Bulk locality and gauge invariance for boundary-bilocal cubic correlators in higher-spin gravity

Higher-spin (HS) gravity Vasiliev:1990en ; Vasiliev:1995dn ; Vasiliev:1999ba is the conjectured interacting theory of an infinite tower of massless gauge fields of all spins. It can be thought of as a “smaller cousin” of string theory. In its simplest version, the theory lacks a realistic GR limit. However, it has the virtues of being native to 4 spacetime dimensions, and consistent with both signs of the cosmological constant. We consider here the “smallest” version of HS gravity in 4d: the so-called minimal type-A theory, which has a single, parity-even field of every even spin. This theory admits a particularly simple holographic dual Klebanov:2002ja ; Sezgin:2002rt ; Sezgin:2003pt via AdS/CFT: a free $O(N)$ vector model on the 3d boundary of AdS₄, whose primary single-trace operators form a tower of conserved HS currents. A major reason to be interested in this particular duality is that it also admits a positive cosmological constant Anninos:2011ui , providing a concrete model of dS₄/CFT₃. In the present paper, we stick for simplicity to AdS₄, in Euclidean signature.

The biggest outstanding question in HS theory concerns its locality properties. In general, since the theory involves infinitely many massless fields interacting at all orders in derivatives, it was always expected to be non-local in some way. Moreover, at the classical level, the only length scale in the theory is the cosmological curvature radius. Thus, the theory was expected to be non-local at the cosmological scale. Though exotic, this still implies a positive expectation of some degree of locality: in particular, at distances much larger than the AdS radius, one expects the couplings to vanish sufficiently fast.

This expectation was put to the test, by a research program to explicitly reproduce the theory’s vertices from its holographic boundary correlators. For 3-point correlators, bulk locality is satisfied automatically: all gauge-invariant cubic vertices for given spins $(s_{1},s_{2},s_{3})$ can be reduced to a finite set of structures with finitely many derivatives Joung:2011ww . Nevertheless, it seems significant that the particular cubic vertex Sleight:2016dba found for the minimal type-A theory takes a remarkably simple form. However, at the 4-point level, disaster strikes: the spin-0 quartic bulk vertex, derived in Bekaert:2015tva , turns out Sleight:2017pcz to be as non-local as an exchange diagram. This result was foreshadowed some years before, in the flat-spacetime context Fotopoulos:2010ay ; Taronna:2011kt . In particular, the authors of Fotopoulos:2010ay conjectured that some additional degrees of freedom should be added to make the theory local. A more subtle resolution is being advocated in e.g. Didenko:2019xzz ; Gelfond:2019tac : to keep the same degrees of freedom, but to extend the ordinary notion of locality to so-called “spin-locality”.

Our own approach to the locality problem is to try and mimic string theory: interactions that appear non-local in terms of field theory may become local when viewed in terms of more appropriate structures, such as the string worldsheet. While HS gravity (in its simplest version) doesn’t give rise to strings, it does contain an analogous object – the Didenko-Vasiliev “BPS black hole” solution Didenko:2008va ; Didenko:2009td . The analogy between this solution and the string is twofold. First, one can view the fundamental string (and all the other branes of string theory) as BPS solutions of supergravity Schwarz:1996bh ; Blumenhagen:2013fgp , with the Didenko-Vasiliev (henceforth, DV) solution playing the analogous role in HS gravity. Second, in AdS/CFT, one can view the string as the bulk dual of boundary Wilson lines or loops Rey:1998ik ; Maldacena:1998im , which contain as a Taylor expansion the whole tower of local single-trace boundary operators (whose bulk duals are the string’s modes). Similarly, in HS holography, the DV solution is the bulk dual David:2020fea ; Lysov:2022zlw of the boundary bilocal operator Das:2003vw ; Douglas:2010rc ; Das:2012dt , which contains as a Taylor expansion the tower of local boundary HS currents (whose bulk duals are the individual HS gauge fields). Due to these analogies, we believe that the key to understanding HS theory lies in the bulk dynamics of not just HS fields, but also DV solutions.

Our focus is on the linearized DV solution Didenko:2008va , which consists simply of linearized HS fields, sourced by a particle-like singularity located on a geodesic “worldline” in the AdS₄ bulk. This particle-like source is charged under the gauge fields of all spins, following a BPS-like proportionality pattern. In David:2020fea ; Lysov:2022zlw , we explored the bulk interaction between two such solutions, showing that it reproduces the CFT correlator of two boundary bilocals. In that case, the “interaction” was simply that of charged particles exchanging (an HS multiplet of) gauge fields, with no self-coupling among the gauge fields themselves. In the present paper, we extend the analysis to three DV solutions, and ask what kind of bulk interactions can reproduce the corresponding cubic CFT correlator. Here, the cubic self-interaction of the HS gauge fields becomes important. In fact, in an appropriate limit, the DV solutions reduce to usual boundary-bulk propagators Lysov:2022zlw , and the boundary correlator is then captured fully by the on-shell cubic vertex found by Sleight and Taronna Sleight:2016dba . Our goal in this paper will be to step away from this limit, and study the locality and gauge-invariance properties of the resulting bulk interactions.

Our eventual goal is to reformulate the entirety of HS theory in terms of cubic interactions between DV solutions FeynmanRules , entirely bypassing the need for quartic or higher vertices. It is this larger project that lends importance to the locality of such cubic interactions.

The formalism we’ll employ is the same as in Sleight:2016dba , combining Fronsdal’s “metric-like” approach to linearized HS fields Fronsdal:1978rb ; Fronsdal:1978vb with the radial-reduction approach to bulk AdS fields Biswas:2002nk , where we choose the scaling weights to match those of the relevant boundary-bulk propagators (as opposed to the more common choice Joung:2011ww , which simplifies the gauge-invariance analysis for general vertices).

We will argue that the cubic correlator of boundary bilocals is reproduced by a set of local Witten diagrams that couple the corresponding DV solutions and their geodesic “worldlines”. These diagrams can be divided into three groups:

1. (a)

   The Sleight-Taronna vertex Sleight:2016dba coupling the three DV solutions.

