Gaudin Models and Multipoint Conformal Blocks: General Theory

Our goal here is to prove that Casimir and vertex operators constructed around different vertices of an OPE diagram commute. More precisely we shall show that

for every pair $(r,p),(r^{\prime},q)$ of Casimir operators and any choice $(\rho,q,\nu)$ of a vertex operator, including the Pfaffian Casimir and vertex operators that appear at order $p,q=d/2+1$ when the dimension $d$ is even. Note that the individual vertex differential operators also depend on the choice $a\in\{12,23,13\}$ of a pair of legs. In addition, we shall also establish that vertex operators associated with different vertices commute,

and all triples $(p,\nu,a)$ and $(q,\mu,b)$. This leaves only the commutativity of operators attached to the same vertex which is deferred to the next subsection.

The properties (17) and (18) are in fact elementary. They require global conformal invariance, the tree structure of OPE diagrams and the commutativity property (7). To begin with, let us recall that we associated two disjoint sets $I_{r,1}$ and $I_{r,2}$ to every link. As we pointed out before, the Casimir differential operators (8) do not depend on whether we used the generators $\mathcal{T}_{\alpha}^{(I_{r,1})}$ or $\mathcal{T}_{\alpha}^{(I_{r,2})}$ to construct them. Let us briefly discuss the details of the proof. Note that we think of $\mathcal{D}$ as an operator acting on correlation functions $\mathcal{G}$. This is signaled by the subscript $|\mathcal{G}$ in the definition of the Casimir differential operators. In evaluating products of first order differential operators, one can only apply the Ward identity (11) to the rightmost operator, which acts directly on the correlation function, and not on some derivative thereof. But once we have converted the rightmost operators $\mathcal{T}^{(I_{r,1})}$ into $-\mathcal{T}^{(I_{r,2})}$, they will commute with all operators to their left, such that we can freely move them all the way to the left and proceed to apply the Ward identity to the next set of first order operators, and so on. If we finally take into account that the invariants $\kappa$ of the conformal Lie algebra are symmetric, we arrive at an expression for the Casimir differential operators in terms of $\mathcal{T}^{(I_{r,2})}$.

A similar analysis can be carried out for vertex operators, see also Subsection 2.2. Without loss of generality we can assume that we have constructed our vertex operators in terms of the generators $\mathcal{T}^{(I_{\rho,1})}$ and $\mathcal{T}^{(I_{\rho,2})}$ and want to switch to constructing them from $\mathcal{T}^{(I_{\rho,1})}$ and $\mathcal{T}^{(I_{\rho,3})}$ instead. To do so we make use of the invariance condition

that follows from relations (5) and (11). After we apply this to the rightmost operator in the vertex differential operator, we use the commutativity property (7) to move the generators $\mathcal{T}^{(I_{\rho,1})}$ and $\mathcal{T}^{(I_{\rho,3})}$ to the left of $\mathcal{T}^{(I_{\rho,2})}$. We continue this replacement process until all the generators $\mathcal{T}^{(I_{\rho,2})}$ are removed and using symmetry of the tensor $\kappa$ we find

along with a similar relation for the Pfaffian vertex operators for $p=d/2+1$ and $d$ even. Since the last term in this sum with $\mu=p-\nu$ is just a Casimir operator, we have managed to express all vertex differential operators that are constructed from the generators associated with $I_{\rho,1}$ and $I_{\rho,2}$ as a linear combination of the vertex operators associated with the pair $I_{\rho,1}$ and $I_{\rho,3}$ and a Casimir operator. This is the main input in proving the commutativity statements (17) and (18).

