Boundary Terms and Quantization in the Einstein–Scalar AdS_4 System

Bulk Action and Necessary Boundary Terms

Setup near the $\text{AdS}_4$ boundary (what we assume on this page)

We work in Poincaré $AdS_4$ with radius $L$,

$$ds^2=\frac{L^2}{r^2}\Big(dr^2+\eta_{ij}dx^i dx^j\Big),\qquad r\to 0\ \text{is the conformal boundary}.$$

We regulate the boundary at $r=\varepsilon$ and only at the end take $\varepsilon\to 0$. On $r=\varepsilon$,

$$\gamma_{ij}=\frac{L^2}{\varepsilon^2}\eta_{ij},\qquad \sqrt{-\gamma}=\frac{L^3}{\varepsilon^3}.$$

A scalar of mass $m^2$ has asymptotic falloffs

$$\phi(r,x)=r^{\Delta_-}\phi_{(0)}(x)+r^{\Delta_+}\phi_{(1)}(x)+\cdots,\qquad \Delta_\pm=\frac{3}{2}\pm\nu,\quad \nu=\sqrt{\frac{9}{4}+m^2L^2},$$

so $\Delta_++\Delta_-=3$.

We consider the Einstein–scalar action in four bulk dimensions with a negative cosmological constant (AdS) and a scalar potential chosen to give mass $m^2$ to the field $\phi$. A convenient form of the action is:

$$S_{\text{bulk}} \;=\; \frac{1}{16\pi G}\int_{M} d^4x\,\sqrt{-g}\Big( R - 2\Lambda \Big)\;-\; \frac{1}{2}\int_{M} d^4x\,\sqrt{-g}\Big(g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi + m^2 \phi^2\Big)\,,$$

where $\Lambda = -\frac{3}{L^2}$ for AdS$_4$ (so that the AdS radius $L$ satisfies $R_{\text{AdS}}=-4\Lambda=12/L^2$). We set $L=1$ units in intermediate formulas for simplicity. This bulk action $S_{\text{bulk}}$ alone is not yet sufficient for a well-defined variational principle because of boundary contributions that arise upon varying the fields. There are two types of boundary terms we must add:

• The Gibbons–Hawking–York (GHY) term for the gravitational part, to ensure a well-posed Dirichlet problem for the metric. This term is added on the boundary $\partial M$ at infinity (conformal boundary of AdS) and is given by $S_{\text{GHY}} = \frac{1}{8\pi G}\int_{\partial M} d^3x\,\sqrt{-\gamma}\,K\,,$ where $\gamma_{ij}$ is the induced metric on the boundary hypersurface and $K$ is the extrinsic curvature of $\partial M$ (the trace of $K_{ij}$). The variation of the Einstein–Hilbert action yields a boundary term proportional to $(K_{ij}-K\,\gamma_{ij})\delta\gamma^{ij}$; the GHY term is engineered to cancel this, so that the net gravitational boundary variation is zero when the boundary metric is held fixed. In other words, $S_{\text{EH}}+S_{\text{GHY}}$ yields a well-posed variational principle for gravity with Dirichlet boundary conditions $\delta\gamma_{ij}=0$ on $\partial M$.

• Scalar boundary terms, which come in two kinds:

1. Boundary terms required for a well-posed variational principle for $\phi$, depending on the choice of boundary condition (fixing either $\phi$ or its normal derivative at the boundary, as we explore below). In certain cases, this may include an “improvement” term added to the action to implement a desired boundary condition (for example, a term that effectively performs a Legendre transform of the action).
2. Counterterms to cancel divergences of the on-shell action (holographic renormalization). These are local functions of boundary fields (and curvature) added to make the action finite as the regulator is removed. They do not affect the equations of motion in the bulk (since they are intrinsic to $\partial M$), but their presence can modify the boundary conditions through their contribution to the boundary variation.

Let us vary the total action step by step. We denote the total action as

$$S_{\text{tot}} = S_{\text{bulk}} + S_{\text{GHY}} + S_{\text{bdy}}\,,$$

where $S_{\text{bdy}}$ will include scalar boundary terms and counterterms introduced later.

Why boundary terms are unavoidable

There are two distinct reasons boundary terms appear:

1. Well-posed variational principle:
   After varying the action and integrating by parts, you get boundary contributions. To ensure $\delta S=0$ on-shell for your chosen boundary conditions, those contributions must vanish—sometimes automatically (e.g. Dirichlet), sometimes only after adding extra boundary terms (e.g. Neumann/alternative quantization).

2. Finiteness (holographic renormalization):
   Even on-shell, the action diverges as the cutoff $\varepsilon\to 0$. You cancel divergences by adding local counterterms on the regulated boundary $r=\varepsilon$. These counterterms also affect the boundary variation, so they must be included when discussing which boundary conditions are “natural”.