2. (b)

   Exchange of two HS gauge fields between the three geodesic “worldlines”. This is just a product of two pairwise interactions between the DV solutions, of the type considered in David:2020fea ; Lysov:2022zlw . In particular, it doesn’t involve self-interaction of HS fields.

3. (c)

   A new vertex, coupling the fields of two DV solutions to the “worldline” of the third.

These different terms (a)-(c) comprising the correlator are depicted in figure 1. Let us now comment on the extent to which each term is known, and the sense in which it is local.

Term (a) – the on-shell cubic coupling of HS fields – is known explicitly Sleight:2016dba , and is local in the traditional sense, i.e. it involves a finite number of derivatives for each set of spins $(s_{1},s_{2},s_{3})$. Note, however, that the DV solutions contain all spins. Therefore, the sum over spins will introduce an infinite tower of derivatives, and with it some degree of non-locality. Fortunately, as we’ll argue in section 4.6, this non-locality is in fact at the scale of $\sim 1$ AdS radius, matching the original expectation for HS theory.

Term (b) consists of simple diagrams whose only “vertices” are the local minimal couplings David:2020fea between an HS-charged particle and an HS gauge field. As such, it is fully known, and manifestly local if we agree to view the DV solutions’ worldlines as HS-charged particles. If one tried instead to express these diagrams as a cubic vertex between HS fields, that vertex would of course be non-local.

Now we turn to term (c) – a new vertex, which will be discussed at length in section 4. One may alternatively view it as an “off-shell” correction to the Sleight-Taronna vertex (i.e. a correction proportional to the free equations of motion), due to the DV fields not being source-free, and thus possessing Fronsdal curvature, concentrated on the corresponding “worldlines”. Even for fixed spins, this new vertex may include an infinite tower of derivatives. The question then is whether the resulting non-locality is restricted to $\sim 1$ AdS radius. We will argue that this question can be reframed as a set of proxy criteria, involving not the vertex formula itself, but rather its contribution to the correlator in certain limits. We will then show that our criteria are indeed satisfied, once the other contributions (a)-(b) to the correlator are taken into account. We won’t evaluate the new vertex explicitly, aside from a numerical study in one simple case (section 5).

An alternative concise way of introducing the three terms (a)-(c) is as follows:

1. (a)

   We draw the most obvious cubic coupling between the three DV solutions, via the Sleight-Taronna vertex. We find that this doesn’t reproduce the boundary cubic correlator of bilocals.

2. (b)

   We add the double-exchange diagrams, still constructed purely from known elements. We find that the boundary correlator is still not reproduced.

3. (c)

   We parameterize the difference between the boundary correlator and terms (a)-(b) in terms of a new vertex (or, alternatively, an off-shell correction to the Sleight-Taronna vertex). Our main result is then that this new vertex has appropriate locality properties.

Finally, note that our terms (a)-(c) don’t include any gauge corrections to the Sleight-Taronna vertex, i.e. corrections due to the DV solutions not being in transverse-traceless gauge. The vanishing of such corrections is one of our results, derived in section 3 and summarized below.

The rest of the paper is structured as follows. In section 2, we review the formalism of Sleight:2016dba for HS fields in Euclidean AdS₄, along with other relevant ingredients: the free vector model on the boundary, asymptotics of bulk fields, boundary-bulk propagators, the DV solution and the Sleight-Taronna vertex.

Section 3 contains our gauge-invariance results for the Sleight-Taronna vertex. We show that, if one merely symmetrizes the original vertex formula from Sleight:2016dba over permutations of its 3 legs, then the vertex’s gauge invariance is extended from source-free fields in transverse-traceless gauge (as originally intended in Sleight:2016dba ) to source-free fields in general traceless gauge. In section 3.3, we prove that this extended gauge-invariance holds up to boundary terms. Then, in section 3.4, we show that the boundary terms also vanish under appropriate assumptions on the fields’ asymptotics, which in particular are satisfied by the DV solution away from its singular worldline.

In section 4, we present our main argument vis. the locality structure of the general cubic correlator and the new vertex. In section 5, we illustrate the locality argument by a numerical analysis in a simple case: a single DV solution coupled to a pair of spin-0 boundary-bulk propagators. In section 6, we outline an alternative technique for calculating the relevant bulk diagram, using a new non-traceless gauge Lysov:2022zlw for the DV solution. Section 7 is devoted to discussion and outlook.

We note that section 3’s gauge-invariance result for the Sleight-Taronna vertex is not essential for the abstract locality argument in section 4. However, the existence of this nice result reinforces our sense that the paper’s main idea – of combining the DV solution with the Sleight-Taronna vertex – is on the right track.

To write the Sleight-Taronna vertex in a simple form, one must use an embedding-space formalism, and in particular the radial reduction approach of Biswas:2002nk . Thus, we describe Euclidean AdS₄ as the hyperboloid of unit timelike radius within 5d flat spacetime $\mathbb{R}^{1,4}$:

$\displaystyle EAdS_{4}=\left\{x^{\mu}\in\mathbb{R}^{1,4}\,|\,x_{\mu}x^{\mu}=-1,\ x^{0}>0\right\}\ .$

(1)

Here, indices $(\mu,\nu,\dots)$ are 5-dimensional, and are raised and lowered with the Minkowski metric $\eta_{\mu\nu}=\operatorname{diag}(-1,1,1,1,1)$. 4d vectors at a point $x^{\mu}\in EAdS_{4}$ are simply 5d vectors $v^{\mu}$ that satisfy $v\cdot x\equiv v_{\mu}x^{\mu}=0$. Covariant derivatives in $EAdS_{4}$ are simply flat $\mathbb{R}^{1,4}$ derivatives, followed by a projection of all indices back into the $EAdS_{4}$ tangent space:

	$\displaystyle\nabla_{\mu}v_{\nu}$	$\displaystyle=P_{\mu}^{\rho}(x)P_{\nu}^{\sigma}(x)\frac{\partial v_{\sigma}}{\partial x^{\rho}}\ ;$		(2)
	$\displaystyle P_{\mu}^{\nu}(x)$	$\displaystyle\equiv\delta_{\mu}^{\nu}-\frac{x_{\mu}x^{\nu}}{x\cdot x}\ .$		(3)

With lowered indices, the projector $P_{\mu}^{\nu}(x)$ becomes the 4d metric of $EAdS_{4}$ at $x$:

$\displaystyle g_{\mu\nu}(x)$

$\displaystyle\equiv P_{\mu\nu}(x)=\eta_{\mu\nu}-\frac{x_{\mu}x_{\nu}}{x\cdot x}\ .$

(4)

Our use of different letters for $P_{\mu}^{\nu}$ and $g_{\mu\nu}$ is purely cosmetic.

