From Gaudin Integrable Models to d -dimensional Multipoint Conformal Blocks
Ilija Buric, Sylvain Lacroix, Jeremy A. Mann, Lorenzo Quintavalle, Volker Schomerus
FFrom Gaudin Integrable Models to d -dimensional Multipoint Conformal Blocks Ilija Buri´c a , Sylvain Lacroix b , Jeremy A. Mann a , Lorenzo Quintavalle a and Volker Schomerus a a DESY Theory Group, DESY Hamburg, Notkestrasse 85, D-22603 Hamburg, b II. Institut f¨ur Theoretische Physik, Universit¨at Hamburg, Luruper Chaussee 149, D-22761 HamburgZentrum f¨ur Mathematische Physik, Universit¨at Hamburg, Bundesstrasse 55, D-20146 Hamburg (Dated: September, 2020)In this work we initiate an integrability-based approach to multipoint conformal blocks for higherdimensional conformal field theories. Our main observation is that conformal blocks for N -pointfunctions may be considered as eigenfunctions of integrable Gaudin Hamiltonians. This provides uswith a complete set of differential equations that can be used to evaluate multipoint blocks. DESY 20-157ZMP-HH/20-25
1. INTRODUCTION
Conformal quantum field theories (CFTs) play an im-portant role for our understanding of phase transitions,quantum field theory and even the quantum physics ofgravity, through Maldacena’s celebrated holographic du-ality. Since they are often strongly coupled, however,they are very difficult to access with traditional pertur-bative methods. Polyakov’s famous conformal bootstrapprogram provides a powerful non-perturbative handlethat allows to calculate critical exponents and other dy-namical observables using only general features such as(conformal) symmetry, locality and unitarity/positivity[1]. The program has had an impressive early success in d = 2 dimensions [2] where it produced numerous exactsolutions. During the last decade, the bootstrap has seena remarkable revival in higher dimensional theories withnew numerical as well as analytical incarnations. Thishas produced many stunning new insights, see e.g. [3]for a recent review and references, including record pre-cision computations of critical exponents in the critical3D Ising model [4, 5]. Despite these advances, it is evi-dent that significant further developments are needed tomake these techniques more widely applicable, beyond afew very special theories.One promising avenue would be to study bootstrap con-sistency conditions for N -point correlators with N > d = 2 is ulti-mately based on the ability to analyze correlation func-tions with any number of stress tensor insertions. But theextension of the bootstrap constraints in d > d , though only after some significanteffort. Here we shall lay the foundations for a system-atic extension to multipoint blocks. Our approach ex- tends a remarkable observation in [19] about a relationbetween 4-point blocks and exactly solvable (integrable)Schroedinger problems.To understand the key challenge in developing a theoryof multipoint conformal blocks for d >
2, let us considera 5-point function of scalar fields, the very simplest caseof an N -point function with N > N = 5 points as longas d >
2. On the other hand, we can reduce the evalu-ation of a 5-point function to a 3-point function by per-forming two operator product expansions (OPEs). Thefields in the OPEs are characterized by a weight ∆ and aspin l . So, from the two expansions we obtain a total offour quantum numbers, two weights and two spins of thefields that propagate in the intermediate channels, seeFig 1. This is not sufficient to resolve the dependenceof the 5-point function on the five cross ratios. In fact,we are missing one additional quantum number. Let usstress that d = 3 is the smallest dimension for which thishappens. In d = 2 a 5-point function depends on fourcross ratios and this matches the quantum numbers ofintermediate fields. It is not difficult to see that the addi-tional quantum numbers that are needed to characterizemultipoint conformal blocks in d > a r X i v : . [ h e p - t h ] S e p