Variation of the bulk action and canonical boundary data

Varying the bulk action gives bulk equations of motion plus boundary terms. Schematically,

$$\delta S_{\text{bulk}} = \int_M d^4x\,\sqrt{-g}\Big( E^{\mu\nu}\,\delta g_{\mu\nu} + \mathcal{E}_\phi\,\delta\phi \Big) \;+\; \int_{\partial M} d^3x\,\Big( \Pi^{ij}\,\delta\gamma_{ij} + \pi_\phi\,\delta\phi\Big)\,,$$

where $E^{\mu\nu}=0$ and $\mathcal{E}_\phi=0$ are the bulk Einstein and Klein–Gordon equations, and $\gamma_{ij}$ is the induced metric on the cutoff surface. The gravitational momentum is

$$\Pi^{ij}=\frac{\sqrt{-\gamma}}{16\pi G}(K^{ij}-K\gamma^{ij})\,,$$

and the (densitized) scalar momentum is

$$\pi_\phi \equiv -\sqrt{-\gamma}\,n^\mu\partial_\mu\phi\,,$$

so that the scalar boundary variation takes the clean form $\delta S_\phi|_{\text{EOM}}=\int_{\partial M} d^3x\,\pi_\phi\,\delta\phi$.

The GHY term is designed so that the combined gravitational action $S_{\text{EH}}+S_{\text{GHY}}$ has no normal-derivative variations of $\delta g_{\mu\nu}$; equivalently, on shell,

$$\delta\big(S_{\text{EH}}+S_{\text{GHY}}\big)\Big|_{\text{EOM}} = \frac12\int_{\partial M} d^3x\,\sqrt{-\gamma}\,T_{\text{BY}}^{ij}\,\delta\gamma_{ij}\,,$$

so Dirichlet boundary conditions $\delta\gamma_{ij}=0$ make the gravitational variation vanish.

A careful near-boundary expansion: why $\delta S_{\text{ren}}\propto \phi_{(1)}\,\delta\phi_{(0)}$

A common point of confusion is the step from the raw boundary term $\int \pi_\phi\,\delta\phi$ to the renormalized on-shell variation written in AdS/CFT as $\delta S_{\text{ren}}=\int \phi_{(1)}\,\delta\phi_{(0)}$ (up to a conventional normalization). The key ingredients are:

1. $\pi_\phi\,\delta\phi$ contains cross terms between $(\phi_{(0)},\phi_{(1)})$, not only $\phi_{(0)}\delta\phi_{(0)}$ and $\phi_{(1)}\delta\phi_{(1)}$.
2. The standard quadratic scalar counterterm cancels both the divergence and the unwanted finite $\phi_{(0)}\delta\phi_{(1)}$ term, leaving only $\phi_{(1)}\delta\phi_{(0)}$.

Take the Poincaré asymptotics stated in the note above and regulate at $r=\varepsilon$. With outward unit normal pointing toward the boundary ($r\to 0$), one has $n^r=-r/L$, so

$$\pi_\phi=-\sqrt{-\gamma}\,n^r\partial_r\phi=\frac{L^2}{r^2}\,\partial_r\phi \qquad\Rightarrow\qquad (\text{set }L=1)\ \ \pi_\phi=r^{-2}\partial_r\phi\,.$$

Using

$$\phi=r^{\Delta_-}\phi_{(0)}+r^{\Delta_+}\phi_{(1)}+\cdots, \qquad \delta\phi=r^{\Delta_-}\delta\phi_{(0)}+r^{\Delta_+}\delta\phi_{(1)}+\cdots,$$

we find

$$\pi_\phi=\Delta_-\,\phi_{(0)}\,r^{\Delta_- -3}+\Delta_+\,\phi_{(1)}\,r^{\Delta_+ -3}+\cdots.$$

Multiplying out (and using $\Delta_-+\Delta_+=3$) gives

$$\pi_\phi\,\delta\phi = \underbrace{\Delta_-\,\phi_{(0)}\delta\phi_{(0)}\,r^{2\Delta_- -3}}_{\text{divergent as }r\to 0} + \underbrace{\Big(\Delta_-\,\phi_{(0)}\delta\phi_{(1)}+\Delta_+\,\phi_{(1)}\delta\phi_{(0)}\Big)}_{\text{finite}} + \underbrace{\Delta_+\,\phi_{(1)}\delta\phi_{(1)}\,r^{2\Delta_+ -3}}_{\sim r^{2\nu}\to 0} +\cdots.$$

So, before renormalization, the on-shell variation contains a divergence (the $r^{2\Delta_- -3}=r^{-2\nu}$ term) and it also contains a finite term proportional to $\phi_{(0)}\delta\phi_{(1)}$.

Now add the standard quadratic scalar counterterm (for flat boundary and $x$-independent leading data; derivative/curvature terms can be added when needed)

$$S_{\text{ct}}^{(\phi)}=-\frac12\int_{r=\varepsilon} d^3x\,\sqrt{-\gamma}\,\frac{\Delta_-}{L}\,\phi^2\,.$$

Its variation contributes

$$\delta S_{\text{ct}}^{(\phi)} = -\int d^3x\,\sqrt{-\gamma}\,\frac{\Delta_-}{L}\,\phi\,\delta\phi = -\int d^3x\Big[ \Delta_-\,\phi_{(0)}\delta\phi_{(0)}\,r^{2\Delta_- -3} +\Delta_-\,\phi_{(0)}\delta\phi_{(1)} +\Delta_-\,\phi_{(1)}\delta\phi_{(0)} +\cdots\Big].$$

Adding $\delta S_\phi|_{\text{EOM}}=\int \pi_\phi\,\delta\phi$ and $\delta S_{\text{ct}}^{(\phi)}$ cancels the divergent piece and the $\phi_{(0)}\delta\phi_{(1)}$ piece, leaving the finite renormalized on-shell variation

$$\delta S_{\text{ren}}\Big|_{\text{EOM}} = (\Delta_+-\Delta_-)\int d^3x\,\phi_{(1)}(x)\,\delta\phi_{(0)}(x) = 2\nu\int d^3x\,\phi_{(1)}(x)\,\delta\phi_{(0)}(x),$$

up to additional local terms when the source $\phi_{(0)}(x)$ varies in $x$ and/or when the boundary metric is curved.