Since HS fields carry many symmetrized tensor indices, it is convenient to package them as functions of an auxiliary “polarization vector” $u^{\mu}\in\mathbb{R}^{1,4}$. Thus, we encode a rank-$p$ symmetric tensor by a function of the form:

$\displaystyle f(x,u)=\frac{1}{p!}\,u^{\mu_{1}}\!\dots u^{\mu_{p}}f_{\mu_{1}\dots\mu_{p}}(x)\ .$

(5)

We denote flat $\mathbb{R}^{1,4}$ derivatives w.r.t. $x^{\mu}$ and $u^{\mu}$ as $\partial_{x}^{\mu}$ and $\partial_{u}^{\mu}$, respectively. The tensor rank of $f_{\mu_{1}\dots\mu_{p}}$, and the fact that it’s tangential to the $EAdS_{4}$ hyperboloid, can be expressed as constraints on $f(x,u)$:

$\displaystyle(u\cdot\partial_{u})f=pf\ ;\quad(x\cdot\partial_{u})f=0\ .$

(6)

Tracing a pair of indices on $f_{\mu_{1}\dots\mu_{p}}$ is encoded by acting on $f(x,u)$ with the operator $\partial_{u}\cdot\partial_{u}$. A factor of the metric $EAdS_{4}$ metric (4) can be encoded as:

$\displaystyle g_{\mu\nu}u^{\mu}u^{\nu}=u\cdot u-\frac{(u\cdot x)^{2}}{x\cdot x}\ .$

(7)

It is convenient to introduce a notation for the traceless part of a symmetric $EAdS_{4}$ tensor $\hat{t}^{\mu_{1}}\dots\hat{t}^{\mu_{s}}$ at a point $x$. This traceless part can be encoded by the function:

$\displaystyle\begin{split}\mathcal{T}^{(p)}(x,\hat{t},u)&\equiv\frac{1}{p!}u^{\mu_{1}}\!\dots u^{\mu_{p}}\mathcal{T}_{\mu_{1}\dots\mu_{p}}(x,\hat{t})=\frac{(\hat{t}\cdot u)^{p}}{p!}-\text{traces}\\ &=\frac{1}{p!}\sum_{n=0}^{\lfloor p/2\rfloor}\binom{p-n}{n}\left(-\frac{1}{4}(\hat{t}\cdot\hat{t})(g_{\mu\nu}(x)u^{\mu}u^{\nu})\right)^{n}(\hat{t}\cdot u)^{p-2n}\\ &=\frac{1}{2^{p}p!}\sum_{n=0}^{\lfloor p/2\rfloor}\binom{p+1}{2n+1}\big{(}{-(\hat{t}\cdot\hat{t})}(q_{\mu\nu}(x,\hat{t})u^{\mu}u^{\nu})\big{)}^{n}(\hat{t}\cdot u)^{p-2n}\ ,\end{split}$

(8)

where, in the third line, we introduced the 3d metric $q_{\mu\nu}=g_{\mu\nu}-\frac{\hat{t}_{\mu}\hat{t}_{\nu}}{\hat{t}\cdot\hat{t}}$ of the subspace orthogonal to both $x^{\mu}$ and $\hat{t}^{\mu}$.

So far, everything was defined on the $EAdS_{4}$ hyperboloid $x\cdot x=-1$. The idea of the radial reduction approach Biswas:2002nk is to define our functions $f(x,u)$ also away from $x\cdot x=-1$, by introducing a scaling law of the form $(x\cdot\partial_{x})f=-\Delta f$ with some weight $\Delta$, usually chosen to match the conformal weight of relevant boundary data. This gives meaning to the 5d flat derivative $\partial_{x}^{\mu}$ in all directions, which can lead to substantial simplifications, in particular for the Sleight-Taronna vertex. Within this formalism, the $EAdS_{4}$ symmetrized gradient, divergence and Laplacian take the form:

	$\displaystyle u\cdot\nabla={}$	$\displaystyle u\cdot\partial_{x}+\frac{u\cdot x}{x\cdot x}(u\cdot\partial_{u}-x\cdot\partial_{x})\ ;$		(9)
	$\displaystyle\partial_{u}\cdot\nabla={}$	$\displaystyle\partial_{u}\cdot\partial_{x}+\frac{u\cdot x}{x\cdot x}(\partial_{u}\cdot\partial_{u})\ ;$		(10)
	$\displaystyle\begin{split}\nabla\cdot\nabla={}&\partial_{x}\cdot\partial_{x}+2\,\frac{u\cdot x}{x\cdot x}(\partial_{u}\cdot\partial_{x})+\left(\frac{u\cdot x}{x\cdot x}\right)^{2}(\partial_{u}\cdot\partial_{u})\\ &-\frac{1}{x\cdot x}\left((x\cdot\partial_{x})^{2}+3(x\cdot\partial_{x})-u\cdot\partial_{u}\right)\ .\end{split}$			(11)

In these expressions, we see two kinds of correction terms:

• •

  The $\sim u\cdot x$ terms serve to project the 5d derivatives back into $EAdS_{4}$.