Let us start with a pair of links $r,r^{\prime}$. Each of these links divides the set of external points into the two disjoint sets $I_{r,1},I_{r,2}$ and $I_{r^{\prime},1},I_{r^{\prime},2}$ respectively. Since the OPE diagram is a tree, it is always possible to find a pair $i,j=1,2$ such that $I_{r,i}\cap I_{r^{\prime},j}=\emptyset$. For this choice

because of the commutativity property (7). This proves our first claim. Note that the same arguments also apply to the case in which $r=r^{\prime}$ and also if one or both operators are Pfaffian, i.e. if $d$ is even and $p,q=d/2+1$.

Let us now extend this argument to include vertex differential operators. In order to prove that the Casimir operators associated to a link $r$ commute with the vertex operators associated to any vertex $\rho$ we recall that any choice of a vertex $\rho$ on an OPE diagram divides the diagram into the three distinct branches that are glued to the vertex, and we denote these by $I_{\rho,j}$ as in Figure 4. A quick glance at Figure 4 suffices to conclude that given $r$ and $\rho$ it is possible to find a pair $i\in{1,2}$ and $j\in{1,2,3}$ such that $I_{r,i}\subset I_{\rho,j}$, since the link $r$ must be in one of the three branches. It follows that $I_{r,i}\cap\left(I_{\rho,j_{1}}\cup I_{\rho,j_{2}}\right)=\emptyset$ for $j_{1}\neq j\neq j_{2}$. Commutativity of Casimir and vertex differential operators then follows since we can construct the Casimir differential operators in terms of the generators for $I_{r,i}$ while using the generators for $I_{\rho,j_{1}}$ and $I_{\rho,j_{2}}$ for the vertex differential operators.

Let us finally consider any two distinct vertices $\rho$ and $\rho^{\prime}$ on an OPE diagram. As we highlight in Figure 5, any configuration of two vertices divides the diagram into five parts: four external branches $I_{\rho,j}$, $I_{\rho^{\prime},j}$, $j=1,2$, attached to only one vertex, and the central part $\mathcal{C}$ of the diagram that is attached to both vertices $\rho$ and $\rho^{\prime}$. Following what we just claimed with focus on one vertex, we can use diagonal conformal symmetry to rewrite the operators around $\rho$ and $\rho^{\prime}$ to depend on disjoint sets of legs $I_{\rho,j}$, $I_{\rho^{\prime},j}$ with $j=1,2$. Since generators associated to these sets commute (7), it follows automatically that operators constructed around different vertices must commute as well.

This implies that to prove commutativity of our set of operators, we can just focus on operators that live around one single vertex. To prove the commutativity of these vertex operators we will now make use of the integrability technology that is provided by Gaudin models.

In this section, we will explain how the operators (14) associated with a vertex $\rho$ in the OPE diagram naturally arise from a specific Gaudin model, which in particular will provide us with a proof of their commutativity. Let us start by reviewing briefly how Gaudin models are defined Gaudin_76a ; Gaudin_book83 . They are integrable systems naturally constructed from a choice of a simple Lie algebra $\mathfrak{g}$. Having in mind applications of these systems to conformal field theories, we will choose $\mathfrak{g}$ to be the conformal Lie algebra $\mathfrak{so}(d+1,1)$ of the Euclidean space $\mathbb{R}^{d}$, with basis $T_{\alpha}$ as in the previous section. The Gaudin model depends in general on $M$ complex numbers $w_{j}$, called its sites, to which are attached $M$ independent representations of the algebra $\mathfrak{g}$. To obtain the vertex system we address in this section, we restrict our attention here to the case $M=3$ and associate with these three sites the representations of $\mathfrak{g}$ corresponding to the three fields attached to the vertex $\rho$. More precisely, using the notation defined in Section 1 and in particular the partition $\underline{N}=I_{\rho,1}\mathbin{\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}}I_{\rho,2}\mathbin{\mathchoice{\leavevmode\vtop{\halign{\hfil$\m@th\displaystyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\textstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}{\leavevmode\vtop{\halign{\hfil$\m@th\scriptscriptstyle#$\hfil\cr\cup\cr\cdot\crcr}}}}I_{\rho,3}$ constructed from the vertex $\rho$, we will attach to the three sites $w_{j}$, $j=1,2,3$, of the Gaudin model the generators $\mathcal{T}_{\alpha}^{(I_{\rho,j})}$ which define representations of $\mathfrak{g}$ in terms of first-order differential operators in the insertion points $x_{i}$.

