aa r X i v : . [ h e p - t h ] O c t First-principles derivation of the AdS/CFT Y-systems
Raphael BenichouSeptember 27, 2018
Theoretische Natuurkunde, Vrije Universiteit Brussel andThe International Solvay Institutes,Pleinlaan 2, B-1050 Brussels, Belgium [email protected]
Abstract
We provide a first-principles, perturbative derivation of the
AdS / CF T Y-systemthat has been proposed to solve the spectrum problem of N = 4 SYM. The proof relieson the computation of quantum effects in the fusion of some loop operators, namely thetransfer matrices. More precisely we show that the leading quantum corrections in thefusion of transfer matrices induce the correct shifts of the spectral parameter in the T-system. As intermediate steps we study UV divergences in line operators up to first orderand compute the fusion of line operators up to second order for the pure spinor string in AdS × S . We also argue that the derivation can be easily extended to other integrablemodels, some of which describe string theory on AdS , AdS and AdS spacetimes. ontents AdS × S r, s ) system . . . . . . . . . . . . . . . . . . . . . 82.4 The line operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 AdS / CF T T-system 165 Generalization to other integrable theories 206 Conclusion 22A Conventions 24B A new look at the gauge covariant current algebra 25
B.1 Derivation of the current-current OPEs . . . . . . . . . . . . . . . . . . . . . 25B.2 The ( r, s ) system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
C Divergences in line operators 31
C.1 Divergences in transition matrices . . . . . . . . . . . . . . . . . . . . . . . . 32C.2 Divergences in monodromy and transfer matrices . . . . . . . . . . . . . . . . 34
D Fusion at second order: computations 35 Introduction
The AdS/CFT correspondence [1][2][3] implies that type IIB string theory in
AdS × S isequivalent to N = 4 Super-Yang-Mills in four dimensions. In the classical string theory limit,or equivalently in the planar gauge theory limit, integrable structures appear. This has leadto impressive progress in the understanding of this system (see [4] for a review).The AdS/CFT dictionary relates the energy of string states in the bulk to the conformaldimensions of operators on the boundary. A set of equations known as the Y-system hasbeen put forward in [5] to solve the spectrum problem of planar N = 4 SYM, or equivalentlyof string theory in AdS × S . The goal of the present article is to make a new step towardsa definite proof of the validity of this set of equations.There is by now solid evidence in favor of the validity of the Y-system. It reproduces theresults of the Asymptotic Bethe Ansatz [6], but it does not suffer from the same limitations.For instance it contains [7][8] the spectrum of the quasi-classical string at large ’t Hooftcoupling (see e.g. [9]). Even more impressively, it lead to correct predictions for the dimensionof the Konishi operator both at large [10] and at small [11] ’t Hooft coupling.In order to claim that the spectrum problem for N = 4 SYM has been definitively solved,it would be comfortable to have a proof of the validity of the Y-system. At that point theonly known derivation of the Y-system relies on the Thermodynamic Bethe Ansatz [12] (seee.g. [13] for a review). This approach was studied in [14][15][16]. The ThermodynamicBethe Ansatz has been very successful and lead to numerous remarkable results. Howeverthis method relies on several crucial assumptions. In the first place, one has to assumequantum integrability of the theory. Then one needs the “string hypothesis”: the spectrumof excitations that contribute to the thermodynamic limit of the theory essentially has to beguessed. Most importantly, this method only gives the ground state energy. The spectrumof excited state can be obtained by analytic continuation, but the reason why this works isnot understood.In this paper, we will initiate a different approach to derive the Y-system from firstprinciples. We will use only elementary tools of two dimensional conformal field theory; inthis aspect this article can be related to the seminal work of [17] where the Y-system wasderived for the minimal models. We will be able to prove the validity of the Y-system up tofirst non-trivial order at large ’t Hooft coupling. The idea of the proof.
Up to a change of variables, the Y-system can be rewritten as aT-system, also known as the Hirota equation: T a,s ( u + 1) T a,s ( u −
1) = T a +1 ,s ( u + 1) T a − ,s ( u −
1) + T a,s +1 ( u − T a,s − ( u + 1) (1.1)In the above equation, u is a spectral parameter and the indices ( a, s ) are integers that labelrepresentations of the global symmetry group of the system. In the case at hand this group is P SU (2 , | P SU (2 , | P Sl ( n | n ) in [19]. Organization of the paper.
In section 2 we describe the features of the pure spinorstring on
AdS × S that are relevant for our purposes. We also introduce the relevant lineoperators. In section 3 we present the central computation of this work: we study the fusionof line operators up to second order in perturbation theory. In section 4 we make good useof this computation to deduce the validity of the T-system up to first order in perturbationtheory. Section 5 contains a discussion of the extension of this method to other integrablemodels. Eventually final remarks are gathered in section 6.In order to keep the bulk of the paper as readable as possible, most of the details ofthe computations are gathered in the appendices. Appendix A contains the conventions.In Appendix B we revisit the computation of the current-current OPEs in the pure spinorformalism using a novel and efficient method. In Appendix C we study the UV divergencesin line operators. Eventually Appendix D contains the computations relevant for the fusionof line operators. AdS × S To describe superstring theory in
AdS × S we will use the pure spinor formalism. Thischoice is a matter of convenience. Indeed the computations of section 3 are simpler in theconformal gauge, where target-space covariance is preserved.In this section we introduce the pure spinor string on AdS × S . We only discuss thefeatures of this formalism that are relevant for the purpose of this paper. A more detaileddiscussion can be found for instance in [20][21][22]. We introduce the flat connection, andshow that the commutator of equal-time connections can be written in the canonical formof a ( r, s ) system. We also introduce the line operators that are defined as the path-orderedexponential of the line integral of the flat connection. Finally we discuss the UV divergencesthat appear in these line operators because of quantum effects. Most of the results discussed inthis section have appeared before in the literature. Some new results are presented concerningthe current algebra, the commutator of equal-time connections and the renormalization ofthe line operators. The target-space
AdS × S is embedded in a superspace with 32 supercharges. It is realizedas the supercoset P SU (2 , | /SO (4 , × SO (5). The Lie superalgebra G = psu (2 , |
4) admits4he action of a Z automorphism. This automorphism induces a Z grading on the elementsof the Lie superalgebra. We can decompose the Lie algebra G according to this grading: G = H ⊕ H ⊕ H ⊕ H (2.1)where the subscript gives the Z grade. Bosonic (respectively fermionic) generators of theLie superalgebra have an even (respectively odd) grade. The action.
Let us introduce the currents J and ¯ J defined in terms of the group element g ∈ P SU (2 , |
4) as: J = g − ∂g ; ¯ J = g − ¯ ∂g (2.2)They take values in the Lie superalgebra G . We decompose the current J according to the Z grading of the Lie superalgebra: J = J + J + J + J (2.3)and similarly for ¯ J . Let us also introduce the bosonic pure spinor ghosts λ, ˆ λ as well astheir conjugate momenta w, ˆ w . They expand on the fermionic generators of the superalgebrawith the following gradings: λ, ˆ w ∈ H and ˆ λ, w ∈ H . The ghosts satisfy the pure spinorconstraint: λγ µ λ = 0 = ˆ λγ µ ˆ λ , where the γ µ ’s are the SO (9 ,
1) gamma matrices. The purespinor Lorentz currents are: N = −{ w, λ } ; ˆ N = −{ ˆ w, ˆ λ } (2.4)The action reads: S = R π ST r Z d z (cid:18) J ¯ J + 32 J ¯ J + 12 ¯ J J (cid:19) + R π ST r Z d z (cid:16) N ¯ J + ˆ N J − N ˆ N + w ¯ ∂λ + ˆ w∂ ˆ λ (cid:17) (2.5)The first line of the action contains a kinetic term both for the bosonic and fermionic tar-get space coordinates. This implies in particular that the model does not exhibit kappa-symmetry, contrary to the Green-Schwarz string. The radius of the target space is denotedby R in units of the string length. Later on we will work perturbatively in a large radiusexpansion: the small parameter is R − . Gauge symmetry.
The action (2.5) admits a H gauge symmetry: δg = gh , δλ = [ λ, h ] , δ ˆ λ = [ˆ λ, h ] , δw = [ w, h ] , δ ˆ w = [ ˆ w, h ] (2.6)The holomorphic currents transform as: i = 0 : δJ i = [ J i , h ] ; δJ = ∂h + [ J , h ] ; δN = [ N, h ] (2.7)and similarly for the anti-holomorphic currents. We introduce the associated covariant deriva-tive: ∇ = ∂ + [ J , · ] ; ¯ ∇ = ¯ ∂ + [ ¯ J , · ] (2.8)5 arity. The model enjoys a Z symmetry that exchanges holomorphic and anti-holomorphicworldsheet coordinates. It also flips the grade of the fermionic elements of the Lie superalge-bra: the subalgebras H and H are exchanged. The Maurer-Cartan equation.
A consequence of (2.2) is that the current satisfies theMaurer-Cartan equation: ∂ ¯ J − ¯ ∂J + [ J, ¯ J ] = 0 (2.9)This is a crucial equation that is essentially responsible for the integrable properties of themodel. Equations of motion.
The equations of motion combined with the Maurer-Cartan equa-tion lead to:¯ ∇ J = [ J , ¯ J ] + [ J , ¯ J ] + [ N, ¯ J ] − [ J , ˆ N ] ∇ ¯ J = [ N, ¯ J ] − [ J , ˆ N ]¯ ∇ J = [ J , ¯ J ] + [ N, ¯ J ] − [ J , ˆ N ] ∇ ¯ J = − [ J , ¯ J ] + [ N, ¯ J ] − [ J , ˆ N ]¯ ∇ J = [ N, ¯ J ] − [ J , ˆ N ] ∇ ¯ J = − [ J , ¯ J ] − [ J , ¯ J ] + [ N, ¯ J ] − [ J , ˆ N ]¯ ∇ N = − [ N, ˆ N ] ∇ ˆ N = [ N, ˆ N ] (2.10) In this section we discuss the current-current OPEs that are the elementary input neededfor the computations of section 3. The set of currents we consider are the currents J , J , J , J as well as the ghost Lorentz current N , together with their anti-holomorphic partners¯ J , ¯ J , ¯ J , ¯ J and ˆ N . In order to simplify the expressions in the following computations, weintroduce the generic notation K m , ¯ K m for the currents. The index m takes the values inthe set { , , , , g } . For m = 0 , , , K m ≡ J m , ¯ K m ≡ ¯ J m . For the particularvalue m = g we define K g ≡ N , ¯ K g ≡ ˆ N , and g stands for “ghost”. The index m codes the Z -grade of the current. The ghost currents have grade zero.The OPEs of the gauge covariant currents have been discussed in various papers. TheOPEs at first-order in the R − expansion have been analyzed in [23][24][25][26]. The R − corrections to the second-order poles have been computed in [27]. For the purpose of thepresent article, the knowledge of the current algebra at order R − is enough.In appendix B we present a new and rather efficient way of computing the current algebra.The idea is to demand compatibility with the Maurer-Cartan equation and with the equationsof motion (more precisely with the reparametrization invariance of the path integral). Noticethat the OPEs involving J and ¯ J generically suffer from some ambiguities because of thegauge freedom. At the end of appendix B we compare the version of the current algebra wecompute with the ones that appeared previously in the literature.It is convenient to expand the current on a basis of the Lie superalgebra that is compatiblewith the Z grading. We write: K m = K A m m t A m (2.11)6he indices A m are adjoint indices restricted to the subspace of the Lie superalgebra ofgrade m . The generators t A m form a basis of the subspace H m .The current algebra takes the form: K A m m ( z ) K B n n ( w ) = R − C mn κ B n A m ( z − w ) + R − X p C pmn f C p B n A m K C p p z − w + R − X p C ¯ pmn f C p B n A m ¯ K C p p ¯ z − ¯ w ( z − w ) + ...K A m m ( z ) ¯ K B n n ( w ) = R − C m ¯ n κ B n A m πδ (2) ( z − w ) + R − X p C pm ¯ n f C p B n A m K C p p ¯ z − ¯ w + R − X p C ¯ pm ¯ n f C p B n A m ¯ K C p p z − w + ... ¯ K A m m ( z ) ¯ K B n n ( w ) = R − C ¯ m ¯ n κ B n A m (¯ z − ¯ w ) + R − X p C p ¯ m ¯ n f C p B n A m K C p p z − w (¯ z − ¯ w ) + R − X p C ¯ p ¯ m ¯ n f C p B n A m ¯ K C p p ¯ z − ¯ w + ... (2.12)The tensors κ AB and f C BA are respectively the metric and the structure constants (seeappendix A for conventions). The non-trivial data in the current algebra (2.12) is coded inthe coefficients C ∗∗ , C ∗∗∗ . These coefficients should be read as follows: the coefficient C give the coefficient of the identity operator in the OPE between the currents J and J , thecoefficient C g give the coefficient of the ghost current N in the OPE between J and ¯ J ,and so on. We introduced a sum over all currents in the first-order poles in order to simplifythe writing, but many of the C ’s are clearly zero since they do not respect the Z grading.The coefficients C ∗∗ , C ∗∗∗ are symmetric in their two lower indices. Also P p C ¯ pmn should beunderstood as C ¯1 mn + C ¯2 mn + ... . The non-zero coefficients are given below. The non-vanishingsecond-order poles are: C = − , C = 1 , C ¯13 = 1 , C ¯1¯3 = − , C = − , C = 1 , C ¯2¯2 = − C = 2 , C ¯211 = 1 , C = 1 , C ¯2¯1¯1 = 1 C = 1 , C ¯23¯3 = 1 , C = 1 , C ¯2¯3¯3 = 2 C = 2 , C ¯312 = 1 , C = 1 , C = 1 , C ¯3¯1¯2 = 1 C = 1 , C ¯1¯32 = 1 , C ¯13¯2 = 1 , C = 1 , C ¯1¯3¯2 = 2 (2.14) In most of the literature the notation are A → [ ab ], A → α , A → a , A → ˆ α . Although slightly lessexplicit, the notations used here allow for a much more compact writing. C g = 1 , C ¯ g = − , C g = 1 , C ¯ g = 1 , C g ¯2¯2 = − , C ¯ g ¯2¯2 = 1 C g = 1 , C ¯ g = − , C g = 1 , C ¯ g = 1 , C g ¯13 = 1 , C ¯ g ¯13 = 1 , C g ¯1¯3 = − , C ¯ g ¯1¯3 = 1 C ggg = − , C ¯ g ¯ g ¯ g = − J , ¯ J are: C = 1 , C = 1 , C ¯1¯01 = 1 , C ¯1¯0¯1 = 1 C = 1 , C = 1 , C ¯2¯02 = 1 , C ¯2¯0¯2 = 1 C = 1 , C = 1 , C ¯3¯03 = 1 , C ¯3¯0¯3 = 1 C = 1 , C ¯013 = 1 , C = 1 , C ¯0¯1¯3 = 1 C = 1 , C ¯022 = 1 , C = 1 , C ¯0¯2¯2 = 1 (2.16)The method we are using to compute the current algebra does not fix completely the self-OPEs of J and ¯ J . We only obtain the following constraints: C = C = − C ; − C ¯000 = C ¯00¯0 = C ¯0¯0¯0 C g = C g = − C g ¯0¯0 ; − C ¯ g = C ¯ g = C ¯ g ¯0¯0 (2.17)It turns out that these constraints are enough to perform explicitly the computations pre-sented in this paper . ( r, s ) system Similarly to the Green-Schwarz string [28][29], the pure spinor string on
AdS × S admits aone-parameter family of flat connections [30]. This implies that the classical theory admitsan infinite number of conserved charges. The flat connection A ( y ) is defined as: A ( y ) =( J + yJ + y J + y J + ( y − N ) dz + ( ¯ J + y − ¯ J + y − ¯ J + y − ¯ J + ( y − −
1) ˆ N ) d ¯ z (2.18)The flat connection is invariant under parity combined with the exchange of y and y − . Theequations of motion together with the Maurer-Cartan equation (2.10) imply that the previousconnection is flat for all values of the spectral parameter y : dA ( y ) + A ( y ) ∧ A ( y ) = 0 (2.19) More precisely, the constraints (2.17) implies that the coefficients cancel against each other in the com-putation of the commutator of equal-times connections (2.20). The reason is essentially that the currents J ,¯ J appear in the flat connection (2.18) with no dependence on the spectral parameter. This in turns impliesthat these coefficients cancel against each other in the computation of the fusion line operators presented insection 3. Notice however that the cancellation of some divergences in the line operators depends on the valueof the coefficients (2.17), see (C.12).