• •

  The terms on the bottom line of (11) are just a constant multiple $\Delta(\Delta-3)-p$ of the $EAdS_{4}$ curvature -$\frac{1}{x\cdot x}$ (which we set equal to 1 in (1)). The 4d Laplacian $\nabla\cdot\nabla$ is then the $EAdS_{4}$ projection of the 5d d’Alembertian $\partial_{x}\cdot\partial_{x}$, shifted by this constant.

Let us review the form of Fronsdal’s field equations for linearized HS fields Fronsdal:1978vb in the above framework. In Fronsdal’s formalism, a spin-$s$ field (more precisely, gauge potential) is a totally symmetric rank-$s$ tensor with vanishing double trace. This can be encoded by a scalar function $h^{(s)}(x,u)$, as in (5). For its scaling weight, we choose $\Delta=1+s$ – the conformal weight of the dual boundary currents. This is the choice of Sleight:2016dba , which brings the Sleight-Taronna vertex into a simple form. Note that this weight is different from that in the general literature on HS cubic vertices Joung:2011ww , where the dual weight choice $\Delta=2-s$ is used. Overall, the constraints on the field $h^{(s)}(x,u)$ read:

	$\displaystyle(u\cdot\partial_{u})h^{(s)}$	$\displaystyle=sh^{(s)}\ ;$	$\displaystyle(x\cdot\partial_{u})h^{(s)}$	$\displaystyle=0\ ;$		(12)
	$\displaystyle(x\cdot\partial_{x})h^{(s)}$	$\displaystyle=-(s+1)h^{(s)}\ ;$	$\displaystyle(\partial_{u}\cdot\partial_{u})^{2}h^{(s)}$	$\displaystyle=0\ .$		(13)

Gauge transformations take the form:

$\displaystyle h^{(s)}\ \rightarrow\ h^{(s)}+(u\cdot\nabla_{x})\,\Lambda^{(s)}=h^{(s)}+\left(u\cdot\partial_{x}+(2s-1)\frac{u\cdot x}{x\cdot x}\right)\Lambda^{(s)}\ ,$

(14)

where $\Lambda^{(s)}$ represents a traceless gauge parameter with $s-1$ tensor indices and weight $\Delta=s$, i.e.:

	$\displaystyle(u\cdot\partial_{u})\Lambda^{(s)}$	$\displaystyle=(s-1)\Lambda^{(s)}\ ;$	$\displaystyle(x\cdot\partial_{u})\Lambda^{(s)}$	$\displaystyle=0\ ;$		(15)
	$\displaystyle(x\cdot\partial_{x})\Lambda^{(s)}$	$\displaystyle=-s\Lambda^{(s)}\ ;$	$\displaystyle(\partial_{u}\cdot\partial_{u})\Lambda^{(s)}$	$\displaystyle=0\ .$		(16)

Out of the field $h^{(s)}$, we can construct a gauge-invariant curvature, which generalizes the $s=2$ linearized Ricci tensor to all spins. This is the Fronsdal tensor $\mathcal{F}h^{(s)}$, where the operator $\mathcal{F}$ is given by:

$\displaystyle\begin{split}\mathcal{F}={}&-\nabla\cdot\nabla+\frac{2+2s-s^{2}}{x\cdot x}+(u\cdot\nabla)(\partial_{u}\cdot\nabla)-\left(\frac{1}{2}(u\cdot\nabla)^{2}+\frac{g_{\mu\nu}u^{\mu}u^{\nu}}{x\cdot x}\right)(\partial_{u}\cdot\partial_{u})\\ ={}&-\partial_{x}\cdot\partial_{x}+\left(u\cdot\partial_{x}+(2s-1)\frac{u\cdot x}{x\cdot x}\right)(\partial_{u}\cdot\partial_{x})\\ &-\left(\frac{u\cdot u}{x\cdot x}+\frac{1}{2}\left(u\cdot\partial_{x}+(2s+1)\frac{u\cdot x}{x\cdot x}\right)\left(u\cdot\partial_{x}+(2s-3)\frac{u\cdot x}{x\cdot x}\right)\right)(\partial_{u}\cdot\partial_{u})\ .\end{split}$

(17)

$\mathcal{F}$ is a second-order differential operator with respect to $x$. The Fronsdal tensor $\mathcal{F}h^{(s)}$ has the same tensor properties as the potential $h^{(s)}$, but with scaling weight increased by 2:

	$\displaystyle(u\cdot\partial_{u})\mathcal{F}h^{(s)}$	$\displaystyle=s\mathcal{F}h^{(s)}\ ;$	$\displaystyle(x\cdot\partial_{u})\mathcal{F}h^{(s)}$	$\displaystyle=0\ ;$		(18)
	$\displaystyle(x\cdot\partial_{x})\mathcal{F}h^{(s)}$	$\displaystyle=-(s+3)\mathcal{F}h^{(s)}\ ;$	$\displaystyle(\partial_{u}\cdot\partial_{u})^{2}\mathcal{F}h^{(s)}$	$\displaystyle=0\ .$		(19)

In analogy with GR, we can rearrange the trace of $\mathcal{F}h^{(s)}$ to obtain the Einstein tensor:

$\displaystyle\mathcal{G}h^{(s)}=\left(1-\frac{1}{4}(g_{\mu\nu}u^{\mu}u^{\nu})(\partial_{u}\cdot\partial_{u})\right)\mathcal{F}h^{(s)}\ .$

(20)

This has the same tensor properties (18)-(19), but also satisfies a conservation law of the form:

$\displaystyle(\partial_{u}\cdot\nabla)\mathcal{G}h^{(s)}=(g_{\mu\nu}u^{\mu}u^{\nu})(\dots)\ ,$

(21)

i.e. the $EAdS_{4}$ divergence of $\mathcal{G}h^{(s)}$ vanishes up to trace terms. This allows us to write a gauge-invariant quadratic action for linearized HS fields:

$\displaystyle S_{s}=\int_{EAdS_{4}}d^{4}x\,s!\,h^{(s)}(x,\partial_{u})\left(\frac{1}{2}\mathcal{G}h^{(s)}(x,u)-J^{(s)}(x,u)\right)\ .$