A key ingredient in the construction of the Gaudin model is its so-called Lax matrix, whose components in the basis $T^{\alpha}$ are defined here as

where $z$ is an auxiliary complex variable called the spectral parameter. In the above equation, we have denoted the Lax matrix as $\mathcal{L}^{\rho}_{\alpha}(z)$ to emphasize that this is the matrix corresponding to the vertex $\rho$. For any elementary symmetric invariant tensor $\kappa_{p}$ of degree $p$ on $\mathfrak{g}$, there is a corresponding $z$-dependent Gaudin Hamiltonian of the form

where $\dots$ represent quantum corrections, involving a smaller number of components of the Lax matrix $\mathcal{L}_{\alpha}^{\rho}$ and their derivatives with respect to $z$. These corrections are chosen specifically to ensure that the Gaudin Hamiltonians commute for all values of the spectral parameter and all degrees:

The existence of such commuting Hamiltonians was first proven in Feigin:1994in , using some previously established results Feigin:1991wy on the so-called Feigin-Frenkel center of affine algebras at the critical level. The explicit expression for the quantum corrections was obtained in Talalaev:2004qi ; Chervov:2006xk for Lie algebras of type A and in Molev:2013 for types B, C, D, which is the case we are concerned with here (indeed, $\mathfrak{g}=\mathfrak{so}(d+1,1)$ is of type B for $d$ odd and of type D for $d$ even). We refer to Molev:2018 for a summary of these results. The properties of the Feigin-Frenkel center were further studied in the recent work Yakimova:2019 , the results of which imply that the quantum corrections in $\mathcal{H}^{(p)}_{\rho}(z)$ are sums of terms of the form

where $q<p$, $\tau^{\alpha_{1}\cdots\alpha_{q}}$ is a completely symmetric invariant tensor of degree $q$ on $\mathfrak{g}$ and $r_{1},\cdots,r_{q}$ are positive integers such that $r_{1}+\cdots+r_{q}=p$. For what follows, it will be useful to consider the leading part of the Hamiltonian (23) alone, without quantum corrections, which we will denote as

Let us finally note that the quantum corrections are absent for both the quadratic Hamiltonian $\mathcal{H}^{(2)}_{\rho}(z)$ and the Pfaffian Hamiltonian $\mathcal{H}^{(d/2+1)}_{\rho}(z)$ that exists for $d$ even, such that these two Hamiltonians coincide with their leading parts.

To make the link with the vertex operators defined in Section 1, we will make a specific choice of the parameters $w_{j}$ of the Gaudin model. More precisely, we set

In particular, the Lax matrix (22) reduces to

Let us now study the Gaudin Hamiltonians $\mathcal{H}^{(p)}_{\rho}(z)$ for this particular choice of parameters. We will first focus on their leading part $\widehat{\mathcal{H}}^{(p)}_{\rho}(z)$. Reinserting the above expression of the Lax matrix in eq. (26), we simply find