8n the following we study line operators that are the path-ordered exponential of the integralof the flat connection along a given contour. We will only consider integration contours thatlie at constant time. Consequently only the spacelike component of the flat connection willappear. For simplicity, we use the same notation A ( y ) for the connection and for its spacelikecomponent.The advantage of the version of the current algebra we are working with is that thecommutator of two equal-time space-component of the flat connection can be written as a( r, s ) system:[ A R ( y ; σ ) , A R ′ ( y ′ ; σ ′ )] =2 s∂ σ δ (2) ( σ − σ ′ ) + [ A R ( y ; σ ) + A R ′ ( y ′ ; σ ′ ) , r ] δ (2) ( σ − σ ′ )+ [ A R ( y ; σ ) − A R ′ ( y ′ ; σ ′ ) , s ] δ (2) ( σ − σ ′ ) (2.20)where R and R ′ denote the representations the two connections are transforming in. Thecommutator transforms in the tensor product R ⊗ R ′ . Only the terms explicitly written downin the OPEs (2.12) contribute to the commutator (2.20). The infinite number of subleadingsingularities contained in the ellipses of (2.12) do not contribute to the commutator of equal-time currents (see e.g. [19]). As shown in appendix B.2, the constant matrices r and s aregiven by: r = iπR − (cid:16) r t RA ⊗ t R ′ B κ B A + r t RA ⊗ t R ′ B κ B A + r t RA ⊗ t R ′ B κ B A + r t RA ⊗ t R ′ B κ B A (cid:17) r = ( y − y − ) + ( y ′ − y ′− ) y − y ′ yy ′ ; r = ( y − y − ) + ( y ′ − y ′− ) y − y ′ y y ′ r = ( y − y − ) + ( y ′ − y ′− ) y − y ′ y y ′ ; r = 2 ( y − y ′ − y − y ′ (2.21)and: s = iπR − (cid:16) s t RA ⊗ t R ′ B κ B A + s t RA ⊗ t R ′ B κ B A + s t RA ⊗ t R ′ B κ B A + s t RA ⊗ t R ′ B κ B A (cid:17) s = 1 y y ′ − yy ′ ; s = 1 y y ′ − y y ′ ; s = 1 yy ′ − y y ′ ; s = 0 (2.22)Later it will be important that the r matrix simplifies in the limit where the difference betweenthe spectral parameters y and y ′ is small: y − y ′ → ⇒ r ∼ iπR − y − y ′ ) y ( y + y − ) t RA ⊗ t R ′ B κ BA (2.23)The ( r, s ) matrices (2.21), (2.22) first appeared in [31]. A detailed study of the ( r, s ) systemfor string theory in AdS × S and its properties can be found in [32][33][34] (see appendixB.2 for more details). 9 .4 The line operators Definitions.
We are interested in studying line operators that are the path-ordered expo-nential of the integral of the flat connection on a given contour. When the contour is aninterval [ a, b ], the line operator is called the transition matrix. We denote it as T b,aR ( y ): T b,aR ( y ) = P exp (cid:18) − Z ba A R ( y ) (cid:19) (2.24)The transition matrix is labelled by the representation R in which the flat connection trans-forms. Flatness of the connection implies that the classical transition matrix does not dependon the integration path chosen. This property has been argued to extend to the quantumtheory in [25]. For simplicity we consider only constant-time contours.For string theory purposes we are lead to define the theory on a cylinder. Then we candefine the monodromy matrix which is the line operator associated with a closed contourwinding once around the cylinder:Ω R ( y ) = P exp (cid:18) − I A R ( y ) (cid:19) (2.25)Flatness of the connection implies that the eigenvalues of the monodromy matrix are indepen-dent on time. Consequently they code an infinite number of conserved charges. Eventuallythe transfer matrix is the supertrace of the monodromy matrix: T R ( y ) = ST r P exp (cid:18) − I A R ( y ) (cid:19) (2.26) Regularization of UV divergences.
In a quantum theory the line operators are generi-cally ill-defined since the collisions of integrated connections lead to divergences. To properlydefine line operators one has to regularize these divergences, and then renormalize the lineoperators. In order to study the UV divergences, we first have to expand the exponentials inthe line operators. We write the transition matrix as: T b,aR ( y ) = ∞ X M =0 ( − M T b,aR, ( M ) ( y ) (2.27)where the M -th term is the path-ordered integral of M connections: T b,aR, ( M ) ( y ) = 1 M ! P (cid:18)Z ba A R ( y ) (cid:19) M = Z b>σ >...>σ M >a dσ ...dσ M A R ( y ; σ ) ...A R ( y ; σ M ) (2.28)and similarly for the monodromy and transfer matrices. Divergences occur when two inte-grated connections collide. It is clear from the current algebra (2.12) that the collision of twoconnections leads to second- and first-order poles . In order to regularize these divergences,we introduce a UV cut-off ǫ . We use a principal-value regularization scheme as suggested in[31]. The OPE between two equal-time connections A ( σ ) and A ( σ ′ ) is regularized by a smallshift in time, in a symmetric way: A ( σ ) A ( σ ′ ) → (cid:0) A ( σ + iǫ ) A ( σ ′ ) + A ( σ ) A ( σ ′ + iǫ ) (cid:1) (2.29) The ellipses in the current algebra (2.12) contain subleading singularities, including possible logarithmicsingularities. Such terms do not lead to any UV divergences in the line operators. Indeed the integral of thesesubleading singularities gives a finite result. σ − σ ′ → P.V. σ − σ ′ = 12 (cid:18) σ + iǫ − σ ′ + 1 σ − iǫ − σ ′ (cid:19) = σ − σ ′ ( σ − σ ′ ) + ǫ (2.30)This regularization scheme turns out to be very convenient to discuss the fusion of lineoperators, as explained in section 3. Divergences at order R − . The first-order divergences in line operators in the pure spinorstring on
AdS × S were first studied in [26]. In this paper the authors used a differentregularization scheme: the OPEs were regularized by imposing that the distance betweentwo connections cannot be smaller than the UV cut-off. The authors of [26] also used aslightly different version of the current algebra. In appendix C we revisit the analysis of [26]using the regularization scheme (2.29) and the current algebra (2.12). The main differencewe obtain with respect to [26] is that the linear divergences do cancel thanks to our choiceof regularization scheme. Below we summarize the results derived in appendix C .The transition matrices contain logarithmic divergences. Schematically, these divergencesread: ∼ log ǫ X i =0 { t A i t A i , T b,a ( y ) } (2.31)The precise expression for these divergences is given in equation (C.14). Consequently thetransition matrices need to be renormalized. These divergences are cancelled by a simplewave-function renormalization.The monodromy matrix also contains logarithmic divergences. These are given in (C.16).Schematically, these divergences read: ∼ log ǫ X i =0 (cid:0) t A i t A i Ω( y ) + Ω( y ) t A i t A i − t A i Ω( y ) t A i (cid:1) (2.32)The new divergences with respect to the transition matrices come from collisions betweenconnections sitting on both sides of the starting point of the integration contour. These di-vergences are cancelled by a simple wave-function renormalization of the monodromy matrix.The most important result for the purpose of this paper is that the transfer matrix iscompletely free of divergences at order R − . This follows simply by taking the supertrace ofequation (2.32). This remarkable property strongly relies on the vanishing of the dual Coxeternumber of the global symmetry group P SU (2 , | In this section we study the fusion of two line operators. The fusion is the process of bringingthe integration contours of two line operators on top of each other. We are interested in thequantum effects that occur in this process. The fusion of line operators for the pure-spinorstring in
AdS × S was studied at first-order in perturbation theory in [31]. In this sectionwe will revisit and extend the first-order computations of [31]. Then we will further extendthe computation of fusion up to second order in perturbation theory.11he structure of the computations is similar to the ones presented in [19], where moredetails can be found. In [19] the computations were performed in the sigma model on thesupergroup P Sl ( n | n ). This theory is a good toy model for the pure spinor string on AdS × S .Indeed the complications coming from the coset structure and the pure spinor ghosts areabsent. Let us consider two transition matrices T b,aR ( y ) and T d,cR ′ ( y ′ ) that transform respectively in therepresentations R and R ′ . The fusion of these two matrices transforms in the tensor product R ⊗ R ′ . In the following we will omit the symbol ⊗ to lighten the formulas. We representthe fusion of these two transition matrices with the symbol ⊲ . This process is defined as: T b,aR ( y ) ⊲ T d,cR ′ ( y ′ ) = lim ǫ → + T b + iǫ,a + iǫR ( y ) T d,cR ′ ( y ′ ) (3.1)Assuming the integration contour of the transition matrices T d,cR ′ ( y ′ ) lies at constant time τ ,then the integration contour of T b + iǫ,a + iǫR ( y ) lies at constant time τ + ǫ . If the intervals [ a, b ]and [ c, d ] do not overlap, the process of fusion is trivial. In the following we assume thatthe overlap of these intervals is non-zero. As the distance between the two contours goes tozero, the OPEs between integrated connections sitting on the two contours produce quantumcorrections to the classical process of fusion. These are the corrections we will evaluate.Let us consider the OPE between two connections A R ( y ; σ + iǫ ) and A R ′ ( y ′ ; σ ′ ) integratedrespectively on the first and on the second contour. We write this OPE as: A R ( y ; σ + iǫ ) A R ′ ( y ′ ; σ ′ ) = 12 (cid:0) A R ( y ; σ + iǫ ) A R ′ ( y ′ ; σ ′ ) + A R ( y ; σ ) A R ′ ( y ′ ; σ ′ + iǫ ) (cid:1) + 12 (cid:0) A R ( y ; σ + iǫ ) A R ′ ( y ′ ; σ ′ ) − A R ( y ; σ ) A R ′ ( y ′ ; σ ′ + iǫ ) (cid:1) (3.2)Comparing with equation (2.29), we notice that the first term in the previous equation shouldbe understood as regularized OPE in the quantum line operator obtained after the fusionhas been completed. On the other hand, the second term in (3.2) should be understood asproducing a quantum correction proper to the process of fusion. This are the corrections wewant to compute.In order to understand better the meaning of (3.2), let us isolate a first-order pole in theOPE between the two connections. Under the decomposition (3.2), it is rewritten as:1 σ + iǫ − σ ′ = 12 (cid:18) σ + iǫ − σ ′ + 1 σ − iǫ − σ ′ (cid:19) + 12 (cid:18) σ + iǫ − σ ′ − σ − iǫ − σ ′ (cid:19) = P.V. σ − σ ′ − iπδ ǫ ( σ − σ ′ ) (3.3)The term we focus on is the second term on the right-hand side. As the notation suggests, it isactually a regularization of the delta-function. Once we integrate upon the free coordinates,it produces a finite quantum corrections to the fusion of line operators. We can performa similar manipulation for all first- and second-order poles appearing in the OPE betweenthe two connections A R ( y ; σ + iǫ ) and A R ′ ( y ′ ; σ ′ ). In order to isolate the contribution tothe quantum corrections associated with fusion, we subtract the principal value from thesingularities. We obtain that all the terms that contribute to the quantum corrections fromfusion come with (derivatives of) regularized delta functions.12he upshot is the following: in order to compute the quantum corrections in the processof fusion, we have to subtract the “principal value” piece from the OPE between connections.What we are left with is essentially the commutator between connections that we computedin (2.20): (1 − P.V. ) A R ( y ; σ + iǫ ) A R ′ ( y ′ ; σ ′ ) ǫ → + = 12 [ A R ( y ; σ ) , A R ′ ( y ′ ; σ ′ )] (3.4) We begin with the corrections of order R − . Since all terms in the current algebra are oforder R − , it is enough to perform one OPE. The computation of the first-order correctionsin the fusion of two transition matrices was performed in [19] using OPE techniques. Thiscomputation holds provided the commutator of connections can be written as a ( r, s ) system,which is the case for the pure spinor string on AdS × S (see (2.20)). The result obtainedin [19] matches the hamiltonian analysis of [37]: T b,aR ( y ) ⊲ T d,cR ′ ( y ′ ) = T b,aR ( y ) T d,cR ′ ( y ′ )+ χ ( b ; c, d ) T d,bR ′ ( y ′ ) r + s T b,aR ( y ) T b,cR ′ ( y ′ ) − χ ( a ; c, d ) T b,aR ( y ) T d,aR ′ ( y ′ ) r + s T b,cR ′ ( y ′ )+ χ ( d ; a, b ) T b,dR ( y ) r − s T d,aR ( y ) T d,cR ′ ( y ′ ) − χ ( c ; a, b ) T b,cR ( y ) T d,cR ′ ( y ′ ) r − s T c,aR ( y )+ O ( R − ) (3.5)The first term on the right-hand side is the zeroth-order result. The remaining terms arethe first-order corrections. The function χ ( a ; b, c ) is the characteristic