Why many texts write $\delta S_{\text{ren}}=\int \phi_{(1)}\,\delta\phi_{(0)}$

The overall coefficient $2\nu=\Delta_+-\Delta_-$ is a matter of convention: it can be absorbed into the normalization of $\phi_{(1)}$ (or, equivalently, into the normalization of the dual operator $\mathcal{O}$). In what follows we often suppress this prefactor and focus on the canonical pairing between the two asymptotic data.

This result means that (after renormalization) the action is functionally differentiable if we restrict variations to those compatible with the chosen boundary condition, as we now explain.

Let’s consider the two natural choices in turn:

• Dirichlet boundary condition (Standard quantization): Here we fix $\phi$ on the boundary, meaning we declare $\phi_{(0)}(x)$ to be a prescribed boundary value (the source) and do not allow it to vary ($\delta\phi_{(0)}=0$). This corresponds to the usual AdS/CFT prescription where $\phi_{(0)}(x)$ sources the dual operator. If we enforce $\delta\phi_{(0)}=0$, the variation $\delta S_{\text{on-shell}} = \int \phi_{(1)}\,\delta\phi_{(0)}$ automatically vanishes without needing any additional term. Thus, a well-posed variational principle is achieved simply by imposing Dirichlet BC for $\phi$ at infinity. No extra boundary term is needed in the action besides the standard GHY (and the counterterms to handle divergences, which we’ll discuss momentarily). The condition that the action be stationary under these variations leads to no further constraint on the fields, meaning $\phi_{(1)}(x)$ is then determined dynamically by the bulk solution (it will be proportional to $\langle \mathcal{O}(x)\rangle$ in the dual theory). The bulk action with GHY is then already stationary for solutions obeying $\phi_{(0)}$ fixed. We will still need to add counterterms to remove divergences, but these can be chosen not to affect the boundary condition (see below).

• Neumann (or alternate) boundary condition (Alternative quantization): Here we want to allow $\phi_{(0)}$ to vary freely, and instead impose a condition on $\phi_{(1)}$. In the simplest case, we fix $\phi_{(1)}(x)$ at the boundary (say specify it to zero or to some value), which is analogous to a Neumann condition (since $\phi_{(1)}$ is related to the normal derivative of the field). In the context of AdS/CFT, fixing $\phi_{(1)}$ corresponds to treating $\phi_{(1)}$ as the source for the dual operator of dimension $\Delta_-$. Equivalently, we want $\delta \phi_{(1)}=0$ (and $\delta\phi_{(0)}$ is arbitrary). Plugging this into the on-shell variation $\delta S_{\text{on-shell}}=\int \phi_{(1)}\,\delta\phi_{(0)}$, we see that without modification, $\delta S$ would not generally vanish because $\delta\phi_{(0)}$ is free while $\phi_{(1)}$ itself is not constrained to zero. To achieve a stationary action for arbitrary $\delta\phi_{(0)}$, we require the coefficient of $\delta\phi_{(0)}$ to vanish, i.e. $\phi_{(1)}(x)=0$ on shell. But $\phi_{(1)}=0$ is only one specific Neumann-like condition (the homogeneous one). In general, we might want $\phi_{(1)}(x)$ to take some fixed form (not necessarily zero) or to satisfy a more general relation. The proper way to enforce a Neumann-type condition in the action is to add a boundary term that swaps the role of source and vev – essentially a Legendre transform of the action with respect to $\phi_{(0)}$.

Concretely, one can add a term $S_{\text{Legendre}} = -\int_{\partial M} d^3x\,\sqrt{-\gamma}\,\phi\,\pi_\phi$ to the action, which has the effect of replacing the boundary term $\pi_\phi\,\delta\phi$ with $-\phi\,\delta\pi_\phi$ in the variation. In the asymptotic expansion, a carefully chosen term of this form becomes (on shell) $-\!\int d^3x\,\phi_{(0)}\,\phi_{(1)}$. Indeed, it is known that adding the boundary term

$$S_{\text{bdy}}^{(\text{alt})} \;=\; -(\Delta_+-\Delta_-)\int_{\partial M} d^3x\,\phi_{(0)}(x)\,\phi_{(1)}(x)\,.$$

to the action yields (after integration by parts) an on-shell variation

$$\delta S_{\text{on-shell}} \;=\; -(\Delta_+-\Delta_-)\int d^3x\,\phi_{(0)}\,\delta\phi_{(1)}\,,$$

so that demanding $\delta S=0$ now implies $\delta\phi_{(1)}=0$ (with $\phi_{(0)}$ free). In other words, by adding this term we have changed the boundary condition enforced by the variational principle: we are now holding $\phi_{(1)}$ fixed instead of $\phi_{(0)}$. This is precisely the scenario for the alternative quantization.

This procedure is often described as a Legendre transform at the level of the action. It “trades” the boundary condition on $\phi$ for the conjugate boundary condition. Physically, this means what was previously the expectation value becomes the source and vice versa. In the dual CFT, this corresponds to defining a new theory in which a double-trace deformation has been added to shift the operator’s dimension from $\Delta_+$ to $\Delta_-$ (or equivalently, one selects the other extremum of the quadratic relation for $\Delta$).