where $\mathcal{D}_{\rho,12}^{p,\nu}$ is the vertex operator defined in eq. (14). To obtain this expression, we have used the fact that $\mathcal{T}_{\alpha}^{(I_{\rho,1})}$ and $\mathcal{T}_{\beta}^{(I_{\rho,2})}$ commute to bring all $\mathcal{T}_{\alpha}^{(I_{\rho,1})}$’s to the left, as well as the symmetry of the tensor $\kappa_{p}^{\alpha_{1}\cdots\alpha_{p}}$ to relabel the Lie algebra indices as in eq. (14). Noting that the fractions $z^{-\nu}(z-1)^{\nu-p}$ for $\nu=0,\cdots,p$ are linearly independent functions of $z$, it is then clear that one can extract all the vertex operators $\mathcal{D}_{\rho,12}^{p,\nu}$ from $\widehat{\mathcal{H}}^{(p)}_{\rho}(z)$. Note that the “extremal” operators $\mathcal{D}_{\rho,12}^{p,0}$ and $\mathcal{D}_{\rho,12}^{p,p}$ coincide with the Casimir operators of the fields at the branches $I_{\rho,1}$ and $I_{\rho,2}$ of the vertex. The same equation also holds for the Pfaffian operators $\mathcal{H}^{(d/2+1)}_{\rho}(z)$.

Our goal in this section is to prove the commutativity of the vertex operators $\mathcal{D}_{\rho,12}^{p,\nu}$ using known results on Gaudin models. This would follow automatically from the commutativity (24) of the Gaudin Hamiltonians $\mathcal{H}_{\rho}^{(p)}(z)$ if these Hamiltonians contained the operators $\mathcal{D}_{\rho,12}^{p,\nu}$. But we have already proven above that the latter are naturally extracted from the leading parts $\widehat{\mathcal{H}}^{(p)}_{\rho}(z)$ of the Gaudin Hamiltonians, without the quantum corrections. These quantum corrections are in general crucial for the commutativity of the Hamiltonians. However, we shall prove below that the quantum corrections of the specific Gaudin model considered here can always be expressed in terms of lower-degree Hamiltonians, and can thus be discarded without breaking the commutativity property. In this case, the non-corrected Hamiltonians $\widehat{\mathcal{H}}^{(p)}_{\rho}(z)$ pairwise commute for all values of the spectral parameter and all degrees, thus demonstrating the desired commutativity of the vertex operators $\mathcal{D}_{\rho,12}^{p,\nu}$.

Let us then analyze these quantum corrections. Recall that they are composed of terms of the form (25). Reinserting the expression (28) of the Lax matrix for the present choice of parameters $w_{j}$ in this equation, we find that the quantum corrections contain only terms of the form

with prefactors composed of powers of $z$ and $z-1$, and where $\tau^{\alpha_{1}\cdots\alpha_{q}}$ is a completely symmetric invariant tensor on $\mathfrak{g}$ of degree $q<p$, as in eq. (25). In particular, $\tau^{\alpha_{1}\cdots\alpha_{q}}$ decomposes as a product of elementary symmetric invariant tensors $\kappa_{k}$, symmetrized over the indices $\alpha_{i}$, with $k\leq q<p$. The correction in the above equation can thus be re-expressed as an algebraic combination of lower-degree vertex operators $\mathcal{D}_{\rho,12}^{k,\nu}$. Since these are the coefficients of the non-corrected Hamiltonian $\widehat{\mathcal{H}}^{(k)}_{\rho}(z)$, recursion on the degrees shows that the quantum corrections can indeed be expressed in terms of lower-degree Hamiltonians, as anticipated.

Let us end this subsection with a brief discussion on the role played by the choice of labeling of the branches $I_{\rho,1}$, $I_{\rho,2}$ and $I_{\rho,3}$ attached to the vertex $\rho$. As mentioned in Section 1, this choice is arbitrary but enters the definition (14) of the vertex operators $\mathcal{D}_{\rho,12}^{p,\nu}$, which in this case contain only the generators $\mathcal{T}_{\alpha}^{(I_{\rho,1})}$ and $\mathcal{T}_{\alpha}^{(I_{\rho,2})}$. In the context of the 3-sites Gaudin model considered in this subsection, this is related to the choice of positions $w_{j}$ of the sites made in eq. (27). In particular, the absence of generators $\mathcal{T}_{\alpha}^{(I_{\rho,3})}$ in the Gaudin Hamiltonians $\mathcal{H}^{(p)}(z)$ is due to the fact that we sent the site $w_{3}$ to infinity. One could have made another choice of labeling and constructed vertex operators $\mathcal{D}_{\rho,23}^{p,\nu}$ from the generators $\mathcal{T}_{\alpha}^{(I_{\rho,3})}$ and $\mathcal{T}_{\alpha}^{(I_{\rho,2})}$, for instance. The corresponding choice of positions of the sites would then be related to the initial one by the Möbius transformation that exchanges $0$ and $\infty$ and fixes $1$, i.e. the inversion $z\mapsto\frac{1}{z}$. More precisely, under such a transformation of the spectral parameter, the Lax matrix of the Gaudin model behaves as a 1-form on the Riemann sphere and satisfies