function of the interval[ b, c ] which takes the value 1 if b > a > c and 0 if a > b or a < c . For the special case wherethe integration intervals of the line operators have coinciding endpoints, that is for a = b or a = c , then the characteristic function χ ( a ; b, c ) has to be evaluated as . This prescriptionessentially had to be guessed in the hamiltonian formalism [36]. In the OPE formalism it isa consequence of the definition (3.1) [19].Notice that the first-order corrections in (3.5) are anti-symmetric in the exchange of thetwo line operators. So they contribute only to the commutator of the line operators. Thisfollows from the fact that the quantum corrections associated with fusion come from theanti-symmetric part in the OPEs (see (3.4)).From equation (3.5) we can deduce the fusion of transfer matrices [19]. We obtain thatthe fusion of transfer matrices at first order is trivial: T R ( y ) ⊲ T R ′ ( y ′ ) = T R ( y ) T R ′ ( y ′ ) + O ( R − ) (3.6)Indeed the first-order corrections associated with fusion in (3.5) take the form of constantmatrices inserted at the endpoints of the overlap of the integration intervals. Since thetransfer matrices have no endpoints, it is not surprising that these corrections vanish. Thisimplies in particular that the commutator of the transfer matrices is zero at first order. Remember that our main goal is to show that the leading quantum correction in the fusionof two transfer matrices gives the shifts in the T-system at first order. Since the fusion oftransfer matrices is trivial at first-order, we need to study the fusion of line operators atsecond-order.There are two different ways we can obtain R − corrections in the process of fusion:13igure 1: A triple collision. The first OPE is taken between two connections sitting ondifferent contours. The second OPE is taken between a third connection and the currentsresulting from the first OPE. • The first way is to take one single OPE between two integrated connections, and in-clude R − corrections to the current algebra (2.12). As argued in section 3.1, only theanti-symmetric part of the current-current OPEs contributes to the process of fusion.Consequently at this order the R − corrections to the current algebra lead to R − cor-rections to the commutator of line operators. We postpone the computation of thesecorrections for future work since the current algebra is only partially known at order R − [27]. • The second way is to perform two OPEs between integrated connections and use thecurrent algebra at order R − . Following the logic explained in section 3.1, we againconsider only the anti-symmetric part in each OPE. Since we perform an even numberof OPEs this time the result will be symmetric under the exchange of the two lineoperators. More precisely we obtain a contribution to the symmetric product of theline operators of order R − . This are the terms that we will compute in this paper.More generally, the arguments of section 3.1 imply that any quantum correction in theprocess of fusion that involve an even (respectively odd) number of OPEs would contributeonly to the symmetric product (respectively commutator) of the line operators. Consequentlythe R − corrections to the current algebra would contribute to the symmetric product of lineoperators at order R − and higher.Below we describe the different steps in the computation of the symmetric fusion of lineoperators at order R − . In an attempt to keep this section readable, the technical details ofthe computation have been gathered in appendix D. More details can also be found in [19].Let us consider two line operators with contours separated in time by a small distance ǫ .We have to take two OPEs between the connections integrated on the contours. The simplestway is to take two OPEs between two distinct pairs of connections. But we can also take oneOPE between two connections, and then take the OPE of the resulting currents with a thirdconnections. We will call this latter process a triple collision (see figure 1).The computation is conveniently decomposed in two steps. • A first part of the total answer is obtained in the way depicted in Figure 2. We startfrom the result of the fusion at first order. We pull the contours away, and then re-fuse14igure 2: We can compute one piece of the result for fusion at second order in the followingway. In step ① we compute the first-order corrections from fusion. In step ② we separatethe contours again. Finally in step ③ we perform a second time the fusion at first order.This computation does not give the full result since the triple collisions are not properlyaccounted for. For transfer matrices the fusion at first order is trivial, so the quantumcorrections obtained in this way actually vanish.the line operators. The result of this procedure was computed in [19] for an arbitrary( r, s ) system. We obtain new insertion of constant matrices at the endpoints of theoverlap of the contours of the line operators. Roughly speaking, the first-order result(3.5) exponentiate . What is important for our purposes is that this procedure givesonce again a vanishing result for the fusion of transfer matrices. This simply followsfrom the fact that there is no first-order correction in the fusion of transfer matrices. • The procedure described above does not capture correctly the quantum correctionscoming from triple collisions. Indeed in this procedure, the intermediate currents inthe triple collisions are distributed in an arbitrary way on the two integration contoursso that they recombine into connections. This induces a source of errors. So we haveto compute separately additional corrections coming from the triple collisions. This isdone in appendix D.The analysis of appendix D shows that there are two types of corrections that we needto add on top of the result obtained by the procedure described in figure 2. The first type ofcorrective terms contain the integration of an operator ˜ K on the overlap of the contours: R − ∞ X M,M ′ =0 ( − ) M + M ′ M X i =0 M ′ X i ′ =0 Z [ a,b ] ∩ [ c,d ] dσ (cid:22)Z bσ A (cid:25) i (cid:22)Z dσ A ′ (cid:25) i ′ ˜ K ( σ ) (cid:22)Z σa A (cid:25) M − i (cid:22)Z σc A ′ (cid:25) M ′ − i ′ (3.7) The precise expression is slightly more complicated than the one given in [19], since the formula (4.15) in[19] does not generalizes to the coset. This implies that the exponentiation observed in [19] is not exact in thecase at hand. ⌊ R ba A ⌉ M to describe the path-ordered integralof M connections on the interval [ a, b ]. The precise expression for the operator ˜ K is explic-itly given in (D.17). Schematically, the operator ˜ K is a linear combination of the currentsmultiplied by three generators of the Lie superalgebra in the representations R or R ′ , andcontracted with structure constants. The second type of corrective terms contain a constantmatrix ˜ tt inserted in between the integrated connections on the overlap of the integrationcontours: R − ∞ X M,M ′ =0 ( − ) M + M ′ M X i =0 M ′ X i ′ =0 Z ba dσ Z dc dσ ′ (cid:22)Z bσ A (cid:25) i (cid:22)Z dσ A ′ (cid:25) i ′ × δ ǫ ( σ − σ ′ ) ˜ tt (cid:22)Z σa A (cid:25) M − i (cid:22)Z σc A ′ (cid:25) M ′ − i ′ (3.8)The precise expression for the matrix ˜ tt is given in (D.19). Schematically, it is a linearcombination of the tensor product of two generators taken in the representation R and R ′ and contracted with structure constants. Notice that the integration over the regularizeddelta function squared produce a linear divergence when the UV regulator ǫ is sent to zero.It would be interesting to perform a complete analysis of the second-order divergences in theline operators along the lines of section 3 in [19], to see whether the transfer matrices as wellas the result of the fusion of transfer matrices are free of divergences up to second-order. Fusion of transfer matrices at second order: upshot.
The detailed expression for thesymmetric fusion of transition and monodromy matrices at second order is quite indigestible,so we refrain from giving an explicit formula for those. On the other hand the symmetricfusion of transfer matrices, that is crucial for the purposes of this paper, turns out to berather simple: T R ( y ) ⊳⊲ T R ′ ( y ′ ) = 12 {T R ( y ) , T R ′ ( y ′ ) } + R − ST r (cid:18)Z π dσ T π,σR ( y ) T π,σR ′ ( y ′ ) ˜ K ( σ ) T σ, R ( y ) T σ, R ′ ( y ′ ) (cid:19) + R − ST r (cid:18)Z π dσ Z π dσ ′ T π,σR ( y ) T π,σR ′ ( y ′ ) δ ǫ ( σ − σ ′ ) ˜ tt T σ, R ( y ) T σ, R ′ ( y ′ ) (cid:19) + O ( R − ) (3.9)where we denoted by ⊳⊲ the symmetrized fusion product. This result is schematically repre-sented in figure 3. The detailed expressions for the operator ˜ K and the constant matrix ˜ tt can be read from equations (D.17) and (D.19). AdS / C F T T-system
In this section we use the previous computations to obtain a first-principle perturbativederivation of the T-system. As explained in the introduction, the idea is to promote theT-system (1.1) to an operator identity, where the product between transfer matrices is un-derstood as the fusion product: T a,s ( u + 1) ⊲ T a,s ( u −
1) = T a +1 ,s ( u + 1) ⊲ T a − ,s ( u −
1) + T a,s − ( u + 1) ⊲ T a,s +1 ( u −
1) (4.1)16igure 3: Schematic representation of the symmetric fusion of transfer matrices at secondorder (3.9). The first term is the classical result. In the second term, an additional operator˜ K is integrated in between the connections. In the third term, a constant matrix ˜ tt is insertedin between the integrated connections.We expect the transfer matrices to commute in the quantum theory. This has been proven insection 3 at order R − . Consequently we can equivalently use the symmetric fusion productto define the T-system (4.1).The integer label a, s label unitary irreducible representation of P SU (2 , | a, s .It is known that these representations satisfy the following supercharacter identity (see e.g.[8][18]): χ ( a, s ) = χ ( a + 1 , s ) χ ( a − , s ) + χ ( a, s + 1) χ ( a, s −
1) (4.2)In the limit where we neglect both the shifts of the spectral parameter as well as the quantumeffects associated with fusion, the T-system (4.1) reduces to the character identity (4.2).Next we want to show that the T-system (4.1) holds at first order. We consider:0 ? = T a,s ( y + δ ) ⊲ T a,s ( y − δ ) − T a +1 ,s ( y + δ ) ⊲ T a − ,s ( y − δ ) − T a,s − ( y + δ ) ⊲ T a,s +1 ( y − δ ) ≡ X R,R ′ T R ( y + δ ) ⊲ T R ′ ( y − δ ) (4.3)where we use the shorthand P R,R ′ to denote the sum over representations that appears in theT-system. We look for a value of the shift of the spectral parameters δ such that the previousquantity does indeed vanish, and the T-system holds. Then we can deduce the relationshipbetween the spectral parameter y used to define the flat connection (2.18), and the spectralparameter u that appears in the T-system (4.1).We assume that δ is of order R − . We will now show that the terms of order R − in (4.3)do vanish. More precisely, the terms of order R − coming from the derivative expansion ofthe transfer matrices cancel against the leading quantum correction coming from the processof fusion. Fusion of transfer matrices when the difference of spectral parameter is small.
Insection 3 we obtained that the leading quantum correction in the fusion of transfer matrices isof order R − . However when the difference of the spectral parameter is of order R − , a pieceof the leading quantum correction actually becomes of order R − . So it has the right orderof magnitude to cancel the first term in the derivative expansion of the transfer matrices in(4.3). More details can be found at the end of appendix D.17he interesting term comes from the second line in (3.9). We assume y − y ′ = O ( R − ).From equations (D.23) we obtain that (3.9) simplifies to: T R ( y ) ⊲ T R ′ ( y ′ ) = T R ( y ) T R ′ ( y ′ )+ R − π y ( y − y − ) y − y ′ ST r (cid:18)Z π dσT π,σR ( y ) T π,σR ′ ( y ′ ) × − X m,n,p,q,r =0 ∂ y A E r ( y ) (cid:16) f C p B n A m f E r C p D q { t RD q , t RA m ] t R ′ B n + f C p B n A m f E r D q C p t RA m { t R ′ B n , t R ′ D q ] (cid:17) × T σ, R ( y ) T σ, R ′ ( y ′ ) (cid:17) + O ( R − ) (4.4)We observe that the linear combination of the currents that appears is now proportional tothe derivative of the flat connection. Useful character identities.