(22)

Here, $J^{(s)}(x,u)$ is an external HS current, which must be conserved in the same sense (21) as $\mathcal{G}h^{(s)}$. The field equations for the action (22) read simply:

$\displaystyle\mathcal{G}h^{(s)}(x,u)=J^{(s)}(x,u)\ .$

(23)

This formalism for HS theory is substantially simplified in a traceless gauge (which can also be viewed as a framework in its own right Skvortsov:2007kz ; Campoleoni:2012th ). In this gauge, the double-traceless condition $(\partial_{u}\cdot\partial_{u})^{2}h^{(s)}=0$ is strengthened into ordinary tracelessness $(\partial_{u}\cdot\partial_{u})h^{(s)}=0$. The remaining gauge freedom is parameterized by (14)-(16), with the further constraint:

$\displaystyle(\partial_{u}\cdot\nabla)\Lambda^{(s)}=0\ .$

(24)

Since $\Lambda^{(s)}$ is traceless, we see from (10) that its 4d divergence $(\partial_{u}\cdot\nabla)\Lambda^{(s)}$ is equal to the 5d one $(\partial_{u}\cdot\partial_{x})\Lambda^{(s)}$. Thus, the constraint (24) can also be written as:

$\displaystyle(\partial_{u}\cdot\partial_{x})\Lambda^{(s)}=0\ .$

(25)

In this gauge, the Fronsdal operator (17) simplifies into:

$\displaystyle\begin{split}\mathcal{F}&=-\nabla\cdot\nabla+\frac{2+2s-s^{2}}{x\cdot x}+(u\cdot\nabla)(\partial_{u}\cdot\nabla)\\ &=-\partial_{x}\cdot\partial_{x}+\left(u\cdot\partial_{x}+(2s-1)\frac{u\cdot x}{x\cdot x}\right)(\partial_{u}\cdot\partial_{x})\ .\end{split}$

(26)

Note also that the trace of the Fronsdal tensor now reads simply:

$\displaystyle(\partial_{u}\cdot\partial_{u})\mathcal{F}h^{(s)}=2(\partial_{u}\cdot\nabla)^{2}h^{(s)}=2(\partial_{u}\cdot\partial_{x})^{2}h^{(s)}\ .$

(27)

With the exception of section 6, we will work in traceless gauge throughout. For source-free fields, one can specialize further to transverse-traceless gauge, by imposing also the zero-divergence condition $(\partial_{u}\cdot\nabla)h^{(s)}=0$, or, equivalently, $(\partial_{u}\cdot\partial_{x})h^{(s)}=0$. A gauge parameter that preserves traceless gauge, i.e. that satisfies (24)-(25), will shift the divergence of $h^{(s)}$ as:

$\displaystyle(\partial_{u}\cdot\nabla)h^{(s)}\ \rightarrow\ (\partial_{u}\cdot\nabla)h^{(s)}+\left(\nabla\cdot\nabla+\frac{s^{2}-1}{x\cdot x}\right)\Lambda^{(s)}\ ,$

(28)

or, equivalently:

$\displaystyle(\partial_{u}\cdot\partial_{x})h^{(s)}\ \rightarrow\ (\partial_{u}\cdot\partial_{x})h^{(s)}+\left(\partial_{x}\cdot\partial_{x}+\frac{2(2s-1)}{x\cdot x}\right)\Lambda^{(s)}\ .$

(29)

The 3d boundary of $EAdS_{4}$ is given by the projective lightcone in $\mathbb{R}^{1,4}$, i.e. by null vectors $\ell^{\mu}\in\mathbb{R}^{1,4}$, $\ell\cdot\ell=0$, modulo rescalings $\ell^{\mu}\cong\rho\ell^{\mu}$. Boundary quantities will transform under such rescalings as $(\ell\cdot\partial_{\ell})f=-\Delta f$, according to their conformal weights $\Delta$. We describe 3d vectors at a boundary point $\ell^{\mu}$ as 5d vectors $\lambda^{\mu}$ that satisfy $\lambda\cdot\ell=0$, modulo shifts $\lambda^{\mu}\cong\lambda^{\mu}+\alpha\ell^{\mu}$. For a boundary scalar $f(\ell)$ with weight $\Delta=\frac{1}{2}$, we can define the conformal Laplacian $\Box_{\ell}f$. In the embedding-space language, this is the same as the 5d d’Alambertian $(\partial_{\ell}\cdot\partial_{\ell})f$, provided that $f$ is extended away from the $\ell\cdot\ell=0$ lightcone in a way that preserves the scaling law $(\ell\cdot\partial_{\ell})f=-\frac{1}{2}f$. The operator $\Box_{\ell}$ itself has conformal weight 2.

The CFT that lives on our 3d boundary is a free $O(N)$ vector model. It is convenient to assume that $N$ is even, and package the vector model’s $N$ real fields as $\frac{N}{2}$ complex fields $\chi^{I}(\ell)$ with complex conjugates $\bar{\chi}_{I}(\ell)$, where $I=1,\dots,\frac{N}{2}$ is a color index. The theory then takes the form of a $U(N/2)$ vector model, whose action reads simply:

$\displaystyle S_{\text{CFT}}=-\int d^{3}\ell\,\bar{\chi}_{I}(\ell)\Box_{\ell}\chi^{I}(\ell)\ ,$

(30)

where $\chi^{I}$ and $\bar{\chi}_{I}$ each have conformal weight $\Delta=\frac{1}{2}$. The propagator for these fundamental fields reads:

$\displaystyle G_{\text{CFT}}(\ell,\ell^{\prime})=\frac{1}{4\pi\sqrt{-2\ell\cdot\ell^{\prime}}}\ ;\quad\Box_{\ell}G_{\text{CFT}}(\ell,\ell^{\prime})=-\delta^{\frac{5}{2},\frac{1}{2}}(\ell,\ell^{\prime})\ ,$

(31)

where the superscripts on the boundary delta function $\delta(\ell,\ell^{\prime})$ denote its conformal weight with respect to each argument.