In summary, to obtain the $\Delta_+$ (standard) quantization, we impose Dirichlet boundary conditions on $\phi$ and need no additional boundary term beyond the usual GHY and counterterms. To obtain the $\Delta_-$ (alternative) quantization, we impose Neumann-type boundary conditions and add a specific finite boundary term (the $\phi_{(0)}\phi_{(1)}$ term above) to make the variational principle well-defined. These choices directly determine which mode is interpreted as the source in AdS/CFT. Table 1 summarizes the situation:

Quantization	Boundary condition on $\phi$	Boundary term in action	CFT interpretation
Standard	Dirichlet: $\phi_{(0)}$ fixed, $\delta\phi_{(0)}=0$	No additional term $(S_{\text{bdy}}=0)$	$\phi_{(0)}$ = source $J$, $\phi_{(1)} \propto \langle O_{\Delta_+}\rangle$
Alternative	Neumann: $\phi_{(1)}$ fixed, $\delta\phi_{(1)}=0$	$S_{\text{bdy}} = -(\Delta_+-\Delta_-)\int \phi_{(0)}\phi_{(1)}$ added	$\phi_{(1)}$ = source $J$, $\phi_{(0)} \propto \langle O_{\Delta_-}\rangle$

One can verify the above by explicitly performing the variation with these boundary terms present. For example, including $S_{\text{bdy}}=-\int \phi_{(0)}\phi_{(1)}$ in the action yields:

$$\delta(S_{\text{bulk}}+S_{\text{bdy}})\Big|_{\text{on-shell}} \;=\; \int d^3x\Big(\phi_{(1)}\,\delta\phi_{(0)} - \phi_{(0)}\,\delta\phi_{(1)}\Big)\,,$$

and since $\delta\phi_{(1)}=0$ under the Neumann condition, we are left with $\int \phi_{(1)}\,\delta\phi_{(0)}$. But now $\phi_{(0)}$ is free to vary, so to have $\delta S=0$ we must have $\phi_{(1)}(x)=0$ on shell. This is indeed the Euler–Lagrange boundary condition arising from the alternative action: the vanishing of the coefficient of $\delta\phi_{(0)}$ gives the Neumann condition $\,\phi_{(1)}=0\,$ (which in turn implies $\langle O_{\Delta_-}\rangle=0$ for the particular solution if we had no source for the $\Delta_-$ operator). In contrast, the standard action ($S_{\text{bdy}}=0$) yields instead $\phi_{(1)}\,\delta\phi_{(0)}$ in $\delta S$, and setting $\delta\phi_{(0)}=0$ (fixed source) makes the variation vanish without forcing $\phi_{(1)}$ to zero.

It is worth emphasizing that at the Breitenlohner–Freedman bound itself ($m^2 = -\frac{9}{4}$ in AdS$_4$, so $\Delta_+ = \Delta_- = 3/2$), the two modes coalesce and one finds a logarithmic branch of the solution. In that case, the distinction between standard and alternative quantization becomes subtle – in fact the two quantizations are related by a marginal double-trace deformation and the choice is “intrinsically ambiguous” at the BF bound. One must then include an additional logarithmic counterterm in the action to handle the asymptotic $\phi^2 \ln r$ behavior. However, for clarity we focus on masses within the BF window but not exactly at the bound, so that $\Delta_+ \neq \Delta_-$.

Holographic Renormalization: Counterterms and Finite Action

Divergences: Both the gravitational and scalar sectors of the action generally diverge as the AdS boundary is approached ($r\to 0$). For example, in pure AdS$_4$, the on-shell Einstein–Hilbert action diverges due to the infinite volume of spacetime. Similarly, the scalar field’s action will diverge if $\phi_{(0)}\neq 0$ because $\phi^2$ falls off too slowly. In our variational analysis above, we glossed over these divergences by implicitly assuming a regulator at $r=\epsilon$ and focusing on finite contributions. In practice, to apply AdS/CFT one must add counterterms $S_{\text{ct}}$ – local functionals of fields on the cutoff surface $r=\epsilon$ – to cancel these infinities and then take $\epsilon\to 0$. These counterterms are fixed by requiring the cancellation of divergences and are conceptually part of $S_{\text{bdy}}$ (though they do not affect the variational principle except through finite contributions to the boundary conditions, which we will note). We now describe the needed counterterms up to the order relevant for the scalar mass window considered.

For AdS$_4$, the necessary counterterms include:

• A cosmological counterterm for the boundary: $S_{\text{ct}}^{(0)} = -\frac{1}{8\pi G}\int_{\partial M} d^3x\,\sqrt{-\gamma}\,\frac{2}{L}$, which cancels the leading $O(\frac{1}{\epsilon^3})$ divergence from the bulk volume (here $L$ is the AdS radius; for $L=1$, the coefficient is $2$). Intuitively, this is proportional to the volume (area) of the boundary and removes the divergent part of the gravitational action.
• A boundary curvature term: $S_{\text{ct}}^{(R)} = \frac{1}{16\pi G}\int_{\partial M} d^3x\,\sqrt{-\gamma}\,\frac{R[\gamma]}{2}$, which cancels the subleading $O(\frac{1}{\epsilon})$ divergence (and the log divergence in even-dimensional boundaries if present). In $d=3$, a $R[\gamma]$ term is actually finite (since the boundary is 3-dimensional, there is a logarithmic divergence associated with $R[\gamma]$, but it cancels automatically for pure AdS; we include it for generality as it would be needed if the boundary metric is curved or in higher dimensions).