To further simplify the quantum corrections that appear in(4.3), we need to use some character identities that holds for the particular combination ofrepresentations that appear in the T-system. In [18] the validity of the T-system was provenfor transfer matrices associated to Gl ( k | m ) spin chains. Then a large family of characteridentities was deduced by expanding the T-system as an infinite series. Here we go the otherway: we try to reconstruct the T-system from a perturbative expansion. Thus it makes sensethat we need to use the character identities of [18]. In the appendix E of [19], it was shownthat some of these character identities imply in particular that for any group element g andfor any function K E : X R,R ′ K E f C BA f ECD
ST r ( { t RD , t RA ] g R ⊗ t R ′ B g R ′ )= 2 X R,R ′ ST r ( g R ⊗ K E t R ′ E g R ′ ) − ST r ( K E t RE g R ⊗ g R ′ ) (4.5)and similarly: X R,R ′ K E f C BA f EDC
ST r ( t RA g R ⊗ { t R ′ B , t D ] g R ′ )= 2 X R,R ′ ST r ( g R ⊗ K E t R ′ E g R ′ ) − ST r ( K E t RE g R ⊗ g R ′ ) (4.6)Essentially, these character identities allow to replace the complicated combination of struc-ture constants and generators appearing in (4.4) by single generators, assuming we considerthe sum of representation that appear in the T-system. So we obtain: X R,R ′ T R ( y + δ ) ⊲ T R ′ ( y − δ ) = X R,R ′ T R ( y + δ ) T R ′ ( y − δ ) + X R,R ′ R − π y ( y − y − ) δ × ST r (cid:18)Z π dσT π,σR ( y ) T π,σR ′ ( y ) (cid:16) − ∂ y A E ( y ; σ )( − t RE + t R ′ E ) (cid:17) T σ, R ( y ) T σ, R ′ ( y ) (cid:19) + O ( R − )= X R,R ′ T R ( y + δ ) T R ′ ( y − δ ) − R − π y ( y − y − ) δ ( ∂ y T R ( y ) T R ′ ( y ) − T R ( y ) ∂ y T R ′ ( y ))+ O ( R − ) (4.7)18 he T-system at first order. Performing a Taylor expansion for the T ’s in (4.7), wededuce: X R,R ′ T R ( y + δ ) ⊲ T R ′ ( y − δ ) = X R,R ′ T R ( y ) T R ′ ( y )+ (cid:18) δ − R − π y ( y − y − ) δ (cid:19) ( ∂ y T R ( y ) T R ′ ( y ) − T R ( y ) ∂ y T R ′ ( y )) + O ( R − ) (4.8)The first term in the previous expression vanishes because of the classical identity (4.2). Inorder for the first-order corrections to vanish as well, we have to take: δ = R − π y ( y − y − ) (4.9)Thus have have shown that: X R,R ′ T R ( y + δ ) ⊲ T R ′ ( y − δ ) = 0 + O ( R − ) (4.10) Redefinition of the spectral parameter.
In order to write the T-system in the canonicalform (4.1) we define: u = R π − y + cst (4.11)such that u ( y ± δ ) = u ( y ) ±
1. Then equation (4.10) is rewritten as: T a,s ( u + 1) ⊲ T a,s ( u −
1) = T a +1 ,s ( u + 1) ⊲ T a − ,s ( u −
1) + T a,s − ( u + 1) ⊲ T a,s +1 ( u −
1) + ... (4.12)This is the canonical form of the T-system (1.1).
Comparison with the T-system obtained via the Thermodynamic Bethe Ansatz.
We can now perform a consistency check with the Thermodynamic Bethe Ansatz derivationof the
AdS × S T-system [14][15][16]. In this context the T-system is obtained in a slightlydifferent form: only the T-functions on the left-hand side have a shifted spectral parameter: T T BAa,s ( u + 1) T T BAa,s ( u −
1) = T T BAa +1 ,s ( u ) T T BAa − ,s ( u ) + T T BAa,s − ( u ) T T BAa,s +1 ( u ) (4.13)This mismatch is easily cured by a redefinition of the T-functions. Let us define the functions T T BA ( u ) as: T T BAa,s ( u ) = T a,s ( u + a − s ) (4.14)Then the functions T T BA ’s satisfy (4.13) if and only if the functions T ’s satisfy the T-system(1.1).However the previous redefinition does not change the magnitude of the shift on the left-hand side of the T-system. Thus the matching of the shifts gives a quantitative check ofthe consistency between the approach taken in this paper, and the TBA approach . Next weperform this matching using the conventions of [5][14]. The flat connection is written in termsof a spectral parameter x as A ( x ) = J dz + ( x − / ( x + 1) J dz + ... . Comparing with (2.18)we deduce that the spectral parameter y that we use is related to x as: y = ( x − / ( x + 1).The variable u T BA that enters the TBA T-system is linked to the spectral parameter x via The author would like to thank N. Gromov for stressing this point. u T BA /g = x + 1 /x , where g is related to the ’t Hooft coupling λ as g = √ λ/ π . The parameter R can be linked to the ’t Hooft coupling λ by identification of theprefactor of the worldsheet action. This gives √ λ/ π = R / π . Consequently the parameter u (4.11) that we obtained is related to the parameter u T BA as: u = 2 u T BA , assuming thefree constant in (4.11) takes the value R / π . The analysis of [14] gives a T-system wherethe parameter u T BA is shifted by ± i/
2. Given that u is shifted in our case by ±
1, there isan apparent mismatch by a factor of i . This comes from the fact that we have been workingon an euclidean worldsheet. If we Wick-rotate the worldsheet to a minkowskian signature,then our analysis produce the T-system with imaginary shifts in perfect agreement with theTBA analysis. Upshot.
As claimed previously, we have derived the T-system up to first order in the largeradius expansion. More precisely, we have sown that the shifts of the spectral parameter inthe T-system come from quantum effects in the fusion of transfer matrices. Moreover we havechecked that the shifts are the same than the ones obtained in the Thermodynamic BetheAnsatz derivation of the T-system.Notice that the vanishing of the divergences in the transfer matrices is important. Indeedif the transfer matrices would need to be renormalized, then the renormalization factor wouldmost likely depend on the representation in which the transfer matrix is taken (see e.g. [35]).It implies that the different terms in the T-system would be renormalized differently, whichwould most likely destroy the balance needed for the previous computation to work.
In this paper we have proven that the T-system is realized in the pure spinor string on
AdS × S up to first order in the large radius expansion. In [19] a similar proof was givenfor the non-linear sigma model on the supergroup P Sl ( n | n ). It is natural to look for othertheories where this derivation can be easily generalized. A close look at the computationleads to the following conclusion: there are only a few necessary and sufficient conditionsthat a given model has to fulfill in order for the derivation to apply. Obviously the theoryhas to exhibit a one-parameter family of flat connections. We assume that the connectiontakes value in a Lie algebra. The other conditions are the following: • The equal-time commutator of the spacelike component of the connection can be writ-ten as a ( r, s ) system. This guarantees that formula (3.9) can be directly reproduced.Moreover the r matrix must satisfy a property similar to equation (2.23): in the limitwhere the difference between the spectral parameter is small, the r matrix needs to beproportional to the Casimir κ BA t A ⊗ t B . This condition is necessary for the simplifica-tion leading to equation (4.4) to occur. • The Lie group needs to possess an equivalent of the character identities (4.5), (4.6).Given the role played by the results of [18], it is tempting to speculate that a sufficientcondition is that there exists a spin chain with the same symmetry group that realizesthe T-system. Let us be a bit more precise on this point. On a minkowskian worldsheet, the imaginary shift iǫ in e.g.(3.3) would be replaced by a real shift ǫ . This implies that in the computation of fusion all OPEs would comewith an additional factor of i . Consequently the second-order corrections from fusion would come with anadditional minus sign. This would in turn induce a factor of i in the shift of the T-system. Eventually the transfer matrix has to be free of divergences at first order in perturbationtheory . A crucial condition here is that the dual Coxeter number of the symmetrygroup vanishes. This condition prevents the renormalization of the transfer matricesfrom destroying the balance needed for the computation to work.Next we discuss several candidate theories that may fulfill these requirements. Candidate theories relevant for the AdS/CFT correspondence.
Let us begin withthe theories that describe superstrings in Anti-de Sitter backgrounds. These theories arebuilt on sigma models on (coset of) supergroups, see e.g. [40] for a classification of therelevant Z cosets. Notice that all the supergroups involved have a vanishing dual Coxeternumber. This should not come as a surprise, since the vanishing of the dual Coxeter numberis tightly related to the vanishing of the spacetime supercurvature, and thus to the fact thatthe equations of motion of supergravity are satisfied.Spacetime covariance was helpful in the previous analysis. Consequently we will mostlydiscuss theories of the pure-spinor type that allow for a covariant quantization. Obviously itwould be interesting to reproduce the previous computations in Green-Schwarz-like theoriesthat realize kappa-symmetry. The structure of the computations would be identical, but thecomputations themselves would be more tedious because the gauge-fixing of kappa symmetryusually comes with a breaking of the target space isometries.The first obvious candidate is string theory on AdS × CP . Indeed in [5] a Y-system wasconjectured to hold in this theory. The TBA derivation of the Y-system was performed in[41][42]. In order to actually reproduce the computations described in the present paper, thepure spinor formulation of superstring theory in AdS × CP developed in [43] is a naturalstarting point (see also [44]).The second candidate is string theory on AdS × S . The analysis of [19] applies tothe sigma-model on P SU (1 , | AdS × S in a superspace with eight supercharges. In this formalism theworldsheet theory is the sigma model on P SU (1 , | AdS × S realizes the T-system up to first order in the large radius expansion, and up tozeroth order in the ghosts expansion. This is valid for AdS × S supported by RR fluxes, NSfluxes or by any mixing of these fluxes. It would be instructive to dress up the computationof [19] with the hybrid ghosts. There has been some interest in the question of integrabilityfor string theory in AdS × S , see e.g. [46]. However the progress have been rather slowerthan in the case of AdS × S , mostly because the dual Conformal Field Theory is not aswell understood. Presumably the approach presented in [19] and in the present article canlead to a faster road to the solution of this problem.Other formulations of string theory on AdS × S involve a Z coset of the supergroup P SU (1 , | × P SU (1 , | AdS × S discussedin [48]. It is also based on a Z coset of the supergroup P SU (1 , | Z coset ofsupergroups with vanishing dual Coxeter number: AdS × S × S , AdS × S × S , AdS × S Actually a weaker condition is that all combinations of transfer matrices that enters the T-system arerenormalized with the same coefficient.
AdS . Quantum integrability is likely to show up at least in some of these examples. Noformalism has been proposed to covariantly quantize string theory in these backgrounds yet. Other candidates.
Other theories that may not be directly relevant for string theorypresumably also realize the T-system in the way described in this paper. These are the sigmamodels on (cosets of) supergroups with vanishing dual Coxeter number, some of which playa role in condensed matter (see e.g. [51][52][53]).A first example is the sigma model on the supergroup
OSp (2 n + 2 | n ). This model sharesmany of the remarkable properties of the sigma model on P Sl ( n | n ) [54], see e.g. [55].Next Z cosets of supergroups with vanishing dual Coxeter number are classically inte-grable [29][55]. Some of them also display nice quantum features [56]. In these model thereis a current that is both flat and conserved. Consequently the current algebra is very similarto the one found in sigma models on supergroups [57]. This follows from the generic methodintroduced in [58] and used in appendix B to compute the current algebra.Classical integrability extends to Z and more generally to Z m cosets [49]. It would beinteresting to understand these models better. In particular it may shed some new light onthe question of the role of the pure spinor ghosts for quantum integrability of the pure spinorstring in AdS × S . Summary of the results.
We have studied the fusion of line operators in the pure spinorstring on
AdS × S up to second order in perturbation theory. We deduced that the purespinor string on AdS × S realizes the T-system as an operator identity, with the fusionproduct, up to first order in the large ’t Hooft coupling expansion. The quantum effects inthe fusion of the transfer matrices give the shifts in the T-system. Comparison with the Thermodynamic Bethe Ansatz.
The T-system was previouslyderived using the TBA machinery [14][15][16]. Here we will compare the advantages of bothapproaches.A weakness of the TBA is that it relies on several assumptions that are notoriouslydifficult to check. In particular one has to assume quantum integrability to start with.Moreover the spectrum of excitations that contribute in the thermodynamic limit essentiallyhas to be guessed through the string hypothesis. The approach of the current article has thebig advantage of starting from first principles.The other drawback of the TBA is that the derivation of the T-system only applies tothe ground state. The fact that the same set of equations also codes the spectrum of excitedstates upon analytic continuation is essentially an empirical observation. In this paper wehave derived the T-system as an operator identity. Thus there is no doubt that all states ofthe theory satisfy the T-system.On the other hand, the approach we are using here is intrinsically perturbative. Thecomputations needed to derive the T-system at first order were already quite heavy. It wouldtake a lot of efforts to go to the next order. The TBA approach is free of this limitation sinceit produces the full T-system in one go.There are also some by-products of the TBA approach that were not reproduced in thepresent work. In particular the TBA gives a explicit formula to extract the spectrum from theT-functions. It also gives some informations about the analytic properties of these functions.22t would be interesting to investigate these questions with the elementary techniques used inthe present paper. We hope to come back to these questions in future work.The computation presented in this paper gives a very strong argument in favor of thevalidity of the T-system. It is not a definite proof since it is perturbative. However theprevious discussion shows that the approach presented here is complementary with the TBAanalysis. Indeed the weak points of the TBA are the the strong points in our approach, andvice-versa. So the combination of both methods leaves little room for doubts.Moreover this works sheds a new light on the T-system. The fact that it should be under-stood as an operator identity where the product is the fusion of line operators, may be helpfulto understand better the integrable structures that appear in the AdS/CFT correspondence.
The role of the pure spinor ghosts.
The pure spinor ghosts are expected to play animportant role in the quantum worldsheet theory. We can ask the question of the role ofthe pure spinor ghosts in the computation described in this paper. Interestingly, the sameresults would be obtained if we would set the pure spinor ghosts to zero from the start. Thereason is that at tree level, the ghosts form a closed subsector. More precisely the OPE of aghost current with any other current can only produce ghost current. In other words, all thecoefficients of the type C mg ∗ and C m ¯ g ∗ in the current algebra (2.12) are zero if the index m isnot g or ¯ g .The fact that the computation does not relies on the pure spinor ghosts can be trackedback to the fact that we only needed the tree-level current algebra to compute the crucialterm that produces the shifts in the T-system. Presumably this is not going to be the caseat higher order. Continuing the computation of [27] to get the full current algebra at secondorder would be interesting. Already the pure spinor should play an important role. Hopefullythey allow for the cancellation of second-order divergences in the transfer matrices, and theyalso insure that transfer matrices commute up to second order in perturbation theory.In this paper we used the pure spinor formalism that allows for a covariant quantization.It may also be instructive to reproduce this computation in the Green-Schwarz formulationof [28]. The algebra of transfer matrices.
