∗ = 0 and n · n ∗ = 1, such that G µ n µ = G + , G µ n ∗ µ = G − . (A.12)While the fermionic generators are obtained with the help of the projectors G ± α = ¯ σ ∓ α ˙ β σ ± ˙ βγ G γ , ¯ G ˙ α ± = σ ∓ ˙ αβ ¯ σ ± β ˙ γ ¯ G ˙ γ . (A.13)One can easily convince oneself that actually only one component in each light-cone Weyl spinor isnonvanishing. This reflect a general phenomenon of the light-cone formalism: a spinor satisfyingthe Weyl condition can be described by a complex Grassmann variable without a Lorentz index.Further, it is convenient to introduce the following combinations of the generators i P + ≡ − L − , i K − ≡ L + , i ( D + M − + ) ≡ L , ( N − R − M ≡ B ,i Q A ≡ √ V − A , i ¯ Q A ˙1 ≡ − i √ V A, − , i S A ≡ − √
32 ¯ V A, + , i ¯ S ˙1 A ≡ i √ V + A . (A.14)The representation (A.7) of these generators in the light-cone superspace then reads for thebosonic sl (2) subalgebra L − = − ∂ z , (A.15) L + = 2 ℓz + ¯ θ A θ B (cid:0) b δ AB − t AB (cid:1) + (cid:0) z + (¯ θ A θ A ) (cid:1) ∂ z + (cid:0) z + ¯ θ B θ B (cid:1) θ A ∂ θ A + (cid:0) z − ¯ θ B θ B (cid:1) ¯ θ A ∂ ¯ θ A ,L = ℓ + z∂ z + θ A ∂ θ A + ¯ θ A ∂ ¯ θ A . V − A = ∂ θ A + ¯ θ A ∂ z , (A.16)¯ V A, − = ∂ ¯ θ A + θ A ∂ z ,V + A = (cid:0) ( ℓ − b ) δ BA + t BA (cid:1) ¯ θ B + (cid:0) z − ¯ θ B θ B (cid:1) ¯ θ A ∂ z + ¯ θ A ¯ θ B ∂ ¯ θ B + (cid:0) z + ¯ θ B θ B (cid:1) ∂ θ A , ¯ V A, + = (cid:0) ( ℓ + b ) δ AB − t AB (cid:1) θ B + (cid:0) z + ¯ θ B θ B (cid:1) θ A ∂ z + θ A θ B ∂ θ B + (cid:0) z − ¯ θ B θ B (cid:1) ∂ ¯ θ A , and the remaining are B = b + (cid:0) − N (cid:1) (cid:0) θ A ∂ θ A − ¯ θ A ∂ ¯ θ A (cid:1) , (A.17) T BA = t AB + (cid:0) θ A ∂ θ B − ¯ θ B ∂ ¯ θ A (cid:1) − N δ AB (cid:0) θ C ∂ θ C − ¯ θ C ∂ ¯ θ C (cid:1) . Here we used a notation z = x − for the coordinate projected on the light cone. The Grassmanncoordinates θ A and ¯ θ A were redefined compared to the ones in the covariant superspace (A.2) asfollows θ A ≡ √ θ A , ¯ θ A ≡ i √ θ ˙1 A . (A.18)The quantum numbers of the superfields are encoded into the conformal spin ℓ and chirality bℓ = ( s + d ) , b = r (cid:0) N − (cid:1) − h , (A.19)with the latter being a linear combination of its helicity h = − Σ and the R − charge r .The commutation relations between the generators in the representation (A.15), (A.16) and(A.17) can be summarized by combining them in an sl (2 |N ) − covariant matrix with components E = L − NN − B , E A = − V + A , E , N +1 = L + ,E A = − ¯ V A, − , E AB = T BA + N − Bδ AB , E A, N +1 = − ¯ V A, + ,E N +1 , = L − , E N +1 ,A = V − A E N +1 , N +1 = − L − NN − B . (A.20)Then the graded commutation relations read in a concise form[ E AB , E CD } ≡ E AB E CD − ( − ( ¯ A + ¯ B )( ¯ C + ¯ D ) E CD E AB = δ CB E AD − ( − ( ¯ A + ¯ B )( ¯ C + ¯ D ) δ AD E CB , (A.21)where the indices A , . . . run over N +2 values (0 , A, N +1) and possess the gradings ¯0 = N + 1 =0 and ¯ A = 1. A.2 Projection to psl (2 | subsector The superfield (2.1) encoding the field content of maximally supersymmetric Yang-Mills theoryis chiral, thus the dependence on the Grassmann variables ¯ θ A is trivial, i.e., Φ( z, θ a , ¯ θ A ) = Φ( z + ¯ θ A θ A , θ A , sl (2 |