The fundamental single-trace operators in the theory (30) are the bilocals:

$\displaystyle\mathcal{O}(\ell,\ell^{\prime})\equiv\frac{2\chi^{I}(\ell)\bar{\chi}_{I}(\ell^{\prime})}{G(\ell,\ell^{\prime})}\ .$

(32)

Here, we made an unconventional normalization choice, which makes $\mathcal{O}(\ell,\ell^{\prime})$ invariant under rescalings of $\ell,\ell^{\prime}$. Thus, our $\mathcal{O}(\ell,\ell^{\prime})$ depends only on the actual choice of two boundary points, which will allow a cleaner interpretation of the bulk dual. The numerical factor in (32) is chosen to ensure the proper relative normalization of the first and second terms in eq. (123) below.

By Taylor-expanding the bilocals (32) around $\ell=\ell^{\prime}$, we obtain the local single-trace primaries, i.e. the tower of HS currents Craigie:1983fb ; Anselmi:1999bb ; David:2020ptn (including the honorary spin-0 “current” $\bar{\chi}_{I}(\ell)\chi^{I}(\ell)$). These local currents can be encoded conveniently by contracting their indices with a null polarization vector $\lambda^{\mu}$ at $\ell^{\mu}$, satisfying $\lambda\cdot\lambda=\lambda\cdot\ell=0$:

$\displaystyle j^{(s)}(\ell,\lambda)=\lambda^{\mu_{1}}\!\dots\lambda^{\mu_{s}}j_{\mu_{1}\dots\mu_{s}}(\ell)\ .$

(33)

The currents’ relation to the bilocal (32) is then expressed compactly via a differential operator $D^{(s)}$, as:

	$\displaystyle\begin{split}j^{(s)}(\ell,\lambda)&=D^{(s)}(\partial_{\ell},\partial_{\ell^{\prime}},\lambda)\big{[}\chi^{I}(\ell)\bar{\chi}_{I}(\ell^{\prime})\big{]}\Big{|}_{\ell=\ell^{\prime}}\\ &=\frac{1}{2}\,D^{(s)}(\partial_{\ell},\partial_{\ell^{\prime}},\lambda)\big{[}G(\ell,\ell^{\prime})\mathcal{O}(\ell,\ell^{\prime})\big{]}\Big{|}_{\ell=\ell^{\prime}}\ ;\end{split}$			(34)
	$\displaystyle D^{(s)}(\partial_{\ell},\partial_{\ell^{\prime}},\lambda)$	$\displaystyle=i^{s}\sum_{m=0}^{s}(-1)^{m}\binom{2s}{2m}(\lambda\cdot\partial_{\ell})^{m}(\lambda\cdot\partial_{\ell^{\prime}})^{s-m}\ .$		(35)

The connected correlators of bilocals (32) are given by simple 1-loop Feynman diagrams composed of propagators (31) (see figure 2), with the normalization factor in (32) simply along for the ride:

$\displaystyle\left<\mathcal{O}(\ell_{1},\ell_{1}^{\prime})\dots\mathcal{O}(\ell_{n},\ell_{n}^{\prime})\right>=\frac{2^{n}}{\prod_{p=1}^{n}G(\ell_{p},\ell_{p}^{\prime})}\times\frac{N}{2}\left(\prod_{p=1}^{n}G(\ell_{p}^{\prime},\ell_{p+1})+\text{permutations}\right)\ ,$

(36)

where the product in the numerator is cyclic, i.e. $\ell_{n+1}\equiv\ell_{1}$, and the sum is over cyclically inequivalent permutations of $(1,\dots,n)$. From these bilocal correlators, one can derive the correlators of local currents $j^{(s)}$, via the Taylor expansion (34).

Up to the boundary field equation $\Box_{\ell}\chi^{I}(\ell)=\Box_{\ell^{\prime}}\chi_{I}(\ell^{\prime})=0$, the local currents (34) span the full space of single-trace operators. This means in particular that, given two points $\ell,\ell^{\prime}$ and a compact boundary region $B$ that includes them, the bilocal $\mathcal{O}(\ell,\ell^{\prime})$ is equivalent to some superposition of local currents (34) inside $B$:

$\displaystyle\mathcal{O}(\ell,\ell^{\prime})\cong\sum_{s=0}^{\infty}\int_{B}d^{3}L\,A^{(s)}_{\ell,\ell^{\prime}}(L,\partial_{\lambda})\,j^{(s)}(L,\lambda)\ ,$

(37)

where $A^{(s)}_{\ell,\ell^{\prime}}(L,\lambda)$ is some configuration of traceless spin-$s$ sources at the boundary point $L$:

$\displaystyle(\lambda\cdot\partial_{\lambda})A^{(s)}=sA^{(s)}\ ;\quad(L\cdot\partial_{L})A^{(s)}=(s-2)A^{(s)}\ ;\quad(\partial_{\lambda}\cdot\partial_{\lambda})A^{(s)}=(L\cdot\partial_{\lambda})A^{(s)}=0\ .$

(38)

The sense in which the equivalence (37) holds is that the LHS and RHS have the same correlators with any number of operators $\mathcal{O}$ or $j^{(s)}$ with support in the complement $\bar{B}$ of $B$. On the other hand, to check that some configuration of local sources $A^{(s)}$ in $B$ satisfies (37), it is sufficient to check just the quadratic correlators with local currents $j^{(s)}$ in $\bar{B}$. This can be seen in two steps. First, in any correlator of one single-trace operator in $B$ and $(n-1)$ such operators in $\bar{B}$, the diagrams (36) are always arranged such that the operator in $B$ effectively couples to a single bilocal in $\bar{B}$ (see figure 2). Thus, it’s enough to match the quadratic correlators with bilocals in $\bar{B}$. But, using now the equivalence (37) for $\bar{B}$, we see that these can be reconstructed from the quadratic correlators with local currents.