The computations we performed allow to addressthe question of the algebra of transfer matrices for the pure spinor string on
AdS × S .Naively, the fusion of two transfer matrices is rather complicated. It seems from (3.9) thatit does not even close on transfer matrices. However by selecting a particular combinationof representations we managed to close the algebra. For these representations, the algebra oftransfer matrices is nothing but the T-system.There might exists a generalization of the T-system that applies for other representations.It would be interesting to further explore this issue. Generalization to other integrable field theories.
It would be also interesting to trythe approach advocated here in other integrable theories that play a role in the AdS/CFTcorrespondence. Some examples were listed in section 5. This approach may be more effi-cient than trying to reproduce the historical steps that were performed for
AdS × S . Moregenerally, developing worldsheet technology for strings in RR background is certainly worth-while. Even if the progress in that direction have been rather slow, the results presented heretogether with other recent works (see e.g. [59][60]) demonstrate that quantum string theoryin some RR backgrounds can be studied with the tools that are currently available.23he results presented here also suggest that the integrable models relevant for the AdS/CFTcorrespondence may belong to a special family. It is not clear that the interpretation of theT-system advocated here applies straightforwardly to generic integrable field theories. Indeedgenerically the transfer matrices have to be renormalized when the dual Coxeter number ofthe symmetry group is non-zero (see e.g. [35]). This would complicate a tentative derivationof the T-system from the fusion of transfer matrices. Acknowledgments
The author would like to thank Gleb Arutyunov, Oscar Bedoya, Denis Bernard, NikolayGromov, Volodya Kazakov, Marc Magro, Valentina Puletti, Sakura Schafer-Nameki, JoergTeschner, Jan Troost, Benoit Vicedo, Dmitro Volin and in particular Pedro Vieira for usefuldiscussions and correspondence. The author is a Postdoctoral researcher of FWO-Vlaanderen.This research is supported in part by the Belgian Federal Science Policy Office throughthe Interuniversity Attraction Pole IAP VI/11 and by FWO-Vlaanderen through projectG011410N.
A Conventions
Let t A be a basis of the generators of the Lie superalgebra. The metric is defined as: κ AB = ST r ( t A t B ) (A.1)where the supertrace ST r is a non-degenerate graded-symmetric inner product. We definethe inverse metric as: κ AB κ AC = δ BC (A.2)The metric and its inverse are graded-symmetric: κ AB = ( − ) AB κ BA (A.3)where ( − ) AB is a minus sign if and only if both indices A and B are fermionic. An element X of the Lie superalgebra is expanded as: J = J A t A (A.4)We adopt “NE-SW” conventions for the contraction of indices. Indices are raised and loweredwith the metric in the following way: X A = κ AB X B ; X A = X B κ BA (A.5) Structure constants.
The graded commutator for the generators is defined as:[ t A , t B } = t A t B − ( − ) AB t B t A (A.6)We define the structure constants f ABC as:[ t A , t B } = f ABC t C (A.7)24he identity ST r ( t A [ t B , t C } ) = Str ([ t A , t B } t C ) follows from the graded-symmetry of thesupertrace. It implies for the structure constants: f BC D κ AD = f ABD κ DC (A.8)In agreement with our conventions we define: f ABC = f ABD κ DC (A.9)and so on. The structure constants are graded-antisymmetric in the 1-2 and 2-3 indices : f ABC = − ( − ) AB f BAC ; f ABC = − ( − ) BC f ACB (A.10)Under the exchange of the first and third indices, we have: f ABC = − ( − ) A + B + C f CBA (A.11)Under cyclic permutation of their indices, the structure constant also satisfy: f ABC = ( − ) A f BCA (A.12)
Tensor product.
The tensor product t RA ⊗ t R ′ B of two generators taken in different repre-sentation R and R ′ is graded: t RA ⊗ t R ′ B = ( − ) AB (1 R ⊗ t R ′ B )( t RA ⊗ R ′ ) (A.13)In order to lighten the expressions in the bulk of the paper we often get rid of the ⊗ symbol: t RA ⊗ t R ′ B ≡ t RA t R ′ B = ( − ) AB t R ′ B t RA (A.14) B A new look at the gauge covariant current algebra
B.1 Derivation of the current-current OPEs
In this appendix we give a new derivation of the tree-level gauge-covariant current algebra forthe pure spinor string on
AdS × S . The method we use is inspired by the analysis of [58] forthe current algebra in sigma-models on supergroups. The different steps are the following.We make a natural ansatz for the current-current OPEs. Then we demand that this ansatzis compatible with reparametrization invariance of the path integral, and the with Maurer-Cartan equation. This typically gives more constraints than the number of free coefficientsin the ansatz. Finally we solve these constraints to get the current algebra.This method can be generalized to compute the quantum corrections to the current al-gebra. This computation can be efficiently organized recursively [58]. Here we will onlycompute the tree-level coefficients since this is sufficient for the purpose of this article. With different conventions (SE-NW), it would be the 1-2 and 1-3 indices.
25e choose the ansatz (2.12) for the current algebra, that we reproduce here for clarity: K A m m ( z ) K B n n ( w ) = R − C mn κ B n A m ( z − w ) + R − X p C pmn f C p B n A m K C p p z − w + R − X p C ¯ pmn f C p B n A m ¯ K C p p ¯ z − ¯ w ( z − w ) + ...K A m m ( z ) ¯ K B n n ( w ) = R − C m ¯ n κ B n A m πδ (2) ( z − w ) + R − X p C pm ¯ n f C p B n A m K C p p ¯ z − ¯ w + R − X p C ¯ pm ¯ n f C p B n A m ¯ K C p p z − w + ... ¯ K A m m ( z ) ¯ K B n n ( w ) = R − C ¯ m ¯ n κ B n A m (¯ z − ¯ w ) + R − X p C p ¯ m ¯ n f C p B n A m K C p p z − w (¯ z − ¯ w ) + R − X p C ¯ p ¯ m ¯ n f C p B n A m ¯ K C p p ¯ z − ¯ w + ... (B.1)This ansatz is based on dimensional analysis and symmetry. We only wrote down the second-and first-order poles, but there is an infinite series of less and less singular terms that comewith operators of (classical) dimension greater or equal to two. Notice also that this ansatz issuitable for the tree-level current algebra, but it should be slightly modified if one is to takeinto account quantum corrections [27]. In the following we will compute the coefficients C ’s.Many of these coefficient vanish trivially because the current algebra has to be compatiblewith the Z grading. Parity also induces some redundancy in the remaining coefficients.These two symmetries leave 57 independent coefficients that we need to compute. Equations of motion and path-integral reparametrization invariance.
In this sub-section we demand that the current algebra (B.1) is compatible with the reparametrizationinvariance of the path integral. In particular this guarantees that the current algebra iscompatible with the equations of motion.Let us consider the action for the pure spinor string in
AdS × S (2.5). We consider asmall variation of the group element g parametrized by a element of the Lie superalgebra X : δg = gX (B.2)The variation of the currents is given by: δJ = ∂X + [ J, X ] δN = 0 δ ¯ J = ¯ ∂X + [ ¯ J , X ] δ ˆ N = 0 (B.3)We can decompose the infinitesimal shift X on the Z subspaces of the Lie superalgebra as X = X + X + X + X . Since X generates gauge transformations that leave the actioninvariant, we set X = 0. We obtain the variation of the J i ’s: δJ = [ J , X ] + [ J , X ] + [ J , X ] δJ = ∂X + [ J , X ] + [ J , X ] + [ J , X ] δJ = ∂X + [ J , X ] + [ J , X ] + [ J , X ]26 J = ∂X + [ J , X ] + [ J , X ] + [ J , X ] (B.4)and similarly for the ¯ J i ’s. We deduce the variation of the action under the infinitesimal shiftof the group element (B.2): δS = R π ST r Z d z (cid:26) X (cid:18) −
32 ¯ ∇ J − ∇ ¯ J −
12 [ J , ¯ J ] −
12 [ J , ¯ J ] + 2[ N, ¯ J ] − J , ˆ N ] (cid:19) + X (cid:16) − ¯ ∇ J − ∇ ¯ J − [ J , ¯ J ] + [ J , ¯ J ] + 2[ N, ¯ J ] − J , ˆ N ] (cid:17) + X (cid:18) −
12 ¯ ∇ J − ∇ ¯ J + 12 [ J , ¯ J ] + 12 [ J , ¯ J ] + 2[ N, ¯ J ] − J , ˆ N ] (cid:19)(cid:27) (B.5)Now we consider the following quantity: h J ( z ) i = Z D Φ J ( z ) e − S (B.6)where D Φ is the path integral measure over the fields. The previous one-point function,whatever its value is, is invariant under the reparametrization of the path integral (B.2). Wefurther assume that the path-integral measure is also invariant under (B.2). Let us mentionat that point that we are simply following the method that would provide a path integralderivation of the Ward identity for a global symmetry, if (B.2) were indeed a global symmetry.We obtain: h δJ ( z ) − J ( z ) δS i = 0 (B.7)It is convenient to rewrite the variation of the current (B.4) as an integral over the worldsheet: δJ ( z ) = Z d w (cid:0) X ( w ) δ ′ ( z − w ) + ([ J ( w ) , X ( w )] + [ J ( w ) , X ( w )] + [ J ( w ) , X ( w )]) δ ( z − w ) (cid:1) (B.8)Projecting equation (B.7) on the Z subspaces, we obtain three operator identities: J A ( z ) (cid:18) −
32 ¯ ∇ J B ( w ) − ∇ ¯ J B ( w ) − f C D B : J C ¯ J D : ( w ) − f C D B : J C ¯ J D : ( w )+2 f C D B : N C ¯ J D : ( w ) − f C D B : J C ˆ N D : ( w ) (cid:17) = 4 πR − κ B A ∂ z δ ( z − w ) J A ( z ) (cid:16) − ¯ ∇ J B ( w ) − ∇ ¯ J B ( w ) − f C D B : J C ¯ J D : ( w ) + f C D B : J C ¯ J D : ( w )+2 f C D B : N C ¯ J D : ( w ) − f C D B : J C ˆ N D : ( w ) (cid:17) = 4 πR − f C B A J C ( w ) δ ( z − w ) J A ( z ) (cid:18) −
12 ¯ ∇ J B ( w ) − ∇ ¯ J B ( w ) + 12 f C D B : J C ¯ J D : ( w ) + 12 f C D B : J C ¯ J D : ( w )+2 f C D B : N C ¯ J D : ( w ) − f C D B : J C ˆ N D : ( w ) (cid:17) = 4 πR − f C B A J C ( w ) δ ( z − w ) (B.9)27here the colons stand for normal ordering. Next we plug the ansatz (B.1) into these equa-tions. More precisely, we use the ansatz (B.1) to perform the OPEs on the left-hand side ofthe identities (B.9). Since we are working at first-order in R − , the OPEs involving compositeoperators are easily dealt with: a single OPE has to be taken with one or the other of thecomponents of the composite operator. We use the equalities: ∂ ¯ w z − w = − πδ ( z − w ) = ∂ w z − ¯ w ; ∂ ¯ w z − w ) = 2 πδ ′ ( z − w ) = ∂ w z − ¯ w ) (B.10)We are left with some identities between operators multiplied by functions that are singularwhen z − w →
0. We demand that the operator identities (B.9) do hold for the singular termsof order two and three: all the terms multiplying either a derivative of a delta function, adelta function, or a second-order pole shall cancel against each other . We obtain a set oflinear equations that the free coefficients in the ansatz (B.1) have to satisfy:2 = − C + 12 C C ¯ g − C ¯ g − C C g + 12 C g − C C ¯013 − C ¯01¯3 + 32 C C + 12 C + 12 C C ¯312 − C ¯31¯2 + C C + C − C C ¯211 − C ¯21¯1 + 12 C C + 32 C − C (B.11)We can play the same game replacing in equation (B.7) J by another current. For eachcurrent we obtain a new set of equations. To get more constraints for the OPEs involvingthe ghosts, we can also vary the ghosts variable instead of (B.2). In total we get 39 linearequations that the coefficients C ’s in the ansatz (B.1) have to satisfy. The Maurer-Cartan equation.
We can further constraint the coefficients in the ansatz(B.1) by demanding compatibility with the Maurer-Cartan equation (2.9). Projecting thisequation according to the Z grading we obtain: ∂ ¯ J − ¯ ∂J + [ J , ¯ J ] + [ J , ¯ J ] + [ J , ¯ J ] + [ J , ¯ J ] = 0 ∂ ¯ J − ¯ ∂J + [ J , ¯ J ] + [ J , ¯ J ] + [ J , ¯ J ] + [ J , ¯ J ] = 0 ∂ ¯ J − ¯ ∂J + [ J , ¯ J ] + [ J , ¯ J ] + [ J , ¯ J ] + [ J , ¯ J ] = 0 ∂ ¯ J − ¯ ∂J + [ J , ¯ J ] + [ J , ¯ J ] + [ J , ¯ J ] + [ J , ¯ J ] = 0 (B.12)The strategy is to demand that the Maurer-Cartan equation does hold as an operator identity.More precisely, we demand that the OPE between a current and the left-hand side of (2.9)does vanish. Let us consider one example for illustrative purposes: we take the OPE betweenthe current J and the left-hand side of the first line in equation (B.12):0 = J A ( z ) (cid:16) ∂ ¯ J B ( w ) − ¯ ∂J B ( w ) + f C D B : J C ¯ J D : ( w ) + f C D B : J C ¯ J D : ( w )+ f C D B : J C ¯ J D : ( w ) + f C D B : J C ¯ J D : ( w ) (cid:17) (B.13)As previously we plug the ansatz (B.1) in the previous equation, and demand that the singularterms of order three and two do vanish. We obtain the following equations:0 = C ¯110 + C ¯11¯0 + C C − C − C (B.14) Demanding that the singular terms of order one or less do also vanish is not consistent with the ansatz(B.1), since the subleading terms in the current algebra that we did not write in (B.1) would contribute [58]. C ’s in the ansatz (B.1)have to satisfy.Let us make a side remark here. There is no doubt that the Maurer-Cartan identitydoes hold at tree level. However it may get quantum corrections. In order to generalize themethod described here to compute quantum corrections to the current algebra, one needsto assume that the Maurer-Cartan identity holds in the quantum theory as well. This maybe interpreted as postulating quantum integrability of the model. This provides an efficientway to use quantum integrability of the model to compute the quantum current algebra. Weleave it for future work. The coefficients of the current algebra.