2) subsector of N = 4 theory are found making use of the superfieldtruncation (2.2), such that E Φ( z, θ A ) | θ =0 = . . . + θ E Ψ WZ ( Z ) , (A.22)23here E are given in Eqs. (A.20) for N = 4 and E are generators in question acting on a functionof Z = ( z, ϑ a ). One finds the following identification of sl (2 |
2) generators E αβ , obeying thecommutation relations [ E αβ , E γδ } ≡ E αβ E γδ − ( − (¯ α + ¯ β )(¯ γ +¯ δ ) E γδ E αβ = δ γβ E αδ − ( − (¯ α + ¯ β )(¯ γ +¯ δ ) δ αδ E γβ , (A.23)to the ones of the truncated N = 2 superconformal light-cone algebra E = L − B , E a = −V + a , E = L + , E a = − ¯ V a, − , E ab = T ba + 2 B δ ab , E a = − ¯ V a, + , E = L − , E a = V − a , E = −L − B . (A.24)As it is obvious from this formula, the indices of E αβ run over α = 0 , a, su (2) indextaking two values a = 1 , a = 1.The above generators are realized on Ψ WZ superfield (2.5) and read L − = − ∂ z , L + = z ∂ z + zϑ a ∂ ϑ a , L = z∂ z + ϑ a ∂ ϑ a , T ba = ϑ a ∂ ϑ b − δ ab ϑ c ∂ ϑ c , ¯ V a, − = ϑ a ∂ z , ¯ V a, + = ϑ a ( z∂ z + ϑ c ∂ ϑ c ) , V − a = ∂ ϑ a , V + a = z∂ ϑ a . (A.25)The u B (1) outer automorphism B , B = − + ϑ a ∂ ϑ a , (A.26)does not enter the right-hand side of the commutation relation of the generators (A.25). Thusthe sl (2 |
2) algebra (A.23) of generators (A.24) is a semidirect product u B (1) ⋉ psu (2 |
2) withgenerators (A.25) forming the projective algebra psl (2 | C =
12 3 X α,β =0 ( − ¯ β E αβ E βα (A.27)= ( L + T )( L − T + 1) + L + L − + T T − V + a ¯ V a, − − ¯ V a, + V − a , where we introduced a notation for T ≡ T = −T . Notice that this su (2) generator is relatedto the one of su (4) internal rotations as T = T − T . B Serre-Chevalley bases for psl (2 | The projective algebra psl (2 |
2) has rank two, but it is described by Kac-Dynkin diagrams havingthree nodes. This implies that the simple roots of the root system are not linearly independent.The root system is expressed in terms of weights v α = ( ε | δ , δ | ε ) which form a basis in thedual space of the Cartan subalgebra. The weights obey the conditions ε + ε = 0 , δ + δ = 0 . (B.1) Our ordering of basis vectors reflects the grading of the matrix of sl (2 |
2) generators E αβ . ε | ε ) = 1 , ( δ | δ ) = − , ( ε | δ ) = 0 . (B.2)The set of nonzero roots ∆ = ∆ ∪ ∆ is divided into the set of even ∆ and odd ∆ roots,∆ = { ε a − ε b , δ a − δ b } , ∆ = { ε a − δ b , δ b − ε a } . (B.3)The root vectors associated to these roots are E αβ ↔ v α − v β , (B.4)and the Cartan subalgebra is spanned by the elements E + E , E − E , E + E . (B.5)It is obvious from explicit realization that these generators are linearly dependent, exhibitingpeculiarities of projective algebras. There are several choices of simple root systems dependingon choices of Borel subalgebras. Let us discuss two simple root systems used in the main text,i.e., corresponding to the distinguished and symmetric Kac-Dynkin diagram with two isotropicfermionic roots. B.1 Distinguished Kac-Dynkin diagram