Most importantly for us, a scalar field counterterm is required to cancel the divergence coming from the $\phi^2$ term in the action. In AdS$_4$, if $\phi_{(0)}\neq 0$, the near-boundary behavior $\phi^2 \sim \phi_{(0)}^2 r^{2\Delta_-}$ leads to a divergence in the radial integral if $2\Delta_- \le 3$. For masses in the BF window, $\Delta_- < \frac{3}{2}$ so indeed $2\Delta_- -3 < 0$ and we get a power-law divergence. The precise form of the necessary counterterm can be found by solving the scalar equation near the boundary. One finds the leading scalar divergences are canceled by adding a local quadratic counterterm on the cutoff surface:

$$S_{\text{ct}}^{(\phi)} \;=\; -\frac{1}{2}\int_{\partial M} d^3x\,\sqrt{-\gamma}\,\frac{\Delta_-}{L}\,\phi^2\,,$$

where $\Delta_-=\frac{3}{2}-\nu$ and $\nu=\sqrt{\frac{9}{4}+m^2L^2}$ (so $\Delta_+-\Delta_-=2\nu$). This counterterm cancels the power-law divergence sourced by the $r^{\Delta_-}$ mode. At the BF bound $\nu=0$ (so $\Delta_+=\Delta_-=3/2$), the falloffs develop logarithms and the corresponding counterterm involves $\phi^2\ln\varepsilon$ instead.

Effect on variational principle: Crucially, we must check that adding these counterterms does not spoil the variational analysis of Effect on variational principle: Counterterms are not just about finiteness; they also contribute to the boundary variation and therefore to the precise definition of “Dirichlet” vs “Neumann” data.

For the scalar, the practical outcome (derived explicitly above for flat boundary and slowly varying data) is that the renormalized on-shell variation takes the form

$$\delta S_{\text{ren}}\Big|_{\text{EOM}} = (\Delta_+-\Delta_-)\int d^3x\,\phi_{(1)}(x)\,\delta\phi_{(0)}(x) \;+\;(\text{local terms}),$$

where the local terms vanish for homogeneous sources on a flat boundary but must be kept when $\phi_{(0)}(x)$ varies and/or when the boundary metric is curved.

• In standard quantization, we fix the source $\phi_{(0)}(x)$ (so $\delta\phi_{(0)}=0$). Then the variational principle is well-posed, and the scalar one-point function is extracted from the coefficient of $\delta\phi_{(0)}$:

  $$\langle \mathcal{O}_{\Delta_+}(x)\rangle \;\propto\; (\Delta_+-\Delta_-)\,\phi_{(1)}(x)\ +\ (\text{local terms}).$$

• In alternative quantization (allowed only in the BF window), we want to fix $\phi_{(1)}$ instead. Starting from $S_{\text{ren}}[\phi_{(0)}]$, this is implemented by a Legendre transform with respect to the renormalized canonical pair. In the simplest (homogeneous, flat) situation one may write, up to the same conventional prefactor,

  $$S_{\text{ren}}^{(\text{alt})}[\phi_{(1)}] = S_{\text{ren}}[\phi_{(0)}]\;-\;(\Delta_+-\Delta_-)\int d^3x\,\phi_{(0)}(x)\,\phi_{(1)}(x),$$

  so that

  $$\delta S_{\text{ren}}^{(\text{alt})}\Big|_{\text{EOM}} = -(\Delta_+-\Delta_-)\int d^3x\,\phi_{(0)}(x)\,\delta\phi_{(1)}(x)\;+\;(\text{local terms}).$$

  Therefore $\delta\phi_{(1)}=0$ gives a well-posed variational principle, and $\phi_{(0)}$ becomes proportional to the vev of the $\Delta_-$ operator.

(For mixed boundary conditions and multi-trace deformations, it is often most transparent to start from the standard-quantization $S_{\text{ren}}[\phi_{(0)}]$ and add a finite boundary functional $W(\phi_{(0)})$; this is discussed below in the RG-flow section.)

In summary, adding the necessary counterterms $S_{\text{ct}}$ for finiteness does not pose a problem: one either chooses them to respect the chosen boundary condition or includes their effect in the renormalized canonical pair $(\phi_{(0)},\phi_{(1)})$. After renormalization, the on-shell action is finite and functionally differentiable with respect to the chosen source, and its functional derivatives generate CFT correlators.

Two-point functions and the two Green’s functions ($\Delta_+$ vs $\Delta_-$)

The AdS/CFT “generating functional” statement (in Euclidean signature) is

$$Z_{\text{CFT}}[J]\;=\;e^{-S_{\text{ren}}[\phi_{\text{cl}}(J)]}, \qquad W[J]\equiv -\log Z_{\text{CFT}}[J]=S_{\text{ren}}[\phi_{\text{cl}}(J)].$$

Here $\phi_{\text{cl}}(J)$ is the bulk solution with boundary condition determined by the source $J$.