21 3 45∆ a , l a ∆ b , l b Figure 1:
OPE diagram for a 5-point function. Thecorresponding 5-point conformal block depends on fivequantum numbers which are measured by four Casimiroperators and one new vertex differential operator. then enabled them to harvest decisive new results on theconformal blocks [23, 24].Shadow integral representations for multipoint blocksare also known. In order to evaluate these, one may wantto follow very much the same strategy that was used for4-point functions. It is indeed relatively straightforwardto write down multipoint generalizations of the Casimiroperators found by Dolan and Osborn. In the case of5-point functions in d >
2. MULTIPOINT SHADOW INTEGRALS INTHE COMB CHANNEL
In order to state our results precisely, we shall brieflyreview some basics of the shadow integral formalism. Theshadow formalism turns the graphical representation of aconformal block, such as that of Fig. 1, into an integralformula. Just as in the case of Feynman integrals, the‘shadow integrand’ is built from relatively simple buildingblocks that are assigned to the links and 3-point verticesin the associated OPE diagram. For a scalar 5-pointfunction, the most complicated vertex contains one scalarleg and two that are carrying symmetric traceless tensor(STT) representations. In order to write this vertex, weshall employ polarization spinors z ∈ C d (see [28–31]) toconvert spinning operators in STT representations intoobjects of the form O ∆ ,l ( x ; z ) = O ν ...ν l ∆ ,l ( x ) z ν ...z ν l ≡ O ν ∆ ,l ( x ) z ν . (1)The usual contraction of the STTs can be re-expressedas an integral over C d as follows [32] O ν ( x ) O ν ( x ) = Z C d d d z δ ( z ) ρ (¯ z · z ) O ( x ; ¯ z ) O ( x ; z ) , (2) ρ ( t ) = (cid:18) π (cid:19) d − (16 t ) − d/ Γ( d/ − K ( d/ − (2 √ t ) , (3)where O and O are fields of equal spin and K is themodified Bessel function of the second kind. In build-ing shadow integrands, the function ρ plays a role anal-ogous to the propagator in Feynman integrals. Havingconverted field multiplets into functions, the 3-point ver-tex with one scalar leg in d = 3 takes the formΦ tacb ( x ; z ) = hO ∆ a ,l a ( x a ; z a ) O ∆ c ( x c ) O ∆ b ,l b ( x b ; z b ) i =( X bc ; a · z a ) l a ( X ca ; b · z b ) l b ( X ab ; c ) − ∆ c ( X ca ; b ) lb − ∆ b ( X bc ; a ) la − ∆ a t ( X ) , (4)if l a − l b ∈ Z and vanishes otherwise. Here we have usedthe following standard notations X µij ; k := x µik x ik − x µjk x jk = − X µji ; k , X ij ; k = x ij x ik x jk , (5)with x ij = x i − x j and X the unique independent cross-ratio that can be constructed from ( x a , x b , x c ; z a , z b ), X = 12 x ab z aµ (cid:0) x ab δ µν − x µab x νab (cid:1) z bν ( z a · X bc ; a )( z b · X ca ; b ) . (6)To a large extent, the function t ( X ) that appears in the3-point vertex is left undetermined by conformal symme-try. The only constraints come from the action of thestabilizer group SO ( d −
