# Bulk solutions for CFTs on non-trivial geometries In [[0001 AdS-CFT|AdS/CFT]] correspondence, an asymptotically locally AdS spacetime is dual to a CFT on its conformal boundary. Usually when people study this duality, the bulk is the global AdS (or the Poincare patch), and the boundary is just a cylinder (time cross a sphere). For more general boundary geometries, the boundary geometry (and topology) can be more non-trivial. Here by 'non-trivial', we mean that the boundary metric is more complicated, and in particular, it can have non-trivial topology. One is usually interested in the lowest-energy solution in the bulk, which corresponds to the ground state, or *vacuum*, or the CFT. But more generally the finite-temperature solutions and other excited states are studied. Below you can find a list of some references on this subject. The list focuses heavily on the cases when the boundary is a product of circles, spheres and planes. There are other cases such as the hyperbolic space, Riemann surfaces (for 2d CFT), or with black holes on the boundary. These are not carefully surveyed yet. ## Guiding principles - ground states are static: [[1992#Sudarsky, Wald]] and [[HertogHollands2005]] - ground states are as symmetric as the conformal boundary metric ## Simple examples - bdry = $R_t\times S^3$ or $R_t\times R^3$ - bulk vacuum is pure AdS (global and Poincare/planar respectively) - bdry = $R_t\times S^1\times R^2$ or $R_t\times T^3$ - bulk vacuum is AdS soliton (see [[1998#Horowitz, Myers]]) - bdry = $R_t \times T^{n} \times S^{m}$ - [[KleihausKunzRadu2010]][](https://arxiv.org/abs/1006.3290) - which generalises Horowitz-Myers soliton to $m>1$ - [[2016#Hickling (Thesis)]] sec. 7.4 - case $n=1$, $m=3$ - they only looked for black hole solutions, not bubbles; however, I think they are related: swapping the roles of time and the spatial $S^1$ gives a bubble - [[2018#Harlow, Ooguri (Long)]] Appendix I (the letter I not 1) - there, it is called vacuum solution for $\alpha>0$ everywhere and wormhole solution for $\alpha(r_0)=0$ for some $r_0>0$ - the focus was on generalisation of the black string, not of the bubble - bdry $R_t\times S^1\times S^2$ - need to solve for static bubbles numerically -> see [[2006#Copsey, Horowitz]] - no way of periodically identifying pure AdS to get this boundary behaviour - bdry $R_t\times S^1\times S^3$ - [[2016#Hickling (Thesis)]] p.127: repeats [[2006#Copsey, Horowitz]] but for $S^3$ now - bdry $R_t \times S^2\times S^2$ - [[2016#Hickling (Thesis)]] sec. 7.4 - BUT they only looked for black hole solutions, not bubbles; in this case you cannot get bubble solutions from the BH solution via analytic continuation - bdry $S^n\times S^m$ (no extra time direction) - [[2019#Aharony, Urbach, Weiss]]: contains an interesting oscillating behaviour in fig.2 - [[2020#Kiritsis, Nitti, Preau]]: another useful reference on this topology. we should see if the results are useful - [[2011#Blackman. McDermott, van Raamsdonk]]: $S^2\times S^d$ - bdry = $M \times S^{1} \times R^{d-1}$, $M \times S^{d}$ - $M=T^2, S^2$ as examples - [[2011#Blackman. McDermott, van Raamsdonk]] - bdry = $S^1\times S^n\times S^m$ - [[2020#Dibitetto, Petri, Schillo]]: only for special ratio between the two spheres; one of the spheres analytically continued to dS - [[2022#Horowitz, Wang, Ye]]: infinitely many solutions as a special ratio of sphere radii is approached - bdry = AdS$_d\times S^n$ - [[2023#Ghodsi, Kiritsis, Nitti]] ## With BH on the boundary - e.g. [[2022#Biggs, Santos]] - more to come ## Related topics - [[0314 Holographic constraints on CFT energies]] - [[0207 Euclidean state preparation]] - [[0448 Scherk-Schwarz compactification]] <!