Standard quantization ($\Delta_+$): $G_+(x,y)$

In standard quantization the source is the leading coefficient:

$$J_+(x)\equiv \phi_{(0)}(x), \qquad \langle \mathcal{O}_+(x)\rangle_{J_+}=\frac{\delta W[J_+]}{\delta J_+(x)} =(\Delta_+-\Delta_-)\,\phi_{(1)}(x)+(\text{local terms}).$$

The connected two-point function is the second functional derivative at vanishing source,

$$G_+(x,y)\equiv \left.\frac{\delta \langle \mathcal{O}_+(x)\rangle}{\delta J_+(y)}\right|_{J_+=0} =\left.\frac{\delta^2 W}{\delta J_+(x)\delta J_+(y)}\right|_{J_+=0}.$$

For linearized fluctuations around a background where translation invariance holds (e.g. pure AdS$_4$ or a homogeneous RG-flow background), it is convenient to Fourier transform. Writing

$$\phi_{(1)}(k)=\mathcal{F}(k)\,\phi_{(0)}(k)\qquad(\text{linear response}),$$

one gets (dropping scheme-dependent contact terms)

$$G_+(k)=(\Delta_+-\Delta_-)\,\mathcal{F}(k)=2\nu\,\frac{\phi_{(1)}(k)}{\phi_{(0)}(k)}.$$

Alternative quantization ($\Delta_-$): $G_-(x,y)$

In alternative quantization (allowed in the BF window), the role of source and vev is swapped. A convenient choice of conventions is

$$J_-(x)\equiv \phi_{(1)}(x), \qquad \langle \mathcal{O}_-(x)\rangle_{J_-}=-(\Delta_+-\Delta_-)\,\phi_{(0)}(x)+(\text{local terms}).$$

(The minus sign is the one naturally produced by the Legendre transform of $S_{\text{ren}}$; different normalizations of $\mathcal{O}_-$ shift overall factors but not the structure.)

The corresponding connected two-point function is

$$G_-(x,y)\equiv \left.\frac{\delta \langle \mathcal{O}_-(x)\rangle}{\delta J_-(y)}\right|_{J_-=0}.$$

In momentum space, using $\phi_{(1)}=\mathcal{F}\phi_{(0)}$ so that $\phi_{(0)}=\mathcal{F}^{-1}\phi_{(1)}$, one finds the characteristic “inverse” relation (up to local/contact terms):

$$G_-(k)=-(\Delta_+-\Delta_-)\,\mathcal{F}(k)^{-1} =-\,2\nu\,\frac{\phi_{(0)}(k)}{\phi_{(1)}(k)}.$$

Equivalently,

$$G_+(k)\,G_-(k)=-(\Delta_+-\Delta_-)^2=-(2\nu)^2,$$

and with a rescaled operator normalization (often used in the literature) this becomes $G_-(k)=-1/G_+(k)$.

Contact terms and scheme dependence

Finite local counterterms on the boundary shift $W[J]$ by local functionals of $J$, and therefore shift $G_\pm(k)$ by polynomials in $k^2$ (“contact terms”). The relations above should be understood for the nonlocal part of the correlator, which is scheme-independent.

Explicit check in pure Euclidean AdS$_4$

In pure Euclidean AdS$_4$ with Poincaré metric $ds^2=L^2(dr^2+dx^2)/r^2$, the linearized scalar equation at momentum $p\equiv |k|$ has a regular interior solution

$$\phi(r,k)=r^{3/2}\,K_\nu(p r)\qquad(\nu\notin \mathbb{Z}),$$

with $K_\nu$ the modified Bessel function. Using the small-$r$ expansion

$$K_\nu(z)\sim 2^{\nu-1}\Gamma(\nu)\,z^{-\nu}+2^{-\nu-1}\Gamma(-\nu)\,z^{\nu}+\cdots,$$

one finds

$$\frac{\phi_{(1)}(k)}{\phi_{(0)}(k)} =2^{-2\nu}\frac{\Gamma(-\nu)}{\Gamma(\nu)}\,p^{2\nu}.$$

Therefore, up to contact terms,

$$G_+(p)=2\nu\,2^{-2\nu}\frac{\Gamma(-\nu)}{\Gamma(\nu)}\,p^{2\nu}, \qquad G_-(p)=-2\nu\,2^{2\nu}\frac{\Gamma(\nu)}{\Gamma(-\nu)}\,p^{-2\nu}.$$

In position space this is consistent with the expected CFT scaling

$$\langle \mathcal{O}_\pm(x)\mathcal{O}_\pm(0)\rangle \propto \frac{1}{|x|^{2\Delta_\pm}}\,.$$

Lorentzian (retarded) Green’s functions in RG-flow backgrounds

In generic holographic RG-flow geometries (domain walls, black branes, etc.), the same logic applies but $\mathcal{F}(k)$ is obtained by solving the linearized bulk wave equation for $\delta\phi$ with:

• the chosen boundary quantization ($\Delta_+$ or $\Delta_-$), and
• an infrared condition (regularity in the deep interior, or ingoing boundary conditions at a horizon for retarded correlators).

For instance, the standard-quantization retarded correlator is still of the form

$$G_{R,+}(\omega,\vec{k})=2\nu\,\frac{\phi_{(1)}(\omega,\vec{k})}{\phi_{(0)}(\omega,\vec{k})}\Bigg|_{\text{ingoing at horizon}},$$

and the alternative-quantization one is obtained by inverting the ratio,

$$G_{R,-}(\omega,\vec{k})=-2\nu\,\frac{\phi_{(0)}(\omega,\vec{k})}{\phi_{(1)}(\omega,\vec{k})}\Bigg|_{\text{ingoing at horizon}},$$

again modulo contact terms and overall conventions.

