# Operator insertions vs boundary conditions in AdS/CFT

This question is motivated by AdS/CFT, but really it's just about AdS quantum gravity. Consider quantum gravity in asymptotically AdS spacetime. For simplicity, assume there are no matter fields: the metric is the only field present. Now, the gravitational partition function $$Z$$ can be "probed" in two, seemingly distinct, ways:

1. Choose an operator $$O(x)$$ at the conformal boundary, and a source function $$J(x)$$, and insert $$e^{i\int J(x)O(x)}$$ into the path integral. The case I'm most interested in is when we choose $$O$$ to be the metric operator $$g_{\mu\nu}$$, in which case we write our source as $$J_{\mu\nu}$$, with indices. The resulting partition function is: $$\tag{1} Z_{sourced}[J_{\mu\nu}]\equiv \int Dg e^{i(S[g]+\int J^{\mu\nu}(x)g_{\mu\nu}(x))}.$$

2. Impose boundary conditions on the subleading behaviour of the metric $$g_{\mu\nu}$$ near the conformal boundary. Schematically, I'll write $$h_{\mu\nu}(x)$$ to denote the appropriate information about the subleading behaviour (perhaps the second coefficient in the Fefferman-Graham expansion?), but I admit I don't know exactly how this works. Anyway, schematically the partition function is: $$Z_{GKPW}[h_{\mu\nu}]\equiv\tag{2} \int_{g_{\mu\nu}\to h_{\mu\nu}} Dg e^{iS[g]}.$$ I named it $$Z_{GKPW}$$ since this is the type of partition function which appears in the GKPW dictionary, but it seems that in that context, usually only the leading saddle-point behaviour is kept: $$\tag{3} Z_{GKPW}[h_{\mu\nu}] \approx Z_{GKPW}^{(0)}[h_{\mu\nu}] \equiv \exp{iS_{cl}[h_{\mu\nu}]},$$ where $$S_{cl}[h_{\mu\nu}]$$ is the value of the action evaluated on the classical solution with boundary conditions $$h_{\mu\nu}$$.

I want to know the relation between these two methods of probing the partition function. My guess is the partition functions are equal if we set $$J_{\mu\nu} = h_{\mu\nu}$$. That is: $$\tag{5} Z_{sourced}[h_{\mu\nu}] = Z_{GKPW}[h_{\mu\nu}].$$ But perhaps this is only true in the saddle point approx, so we can only say $$\tag{6}Z_{sourced}[h_{\mu\nu}] = Z_{GKPW}^{(0)}[h_{\mu\nu}] + ...$$ where $$...$$ denotes quantum corrections. I'd like someone to confirm and justify whether or not (5) or (6) is true. If they're both false, is there some other way to relate $$Z_{sourced}$$ and $$Z_{GKPW}$$?

• I believe the following paper discusses exactly what you want: arxiv.org/abs/0808.2054
– Gold
Commented Sep 5, 2022 at 16:02
• There's no metric opetator. Only a stress tensor operator sourced by the metric. Commented Sep 5, 2022 at 16:12
• @ConnorBehan why can't there be a metric operator acting at the boundary? Sure, it's not invariant under large diffeos, but we don't care since these aren't gauge transformations - they're genuine symmetries. Commented Sep 5, 2022 at 21:06
• Let me put it this way. A two point function of the metric (or any other bulk field) will approach a power law at distances much larger than the AdS radius. Therefore, if (1) were correct, the things computed from it by taking $J$ derivatives would vanish identically. Commented Sep 6, 2022 at 0:40
• Ok so evidently we must use some limiting procedure where we scale $J(x)$ by the correct power as $x$ is taken to the conformal boundary, so as to get a sensible finite limit for the two point function. If not, then how else are we going to insert operators at the conformal boundary? There surely should be some way to do so, since we'd like to be able to talk about operators living at infinity (e.g. the ADM energy). Commented Sep 6, 2022 at 2:07