1) of three points in R d . To de-scribe these explicitly, let us restrict to d = 3 from nowon. In this last setting, we are required to distinguishtwo cases, depending on the behavior of t ( X ) under theaction of the parity operator in O (3): W + t : For parity-even functions, t ( X ) must be a polyno-mial of order at most min ( l a , l b ). W − t : For parity-odd functions, t ( X ) / p X (1 − X ) mustbe a polynomial of order at most min ( l a , l b ) − W ± t of functions t ( X ) form a vector space of dimension n ab = 2 min ( l a , l b ) + 1 = min (2 l a + 1 , l b + 1) . (7)The integer n ab counts the number of 3-point tensor struc-tures [31]. Note that n ab = 1 if either l a = 0 or l b = 0which means that t is a constant factor if there are twoor three scalar legs. We shall therefore simply drop thecorresponding vertex factors t when using formula (4) forvertices with two scalar legs.Having described the vertex, we can now write down(shadow) integrals for any desired N -point function inthe so-called comb channel, in which every OPE includesat least one of the external scalar fields. For N = 5external scalar fields of weight ∆ i , i = 1 , . . . , (∆ ,..., ∆ )(∆ a , ∆ b ; l a ,l b ; t ) ( x , ..., x ) = (8)= Y s = a,b Z R d d d x s Z C d d d z s δ ( z s ) ρ (¯ z s · z s )Φ a ( x , x , x a ; ¯ z a ) × Φ ta b ( x a , x , x b ; z a , z b )Φ ˜ b ( x b , x , x ; ¯ z b ) . Here the tilde on the indices of the first and third vertexmeans that we use eq. (4) for two scalar legs but with ∆ a and ∆ b replaced by d − ∆ a and d − ∆ b , respectively.After splitting off some factor Ω that accounts for thenontrivial covariance law of the scalar fields under con-formal transformations,Ψ (∆ i )(∆ a , ∆ b ; l a ,l b ; t ) ( x i ) = Ω (∆ i ) ( x i ) ψ (∆ , ∆ , ∆ )(∆ a , ∆ b ; l a ,l b ; t ) ( u , ..., u ) , Ω (∆ i ) ( x i ) := ( X ) ∆12 Y i =2 ( X i +1 ,i − i ) ∆ i ( X ) ∆52 , with ∆ ij = ∆ i − ∆ j as usual, the shadow integral (8)gives rise to a finite conformal integral that defines theconformal block ψ as a function of five conformally invari-ant cross ratios u i . These integrals depend on the choiceof (∆ a , l a ), (∆ b , l b ) and the function t ( X ) at the middlevertex. Our goal is to compute this uninviting lookingintegral.The strategy we have sketched in the introduction isto write down five differential equations for these blocks.Four of these are given by the eigenvalue equations forthe second and fourth order Casimir operators for theintermediate channels, D sp ψ (∆ , ∆ , ∆ )(∆ a , ∆ b ; l a ,l b ; t ) ( u ) = C sp ψ (∆ , ∆ , ∆ )(∆ a , ∆ b ; l a ,l b ; t ) , (9)where p = 2 , C sp denotes the eigenvalue of the p -thorder Casimir operator in the representation (∆ s , l s ) for s = a, b . The explicit form of the differential operators D sp can be worked out and the resulting expressions resemblethose found by Dolan and Osborn for N = 4.But we are missing one more differential equationwhich we shall construct in the next section. It will turnout that shadow integrals are eigenfunctions of a fifthdifferential operator provided we prepare a very specialbasis t n ( X ) , n = 1 , . . . , n ab , in the space of 3-point tensorstructures. We can characterize these functions t n ( X ) aseigenfunctions of a particular fourth order differential op-erator H (∆ i ,l i ) = h ( X ) + X q =1 h q ( X ) X q − (1 − X ) q − ∂ qX , (10)where h q = h (∆ i ,l i ) q are polynomials of order at mostthree, see Appendix A for concrete expressions. The op-erator H , which has several remarkable properties, ap-pears to be new. For our discussion it is most importantto note that H leaves the two subspaces W ± t invariantwhenever both l a and l b are integer. Consequently, itspecifies a special basis t n of functions t ( X ) in the spaceof tensor structures, H (∆ i ,l i ) t n ( X ) = τ n t n ( X ) , n = 0 , . . . , n ab . (11)Explicit formulas for the eigenvalues τ n and the eigen-functions t n ( X ) can be worked out, and it is this basis of3-point tensor structures that we will use to write downdifferential equations for the associated shadow integrals.