The distinguished Kac-Dynkin diagram corresponding to the BBFF grading possesses the fol-lowing ordering of the basis elements of the dual Cartan subalgebra ( ε , ε | δ , δ ) and yields thesimple root system α = √ ( ε − ε ) , α = √ ( ε − δ ) , α = √ ( δ − δ ) . (B.6)The Cartan subalgebra is formed by the generators h = E − E = 2 L , h = E + E = T − L , h = −E + E = − T . (B.7)and together with positive and negative root vectors e +1 = E = L + , e +2 = E = V − , e +3 = E = T ,e − = E = L − , e − = E = − ¯ V , + , e − = E = T , (B.8)form the Serre-Chevalley basis obeying the algebra[ h p , e ± q ] = ± A pq e ± q , [ h p , h q ] = 0 , [ e ± p , e ∓ q ] p = q = 0 , [ e +1 , e − ] = h , { e +2 , e − } = h , [ e +3 , e − ] = − h , (B.9)with the Cartan matrix given in Eq. (3.2). From these one can generate the entire algebra. Thelinear dependence of Cartan generators results in linear dependence of the roots α + 2 α + α = 0 . (B.10) Note that the Kac-Dynkin diagrams for superalgebras are ambiguous. There is yet another distinguisheddiagram with FFBB grading. The normalization factors reflect linear dependence (B.1) of basis elements and are introduced in order tohave conventional definition of the Cartan matrix A pq = ( α p | α q ). .2 Symmetric Kac-Dynkin diagram The symmetric Kac-Dynkin diagram with two isotropic fermionic roots corresponding to FBBFgrading with the dual basis ( δ | ε , ε | δ ) possesses the simple root system α = √ ( δ − ε ) , α = √ ( ε − ε ) , α = √ ( ε − δ ) . (B.11)The Serre-Chevalley basis is h = −E − E = −L − T , h = E − E = 2 L , h = E + E = −L − T ,e +1 = E = − ¯ V , − , e +2 = E = L + , e +3 = E = V − ,e − = E = −V +1 , e − = E = L − , e − = E = − ¯ V , + , (B.12)with their commutation relations determined by the Cartan matrix (3.25),[ h p , e ± q ] = ± A pq e ± q , [ h p , h q ] = 0 , [ e ± p , e ∓ q } p = q = 0 , { e +1 , e − } = − h , [ e +2 , e − ] = h , { e +3 , e − } = h . (B.13) C Excitation numbers
Here we will present the oscillator realization of sl (2 |
2) algebra which is useful in relating thenumber of excitations in nested Bethe Absatz to the eigenvalues of Cartan generators [46]. Oneintroduces [65, 43, 17, 23] bosonic and fermionic raising ( a † , b † , c a † ) and lowering ( a , b , c a ) opera-tors, which obey the commutation relations[ a , a † ] = 1 , [ b , b † ] = 1 , { c a , c b † } = δ ba . (C.1)The generators then read L − = − ab , L + = a † b † , L = a † a + b † b + , T ba = c a † c a − δ ab c c † c c , ¯ V a, − = ac a † , ¯ V a, + = b † c a † , V − a = bc a , V + a = a † c a , B = − a † a + b † b , (C.2)and the oscillators obey the vanishing central charge condition a † a − b † b + c a † c a − . (C.3)The nested Bethe Ansatz for symmetric Kac-Dynkin diagram is built on an L − site first levelpseudovacuum state | Ω i L = | Z Z . . . Z L i , (C.4)which in the basis of local Wilson operators corresponds to the product Z L (0). There are fourdifferent types of excitations on each spin chain site | Z i identified with particle content of the sl (2 |
2) subsector as follows, a † b † | Z i = |D + Z i , c † c | Z i = | X i , a † c | Z i = | ¯ χ i , b † c † | Z i = | ψ i . (C.5) These have hermitian conjugation properties identical to ones in differential representation (A.25) endowedwith sl (2 |