Acting on the correlation function $\mathcal{G}_{N}$, which satisfies the Ward identities (11), this Lax matrix then becomes

and thus coincides with the Lax matrix from which one would build the vertex operators $\mathcal{D}_{\rho,23}^{p,\nu}$ with generators $\mathcal{T}_{\alpha}^{(I_{\rho,3})}$ and $\mathcal{T}_{\alpha}^{(I_{\rho,2})}$. This proves that the operators $\mathcal{D}_{\rho,23}^{p,\nu}$ can naturally be extracted from the generating functions

Using the expression (29) of $\widehat{\mathcal{H}}_{\rho}^{(p)}\left(z\right)$ in terms of the initial vertex operators $\mathcal{D}_{\rho,12}^{p,\nu}$, we thus get that the $\mathcal{D}_{\rho,23}^{p,\nu}$’s are linear combinations of the $\mathcal{D}_{\rho,12}^{p,\nu}$’s, as was demonstrated by direct computation in the previous subsection. Let us note that the use of the Ward identities was a crucial step in the above reasoning, as highlighted for instance by the subscript $\mathcal{G}$ in eq. (33). This step should be performed with care, in particular when using the Ward identities to replace $-\mathcal{T}_{\alpha}^{(I_{\rho,1})}-\mathcal{T}_{\alpha}^{(I_{\rho,2})}$ by $\mathcal{T}_{\alpha}^{(I_{\rho,3})}$ in the Hamiltonians (33). Indeed, one can use the Ward identities only for generators on the right. In order to do so, one thus has to commute generators to bring them to the right, replace them through the Ward identities and commute them back to their original place. Although this procedure can in general create non-trivial corrections, it can in fact be done freely in the case at hand: indeed, commuting operators $\mathcal{T}_{\alpha_{k}}^{(I_{\rho,j})}$ and $\mathcal{T}_{\alpha_{l}}^{(I_{\rho,j})}$ within the Hamiltonian $\widehat{\mathcal{H}}_{\rho}^{(p)}$ creates a term proportional to the structure constant $f_{\alpha_{k}\alpha_{l}}^{\hskip 17.0pt\beta}$, which vanishes when contracted with the symmetric tensor $\kappa_{p}^{\alpha_{1}\cdots\alpha_{p}}$. This ensures that the Hamiltonian (33) indeed serves as a generating function of the operators $\mathcal{D}_{\rho,23}^{p,\nu}$ built from $\mathcal{T}_{\alpha}^{(I_{\rho,3})}$ and $\mathcal{T}_{\alpha}^{(I_{\rho,2})}$. A similar reasoning applies for the other choices of labeling, by considering the appropriate Möbius transformations that permute the sites $0$, $1$ and $\infty$ of our 3-site Gaudin model.

