Asymptotically AdS wormhole solutions are considered in the context of holography. Correlation functions of local operators on distinct boundaries are studied. It is found that such correlators are finite at short distances. Correlation functions of non-local operators (Wilson loops) on distinct boundaries are also studied, with similar conclusions. Deformations of the theory with multi-trace operators on distinct boundaries are considered and studied. As a consequence of these results, the dual theory is expected to factorize in the UV, and the two sectors to be coupled by a soft non-local interaction. A simple field theory model with such behavior is presented.

Article funded by SCOAP3

0$). Since the last term of~\eqref{4} is positive, then one needs an imaginary $\phi'$ to satisfy this equation, or equivalently a kinetic term for the scalar with the opposite sign --- to provide for negative Euclidean energy ---. Such a theory would violate reflection positivity and we therefore disregard such a possibility in the rest. Nevertheless, conformally coupled scalars can give rise to wormhole solutions as found in~\cite{Coule-ml-1989xu,Halliwell-ml-1989ky}. We provide also an explicit demonstration of the pathologies encountered when \begin{equation} V=-{d(d-1)\over \alpha^2} \end{equation} in appendix~\ref{scalarpathologies}. \boldmath ]]>

0$ and $\tilde{C}<0$ as well as the special case $\tilde{C}=1$. \begin{itemize} \item $\tilde C>0$. In this case the scale factor never vanishes, but it bounces at \begin{equation} e^{2A_{+}} = {\alpha^2\over 2 }+ {1\over 2}\sqrt{\tilde C} \label{14a} \end{equation} The second boundary is near $Z\to 0$ (or $r\to -\infty$) where the metric asymptotes to~\eqref{12a} but with the slice curvature $ R_{\rm uv}'$ given by \begin{equation}\label{15a} R_{\rm uv}'=\sqrt{\tilde C} ~ R_{\rm uv} \end{equation} The scalar interpolates between $\phi_0$ at the first boundary and \begin{equation}\label{16a} \phi\simeq \phi_1+16{C\over \tilde C}Z+{\cal O}(Z^2)\sp \phi_1=\phi_0+ \log{-2C+{\alpha^2}\over 2C+{\alpha^2\over r^2}} \end{equation} \item $\tilde{C}=1$. A special case that we shall examine in more detail in the next sections is when $\tilde{C}=1$. Then the slices acquire the same asymptotic curvature on both sides and we find that \begin{equation}\label{17a} ds^2 = d r^2 + \frac{\alpha^2}{2 } \left( \cosh \left(\frac{2 r}{\alpha} \right) + 1 \right) dH_2^2 \end{equation} This is a symmetric wormhole centered around $r=0$. The metric is similar to the ones that we shall encounter in the Einstein Yang-Mills system as well as in global ${\rm AdS}_2$. \item $\tilde C<0$. In this case, the scale factor vanishes at an intermediate point where $\phi\to -\infty$. This is a solution with a single boundary and the point where $e^A$ vanishes is a singularity. Moreover on can observe that at this point $(\p \phi)^2$ diverges. \end{itemize} ]]>