3. MULTIPOINT BLOCKS AND GAUDINHAMILTONIANS
Our goal now is to characterize the shadow integralsthrough a complete set of five differential equations.These will take the form of eigenvalue equations for a setof commuting Gaudin Hamiltonians. In order to stateprecise formulas we need a bit of background on Gaudinmodels [25, 26]. Let us begin with a central object, theso-called Lax matrix, L ( w ) = N X i =1 T ( i ) α T α w − w i = L α ( w ) T α . (12)Here w i are a set of complex numbers, T α denotes a ba-sis of generators of the conformal Lie algebra and T α its dual basis with respect to an invariant bilinear form.The object T ( i ) α is the standard first order differentialoperator that describes the behavior of a scalar primaryfield O ( x i ) of weight ∆ i under the infinitesimal conformaltransformation generated by T α .Given some conformally invariant symmetric tensor κ p of degree p one can construct a family H p ( w ) of commut-ing operators as [33–35] H p ( w ) = κ α ··· α p p L α ( w ) · · · L α p ( w ) + . . . , (13)where the dots represent correction terms expressible aslower degree combinations of the Lax matrix components L α ( w ) and their derivatives with respect to w . For p = 2such correction terms are absent. The correction termsare necessary to ensure that the families commute,[ H p ( w ) , H q ( w ) ] = 0 , (14)for all p, q and all w, w ∈ C . In the case where d = 3, theconformal algebra possesses two independent invarianttensors of second and fourth degree. Hence, we obtaintwo families of commuting differential operators that acton functions of the coordinates x i .It is a well-known fact that these families commutewith the diagonal action of the conformal algebra, i.e.[ T α , H p ( w ) ] = 0 where T α = N X i =1 T ( i ) α . (15)Hence the commuting families H p ( w ) of operators de-scend to differential operators on functions ψ ( u ) of theconformally invariant cross ratios u .The functions H p ( w ) provide several continuous fam-ilies of commuting operators. Only a finite set of theseoperators are independent. There are many ways of con-structing such sets of independent operators, e.g. by tak-ing residues of H p ( w ) at the singular points to give justone example. For the moment any such set still contains N parameters w i , i = 1 , . . . , N . Without loss of general-ity we can set three of these complex numbers to somespecific value, e.g. w = 0 , w N − = 1 , w N = ∞ so thatwe remain with N − l of fields that are ex-changed in intermediate channels, as do the multipointCasimir operators. So, in order for the Gaudin Hamilto-nians to be of any use to us, we must ensure that theyinclude all such Casimir operators. For this to be thecase, we are forced to make a very special choice of theremaining parameters w r and to consider specific limitsof these parameters (such limits have also been consid-ered in [36, 37] to study bending flow Hamiltonians andtheir generalisations [38–41]). Let us explain this herefor N = 5. Setting w = $ and w = $ we can define e H p ( w ) := lim $ → $ p H p ( $w ) , p = 2 , . (16)The new functions e H p take values in the space of p th order differential operators on cross ratios. They possesssingularities at three points only, namely at w = 0 , , ∞ .Let us note that taking the limit $ → w r we can now extract the multipoint Casimir operatorsrather easily. In fact, it is not difficult to check that D ap = lim w → w p e H p ( w ) , D bp = lim w →∞ w p e H p ( w ) (17) for p = 2 ,
4. Any additional independent operator wecan obtain from e H p ( w ) may be used to measure a fifthquantum number. One can show that the two secondorder Casimir operators D s , s = a, b exhaust all the in-dependent operators that can be obtained from e H ( w ).The family e H ( w ), on the other hand, indeed suppliesone independent operator in addition to the fourth orderCasimir operators D s , s = a, b . We propose to use theoperator V defined through e H (cid:18) w = 12 (cid:19) = 16 V + . . . , (18)where the dots represent quadratic terms coming fromthe corrections in eq. (13). In the particular limit $ → D s , s = a, b ,and can thus be discarded without spoiling commutativ-ity of V with the Casimirs. An explicit computationthen shows that V is expressed in terms of the confor-mal generators T ( i ) α as V = κ α ··· α S α · · · S α , S α = T (1) α + T (2) α − T (3) α . (19)The explicit form of V as a differential operator actingon functions ψ ( u ) of five cross ratios will be spelled outin our forthcoming publication [27]. Our central claim isthat the 5-point shadow integrals ψ we discussed in theprevious subsection are joint eigenfunctions of the fourCasimir operators, see eq. (9), and of the vertex operatorwe defined through eq. (18), V ψ (∆ , ∆ , ∆ )(∆ a , ∆ b ; l a ,l b ; t n ) ( u ) = τ n ψ (∆ , ∆ , ∆ )(∆ a , ∆ b ; l a ,l b ; t n ) ( u ) , (20)where the eigenvalues τ n coincide with those that ap-peared in eq. (11) when describing the particular choice ofa basis t n ( X ) of tensor structures. These five differentialequations characterize the shadow integral completely.