Using the Maurer-Cartan identity and reparametriza-tion invariance of the path integral, we find in total 82 equations that constrain the 57 in-dependent coefficients of the current algebra (B.1). This system of equation can be easilydecomposed in subsystems of eight equations or less. It is remarkable that there exists asolution to this set of equations. The non-zero coefficients are given in section 2.2.There is however one exception for the OPEs between two of the currents J and ¯ J . Inthat case the equations we obtain only provide the constraints (2.17). Associativity.
The current algebra has to be associative. Associativity of the currentalgebra can be tested in the following way. Let us consider a 3-points function, for instance: h J ( x ) J ( y ) J ( z ) i (B.15)It can be computed by taking first the OPE between J ( x ) and J ( y ), and then take the OPEbetween the resulting current and J ( z ). But one can also start by taking the OPE between J ( y ) and J ( z ), and then take the OPE of the result with J ( x ). The two methods lead tothe same result if the coefficients of the current algebra satisfy: C C = C C (B.16)We can play the same game with any three-points functions. We find a large set of constraintsthat are all satisfied by the current algebra obtained previously. B.2 The ( r, s ) system. In this section we give some details on the derivations of equations (2.20), (2.21) and (2.22).We want to compute the commutator of two equal-time connections A R ( y ; σ ) and A R ′ ( y ′ ; σ ′ )evaluated for different values of the spectral parameter y and y ′ and taken in possibly differentrepresentation R and R ′ . We can deduce this commutator from the current algebra. We definethe commutator of equal-time operators as:[ A ( σ ) , B (0)] = lim ǫ → + ( A ( σ + iǫ ) B (0) − B ( iǫ ) A ( σ )) (B.17)From this definition we extract an operative dictionary between OPEs and commutators. Letus consider for instance the following OPE: A ( z ) B (0) = Cz + D ¯ z + Eδ (2) ( z ) + F (0) z + G (0)¯ zz + H (0)¯ z + I (0) z ¯ z + ... (B.18)29e deduce the commutator:12 πi [ A ( σ ) , B (0)] = Cδ ′ ( σ ) − Dδ ′ ( σ ) − F (0) δ ( σ ) − G (0) δ ( σ ) + H (0) δ ( σ ) + I (0) δ ( σ ) (B.19)This dictionary shows that the first-order computation of fusion presented in section 3.5 isequivalent to the computation of the Poisson bracket of line operators in the Hamiltonianformalism. Notice that the OPE (B.18) generically contains additional sub-leading singular-ities, for instance ¯ zz , or even logarithms. They do not contribute to the commutator [19].Generically, the OPE contains more information than the commutator.In order to simplify the following expressions, we write the (spacelike component of the)flat connection A ( y ) defined in (2.18) as: A ( y ) = X m F m ( y ) K m + ¯ F m ( y ) ¯ K m (B.20)Using the previous dictionary, we obtain for the commutator of two connections:[ A R ( y ; σ ) , A R ′ ( y ′ ; σ ′ )] =2 πiR − ∂ σ δ ( σ − σ ′ ) X m,n κ B n A m t RA m t R ′ B n ( F m ( y ) F n ( y ′ ) C mn − ¯ F m ( y ) ¯ F n ( y ′ ) C ¯ m ¯ n )+ 2 πiR − δ ( σ − σ ′ ) X m,n,p f C p B n A m t RA m t R ′ B n K C p p ( − F m ( y ) F n ( y ′ ) C pmn + F m ( y ) ¯ F n ( y ′ ) C pm ¯ n + ¯ F m ( y ) F n ( y ′ ) C p ¯ mn + ¯ F m ( y ) ¯ F n ( y ′ ) C p ¯ m ¯ n )+ 2 πiR − δ ( σ − σ ′ ) X m,n,p f C p B n A m t RA m t R ′ B n ¯ K C p p ( − F m ( y ) F n ( y ′ ) C ¯ pmn − F m ( y ) ¯ F n ( y ′ ) C ¯ pm ¯ n − ¯ F m ( y ) F n ( y ′ ) C ¯ p ¯ mn + ¯ F m ( y ) ¯ F n ( y ′ ) C ¯ p ¯ m ¯ n ) (B.21)We wish to write this commutator as a ( r, s ) system (2.20). From the terms coming with aderivative of the delta function in the commutator (B.21), we can read directly the s -matrix.We obtain: s = πiR − X m,n κ B n A m t RA m t R ′ B n (cid:0) F m ( y ) F n ( y ′ ) C mn − ¯ F m ( y ) ¯ F n ( y ′ ) C ¯ m ¯ n (cid:1) (B.22)Plugging in the value of the coefficients, we obtain (2.22). To obtain the r matrix, we haveto compare the terms coming with a delta function in (B.21) and (2.20). This leads to thefollowing equations for the components of the r and s matrices: ∀ m, n, p : F p ( y ) r − n,n − F p ( y ′ ) r m, − n = − F p ( y ) s − n,n − F p ( y ′ ) s m, − n − F m ( y ) F n ( y ′ ) C pmn + 2 F m ( y ) ¯ F n ( y ′ ) C pm ¯ n + 2 ¯ F m ( y ) F n ( y ′ ) C p ¯ mn + 2 ¯ F m ( y ) ¯ F n ( y ′ ) C p ¯ m ¯ n ¯ F p ( y ) r − n,n − ¯ F p ( y ′ ) r m, − n = − ¯ F p ( y ) s − n,n − ¯ F p ( y ′ ) s m, − n − F m ( y ) F n ( y ′ ) C ¯ pmn − F m ( y ) ¯ F n ( y ′ ) C ¯ pm ¯ n − F m ( y ) F n ( y ′ ) C ¯ p ¯ mn + 2 ¯ F m ( y ) ¯ F n ( y ′ ) C ¯ p ¯ m ¯ n (B.23)In the previous equations, when the indices m, n take the value 0 or g , one should understand“ r , ”and “ r − g,g ” as being r , , etc. Remarkably, this largely over-constrained system is solved bythe r and s matrices (2.21) and (2.22). 30 omparison with previous analyses. The current-current OPEs were previously dis-cussed in the literature. In [24] the OPEs for the currents of non-zero grade were computedusing the background field methods. Some of the OPEs involving the grade zero currentswere further given in [25]. The results we obtained here agree with these papers.In [26] the current algebra was also computed using Feynman diagram technology. TheOPEs do match the ones we derived here except for those involving the currents J , ¯ J . Thisis not surprising given the gauge choice that was explicitly made for the coset element in [26].A consequence of this discrepancy is that the commutator of equal-time connections can notbe written as a ( r, s ) system with the OPEs of [26]. However one should keep in mind thatit is only an issue of gauge fixing. Indeed in [31] the OPEs of [26] were used to compute thefusion of line operators at first order. Then the r and s matrices were deduced by comparisonwith the expectations from the Hamiltonian formalism. These matrices agree with the onesthat we derived in this paper.In [32] the hamiltonian formalism was used to compute the commutator of equal timeconnections. A careful treatment of the constraints was performed. It was argued that in theHamiltonian formalism, the flat connection (2.18) realizes a ( r, s ) system up to constraintsgenerating gauge transformations. It was shown that one should add to the flat connection aterm proportional to the constraints so that the commutator of connections take exactly theform of a ( r, s ) system. The resulting ( r, s ) system is slightly different than the one used hereand in [26]. The flat connection obtained in [32], including the additional term proportionalto the constraints, was derived from first principles in [33] in the Green-Schwarz formalism.It is remarkable that the analysis of [33] leads to the pure spinor-like flat connections of [30](without the pure spinor ghosts contribution) and not to the Bena-Polchinski-Roiban flatconnections [29]. This provides some evidence for the equivalence of the pure spinor andGreen-Schwarz formulations of string theory on AdS × S . In [34] it was shown that the( r, s ) system of [32] has a nice algebraic interpretation.For the purposes of this paper it is important that the r matrix found in [32] is identical tothe one we worked with in the limit where the difference of spectral parameter is small (2.23).This guarantees that the results derived in the present paper would also hold if one were towork with the ( r, s ) system of [32]. In order to reproduce the ( r, s ) system found in [32] usingOPEs technology, the first step would be to gauge-fix the H gauge symmetry via a BRSTprocedure. Then one should generalize the analysis of [30] by including in the flat connectionsadditional terms written in terms of the ghosts resulting from the H gauge-fixing. Thesenew connections should realize the ( r, s ) system of [32] . C Divergences in line operators
In this appendix we give some details about the computations of the first-order divergencesin line operators. As explained in section 2, we use a principal-value regularization scheme.A first-order pole is regularized as:1 σ − σ ′ → P.V. σ − σ ′ = 12 (cid:18) σ + iǫ − σ ′ + 1 σ − iǫ − σ ′ (cid:19) = σ − σ ′ ( σ − σ ′ ) + ǫ (C.1) The author would like to thank B. Vicedo for illuminating discussions on this point. ① the first-order poles in the OPE between two connections are considered. In ② and ③ thesecond-order poles are considered.and a second-order pole is regularized as:1( σ − σ ′ ) → P.V. σ − σ ′ ) = 12 (cid:18) σ + iǫ − σ ′ ) + 1( σ − iǫ − σ ′ ) (cid:19) = ( σ − σ ′ ) − ǫ (( σ − σ ′ ) + ǫ ) (C.2) C.1 Divergences in transition matrices
There are three sources of divergences in the transition matrices. They are depicted infigure 4. The first divergences come from the first order poles in the OPE of two neighboringconnections, say A ( y ; σ ) and A ( y ; σ ) (case ① in figuredivergences). We evaluate the resultingcurrents at the point σ and perform the integration over σ . We obtain a logarithmicdivergences:( − log ǫ ) X m,n,p (cid:16) K C p p ( σ )( F m F n C pmn + ¯ F m F n C p ¯ mn + F m ¯ F n C pm ¯ n + ¯ F m ¯ F n C p ¯ m ¯ n )+ ¯ K C p p ( σ )( F m F n C ¯ pmn + ¯ F m F n C ¯ p ¯ mn + F m ¯ F n C ¯ pm ¯ n + ¯ F m ¯ F n C ¯ p ¯ m ¯ n ) (cid:17) f C p B n A m t A m t B n (C.3)where the functions F ’s were defined in (B.20).The second type of divergences come from the second-order poles in the OPE betweentwo neighboring connections (case ② in figuredivergences). After performing the integrationover the positions of the two connections, we obtain a logarithmic divergence:log ǫ X m,n ( F m F n C mn + ¯ F m ¯ F n C ¯ m ¯ n ) κ B n A m t A m t B n (C.4)Notice that there is no linear divergences. This is a pleasant feature of the regularizationscheme that we are using. Eventually the third type of divergences come from the second-order poles in the OPE between two connections that are separated by a third one sitting inbetween (case ③ in figuredivergences). Let us denote this third connection by A ( y ; σ ). Afterperforming the integrations, we obtain another logarithmic divergence:( − log ǫ ) X m,n ( F m F n C mn + ¯ F m ¯ F n C ¯ m ¯ n ) (cid:18) (cid:8) κ B n A m t A m t B n , A ( y ; σ ) (cid:9) − X p,q (cid:0) f C p D q A m t A m t D q + f C p B n D q t D q t B n (cid:1) (cid:16) F p K C p p ( σ ) + ¯ F p ¯ K C p p ( σ ) (cid:17)! (C.5)32here is some freedom in how we write the last expression since we can commute the gen-erators in different ways. We choose a writing that is symmetric with respect to the centralconnection A ( y ; σ ).Starting from a transition matrix, we compute all the different OPEs of the types describedpreviously that lead to divergences. Next we sum all these terms. Most of the terms of thetype (C.4) cancel against the first terms in (C.5). We obtain: ∞ X M =0 ( − ) M log ǫ X m,n ( F m F n C mn + ¯ F m ¯ F n C ¯ m ¯ n ) ( κ B n A m t A m t B n , (cid:22)Z ba A (cid:25) M ) + ( − ) M +1 log ǫ M − X i =0 Z ba dσ (cid:22)Z bσ A (cid:25) i X m,n,p f C p B n A m t A m t B n × ( K C p p ( σ )( F m F n C pmn + ¯ F m F n C p ¯ mn + F m ¯ F n C pm ¯ n + ¯ F m ¯ F n C p ¯ m ¯ n + 12 F p X q ( F m F q C mq + ¯ F m ¯ F q C ¯ m ¯ q + F n F q C nq + ¯ F n ¯ F q C ¯ n ¯ q ))+ ¯ K C p p ( σ )( F m F n C ¯ pmn + ¯ F m F n C ¯ p ¯ mn + F m ¯ F n C ¯ pm ¯ n + ¯ F m ¯ F n C ¯ p ¯ m ¯ n (C.6)+ 12 ¯ F p X q ( F m F q C mq + ¯ F m ¯ F q C ¯ m ¯ q + F n F q C nq + ¯ F n ¯ F q C ¯ n ¯ q ))) (cid:22)Z σa A (cid:25) M − i − ! Consequences of the vanishing of the dual Coxeter number.