Again, the theory described above is not quite the $O(N)$ vector model, but the $U(N/2)$ one. However, we can obtain the $O(N)$ model by simply truncating the single-trace operators (32),(34) from all those invariant under $U(N/2)$ to those invariant under the larger group $O(N)$. For the bilocals (32), this requires symmetrizing over $\ell\leftrightarrow\ell^{\prime}$:

$\displaystyle\mathcal{O}^{+}(\ell,\ell^{\prime})=\frac{1}{2}\big{(}\mathcal{O}(\ell,\ell^{\prime})+\mathcal{O}(\ell^{\prime},\ell)\big{)}\ ,$

(39)

whereas for the local currents (34), it requires restricting to even spins $s$. It’s easy to see that the even-spin currents $j^{(s)}$ can indeed be constructed from the symmetrized bilocal (39). For odd $N$, the above construction starting from $U(N/2)$ doesn’t directly apply. However, the end results for the correlators are the same, with $N$ simply an overall prefactor, as in (36).

In this subsection, we set up a framework for discussing the asymptotic behavior of fields in $EAdS_{4}$. For this purpose, it’s convenient to use Poincare coordinates $(z,y^{a})$ for $EAdS_{4}$:

$\displaystyle x^{\mu}(z,y^{a})=\frac{1}{z}\left(\frac{1+z^{2}+y^{2}}{2},\frac{1-z^{2}-y^{2}}{2},y^{a}\right)\ ;\quad dx\cdot dx=\frac{dz^{2}+dy^{2}}{z^{2}}\ ,$

(40)

where $y^{2}\equiv\delta_{ab}y^{a}y^{b}$. The boundary of $EAdS_{4}$ can be similarly parameterized as:

$\displaystyle\ell^{\mu}(y^{a})=\left(\frac{1+y^{2}}{2},\frac{1-y^{2}}{2},y^{a}\right)\ ;\quad d\ell\cdot d\ell=dy^{2}\ .$

(41)

The parameterization (41) chooses a flat section of the $\mathbb{R}^{1,4}$ lightcone, defined by $\ell\cdot n=-\frac{1}{2}$, where $n^{\mu}=\left(\frac{1}{2},-\frac{1}{2},\vec{0}\right)$. The bulk and boundary coordinates (40)-(41) are related by:

$\displaystyle x^{\mu}(z,y^{a})=\frac{1}{z}\ell^{\mu}(y^{a})+zn^{\mu}\ .$

(42)

In the limit $z\rightarrow 0$, the bulk point $x(z,y^{a})$ asymptotes to the boundary point $\ell(y^{a})$, in the precise manner defined by (42).

To study the asymptotics of tensor fields, it is convenient to use an orthonormal basis $(e_{0},e_{a})$ along the $(z,y^{a})$ coordinate axes:

$\displaystyle e_{0}^{\mu}(z,y^{a})=-z\frac{\partial x^{\mu}}{\partial z}\ ;\quad e_{a}^{\mu}(y^{a})=z\frac{\partial x^{\mu}}{\partial y^{a}}\ .$

(43)

In the boundary limit $z\rightarrow 0$, the $\mathbb{R}^{1,4}$ components of the “tangential” basis vectors $e_{a}^{\mu}$ are $z$-independent, while those of the “radial” vector $e_{0}^{\mu}$ behave as:

$\displaystyle e_{0}^{\mu}(z,y^{a})=x^{\mu}(z,y^{a})+O(z)=\frac{1}{z}\ell^{\mu}(y^{a})+O(z)\ .$

(44)

We can now discuss the asymptotics of symmetric bulk tensor fields (5) by describing the $z\rightarrow 0$ scaling of their different components in the orthonormal $(e_{0},e_{a})$ basis. For a rank-$p$ field $f(x,u)$, we’ll use the compact notation $[f]_{q,p-q}$ to refer to its components with $q$ indices along $e_{0}$ and $p-q$ indices along $e_{a}$.

The boundary-bulk propagators dual to the boundary HS currents (34) read Mikhailov:2002bp :

$\displaystyle\Pi^{(s)}(x,u;\ell,\lambda)=-\frac{(\sqrt{2})^{s}(m\cdot u)^{s}}{16\pi^{2}(\ell\cdot x)^{2s+1}}\ ;\quad m^{\mu}(x;\ell,\lambda)\equiv(\lambda\cdot x)\ell^{\mu}-(\ell\cdot x)\lambda^{\mu}\ ,$

(45)

where we chose a non-standard normalization for later convenience. With respect to its bulk arguments $(x,u)$, the propagator $\Pi^{(s)}$ satisfies the standard constraints (12)-(13) for a Fronsdal field, as well as the traceless and transverse gauge conditions $(\partial_{u}\cdot\partial_{u})\Pi^{(s)}=(\partial_{u}\cdot\nabla)\Pi^{(s)}=0$. With respect to its boundary arguments $(\ell,\lambda)$, $\Pi^{(s)}$ has the same conformal weight $(\ell\cdot\partial_{\ell})\Pi^{(s)}=-(s+1)\Pi^{(s)}$ and tensor rank $(\lambda\cdot\partial_{\lambda})\Pi^{(s)}=s\Pi^{(s)}$ as the boundary currents (34), and is invariant under the shift symmetry $\lambda^{\mu}\rightarrow\lambda^{\mu}+\alpha\ell^{\mu}$.