2) invariant scalar product (2.8), see Ref. [29]. This state is created from true vacuum with su (4) oscillators ( d , c , c , d ) as | Z i = | ¯ φ i = c † d † | i . B| ¯ χ i = − | ¯ χ i , B| ψ i = | ψ i , T | X i = | X i , T | Z i = − | Z i . (C.6)The ground state of the second level in the nested Bethe Ansatz is chosen in terms of non-compact primary excitations D + Z (0) on each site of the chain. The third vacuum state can bechosen in term of fermions, either ¯ χ or ψ . Thus the excitations on the spin chain sites are builtin terms of raising operators corresponding to simple roots acting on nodes of the Kac-Dynkindiagram such that a general state reads schematically |O ω i = ( e +3 ) n ( e +1 ) n ( e +2 ) n | Ω i L = ( V − ) n ( − ¯ V , − ) n ( L + ) n | Ω i L (C.7) ∼ ( c ) n ( c † ) n ( a † ) n − n ( b † ) n − n | Ω i L . This formula requires clarifications. First, since the fermionic generators are nilpotent they allhave to act on different spin chain sites, thus their number cannot exceed the number of sites, i.e.,0 ≤ n , ≤ L . For instance, for n = 0 and n = L and n ≥ n , the resulting state correspondsto the L − fermion operator ( D + ) n − n ψ L (0). Second, since both e +1 and e +3 annihilate the vacuumstate | Ω i L , this gives a natural restriction on the number of excitations acting on the same sites,i.e., n ≤ n ≥ n . Third, the number of noncompact excitations n is unrestricted from above.Now, the number-of-excitation operators N a = a † a , N b = b † b , N c = c † c , N c = c c † , (C.8)can be related to generators of the Cartan subalgebra of psl (2 |
2) and u B (1) automorphism N a = L − B− L , N b = L +2 B− L , N c = T +2 B + L , N c = T − B + L , (C.9)and yield the following relations between the excitation numbers n p and eigenvalues ℓ , t and b of L , B and T , respectively, for the state (C.7) n = n c = t + 2 b + L , n = n a + n c = n b + n c = t + ℓ , n = n c = t − b + L . (C.10)
D Transfer matrices for low-dimensional representations
We give here explicit expressions for transfer matrices with low-dimensional auxiliary space. Thetransfer matrix conjugate to Eq. (4.17) reads¯ t [1] ( x ) = ( x + ) L b Q (3) (cid:0) u − i (cid:1)b Q (3) (cid:0) u + i (cid:1) (cid:18) e ∆ + ( x + ) Q (2) ( u + i ) Q (2) ( u ) − e ∆ − ( x + )+ ∆ + ( x + ) (cid:19) + ( x − ) L b Q (1) (cid:0) u + i (cid:1)b Q (1) (cid:0) u − i (cid:1) (cid:18) e ∆ − ( x − ) Q (2) ( u − i ) Q (2) ( u ) − e ∆ − ( x − )+ ∆ + ( x − ) (cid:19) . (D.1)27he generating function (4.11) yields the eigenvalues of antisymmetric transfer matrix t [2] e −
12 ∆ − ( x ) −
12 ∆ + ( x ) t [2] ( x ) (D.2)= (cid:0) x x [ − (cid:1) L Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u + i (cid:1) b Q (3) ( u + i ) b Q (3) ( u − i ) e ∆ + ( x [ − ) + ∆ − ( x [ − ) − e ∆ − ( x [ − ) Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u − i (cid:1) ! + (cid:0) x x [+2] (cid:1) L Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u − i (cid:1) b Q (1) ( u − i ) b Q (1) ( u + i ) e ∆ + ( x [+2] ) + ∆ − ( x [+2] ) − e ∆ + ( x [+2] ) Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u + i (cid:1) ! − x L b Q (3) ( u + i ) b Q (3) ( u ) b Q (1) ( u − i ) b Q (1) ( u ) (cid:16) e
12 ∆ + ( x ) Q (2) (cid:0) u + i (cid:1) − e
12 ∆ − ( x ) Q (2) (cid:0) u − i (cid:1) (cid:17) Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u − i (cid:1) , and its conjugate ¯ t [2] e −
12 ∆ − ( x ) −