4. CONCLUSIONS AND OUTLOOK
In this work we initiated a systematic construction ofmultipoint conformal blocks in d ≥
3. Our advance re-lies on a characterization of multipoint conformal blocksas wave functions of Gaudin integrable models, whichextends a similar relation between 4-point blocks and in-tegrable Calogero-Sutherland models uncovered in [19].More specifically, we have explained that for a very spe-cial choice of tensor structures at the 3-vertices Φ in theshadow integrand of eq. (8), the corresponding shadow in-tegral becomes a joint eigenfunction of a complete set ofcommuting differential operators. The latter are Hamil-tonians of special limits of the Gaudin model.While we have explained the main ideas within the ex-ample of 5-point functions in d = 3, the strategy and inparticular the relation with Gaudin models is completelygeneral, i.e. it extends to N > d > N = 6, there exist topologically distinct channelsthat can include vertices in which all three legs carryspin, such as the so-called snowflake channel for N = 6[12]. Such vertices involve functions t of more than onevariable and hence the choice of basis in the space of ten-sor structures needs to be extended. Tensor structuresfor the generic vertex in d = 3, for example, are charac-terized as eigenfunctions of two commuting differentialoperators of fourth order in two variables X, Y ratherthan a single such operator acting on X . As we go tohigher dimension d , the vertices can become more andmore involved and links can carry more than just STTrepresentations. Treating such more general links onlyrequires us to consider higher order Casimir operators.Through the relation to Calogero-Sutherland models [19],their solution theory is well known, see e.g. [42]. In thissense, links do not pose a significant new complicationfor the construction of multipoint blocks even if d > d = 3. This can then serve as a startingpoint to evaluate 5-point blocks explicitly, e.g. throughseries expansions or Zamolodchikov-like recursion formu-las, similar to those used for 4-point blocks [24, 42–46].Obviously, it would be very interesting to extend theseconstructions of differential operators to 6-point blocks,to develop an evaluation theory and to initiate a multi-point bootstrap for d >
2. As we have argued in the in-troduction, taking bootstrap constraints from multipointcorrelation functions seems like a good strategy that iswell aligned with the success of the d = 2 bootstrap.Key examples for initial studies include the O ( n ) Wilson-Fisher fixed points with n = 2 , λ -pointin Helium or the ferromagnetic phase transition, respec-tively. The current state-of-the-art for n = 2 was setrecently in [47, 48], using 4-point mixed correlator andanalytic bootstrap, respectively. Since 6-point functionsof a single scalar field contain the same information as in-finitely many mixed 4-point functions, including fields ofarbitrarily high spin, the multipoint bootstrap for N = 6can be expected to provide significantly stronger bounds.While we were completing this letter Vieira et al. is-sued the paper [49] in which they initiate a multipointlight-cone bootstrap. With the techniques we proposehere, it should be possible to study light-cone blocksalong with systematic corrections in the vicinity of thestrict light-cone limit and for any desired channel. Wewill come back to these topics in future work. Acknowledgements:
We are grateful to Gleb Aru-tyunov, Mikhail Isachenkov, Madalena Lemos, PedroLiendo, Junchen Rong, Joerg Teschner and BenoˆıtVicedo for useful discussions. Part of this project re-ceived funding from the Deutsche Forschungsgemein-schaft (DFG, German Research Foundation) under Ger-many’s Excellence Strategy – EXC 2121 „Quantum Uni- verse” – 390833306 and from the European Union’s Hori-zon 2020 research and innovation programme under theMarie Sklodowska-Curie grant agreement No. 764850“SAGEX”.