Mixed boundary conditions and double-trace deformations

If you impose a mixed condition $\phi_{(1)}=\kappa\,\phi_{(0)}$ (which is the bulk avatar of a double-trace deformation), the deformed two-point function in standard quantization takes the universal form

$$G_{\kappa}(k)=\frac{1}{G_+(k)^{-1}+\kappa},$$

up to contact terms. This is the basic holographic mechanism behind RG flows triggered by double-trace operators.

Holographic RG Flows and Mixed Boundary Conditions

So far we discussed the two extreme choices of boundary conditions (fixing $\phi_{(0)}$ or fixing $\phi_{(1)}$). More generally, one can impose a mixed (Robin) boundary condition of the form

$$\phi_{(1)}(x) = f\big(\phi_{(0)}(x)\big)\,,$$

for some functional relation $f$. From the dual CFT perspective, this corresponds to adding a multi-trace deformation to the action that ties the source and vev of the operator. For example, a simple mixed condition $\phi_{(1)} = \kappa\,\phi_{(0)}$ corresponds to adding a double-trace term $\frac{\kappa}{2}\int d^3x\,\mathcal{O}^2$ to the standard quantization CFT, which shifts the scaling dimension of $\mathcal{O}$ continuously (triggering an RG flow). At the endpoints of the flow (fixed points), one recovers either the standard or alternate quantization as the IR or UV limit. In the bulk, such a mixed boundary condition can be implemented by adding a corresponding boundary potential term $S_{\text{bdy}} = \int d^3x\, \frac{\kappa}{2}\sqrt{-\gamma}\,\phi^2$ (or more complicated nonlinear functionals for general $f$). The variational principle will then enforce $\phi_{(1)} = \kappa\,\phi_{(0)}$ as the boundary Euler–Lagrange relation. This is often referred to as “designer gravity,” where one specifies boundary conditions to engineer particular IR behavior in the bulk solution. Notably, if $\kappa$ is tuned to $\kappa \to \infty$, the mixed condition $\phi_{(1)}=\kappa\phi_{(0)}$ reduces to $\phi_{(0)}=0$ (for finite $\phi_{(1)}$), which is precisely the alternate quantization (since $\phi_{(0)}=0$ means the leading mode vanishes, leaving the subleading mode free). Conversely, $\kappa=0$ corresponds to $\phi_{(1)}=0$, i.e. the standard quantization. Any intermediate $\kappa$ yields a relevant deformation that causes an RG flow between the two fixed points (with $\kappa=0$ in the UV and $\kappa=\infty$ in the IR, for a positive $\kappa$).

These considerations become important in holographic RG flow solutions, where one may start in the UV with alternative quantization (if the operator is relevant in that quantization) and flow to an IR fixed point with standard quantization (or vice versa). For instance, in a famous example, the 3d $O(N)$ vector model has a free UV fixed point (operator of dimension $\Delta_- \approx 1$) and an interacting IR fixed point (operator of dimension $\Delta_+ \approx 2$). These two are related by exactly the two possible quantizations of a bulk scalar with $m^2 = -2$ (since $\Delta_-=1$, $\Delta_+=2$ for $m^2=-2$). A double-trace deformation $g \mathcal{O}^2$ triggers the flow from the $\Delta_-$ theory to the $\Delta_+$ theory. In the bulk, this corresponds to choosing a mixed boundary condition $\phi_{(1)} = g\,\phi_{(0)}$ in the UV. The sign of $g$ determines which direction the flow goes (relevant vs irrelevant perturbation).

Extension: Fermionic Fields and Quantization Choices

The story of boundary terms and quantization has a direct parallel for fermionic (spinor) fields in AdS, though with a few twists. A Dirac field in AdS$_{d+1}$ has two independent components (often termed the “upper” and “lower” components, or equivalently the chiral components in Euclidean signature) whose asymptotic fall-offs are related to the mass. For a 4D bulk Dirac fermion of mass $m$, the two asymptotic solutions behave like $r^{\Delta_+ - 3/2}$ and $r^{\Delta_- - 3/2}$ (since the Dirac equation is first order, the half-integer shift appears). Here $\Delta_{\pm} = \frac{3}{2} \pm \sqrt{m^2 + \frac{1}{4}}$ are analogous to the scalar dimensions (for example, at $m=0$, one gets $\Delta_+=\Delta_- =3/2$). If $0 \le m < \frac{1}{2}$, then $\Delta_- > \frac{1}{2}$ and an alternate quantization exists for the spinor. The two quantizations correspond to imposing either a chirality (boundary) projection that effectively fixes the leading component or the subleading component of the spinor at the boundary. In the standard case, one typically imposes the condition that the slower-decaying part of the spinor vanishes (this is analogous to Dirichlet for bosons). In the alternate case, one imposes that the faster-decaying part vanishes (analogous to Neumann). For example, if we write a 4-component Dirac spinor as $(\psi_+,\psi_-)$ (two-component spinors) such that $\psi_+$ is the leading piece, then standard quantization corresponds to requiring $\psi_+ = 0$ at the boundary, whereas alternative quantization would require $\psi_- = 0$ at the boundary (when allowed by the mass range).

