gbsn

More precisely, we consider the contact Witten diagrams depicted in Fig 1, which appear naturally in holographic models of boundary CFTs. The vertical co-dimension 1 surface is the holographic dual of the boundary. When all insertions are moved to the boundary, the diagrams are fully within the $AdS_{d}$ subspace and reduce to the so-called $D$-functions in the AdS/CFT literature. These contact Witten diagrams are the building blocks of holographic correlators. As we will show, these diagrams can be identified with the following conformal Feynman integral in $D$-dimensional flat space

where $x_{ij}^{\mu}=x_{i}^{\mu}-x_{j}^{\mu}$, $x_{ij}^{2}=x_{ij}^{\mu}x_{ij,\mu}$ and $\sum_{i=1}^{n}\Delta_{i}=D$. The perpendicular distances $x_{i,\perp}$ are identified with the masses $m_{i}$. These integrals, which generalize box diagrams, are remarkably invariant under the conformal Yangian algebra. The discovery of this property was motivated by special cases of such diagrams appearing in the so-called fishnet theories which are known to be integrable Gurdogan:2015csr ; Caetano:2016ydc ; Chicherin:2017frs ; Chicherin:2017cns . Integrability of (1) was first proven in the massless case, for integrals with $n=4,6$. The proof was streamlined and extended to the massive case in Loebbert:2020hxk ; Loebbert:2020glj where it was shown that all such integrals are Yangian invariant. Since contact Witten diagrams are essentially $I_{n}$, it follows that they are Yangian invariant as well. On the other hand, we will show that the contact Witten diagrams satisfy an intricate web of differential recursion relations shifting the weights $\Delta_{i}$. For example, there are differential operators $\mathbb{O}_{ij}$ which shift $\Delta_{i}$ and $\Delta_{j}$ by 1

For these relations to be consistent, the action of $\mathbb{O}_{ij}\mathbb{O}_{kl}$ must be equal to that of $\mathbb{O}_{ik}\mathbb{O}_{jl}$ as they lead to the same contact Witten diagram. This imposes nontrivial constraints on $W$. Remarkably, we find that the full set of consistency conditions is precisely the Yangian invariance condition.

The massive Yangian is generated by the above level-zero generators and the following level-one generators ³³3See Loebbert:2016cdm for a pedagogical introduction.

where $f^{a}{}_{bc}$ are the structure constants and $s_{i}$ are the evaluation parameters. The integrals $I_{n}$ are annihilated by the level-one generators, and consequently the entire Yangian. Moreover, $I_{n}$ are invariant under level-zero generators and the level-one generators are in the adjoint representation of the level-zero algebra. It is therefore sufficient to require that $I_{n}$ is annihilated by the level-one momentum operators $\widehat{{\rm P}}^{\mu}$. Furthermore, because (1) also has permutation symmetry, invariance under $\widehat{{\rm J}}^{a}$ is equivalent to invariance under any two-site operators Loebbert:2020glj

In terms of $\widehat{{\rm J}}^{a}_{jk}$, $\widehat{{\rm J}}^{a}$ can be written as $\widehat{{\rm J}}^{a}=\sum_{k>j=1}^{n}\widehat{{\rm J}}^{a}_{jk}$. The momentum operator is given by

Using (Yangian Symmetry in Holographic Correlators), we can write it explicitly as

where

In addition to the above operators $\widehat{{\rm J}}^{a}_{jk}$, it was observed in Loebbert:2020hxk that $I_{n}$ are also annihilated by an extra set of bi-local operators $\widehat{{\rm J}}^{a}_{{\rm extra},jk}$. For example,

Here, we have written down a mass $m_{i}$ for each site $i$. The massless (or partially massless) case is obtained by just setting the masses to zero.

A particularly useful representation of $W$ is Rastelli:2017ecj

which is obtained by using the Schwinger parameterization and integrating out the AdS coordinates. Here, $C_{n}=\pi^{\frac{d-1}{2}}\Gamma[\frac{\sum_{i=1}^{n}\Delta_{i}-d+1}{2}]\prod_{i=1}^{n}\Gamma^{-1}[\Delta_{i}]$ is a coefficient and we have defined

An important consequence of (12) is that contact Witten diagrams are dimension-independent after factoring out a numerical coefficient

On the other hand, if we integrate out only the radial coordinate $z_{0}$, we find

Using the $d$-independence of $\widetilde{W}$, we can conveniently set $d=D+1$. Then (15) is nothing but the conformal integral $I_{n}$ after using the Schwinger parameterization

Since the integrals $I_{n}$ are invariant under the Yangian Loebbert:2019vcj ; Loebbert:2020hxk ; Loebbert:2020glj , the contact Witten diagrams $W$ are Yangian invariant as well.