Appendix A: The Vertex Operator H In this appendix we list all the coefficients h q ( X ) ofthe Hamiltonian (10) that is used to define our basis t n of 3-point tensor structures. Except for a constant termin h which depends a bit on the precise choice of thefifth Gaudin Hamiltonian we extract, all coefficients aresymmetric w.r.t. exchange of a and b . Hence we will splitthem as h (∆ a ,l a ;∆ c ;∆ b ,l b ) ( X ) = χ (∆ a ,l a ;∆ c ;∆ b ,l b ) ( X ) + a ↔ b and display the polynomials χ ( X ) instead of h ( X ), χ = 8 ,χ = 32 X ( l a − − l a + 2∆ c − ,χ = 16 X (cid:16) l a + 2 l a l b − l a + 7 (cid:17) − X (cid:16) l a + 8 l a l b + 2 l a (2∆ c − a ∆ b − a − c + 43 (cid:17) +2 (cid:16) ( l a + l b ) + 2 l a (2∆ c − − a + ∆ c + 2∆ a ∆ b − c + 22 (cid:17) ,χ = 16 X ( l a −
1) ( l b −
1) ( l a + l b − − X (cid:16) l a ( l b −
1) + 2 l a l b (2∆ c −
27) + 12+(4 l a − a ∆ b − a − b − c + 27) (cid:17) +2 X (cid:16) l a (4 l b −
5) + 2 l a l b (2∆ c − l a (4∆ a ∆ b − a − b − c + 31)+2∆ a − ∆ c − a ∆ b + 6∆ a + 9∆ c − (cid:17) + (cid:16) ( l a + l b ) + 2 l a (2∆ c − − a + ∆ c + 2∆ a ∆ b − c + 4 (cid:17) ,χ = − X l a ( l a − l b ( l b − Xl a l b (cid:16) l a l b − l a + 2∆ a ∆ b − a − ∆ c + 8 (cid:17) + const . Despite its relevance for representation theory, we havenot found the fourth order operator (10) in the existingliterature on orthogonal polynomials, except for some spe-cial cases. [1] A. Polyakov, “Nonhamiltonian approach to conformalquantum field theory” , Zh. Eksp. Teor. Fiz. 66, 23 (1974) .[2] A. Belavin, A. M. Polyakov and A. Zamolodchikov, “InfiniteConformal Symmetry in Two-Dimensional Quantum FieldTheory” , Nucl. Phys. B 241, 333 (1984) .[3] D. Poland, S. Rychkov and A. Vichi, “The conformalbootstrap: Theory, numerical techniques, and applications” , Reviews of Modern Physics 91, A. Vichi (2019) , http://dx.doi.org/10.1103/RevModPhys.91.015002 .[4] F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, “Precision Islands in the Ising and O ( N ) Models” , JHEP 1608, 036 (2016) , arxiv:1603.04436 .[5] D. Simmons-Duffin, “The Lightcone Bootstrap and theSpectrum of the 3d Ising CFT” , JHEP 1703, 086 (2017) , arxiv:1612.08471 .[6] V. Rosenhaus, “Multipoint Conformal Blocks in the CombChannel” , JHEP 1902, 142 (2019) , arxiv:1810.03244 .[7] S. Parikh, “Holographic dual of the five-point conformalblock” , JHEP 1905, 051 (2019) , arxiv:1901.01267 .[8] J.-F. Fortin and W. Skiba, “New methods for conformalcorrelation functions” , JHEP 2006, 028 (2020) , arxiv:1905.00434 .[9] S. Parikh, “A multipoint conformal block chain in d dimensions” , JHEP 2005, 120 (2020) , arxiv:1911.09190 .[10] J.-F. Fortin, W. Ma and W. Skiba, “Higher-Point ConformalBlocks in the Comb Channel” , arxiv:1911.11046 .