-- ## Some email correspondence ###### Canonical ensemble for boundary topology $S^1 \times S^1 \times S^2$ In Lorentzian space with a boundary topology of $R \times S^1 \times S^2$, where R is the time direction, in your paper with Copsey you found three static solutions, one with $S^2$ contracting (Copsey-Horowitz solution, or CH for short) which is numerically constructed and two with $S^1$ contracting (bubbles). For small $S^1$, the large bubble solution has lower energy than both the small bubble and the CH solution, while for large $S^1$, CH is the ground state. Let us call this direction $\chi$. To study canonical ensemble, we just go to Euclidean time and compactly the time direction. Then we look for solutions with boundary topology $S^1 \times S^1 \times S^2$. Obviously, if we believe that all saddles have $SO(2)\times SO(2)\times SO(3)$ symmetry, then there are five saddles: big and small black strings (analytically continuation of big and small bubbles so that the time circle contracts), big and small bubbles, and CH. Now my conjecture is the following, based on the intuition from the original Hawking-Page study.  For large $\chi$-circle, we have three candidate solutions in the canonical ensemble: big BS, small BS and CH. Then I believe that this is just the same as HP, where small BS never dominates,  big BS dominates at high temperature, and thermal CH dominates at low temperature. For small $\chi$-circle, we have five candidate solutions, but CH and small bubble will always have lower energy than big bubble. So I will guess that there is again something similar to original HP, where now big bubble replaces thermal CH. In conclusion, I believe that the phase diagram in the plane of temperature and period of $\chi$ would have three phases: high temperature = big BS, low temperature & small $\chi$-circle = big bubble, low temperature & big $\chi$-circle = thermal CH.  Now, on one hand, this may be interesting, because it is a three-phase diagram like a liquid-gas system. The famous paper you mentioned \[Chamblin, Emparan, Johnson, Myers 1999\]([https://arxiv.org/pdf/hep-th/9902170.pdf](https://arxiv.org/pdf/hep-th/9902170.pdf)) found something similar to this in their figure 1b in the grand canonical ensemble for boundary topology R x S^{d-1}, certainly for a different reason. On the other hand though, while I have not found such a phase diagram anywhere, it is obvious that people have expected this. There are comments like “in XX situations, we believe there is something similar to HP where YY plays the role of thermal AdS”. ###### Grand canonical ensemble for boundary topology $S^1 \times S^1 \times S^2$ In a grand canonical ensemble in the same topology as above, if I were to guess, I would guess that the result would be the following: with big $\chi$-circle, the story is the same as Chamblin et al above, where now thermal CH replaces thermal AdS. For small $\chi$-circle, big bubble replaces thermal AdS. In other words, there may not be a complex interplay between all these candidates. However, there is a paper that discusses this explicitly: [https://arxiv.org/pdf/0708.2402.pdf](https://arxiv.org/pdf/0708.2402.pdf). In particular, I quote, "we show that the static magnetically charged bubbles exist for any size of the S1 at infinity in contrast to the uncharged case where bubbles exist only below a critical size of the S1”. Their introduction claims that they found something similar to Chamblin et al, so I guess my naive guess was not too far off.  ###### Some questions In summary, I think the idea we had in mind was interesting, but people have done it. Now I don’t immediately see any reason to pursue this further, but there are two minor questions I will just throw out there. - There are many more inequivalent topologies in higher dimensions. What can we say about the phase diagrams, or, even before that, static solutions? Will the more complicated topologies just lead to some trivial generalisations, or will there be surprises? - How is this related to replica symmetries? If we have e.g. S^1 x S^2 x S^2 x T^4, we can require the two sphere to have the same size and T^4 to have Z^4 symmetry. There must be bubble solutions that breaks Z^2 x Z^4 symmetry. Does this lead to any insight? (Xi once said in gravity lunch that it would be interesting to find a case where a non-replica-symmetric saddle dominates) Conversely, what does replica story tell us about HP here? Don, Zhencheng, Shannon, Xi and Huajia showed that including them modifies the Page curve near phase transition. I believe that they will also modify the story near e.g. the critical point in the 3-phase diagram. -->