Here we derive someidentities that are useful to show the vanishing of some divergences in the line operators.These identities were first derived in [26]. The vanishing of the dual Coxeter number can bewritten as: f C BA [ t A , t B } = 0 (C.7)The super-Jacobi identity together with the fact that κ A B { t A , t B } = 0 implies: κ B A [ t A , [ t B , t C }} = κ A B [ t B , [ t A , t C }} (C.8)The identities (C.7) and (C.8) further imply: f C D B [ t B , t D } = f C D A [ t A , t D } = 0 (C.9) f C D B [ t B , t D } = f C D A [ t A , t D } = 0 (C.10) f C D A [ t A , t D } = f C D A [ t A , t D } = − f C D B [ t B , t D } (C.11) Cancellation of divergences.
Let us come back to the expression (C.6). We will nowargue that the second piece of (C.6) vanishes, as first shown in [26]. Using the identities(C.9) and (C.10), we observe that the terms proportional to J , ¯ J , J and ¯ J vanish straightaway. Then using the identities (C.11) together with the actual value of the coefficients ofthe current algebra, it is straightforward to check that the terms proportional to J and ¯ J also drop out.The vanishing of the terms proportional to J , ¯ J , N and ˆ N depends on the value ofthe simple poles in the OPEs J .J , J . ¯ J and ¯ J . ¯ J . The method explained in appendix33 to compute the current algebra does not fix completely these OPEs, but only gives theconstraint (2.17). The identity (C.7) combined with the value of the other coefficients of thecurrent algebra implies the vanishing of all terms proportional to J , ¯ J , N and ˆ N providedwe have: C = C = − C = 0 ; − C ¯000 = C ¯00¯0 = C ¯0¯0¯0 = 0 C g = C g = − C g ¯0¯0 = 2 ; − C ¯ g = C ¯ g = C ¯ g ¯0¯0 = 2 (C.12)Demanding consistency with the analysis of [26] implies the previous equations. We deducethat only the first term in (C.6) survives. We can write it as:( − ) M +1 log ǫ y + y − ( κ B A t A t B + κ B A t A t B + κ B A t A t B , (cid:22)Z ba A (cid:25) M ) (C.13)So the first-order divergences in the transition matrix can be rewritten as: − log ǫ y + y − n κ B A t A t B + κ B A t A t B + κ B A t A t B , T b,a ( x ) o (C.14) C.2 Divergences in monodromy and transfer matrices
In loop operators we have additional divergences coming from the collisions between twoconnections sitting on either side of the starting point of the integration contour. Only thesecond-order pole in such a collision lead to a divergence. These divergences read: ∞ X M =0 ( − ) M ( − log ǫ ) Z π>σ >...>σ M > dσ ...dσ M A B (1) ( σ ) ...A B ( M ) ( σ M ) × X m,n ( F m F n C mn + ¯ F m ¯ F n C ¯ m ¯ n ) κ C n D m t D m t B (1) ...t B ( M ) t C n M Y i =1 ( − ) C n B ( i ) (C.15)So the first-order divergences in the monodromy matrix add up to:log ǫ y + y − ∞ X M =0 ( − ) M Z π>σ >...>σ M > dσ ...dσ M A B (1) ( σ ) ...A B ( M ) ( σ M ) × ( − ( κ C D t D t C + κ C D t D t C + κ C D t D t C ) t B (1) ...t B ( M ) − t B (1) ...t B ( M ) ( κ C D t D t C + κ C D t D t C + κ C D t D t C )+ 2 κ C D t D t B (1) ...t B ( M ) t C M Y i =1 ( − ) D B ( i ) + 2 κ C D t D t B (1) ...t B ( M ) t C M Y i =1 ( − ) D B ( i ) + 2 κ C D t D t B (1) ...t B ( M ) t C M Y i =1 ( − ) D B ( i ) ) (C.16)Taking the supertrace, we see that the transfer matrix is free of divergences at first order.34 Fusion at second order: computations
In this appendix we give some details concerning the computation of the fusion of line opera-tors at second order. In particular we describe the computation that leads to (3.7) and (3.8).These terms are produced by triple collisions of connections. A triple collision means thatwe take one OPE between two connections, and then take the OPE of the resulting currentswith a third connection.
Treatment of the OPEs.
As explain in section 3.1, one needs to disentangle two contribu-tions from the OPEs. On one hand there is the contribution that gives a quantum correctionassociated with fusion. On the other hand there is the contribution that is interpreted as aregularized OPE in the double line operator resulting from the process of fusion. In order toisolate the interesting part associated with fusion, we subtract the principal value from thesingularities. We obtain : 1( σ ± iǫ − σ ′ ) − P.V. σ − σ ′ ) = ± iπδ ′ ǫ ( σ − σ ′ )1 σ ± iǫ − σ ′ − P.V. σ − σ ′ = ∓ iπδ ǫ ( σ − σ ′ ) σ ∓ iǫ − σ ′ ( σ ± iǫ − σ ′ ) − P.V. σ − σ ′ = ∓ iπδ ǫ ( σ − σ ′ ) (D.1) Computation of the individual terms.
Let us now face the computation of the individ-ual quantum corrections that add up to (3.7) and (3.8). In the first step of the computationof a triple collision we perform an OPE between two connections sitting on different con-tours. We obtain intermediate currents that we evaluate on one of the two contours . Thecontour on which these intermediate currents are evaluated matters for the second step ofthe computation. The relevant OPE for the first step of the computation is thus:(1 − P.V. ) A R ( y ; σ + iǫ ) A R ′ ( y ′ ; σ ′ ) ⊃ πiR − δ ǫ ( σ − σ ′ ) X m,n =0 X p f C p b n A m t RA m t R ′ B n × ( D pmn K C p p ( σ + iǫ ) + D ′ pmn K C p p ( σ ′ ) + D ¯ pmn ¯ K C p p ( σ + iǫ ) + D ′ ¯ pmn ¯ K C p p ( σ ′ )) (D.2)where the index p can take the values { , , , , g } . In the following when the range of the sumfor some index is not specified, it is understood that the sum runs over the set { , , , , g } .The coefficients D ∗∗∗ depends on the precise location where the currents are evaluated in thesimple poles of the current algebra (2.12). From the coefficient given in section 2.2 we canonly deduce the sums D pmn + D ′ pmn and D ¯ pmn + D ′ ¯ pmn . It turns out that we don’t need moreinformation about the coefficients D ∗∗∗ for the purpose of this article (see equation (D.20)).Actually what we want to compute here is not exactly the contribution of the triplecollisions to the fusion of line operators. It is rather the part of this contribution that hasnot been taken into account by the first part of the computation described in section 3.3 and The regularized delta-function in the third line of (D.1) is not exactly the same one as in the first twolines. However to keep the formulas simple we will adopt the same notations for both regularizations of thedelta-function. We can also choose to evaluate these intermediary currents in between the two contours. This would notchange equations (D.23) and (D.24). K ′ s . In ① and ② the contribution from the first-order poles in the second OPE is singled out. In the four other cases the contribution fromthe second-order poles in the second OPE is considered.depicted in figure 2. Removing the piece of the triple collisions already taken into accountamounts to perform the following replacement in the first OPE (D.2): D pmn → ˜ D pmn = D pmn − F p ( y )( s − n,n + r − n,n ) D ¯ pmn → ˜ D ¯ pmn = D ¯ pmn −
12 ¯ F p ( y )( s − n,n + r − n,n ) D ′ pmn → ˜ D ′ pmn = D ′ pmn − F p ( y ′ )( s m, − m − r m, − m ) D ′ ¯ pmn → ˜ D ′ ¯ pmn = D ′ ¯ pmn −
12 ¯ F p ( y ′ )( s m, − m − r m, − m ) (D.3)Now we can compute the individual terms that add up to (3.7) and (3.8). These differentterms are schematically depicted in figure 5. We compute separately the contribution of thefirst- and second-order poles in the second OPE (obviously only the first order poles have tobe taken into account in the first OPE). Let us begin with the first-order poles in the second36PE. The triple collision involves three connections, two of which are integrated on thesame contour. There are two different cases. For a triple collision involving two neighboringconnections on the first contour (case ① in figure 5), we obtain:( iπR − ) δ ǫ ( σ − σ ′ ) δ ǫ ( σ − σ ′ ) X m,n =0 X p,q,r f C p B n A m f E r C p D q { t RD q , t RA m ] t R ′ B n × ( K E r r ( ˜ D ′ pmn F q C rpq − ˜ D ′ pmn ¯ F q C rp ¯ q − ˜ D ′ ¯ pmn F q C r ¯ pq − ˜ D ′ ¯ pmn ¯ F q C r ¯ p ¯ q )+ ¯ K E r r ( ˜ D ′ pmn F q C ¯ rpq + ˜ D ′ pmn ¯ F q C ¯ rp ¯ q + ˜ D ′ ¯ pmn F q C ¯ r ¯ pq − ˜ D ′ ¯ pmn ¯ F q C ¯ r ¯ p ¯ q )) (D.4)where we wrote F q as a shorthand for F q ( y ). Similarly we will write F ′ q for F q ( y ′ ). For atriple collision involving two neighboring connections on the second contour (case ② in figure5), we obtain:( iπR − ) δ ǫ ( σ ′ − σ ) δ ǫ ( σ ′ − σ ) X m,n =0 X p,q,r f C p B n A m f E r D q C p t RA m { t R ′ B n , t R ′ D q ] × ( K E r r ( − ˜ D pmn F ′ q C rpq + ˜ D pmn ¯ F ′ q C rp ¯ q + ˜ D ¯ pmn F ′ q C r ¯ pq + ˜ D ¯ pmn ¯ F ′ q C r ¯ p ¯ q )+ ¯ K E r r ( − ˜ D pmn F ′ q C ¯ rpq − ˜ D pmn ¯ F ′ q C ¯ rp ¯ q − ˜ D ¯ pmn F ′ q C ¯ r ¯ pq + ˜ D ¯ pmn ¯ F ′ q C ¯ r ¯ p ¯ q )) (D.5)Next we consider the second-order poles in the second OPE. There are now four differentcases. For a triple collision involving two neighboring connections on the first contour (case ③ in figure 5), we obtain:( iπR − ) δ ′ ǫ ( σ − σ ′ ) δ ǫ ( σ − σ ′ ) X m,n =0 X p,q f B n A m D q t RD q t RA m t R ′ B n ( − ˜ D ′ pmn F q C pq + ˜ D ′ ¯ pmn ¯ F q C ¯ p ¯ q )+ ( iπR − ) δ ǫ ( σ − σ ′ ) δ ′ ǫ ( σ − σ ′ ) X m,n =0 X p,q f B n A m D q ( − ) A m D q t RA m t RD q t R ′ B n ( − ˜ D ′ pmn F q C pq + ˜ D ′ ¯ pmn ¯ F q C ¯ p ¯ q )(D.6)and for a triple collision involving two neighboring connections on the second contour (case ④ in figure 5), we obtain: − ( iπR − ) δ ǫ ( σ ′ − σ ) δ ′ ǫ ( σ ′ − σ ) X m,n =0 X p,q f D q B n A m t RA m t R ′ B n t R ′ D q ( − ˜ D pmn F ′ q C pq + ˜ D ¯ pmn ¯ F ′ q C ¯ p ¯ q ) − ( iπR − ) δ ′ ǫ ( σ ′ − σ ) δ ǫ ( σ ′ − σ ) X m,n =0 X p,q f D q B n A m t RA m ( − ) D q B n t R ′ D q t R ′ B n ( − ˜ D pmn F ′ q C pq + ˜ D ¯ pmn ¯ F ′ q C ¯ p ¯ q )(D.7)We also obtain a non-zero contribution if the two connections that are on the same contourare separated by a third one. When this happens on the first contour (case ⑤ in figure 5),37e obtain:( iπR − ) δ ′ ǫ ( σ − σ ′ ) δ ǫ ( σ − σ ′ ) 12 X m,n =0 X p,q { A R ( σ ) , f B n A m D q [ t RD q , t RA m } t R ′ B n }× ( − ˜ D ′ pmn F q C pq + ˜ D ′ ¯ pmn ¯ F q C ¯ p ¯ q )+( iπR − ) δ ′ ǫ ( σ − σ ′ ) δ ǫ ( σ − σ ′ ) 12 X m,n,q =0 X p,r,s f C p B n A m f E r C p D q { t RD q , t RA m ] t R ′ B n × ( ˜ D ′ smn F p C sp − ˜ D ′ ¯ smn ¯ F p C ¯ s ¯ p )( F r K E r r + ¯ F r ¯ K E r r )+( iπR − ) δ ′ ǫ ( σ − σ ′ ) δ ǫ ( σ − σ ′ ) 12 X n,p,q =0 X m,r,s f C p B n A m f E r C p D q { t RD q , t RA m ] t R ′ B n × ( ˜ D ′ spn F m C sm − ˜ D ′ ¯ spn ¯ F m C ¯ s ¯ m )( F r K E r r + ¯ F r ¯ K E r r ) (D.8)Finally for two connections separated by a third one in the second contour (case ⑥ in figure5), we obtain:( iπR − ) δ ′ ǫ ( σ ′ − σ ) δ ǫ ( σ ′ − σ ) 12 X m,n =0 X p,q { A R ′ ( σ ′ ) , f D q B n A m t RA m [ t R ′ B n , t R ′ D q }}× ( − ˜ D pmn F ′ q C pq + ˜ D ¯ pmn ¯ F ′ q C ¯ p ¯ q )+( iπR − ) δ ′ ǫ ( σ ′ − σ ) δ ǫ ( σ ′ − σ ) 12 X m,n,q =0 X p,r,s f C p B n A m f E r D q C p t RA m { t R ′ B n , t R ′ D q ] × ( ˜ D smn F ′ p C sp − ˜ D ¯ smn ¯ F ′ p C ¯ s ¯ p )( F ′ r K E r r + ¯ F ′ r ¯ K E r r )+( iπR − ) δ ′ ǫ ( σ ′ − σ ) δ ǫ ( σ ′ − σ ) 12 X n,p,q =0 X m,r,s f C p B n A m f E r D q C p t RA m { t R ′ B n , t R ′ D q ] × ( ˜ D spn F ′ m C sm − ˜ D ¯ spn ¯ F ′ m C ¯ s ¯ m )( F ′ r K E r r + ¯ F ′ r ¯ K E r r ) (D.9) Performing the integration.