To achieve these conditions from an action perspective, one must add appropriate boundary terms to the Dirac action. The Dirac action is first order, and its variation yields a boundary term $\sim \int \bar\psi_+ \delta\psi_- - \bar\psi_- \delta\psi_+$ (in a certain chiral basis) after integration by parts. To impose, say, $\delta\psi_-=0$ (standard quantization, fixing $\psi_-$), one finds that $\bar\psi_+ \delta\psi_-$ must vanish, which it does without additional terms if $\delta\psi_-=0$. However, for the opposite choice $\delta\psi_+=0$ (alternate quantization, fixing the other component), the term $\bar\psi_+ \delta\psi_-$ is problematic since $\delta\psi_-$ is free. In that case, one can add a boundary term of the form

$$S_{\text{bdy}}^{(\psi)} \;=\; -\,i\int_{\partial M} d^3x\,\sqrt{-\gamma}\;\bar\psi_+ \psi_-\,,$$

which modifies the boundary variation. Varying this term produces $-i\int \bar\psi_+ \delta\psi_- - i\int \delta\bar\psi_+ \psi_-$, which can cancel the original term from $\delta S_{\text{Dirac}}$ for a suitable choice of phase conventions. The end result is that with $S_{\text{bdy}}^{(\psi)}$ included, the combined variation yields a term $\propto (\bar\psi_+ \delta\psi_- - \bar\psi_- \delta\psi_+)$ plus additional pieces that can enforce either $\delta\psi_+=0$ or a specific linear combination. In essence, this $-i\bar\psi_+\psi_-$ term is the analog of the $-\phi_{(0)}\phi_{(1)}$ term for scalars: it performs a Legendre transform for the spinor boundary condition. Imposing $\delta\psi_+=0$ with this term present leads to the alternative quantization boundary condition and removes the unwanted $\psi_+\delta\psi_-$ variation term.

From the CFT viewpoint, the two spinor quantizations correspond to choosing the spinor operator in the dual 3d CFT to have dimension $\Delta_+$ or $\Delta_-$. For example, a free 3d fermion has dimension $1$ (if we consider it as the alternate quantization of a bulk field with $m=0$, since $\Delta_- = 3/2 - 1/2 = 1$), whereas the standard quantization at $m=0$ would correspond to a conserved spinor current of dimension $2$ (which is not unitary for a spinor, so indeed one only allows alternate quantization for $m=0$). The consistent stability bound for fermions is $m \ge 0$ (there is no negative-$m^2$ instability like for scalars, instead an $m<0$ would relate to the other chirality). In any case, similar to the scalar, if the mass of the Dirac field is in the appropriate range, one can choose to impose the alternative boundary condition and thereby define a different boundary CFT. Counterterms (like a mass term on the boundary for the spinor) may be needed to cancel divergences and must be chosen consistently with the boundary condition to ensure finiteness and a well-defined variational principle (e.g. a term like $\frac{1}{2}\int \sqrt{\gamma}\,\bar\psi\psi$ might be added as a counterterm, and its variation would alter the boundary condition slightly, analogous to the scalar case).

In conclusion, the treatment of boundary terms and quantization choices in AdS is a general theme: for each bulk field (scalar, spinor, vector, etc.) that has two normalizable fall-offs, one must specify a boundary condition (which mode to fix) and include any necessary boundary term so that the action’s variation is consistent with that choice. This choice determines the identification of source vs operator in the dual CFT. As we saw, for scalars the choice is between Dirichlet and Neumann boundary conditions, corresponding to standard or alternative quantization. For fermions, the choice is between two chiral Dirichlet conditions (fixing one spinor component or the other) – again yielding two possible CFT operators. In all cases, a well-posed variational principle and a finite on-shell action can be achieved by adding the appropriate boundary terms (GHY for gravity, plus field-specific terms like $\frac{1}{2}\phi^2$ or $-i\bar\psi_+\psi_-$, and counterterms as required). These ensure that the AdS/CFT dictionary is consistent: the leading mode of the bulk field is treated as the source for the dual operator if and only if the action has been adjusted (by boundary terms) to make that mode the one held fixed in the variational principle.

Sources:

• E. Witten, “Multi-trace operators and the AdS/CFT correspondence” (hep-th/0112258), which introduced alternative boundary conditions for bulk fields.
• I. Klebanov & E. Witten, “AdS dual of the critical O(N) vector model” (2002) – classic example of standard vs. alternative quantization for a scalar with $m^2=-2$ in AdS$_4$.
• M. Breitenlohner & D. Z. Freedman, “Positive Energy in Anti–de Sitter Backgrounds and Gauged Extended Supergravity”, Phys. Lett. B115 (1982) 197 – BF stability bound and quantization in AdS.
• S. de Haro, S. Solodukhin, K. Skenderis, “Holographic Renormalization”, Commun. Math. Phys. 217 (2001) 595 – systematic derivation of counterterms and identification of source/vev.
• Reference [1] in the footnotes of Ro Jefferson’s blog post on “Boundary conditions in AdS/CFT” for additional technical details.
• Discussions of variational principles with boundaries in: R. McNees’s StackExchange answer, and the papers of Papadimitriou & Skenderis on asymptotically locally AdS spacetimes. These cover gravity and scalar boundary terms in detail.
• For fermions: M. Henneaux, “Boundary terms in the AdS/CFT correspondence for spinor fields” (hep-th/9902137) – which classifies allowed spinor boundary conditions; and recent analyses like A. Ishibashi & R. Wald, “Dynamics in Non-globally-hyperbolic Static Spacetimes: II. General Analysis of Prescriptions for Evolution”, which discuss self-adjoint extensions corresponding to different boundary conditions for fields.