[11] N. Irges, F. Koutroulis and D. Theofilopoulos, “Theconformal N -point scalar correlator in coordinate space” , arxiv:2001.07171 .[12] J.-F. Fortin, W.-J. Ma and W. Skiba, “Six-Point ConformalBlocks in the Snowflake Channel” , arxiv:2004.02824 .[13] J.-F. Fortin, W.-J. Ma, V. Prilepina and W. Skiba, “EfficientRules for All Conformal Blocks” , arxiv:2002.09007 .[14] X. Zhou, “How to Succeed at Witten Diagram Recursionswithout Really Trying” , arxiv:2005.03031 .[15] A. Pal and K. Ray, “Conformal Correlation functions infour dimensions from Quaternionic Lauricella system” , arxiv:2005.12523 .[16] J.-F. Fortin, W.-J. Ma and W. Skiba, “Seven-PointConformal Blocks in the Extended Snowflake Channel andBeyond” , arxiv:2006.13964 .[17] S. Hoback and S. Parikh, “Towards Feynman rules forconformal blocks” , arxiv:2006.14736 .[18] S. Ferrara, A. Grillo, G. Parisi and R. Gatto, “Covariantexpansion of the conformal four-point function” , Nucl. Phys. B 49, 77 (1972) , [Erratum: Nucl.Phys.B 53,643–643 (1973)].[19] M. Isachenkov and V. Schomerus, “Superintegrability of d -dimensional Conformal Blocks” , Phys. Rev. Lett. 117, 071602 (2016) , arxiv:1602.01858 .[20] T. Anous and F. M. Haehl, “On the Virasoro six-pointidentity block and chaos” , JHEP 2008, 002 (2020) , arxiv:2005.06440 .[21] J.-F. Fortin, W.-J. Ma and W. Skiba, “All Global One- andTwo-Dimensional Higher-Point Conformal Blocks” , arxiv:2009.07674 .[22] S. Ferrara, A. Grillo, G. Parisi and R. Gatto, “The shadowoperator formalism for conformal algebra. Vacuumexpectation values and operator products” , Lett. Nuovo Cim. 4S2, 115 (1972) .[23] F. Dolan and H. Osborn, “Conformal partial waves and the operator product expansion” , Nucl. Phys. B 678, 491 (2004) , hep-th/0309180 .[24] F. Dolan and H. Osborn, “Conformal Partial Waves:Further Mathematical Results” , arxiv:1108.6194 .[25] M. Gaudin, “Diagonalisation d’une classe d’hamiltoniens despin” , Journal de Physique 37, 1087 (1976) .[26] M. Gaudin, “La fonction d’onde de Bethe” , Masson (1983).[27] I. Buri´c, S. Lacroix, L. Quintavalle, J. A. Mann andV. Schomerus, In preparation.[28] V. Dobrev, G. Mack, V. Petkova, S. Petrova and I. Todorov, “Dynamical derivation of vacuum operator-product expansionin Euclidean conformal quantum field theory” , Physical Review D 13, 887 (1976) .[29] V. Dobrev, G. Mack, V. Petkova, S. Petrova and I. Todorov, “Harmonic Analysis on the n-Dimensional Lorentz Groupand Its Application to Conformal Quantum Field Theory” , “volume 63” .[30] M. S. Costa, J. Penedones, D. Poland and S. Rychkov, “Spinning Conformal Blocks” , JHEP 1111, 154 (2011) , arxiv:1109.6321 .[31] M. S. Costa, J. Penedones, D. Poland and S. Rychkov, “Spinning Conformal Correlators” , JHEP 1111, 71 (2011) , arxiv:1107.3554 .