In the previous subsection we have shown that all of the operators listed in eq. (14) commute with each other. As we have pointed out before, we did not include operators with $\nu=0$ and $\nu=p$ in the list since these coincide with Casimir differential operators,

Here $r_{i}$ denotes the link that is attached to the $i^{th}$ leg of the vertex $\rho$, i.e. for which $I_{r_{i},j}=I_{\rho,i}$ with either $j=1$ or $j=2$. The remaining $p-1$ operators satisfy one more linear relation since

Let us note that this relation also applies to the Pfaffian vertex operators that exist for $p=d/2+1$ when $d$ is even. We have used this relation to drop one of the vertex differential operators. Once these obvious relations are taken care of, the total number of commuting vertex differential operators is given by eq. (15) and matches precisely the maximal number of cross ratios that can be associated to a single (generic) vertex, see upper bound of eq. (16) in the introduction. But restricted vertices carry fewer variables, so their corresponding differential operators (constructed in the previous section) must obey further relations. It is the main goal of this subsection to discuss these relations. We will also check that, once these are taken into account, the number of remaining vertex differential operators matches the number (16) of cross ratios at restricted vertices.

Our arguments are based on an important auxiliary result concerning the differential operators $\mathcal{T}^{(I)}_{\alpha}$ that are associated to some subset $I\subset\underline{N}$ of order $|I|\leq N/2$. To present this requires a bit of preparation. Up to this point there was no need to spell out the precise form of the symmetric invariant tensors $\kappa_{p}$ that we used to construct our differential operators. Now we need to be a bit more specific. As is well known, such tensors can be realized as symmetrized traces,

Here $T^{\alpha}$ denote the generators of the conformal Lie algebra and $(\dots)$ signal symmetrization with respect to the indices. In the following we shall use the symbol str to denote this symmetrized trace. The trace can be taken in any faithful representation. The simplest of such choices is to use the fundamental representation. In order to construct the associated symmetric invariants more explicitly, we shall replace the index $\alpha$ that enumerates the basis of the conformal algebra by a pair $\alpha=[AB]$ where $A,B$ run through $A,B=0,1,\dots,d+1$ with $T^{[AB]}=-T^{[BA]}$. In the fundamental representation, the matrix elements of these generators take the form

where $\eta_{AB}$ is the Minkowski metric with signature $(d+1,1)$ and $\eta^{AB}$ is its inverse. This makes it now easy to compute $\kappa_{p}$ explicitly. The only issue arises in even $d$. In this case the symmetrized traces in the fundamental representation do not generate all the invariants. In order to obtain the missing invariant, one has to include the trace in a chiral representation. The standard construction employs the spinor representation in which generators $T^{[AB]}$ are represented as

where $\gamma^{A}$ are the $d+2$-dimensional $\gamma$ matrices and the matrix indices are $\sigma,\tau=1,\dots,2^{d/2+1}$. One can then project to a chiral spinor representation with the help of $\gamma_{c}\sim\gamma^{0}\cdots\gamma^{d+1}$.

Let us now introduce the symbol $\mathcal{T}^{(I)}$ to denote the following Lie-algebra valued differential operators

Upon evaluation in some finite-dimensional representation, such as the fundamental or the spinor representation, these become matrix valued differential operators. With this notation we write our set (14) as

when $d$ is odd and the parameters $p$ and $\nu$ assume the values $p=2,4,\dots,d+1$ and $\nu=1,\dots,p-1$, as usual. For even dimension $d$, on the other hand, we use the symmetrized trace in the fundamental representation for $p=2,4,\dots,d$ and construct the missing Pfaffian vertex differential operators as

where we take the trace in the spinor representation and include the factor $\gamma_{c}$ in the argument.

In finding relations between the vertex differential operators for restricted vertices we actually work with the total symbols of the differential operators rather than the operators themselves. This means that we replace the partial derivatives $\partial^{(i)}_{\mu}$ with commuting coordinates $p^{i}_{\mu}$. The associated matrices of functions of $x_{i}^{\mu}$ and $p^{i}_{\mu}$ will be denoted by $\bar{\mathcal{T}}^{(I)}$. As before we shall add a subscript $f,s$ to denote the matrices in the fundamental and the spinor representation. After passing to the total symbol the entries of the matrices commute and we can drop the symmetrization prescription when taking traces. As a result, the total symbols of the vertex differential operators are simply traces of powers of the matrices $\bar{\mathcal{T}}^{(I)}$.