12 ∆ + ( x ) ¯ t [2] ( x ) (D.3)= (cid:0) x x [+2] (cid:1) L Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u − i (cid:1) b Q (3) ( u − i ) b Q (3) ( u + i ) e ∆ − ( x [+2] ) + ∆ + ( x [+2] ) − e ∆ + ( x [+2] ) Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u + i (cid:1) ! + (cid:0) x x [ − (cid:1) L Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u + i (cid:1) b Q (1) ( u + i ) b Q (1) ( u − i ) e ∆ − ( x [ − ) + ∆ + ( x [ − ) − e ∆ − ( x [ − ) Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u − i (cid:1) ! − x L b Q (3) ( u − i ) b Q (3) ( u ) b Q (1) ( u + i ) b Q (1) ( u ) (cid:16) e
12 ∆ − ( x ) Q (2) (cid:0) u − i (cid:1) − e
12 ∆ + ( x ) Q (2) (cid:0) u + i (cid:1) (cid:17) Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u + i (cid:1) . Similarly, one finds from Eq. (4.12) for symmetric t { } t { } ( x ) = (cid:0) x [ − x [+2] (cid:1) L e ∆ − ( x [ − ) Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u − i (cid:1) − e ∆ + ( x [ − ) + ∆ − ( x [ − ) ! (D.4) × e ∆ + ( x [+2] ) Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u + i (cid:1) − e ∆ + ( x [+2] ) + ∆ − ( x [+2] ) ! b Q (3) ( u ) b Q (3) ( u − i ) b Q (1) ( u ) b Q (1) ( u + i )+ (cid:0) x x [ − (cid:1) L e ∆ − ( x ) e ∆ − ( x [ − ) Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u − i (cid:1) − e ∆ + ( x [ − ) + ∆ − ( x [ − ) ! Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u + i (cid:1) b Q (3) ( u + i ) b Q (3) ( u − i )+ (cid:0) x x [+2] (cid:1) L e ∆ + ( x ) e ∆ + ( x [+2] ) Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u + i (cid:1) − e ∆ + ( x [+2] ) + ∆ − ( x [+2] ) ! Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u − i (cid:1) b Q (1) ( u − i ) b Q (1) ( u + i ) , t { } ¯ t { } ( x ) = (cid:0) x [+2] x [ − (cid:1) L e ∆ + ( x [+2] ) Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u + i (cid:1) − e ∆ − ( x [+2] ) + ∆ + ( x [+2] ) ! (D.5) × e ∆ − ( x [ − ) Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u − i (cid:1) − e ∆ − ( x [ − ) + ∆ + ( x [ − ) ! b Q (3) ( u ) b Q (3) ( u + i ) b Q (1) ( u ) b Q (1) ( u − i )+ (cid:0) x x [+2] (cid:1) L e ∆ + ( x ) e ∆ + ( x [+2] ) Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u + i (cid:1) − e ∆ − ( x [+2] ) − ∆ + ( x [+2] ) ! Q (2) (cid:0) u + i (cid:1) Q (2) (cid:0) u − i (cid:1) b Q (3) ( u − i ) b Q (3) ( u + i )+ (cid:0) x x [ − (cid:1) L e ∆ − ( x ) e ∆ − ( x [ − ) Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u − i (cid:1) − e ∆ − ( x [ − ) + ∆ + ( x [ − ) ! Q (2) (cid:0) u − i (cid:1) Q (2) (cid:0) u + i (cid:1) b Q (1) ( u + i ) b Q (1) ( u − i ) . References [1] L.N. Lipatov, Phys. Lett. B 309 (1993) 394; JETP Lett. 59 (1994) 596.[2] L.D. Faddeev, G.P. Korchemsky, Phys. Lett. B 342 (1995) 311.[3] A.P. Bukhvostov, G.V. Frolov, E.A. Kuraev, L.N. Lipatov, Nucl. Phys. B 258 (1985) 601.[4] V.M. Braun, S.E. Derkachov, A.N. Manashov, Phys. Rev. Lett. 81 (1998) 2020.[5] V.M. Braun, S.E. Derkachov, G.P. Korchemsky, A.N. Manashov, Nucl. Phys. B 553 (1999)355.[6] A.V. Belitsky, Phys. Lett. B 453 (1999) 59; Nucl. Phys. B 574 (2000) 407.[7] A.V. Belitsky, A.S. Gorsky, G.P. Korchemsky, Nucl. Phys. B 667 (2003) 3.[8] A.V. Belitsky, S.E. Derkachov, G.P. Korchemsky, A.N. Manashov, Phys. Lett. B 594 (2004)385; Nucl. Phys. B 708 (2005) 115.[9] L.N. Lipatov,
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The su (2 | dynamic S-matrixdynamic S-matrix