The propagator (45) is a special case $(p,w)=(s,s+1)$ of the general formula:

$\displaystyle f(x,u;\ell,\lambda)\sim\frac{(m\cdot u)^{p}}{(\ell\cdot x)^{p+w}}\ ,$

(46)

which spans the solution space of the free field equations for rank-$p$ symmetric, transverse-traceless fields with arbitrary mass parameterized by $w$:

$\displaystyle(\partial_{u}\cdot\partial_{u})f=(\partial_{u}\cdot\nabla)f=\left(\nabla\cdot\nabla-\frac{w(3-w)+p}{x\cdot x}\right)f=0\ .$

(47)

Let’s now apply the formalism of section 2.4 to discuss the asymptotic behavior of the general propagator (46). Let us choose Poincare coordinates (40)-(41) such that the boundary source point $\ell^{\mu}$ in (46) is at $y^{a}=0$, i.e. $\ell^{\mu}=\left(\frac{1}{2},\frac{1}{2},\vec{0}\right)$. We can also choose the polarization vector $\lambda^{\mu}$ as $\lambda^{\mu}=(0,0,\lambda^{a})$, which becomes $\lambda^{\mu}=\lambda^{a}e_{a}^{\mu}$ in terms of the orthonormal basis at $y^{a}=0$. The ingredients of the tensor field (46) at an arbitrary bulk point $x^{\mu}(z,y^{a})$ now read:

$\displaystyle\ell\cdot x=-\frac{z^{2}+y^{2}}{2z}\ ;\quad m^{\mu}=(\lambda\cdot y)e_{0}^{\mu}+\frac{1}{z}\left(\frac{z^{2}+y^{2}}{2}\lambda^{a}-(\lambda\cdot y)y^{a}\right)e_{a}^{\mu}\ ,$

(48)

where $\lambda\cdot y\equiv\delta_{ab}\lambda^{a}y^{b}$. Assuming $y^{a}\neq 0$, we see that in the small-$z$ limit $\ell\cdot x$ scales as $z^{-1}$, while $m^{\mu}$ has $\sim z^{-1}$ components along $e_{a}^{\mu}$ and a $\sim z^{0}$ component along $e_{0}^{\mu}$. Thus, at $y^{a}\neq 0$, the various components of $f$ scale at small $z$ as:

$\displaystyle y^{a}\neq 0:\quad[f]_{q,p-q}\sim z^{w+q}\ .$

(49)

Now, note that under $w\rightarrow 3-w$, the field equations (47) do not change. Therefore, the same field equations must also support the asymptotics $[f]_{q,p-q}\sim z^{3-w+q}$. In a neighborhood of the boundary, the two asymptotics $\sim z^{w+q}$ and $\sim z^{3-w+q}$ constitute a pair of independent boundary data (more precisely, within each set, it is the $q=0$ data that’s independent, with the $q>0$ data determined from it). For a regular solution in all of $EAdS_{4}$, these two boundary data cease to be independent, i.e. one becomes linearly determined by the other. In particular, a closer inspection of the solution (46) reveals that it also contains the “other” asymptotics $\sim z^{3-w+q}$, as a delta-function-like distribution with support at $y^{a}=0$. Rotational invariance and the dilatation symmetry $(z,y^{a})\rightarrow(\rho z,\rho y^{a})$ fix this delta-function-like piece to take the form:

$\displaystyle y^{a}=0:\quad[f]_{q,p-q}\sim z^{3-w+q}(e_{0}\cdot u)^{q}(\lambda\cdot u)^{p-q}(\lambda\cdot\partial_{y})^{q}\,\delta^{3}(y)\ ,$

(50)

where $\lambda\cdot\partial_{y}\equiv\lambda^{a}\frac{\partial}{\partial y^{a}}$. Specializing back to $(p,w)=(s,s+1)$, we obtain, for our original propagator (45):

	$\displaystyle y^{a}\neq 0:\quad$	$\displaystyle[\Pi^{(s)}]_{q,s-q}\sim z^{s+1+q}\ ;$		(51)
	$\displaystyle y^{a}=0:\quad$	$\displaystyle[\Pi^{(s)}]_{q,s-q}\sim z^{2-s+q}(e_{0}\cdot u)^{q}(\lambda\cdot u)^{s-q}(\lambda\cdot\partial_{y})^{q}\,\delta^{3}(y)\ .$		(52)

The Didenko-Vasiliev solution is the field of an HS-charged source concentrated on a bulk geodesic. Before describing the solution and its properties, it is useful to discuss bulk geodesics in their own right.

A geodesic in $EAdS_{4}$ is a hyperbola in the $\mathbb{R}^{1,4}$ embedding space. The hyperbola’s asymptotes are two lightrays through the origin in $\mathbb{R}^{1,4}$, or, equivalently, two points on the conformal boundary of $EAdS_{4}$. In fact, (oriented) bulk geodesics are in one-to-one correspondence with (ordered) pairs of boundary points. We can parameterize a geodesic’s boundary endpoints by two lightlike vectors $\ell^{\mu},\ell^{\prime\mu}$, keeping in mind the usual redundancy of such vectors under rescalings. The geodesic itself can then be parameterized as:

$\displaystyle\gamma(\ell,\ell^{\prime}):\quad x^{\mu}(\tau;\ell,\ell^{\prime})=\frac{e^{\tau}\ell^{\mu}+e^{-\tau}\ell^{\prime\mu}}{\sqrt{-2\ell\cdot\ell^{\prime}}}\ ,$

(53)

where $\tau$ is a proper-length parameter. If we allow rescalings of $x^{\mu}$ away from the $EAdS_{4}$ hyperboloid $x\cdot x=-1$, then the geodesic (53) becomes just a 2d plane in the $\mathbb{R}^{1,4}$ embedding space – the plane spanned by $\ell^{\mu},\ell^{\prime\mu}$.

The distance of a bulk point $x\in EAdS_{4}$ from a geodesic $\gamma(\ell,\ell^{\prime})$ can be parameterized by the function:

$\displaystyle R(x;\ell,\ell^{\prime})=\sqrt{\frac{2(\ell\cdot x)(\ell^{\prime}\cdot x)}{(\ell\cdot\ell^{\prime})(x\cdot x)}-1}\ .$

(54)

This has weight 0 (i.e. is invariant) under rescalings of $\ell^{\mu},\ell^{\prime\mu}$, as well as rescalings of $x^{\mu}$. For $x^{\mu}$ on the $x\cdot x=-1$ hyperboloid, $R(x;\ell,\ell^{\prime})$ is just the flat $\mathbb{R}^{1,4}$ distance between $x^{\mu}$ and the $(\ell,\ell^{\prime})$ plane. This is related to the geodesic $EAdS_{4}$ distance $\chi$ as $R=\sinh\chi$.