We are now ready to state the main result needed to elucidate the relations between vertex differential operators. It concerns the matrix elements of the $n^{\textit{th}}$ power of the matrices $\bar{\mathcal{T}}^{(I)}_{f,s}$ for the fundamental and the spinor representations. In both cases, these matrix elements are functions of $x_{i}^{\mu}$ and $p_{\mu}^{i}$ with $i\in I$. Our main claim is that these matrix elements can be expressed in terms of lower order ones of the same form whenever $n>2\mathfrak{d}_{I}$, where $\mathfrak{d}_{I}=\mathfrak{d}(I,d)$ is the integer defined in eq. (12). More precisely, for each matrix element $AB$ there exist coefficients $\varrho^{(n,m)}_{AB}$ such that

Note that there is no summation over $A,B$ on the right hand side. The coefficients $\varrho_{AB}$ depend only on the external conformal weights $\Delta_{i}$ and the total symbols $\bar{\mathcal{D}}^{p}_{(I)}$ of the Casimir operators associated with the index set $I$. If the index set $I$ has depth $\mathfrak{d}_{I}=1$, for example, i.e.  if the $\bar{\mathcal{T}}^{(I)}$ describe the action of the conformal algebra on a single scalar primary, then starting from $n=3$ all matrix elements can be expressed in terms of lower order ones. In the case of the spinor representation one has a very similar relation

which now applies for $n>\mathfrak{d}_{I}$ and involves a summation over $m$ that ends at $\mathfrak{d}_{I}=\mathfrak{d}(I,d)$, see definition (12). So if $\mathfrak{d}_{I}=1$, for example, the matrix elements of the square are expressible in terms of the matrix elements of $\bar{\mathcal{T}}^{(I)}_{s}$. We verify both statements (39) and (40) in Appendix C using embedding space formalism, see Appendix B.

We are now prepared to discuss relations between vertex differential operators. Let us consider a vertex $\rho$ inside our OPE diagram. As we have explained before, $\rho$ splits the set $\underline{N}$ into three subsets $I_{\rho,i}$ with $i=1,2,3$. Each of these sets determines an integer $\mathfrak{d}_{i}=\mathfrak{d}(I_{\rho,i},d)$. Let us suppose that we construct the vertex differential operators using $\mathcal{T}^{(I_{\rho,i})}$ for $i=1,2$ as in eq. (37). If one of the integers $\mathfrak{d}_{1}$ or $\mathfrak{d}_{2}$ is smaller than $r_{d}$ we immediately obtain relations among the vertex differential operators. In fact, when applied to the matrices $\bar{\mathcal{T}}^{(I_{\rho},1)}$, our claim (39) implies that all operators we obtain when $\nu>2\mathfrak{d}_{1}$ can be expressed in terms of Casimir and vertex differential operators of lower order. The same is true when $p-\nu>2\mathfrak{d}_{2}$, as follows again from eq. (39), but this time applied to $\bar{\mathcal{T}}^{(I_{\rho},2)}$. Consequently, for any $p\geq 2\mathfrak{d}_{1},2\mathfrak{d}_{2}$, we can restrict the range of the index $\nu$ to be $p-2\mathfrak{d}_{2}\leq\nu\leq 2\mathfrak{d}_{1}$, with $\nu=0,p$ excluded as before.

But this does not yet include the full set of relations that appears whenever $\mathfrak{d}_{3}$ is smaller than $\textit{min}(\mathfrak{d}_{1}+\mathfrak{d}_{2},r_{d})$. One of the simplest examples of this occurs in the 6-point snowflake channel in $d>3$, see Figure 6, where two symmetric traceless tensor operators on two branches are combined at the central vertex $\rho$ to form another symmetric traceless tensor on the third branch.