[32] V. Bargmann and I. T. Todorov, “Spaces of analyticfunctions on a complex cone as carriers for the symmetrictensor representations of SO(n)” , Journal of Mathematical Physics 18, 1141 (1977) .[33] B. Feigin, E. Frenkel and N. Reshetikhin, “Gaudin model,Bethe ansatz and correlation functions at the critical level” , Commun. Math. Phys. 166, 27 (1994) , hep-th/9402022 .[34] D. Talalaev, “Quantization of the Gaudin system” , hep-th/0404153 .[35] A. I. Molev, “Feigin-Frenkel center in types B, C and D” , Inventiones Mathematicae 191, 1 (2013) , arxiv:1105.2341 .[36] A. Chervov, G. Falqui and L. Rybnikov, “Limits of Gaudinalgebras, quantization of bending flows, Jucys-Murphyelements and Gelfand-Tsetlin bases” , Lett. Math. Phys. 91, 129 (2010) , arxiv:0710.4971 .[37] A. Chervov, G. Falqui and L. Rybnikov, “Limits of GaudinSystems: Classical and Quantum Cases” , SIGMA 5, 029 (2009) , arxiv:0903.1604 .[38] M. Kapovich and J. Millson, “On the moduli space ofpolygons in the Euclidean plane” , J. Differential Geom. 42, 430 (1995) .[39] M. Kapovich and J. J. Millson, “The symplectic geometry ofpolygons in Euclidean space” , J. Differential Geom. 44, 479 (1996) .[40] H. Flaschka and J. Millson, “Bending flows for sums of rankone matrices” , Canadian Journal of Mathematics 57, 114–158 (2005) , math/0108191 .[41] G. Falqui and F. Musso, “Gaudin models and bending flows:a geometrical point of view” , J. of Phys. A 36, 11655 , nlin/0306005 .[42] M. Isachenkov and V. Schomerus, “Integrability of conformalblocks. Part I. Calogero-Sutherland scattering theory” , JHEP 1807, 180 (2018) , arxiv:1711.06609 .[43] M. Hogervorst and S. Rychkov, “Radial Coordinates forConformal Blocks” , Phys. Rev. D 87, 106004 (2013) , arxiv:1303.1111 .[44] F. Kos, D. Poland and D. Simmons-Duffin, “Bootstrapping the O ( N ) vector models” , JHEP 1406, 091 (2014) , arxiv:1307.6856 .[45] F. Kos, D. Poland and D. Simmons-Duffin, “BootstrappingMixed Correlators in the 3D Ising Model” , JHEP 1411, 109 (2014) , arxiv:1406.4858 .[46] J. Penedones, E. Trevisani and M. Yamazaki, “RecursionRelations for Conformal Blocks” , JHEP 1609, 070 (2016) , arxiv:1509.00428 .[47] S. M. Chester, W. Landry, J. Liu, D. Poland,D. Simmons-Duffin, N. Su and A. Vichi, “Carving out OPEspace and precise O(2) model critical exponents” , Journal of High Energy Physics 2020, A. Vichi (2020) , http://dx.doi.org/10.1007/JHEP06(2020)142 .[48] J. Liu, D. Meltzer, D. Poland and D. Simmons-Duffin, “TheLorentzian inversion formula and the spectrum of the 3dO(2) CFT” , arxiv:2007.07914 .[49] P. Vieira, V. Gon¸calves and C. Bercini, “MultipointBootstrap I: Light-Cone Snowflake OPE and the WLOrigin” , arxiv:2008.10407arxiv:2008.10407