Let us also mention that the conformal invariance of Witten diagrams implies the following relations

These conditions can be easily derived in the embedding space formalism and they follow from requiring $W$ to scale correctly when independently rescaling the embedding vector of each operator Rastelli:2017ecj . The details can be found in the Supplemental Material.

The massless case. For simplicity, let us first demonstrate the equivalence for the massless case, i.e., $m_{i}=0$, which is relevant for $D$-functions in pure AdS. Note that $\widehat{{\rm P}}^{\mu}_{jk,{\rm extra}}$ vanishes in this case so we have only (26) with $m_{i}$ set to zero. From (7) it is not difficult to see that almost all terms are already in the form of (29), except for those coming from the contraction with $X^{\nu\mu\rho}$. To proceed, we note the following useful identity

We then find all the coefficient functions are given by

where $i,l\neq j,k$ ⁵⁵5The indices $i$ and $l$ are dummy indices. Therefore, $E_{kl}$ and $E_{ji}$ are not new coefficient functions.. From $E_{il}=0$, we reproduce the consistency condition (22). Since (23) and (24) are identically zero on both sides in the massless limit, the remaining conditions must not produce nontrivial constraints. To show $E_{ki}=0$, we first use (22) to write $E_{ki}$ as

Then using the massless limit of (25) we find that $E_{ki}$ vanishes. Symmetry implies that $E_{jl}=0$ as well. To see $E_{jk}=0$, we use permutation symmetry and (22) to write

From (25) we also have

where we have used $D=\sum_{i=1}^{n}\Delta_{i}$. It is then clear that $E_{jk}$ also vanishes.

The massive case. Having proven the equivalence in the massless limit, let us now move on to the general case. We first focus on the condition (26) where the proof is similar to the massless case above. To cast the action of $\widehat{{\rm P}}^{\mu}_{jk}$ in the form of (29), let us use the following massive version of (30)

We find the coefficient functions are

with $i,l\neq j,k$. Requiring (36) to vanish, we recover the first consistency condition (22). Following similar manipulations as in the massless case, which are detailed in the Supplemental Material, we find from $E_{ki}=0$ the second consistency condition (23). However, the condition from the coefficient (39) yields no further constraint. In fact, we find that $E_{jk}=0$ follows from (22) and (23). To derive the last consistency condition (24), we must examine the extra contraint (27). The explicit operator action reads

where $m_{j}\partial_{m_{j}}$ should be expressed in terms of $\mathbb{D}_{j}$ and $\mathbb{O}_{ij}$ using (18). Naively, the form of (40) seems to be in contradiction with (29). However, this expression can be greatly simplified upon using (22), (23) and the conformal invariance condition (25) (see Supplemental Material). We find that all $E_{ab,{\rm extra}}$ vanish except for $E_{jk,{\rm extra}}$

which gives the last condition (24).

There are plenty of future directions worth exploring. Firstly, we only focused on contact Witten diagrams which correspond to one-loop Feynman integrals. It would be interesting to study Yangian symmetry in exchange Witten diagrams. Certain two-loop Feynman integrals are also known to be Yangian invariant Loebbert:2019vcj ; Loebbert:2020hxk ; Loebbert:2020glj and coincide with exchange Witten diagrams when conformal dimensions satisfy special conditions Paulos:2012nu ; Ma:2022ihn . However, the general story is still unclear at the moment. Secondly, another exciting research avenue is to extend the analysis to include supersymmetry. The superconformal Yangian constraints should be highly nontrivial and will presumably select “superspace $D$-functions” with quantized dimensions as their solutions. It would be extremely interesting to see if these superconformal Yangian constraints can be used as an alternative method to rederive the general results of holographic four-point correlators of IIB supergravity in $AdS_{5}\times S^{5}$ Rastelli:2016nze ; Rastelli:2017udc . Finally, Witten diagrams also play an important role in the analytic functional approach to the conformal bootstrap where they serve as generating functions for the analytic functionals Mazac:2018mdx ; Mazac:2018ycv ; Kaviraj:2018tfd ; Mazac:2018biw ; Mazac:2019shk ; Caron-Huot:2020adz ; Giombi:2020xah . It would be interesting to explore the consequence of Yangian symmetry in that context.