Next we have to perform the integration over the free co-ordinates in the previous results. The integrals over the regularized delta functions provide awell-defined answer. This is an advantage of the OPE formalism with respect to the Hamil-tonian formalism. The integrals needed are given below. The results are given in the limit ǫ → Z b>σ >σ >a dσ dσ δ ǫ ( σ − σ ′ ) δ ǫ ( σ − σ ′ ) = 12 χ ( σ ′ ; a, b ) (D.10) Z b>σ >σ>σ >a dσ dσ Z dc dσ ′ δ ′ ǫ ( σ − σ ′ ) δ ǫ ( σ − σ ′ ) = − χ ( σ ; c, d ) (D.11) Z b>σ >σ >a dσ dσ Z dc dσ ′ δ ′ ǫ ( σ − σ ′ ) δ ǫ ( σ − σ ′ )= 12 χ ( b ; c, d ) − Z ba dσ Z dc dσ ′ δ ǫ ( σ − σ ′ ) (D.12)38t is sometimes convenient to write χ ( b ; c, d ) as ( χ ( b ; c, d ) + χ ( a ; c, d ) + χ ( c ; a, b ) − χ ( d ; a, b )).Notice that the integral over the squared regularized delta function is divergent in the limit ǫ → Z ba dσ Z dc dσ ′ δ ǫ ( σ − σ ′ ) = 12 πǫ | [ a, b ] ∩ [ c, d ] | (D.13)where we denoted by | [ a, b ] ∩ [ c, d ] | the length of the overlap of the intervals [ a, b ] and [ c, d ].Similarly we have: Z b>σ >σ >a dσ dσ Z dc dσ ′ δ ǫ ( σ − σ ′ ) δ ′ ǫ ( σ − σ ′ ) = − χ ( a ; c, d ) + Z ba dσ Z dc dσ ′ δ ǫ ( σ − σ ′ )(D.14)where we can also replace χ ( a ; c, d ) by ( χ ( b ; c, d ) + χ ( a ; c, d ) − χ ( c ; a, b ) + χ ( d ; a, b )). Summing the terms.
Finally we can sum the various contributions from triple collisions.The terms of the form (D.4) combined with the second and third terms of (D.8) lead to: ∞ X M =0 ( − ) M + M ′ +3 M X i =0 M ′ X i ′ =0 Z [ a,b ] ∩ [ c,d ] dσ (cid:22)Z bσ A (cid:25) i (cid:22)Z dσ A ′ (cid:25) i ′ × ( iπR − ) ˜ X m,n,p,q,r f C p B n A m f E r C p D q { t RD q , t RA m ] t R ′ B n × ( K E r r ( ˜ D ′ pmn F q C rpq − ˜ D ′ pmn ¯ F q C rp ¯ q − ˜ D ′ ¯ pmn F q C r ¯ pq − ˜ D ′ ¯ pmn ¯ F q C r ¯ p ¯ q + 12 F r X s ( ˜ D ′ smn F p C sp + ˜ D ′ spn F m C sm − ˜ D ′ ¯ smn ¯ F p C ¯ s ¯ p − ˜ D ′ ¯ spn ¯ F m C ¯ s ¯ m ))+ ¯ K E r r ( ˜ D ′ pmn F q C ¯ rpq + ˜ D ′ pmn ¯ F q C ¯ rp ¯ q + ˜ D ′ ¯ pmn F q C ¯ r ¯ pq − ˜ D ′ ¯ pmn ¯ F q C ¯ r ¯ p ¯ q + 12 ¯ F r X s ( ˜ D ′ smn F p C sp + ˜ D ′ spn F m C sm − ˜ D ′ ¯ smn ¯ F p C ¯ s ¯ p − ˜ D ′ ¯ spn ¯ F m C ¯ s ¯ m ))) × (cid:22)Z σa A (cid:25) M − i (cid:22)Z σc A ′ (cid:25) M ′ − i ′ (D.15)In order to shorten the previous expression we introduce the symbol ˜ P with the followingmeaning: for each term in the expression, the lower indices of a coefficient D have to besummed over the values { , , , } , while all other indices have to be summed over the values { , , , , g } . Similarly the terms of the form (D.5) combined with the second and third terms39f (D.9) lead to: ∞ X M =0 ( − ) M + M ′ +3 M X i =0 M ′ X i ′ =0 Z [ a,b ] ∩ [ c,d ] dσ (cid:22)Z bσ A (cid:25) i (cid:22)Z dσ A ′ (cid:25) i ′ × ( iπR − ) ˜ X m,n,p,q,r f C p B n A m f E r D q C p t RA m { t R ′ B n , t R ′ D q ] × ( K E r r ( − ˜ D pmn F ′ q C rpq + ˜ D pmn ¯ F ′ q C rp ¯ q + ˜ D ¯ pmn F ′ q C r ¯ pq + ˜ D ¯ pmn ¯ F ′ q C r ¯ p ¯ q − F ′ r X s ( ˜ D smn F ′ p C sp + ˜ D spn F ′ m C sm − ˜ D ¯ smn ¯ F ′ p C ¯ s ¯ p − ˜ D ¯ spn ¯ F ′ m C ¯ s ¯ m ))+ ¯ K E r r ( − ˜ D pmn F ′ q C ¯ rpq − ˜ D pmn ¯ F ′ q C ¯ rp ¯ q − ˜ D ¯ pmn F ′ q C ¯ r ¯ pq + ˜ D ¯ pmn ¯ F ′ q C ¯ r ¯ p ¯ q ) −
12 ¯ F ′ r X s ( ˜ D smn F ′ p C sp + ˜ D spn F ′ m C sm − ˜ D ¯ smn ¯ F ′ p C ¯ s ¯ p − ˜ D ¯ spn ¯ F ′ m C ¯ s ¯ m ))) × (cid:22)Z σa A (cid:25) M − i (cid:22)Z σc A ′ (cid:25) M ′ − i ′ (D.16)The sum of (D.15) and (D.16) leads to (3.7) where the operator ˜ K is given by:˜ K = π X m,n,p,q,r f C p B n A m f E r C p D q { t RD q , t RA m ] t R ′ B n × ( K E r r ( ˜ D ′ pmn F q C rpq − ˜ D ′ pmn ¯ F q C rp ¯ q − ˜ D ′ ¯ pmn F q C r ¯ pq − ˜ D ′ ¯ pmn ¯ F q C r ¯ p ¯ q + 12 F r X s ( ˜ D ′ smn F p C sp + ˜ D ′ spn F m C sm − ˜ D ′ ¯ smn ¯ F p C ¯ s ¯ p − ˜ D ′ ¯ spn ¯ F m C ¯ s ¯ m ))+ ¯ K E r r ( ˜ D ′ pmn F q C ¯ rpq + ˜ D ′ pmn ¯ F q C ¯ rp ¯ q + ˜ D ′ ¯ pmn F q C ¯ r ¯ pq − ˜ D ′ ¯ pmn ¯ F q C ¯ r ¯ p ¯ q + 12 ¯ F r X s ( ˜ D ′ smn F p C sp + ˜ D ′ spn F m C sm − ˜ D ′ ¯ smn ¯ F p C ¯ s ¯ p − ˜ D ′ ¯ spn ¯ F m C ¯ s ¯ m )))+ f C p B n A m f E r D q C p t RA m { t R ′ B n , t R ′ D q ] × ( K E r r ( − ˜ D pmn F ′ q C rpq + ˜ D pmn ¯ F ′ q C rp ¯ q + ˜ D ¯ pmn F ′ q C r ¯ pq + ˜ D ¯ pmn ¯ F ′ q C r ¯ p ¯ q − F ′ r X s ( ˜ D smn F ′ p C sp + ˜ D spn F ′ m C sm − ˜ D ¯ smn ¯ F ′ p C ¯ s ¯ p − ˜ D ¯ spn ¯ F ′ m C ¯ s ¯ m ))+ ¯ K E r r ( − ˜ D pmn F ′ q C ¯ rpq − ˜ D pmn ¯ F ′ q C ¯ rp ¯ q − ˜ D ¯ pmn F ′ q C ¯ r ¯ pq + ˜ D ¯ pmn ¯ F ′ q C ¯ r ¯ p ¯ q ) −
12 ¯ F ′ r X s ( ˜ D smn F ′ p C sp + ˜ D spn F ′ m C sm − ˜ D ¯ smn ¯ F ′ p C ¯ s ¯ p − ˜ D ¯ spn ¯ F ′ m C ¯ s ¯ m ))) (D.17)40he combination of the terms (D.6) and (D.7) together with the first terms of (D.8) and(D.9) simplifies to: ∞ X M =0 ( − ) M + M ′ +3 M X i =0 M ′ X i ′ =0 Z ba dσ Z dc dσ ′ (cid:22)Z bσ A (cid:25) i (cid:22)Z dσ ′ A ′ (cid:25) i ′ × ( iπR − ) δ ǫ ( σ − σ ′ ) ˜ X m,n,p,q,r f B n A m D q f D q A m E r t RE r t R ′ B n ( − ˜ D ′ pmn F q C pq + ˜ D ′ ¯ pmn ¯ F q C ¯ p ¯ q − ˜ D prm F ′ q C pq + ˜ D ¯ prm ¯ F ′ q C ¯ p ¯ q ) × (cid:22)Z σa A (cid:25) M − i $Z σ ′ c A ′ ' M ′ − i ′ (D.18)This can be rewritten as (3.8) where the matrix ˜ tt is given by:˜ tt = π X m,n,p,q,r f B n A m D q f D q A m E r t RE r t R ′ B n × ( − ˜ D ′ pmn F q C pq + ˜ D ′ ¯ pmn ¯ F q C ¯ p ¯ q − ˜ D prm F ′ q C pq + ˜ D ¯ prm ¯ F ′ q C ¯ p ¯ q ) (D.19) Simplifications in the limit y − y ′ = O ( R − ) . For the purposes of this paper it isinteresting to take the limit where the difference of spectral parameters is small. Moreprecisely we assume that the difference y − y ′ is of order O ( R − ). In this limit one termdominates the previous result. This follows essentially from the observation that the r matrixsatisfies (2.23). In the limit y − y ′ = O ( R − ), the r matrix is no longer of order R − butrather of order R . Consequently the coefficients ˜ D ∗∗∗ introduced in (D.3) behave like:˜ D pmn = − F p ( y ) y ( y + y − ) y − y ′ + O ( R )˜ D ¯ pmn = −
14 ¯ F p ( y ) y ( y + y − ) y − y ′ + O ( R )˜ D ′ pmn = + 14 F p ( y ) y ( y + y − ) y − y ′ + O ( R )˜ D ¯ pmn = + 14 ¯ F p ( y ) y ( y + y − ) y − y ′ + O ( R ) (D.20)Using equation (B.23), we deduce that (D.15) simplifies to:( − ) M + M ′ +3 M X i =0 M ′ X i ′ =0 Z [ a,b ] ∩ [ c,d ] dσ (cid:22)Z bσ A (cid:25) i (cid:22)Z dσ A ′ (cid:25) i ′ × ( iπR − ) X m,n,p,q =0 X r f C p B n A m f E r C p D q { t RD q , t RA m ] t R ′ B n × (cid:18) − y ( y + y − ) y − y ′ (cid:19) (cid:0) ∂ y F r ( y ) K E r r + ∂ y ¯ F r ( y ) ¯ K E r r (cid:1) × (cid:22)Z σa A (cid:25) M − i (cid:22)Z σc A ′ (cid:25) M ′ − i ′ + O ( R − ) (D.21)41emarkably the combination of currents that factors out is the derivative of the components ofthe flat connection (B.20) with respect to the spectral parameter. Similarly (D.16) simplifiesto: ( − ) M + M ′ +3 M X i =0 M ′ X i ′ =0 Z [ a,b ] ∩ [ c,d ] dσ (cid:22)Z bσ A (cid:25) i (cid:22)Z dσ A ′ (cid:25) i ′ × ( iπR − ) X m,n,p,q =0 X r f C p B n A m f E r D q C p t RA m { t R ′ B n , t R ′ D q ] × (cid:18) − y ( y + y − ) y − y ′ (cid:19) (cid:0) ∂ y F r ( y ) K E r r + ∂ y ¯ F r ( y ) ¯ K E r r (cid:1) × (cid:22)Z σa A (cid:25) M − i (cid:22)Z σc A ′ (cid:25) M ′ − i ′ + O ( R − ) (D.22)So the operator ˜ K becomes:˜ K = − π y ( y + y − ) y − y ′ X m,n,p,q =0 X r ∂ y A E r ( y ) × (cid:16) f C p B n A m f E r C p D q { t RD q , t RA m ] t R ′ B n + f C p B n A m f E r D q C p t RA m { t R ′ B n , t R ′ D q ] (cid:17) + O ( R )(D.23)Eventually the term (D.18) remains of order R − , since: − ˜ D ′ pmn F q C pq + ˜ D ′ ¯ pmn ¯ F q C ¯ p ¯ q − ˜ D prm F ′ q C pq + ˜ D ¯ prm ¯ F ′ q C ¯ p ¯ q = 0 + O ( R − ) (D.24)Consequently the matrix ˜ tt defined in (D.19) remains of order one. 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