0$ for the cases of AdS/dS\@. We observe that the two cases are related by changing the sign of the cosmological constant $\Lambda$ and the size of the wormhole throat remains positive for $B \leq \half$ or $B \geq \half$ respectively. From here on we focus on the AdS case ($\Lambda<0$). The geometry is conformally flat and has two asymptotic boundaries, one at $r\to\infty$ with \begin{equation}\label{8b} e^{2 w}\simeq {B \over 2}e^{2 r} \end{equation} and one at $r\to-\infty$ with \begin{equation}\label{8b1} e^{2 w}\simeq {B \over 2}e^{- 2 r}\end{equation} Therefore, the space-time approaches AdS and the three-sphere slices have the same asymptotic radii. The throat connecting them obtains a minimum size equal to \begin{equation}\label{9b} r_{\rm min}^2 = B - \half . \end{equation} For $B=\half$. the two sides pinch off in a smooth fashion and the metric becomes exactly that of two disconnected Euclidean ${\rm AdS}_4$ (${\rm EAdS}_4$) copies \begin{equation}\label{10b} ds^2_{\rm EAdS} = dr^2 + \sinh^2 r d \Omega_3^2\, , \qquad r \in [0, \infty) \, . \end{equation} We notice that in order to achieve this one needs to send $g_{YM} \rightarrow \infty$. This wormhole background also involves a non-trivial non-abelian magnetic field configuration also known as the meron~\cite{deAlfaro-ml-1976qet,Callan-ml-1977gz} whose magnetic flux is responsible for supporting the wormhole throat. In order to understand this configuration, it is useful to define the metric on the three sphere in terms of Euler angles as \begin{equation}\label{6b} d\Omega^2_3 = {1 \over 4} \biggl( d t_1^2 + d t_2^2 + d t_3^2 + 2 \cos\, t_1 \, d t_2 d t_3 \biggr) = \frac{1}{4} \omega^a \omega^a \, , \end{equation} where $\omega^a$ are the Maurer-Cartan (left-invariant) forms on $S^3$ and $t_i$ the Euler angles, for more details see appendix~\ref{MaurerHopf}. The non-trivial gauge field then can be described as ($A^a = A^a_\mu dx^\mu$) \begin{equation}\label{7b} A = \half g^{-1} d g \, , \qquad \text{or} \quad A^a = \half \omega^a \, , \qquad \text{with} \quad F^a = \frac{1}{8} \epsilon^{a b c} \omega^b \wedge \omega^c \, , \end{equation} with $g$ an $\SU(2)$ group element. This is the \emph{meron} configuration, which provides the appropriate magnetic flux to support the throat from collapsing.\footnote{We should also note that there exists also an anti-meron wormhole solution for which $A =\half g d g^{-1}$, which is expressed in terms of right-invariant forms. It would be interesting if there exists a solution that asymptotes to a meron on the first boundary and to an antimeron on the second one.} In order to understand better the holographic interpetation of this solution, we pick a radial gauge $A^a_r = 0$ and use the explicit expression for the Maurer-Cartan forms given in~\eqref{m12} to read off the components of the $\SU(2)$ gauge field from the matrix \begin{equation}\label{gauge_f} A^{a}_{\mu}= \begin{pmatrix} 0\,\, & \frac{1}{2}\cos t_{3}\,\, & \frac{1}{2}\sin t_{3}\, \sin t_{1}\,\, & 0\\ 0\,\, & \frac{1}{2}\sin t_{3}\,\, & -\frac{1}{2}\cos t_{3}\, \sin t_{1}\,\, & 0\\ 0\,\, & 0\,\, & \frac{1}{2}\cos t_{1}\,\, & \frac{1}{2} \end{pmatrix} \end{equation} where the index $a = 1,2,3$ labels the rows of the matrix and is the gauge group index and $\mu$ is the bulk space index that labels the columns of the matrix. One can notice that the gauge field is independent of the radial direction. This corresponds to having a constant magnetic source turned on in the dual theory. This source breaks explicitly the global $\SU(2)$ defined on each boundary. In addition if we compute the total topological charge on the wormhole manifold $\mathcal{M}_4$ it reduces to a sum of two integrals over the boundaries of the topological current $J_\mu^T$ \begin{align}\label{gauge_g} q_T & = \int_{\mathcal{M}_4} \Tr ( \epsilon^{\mu \nu \rho \sigma} F_{\mu \nu} F_{\rho \sigma}) = \oint_{\sum_{i=1}^2 S_i^3} d \Omega_\mu J_\mu^T \, \nn \\ & = \oint_{\sum_{i=1}^2 S_i^3} d \Omega_\mu \Tr \epsilon^{\mu \nu \rho \sigma} \left( A_\nu \partial_\rho A_\sigma + \frac{2}{3} A_\nu A_\rho A_\sigma \right) = 0 \, . \end{align} This means that the total topological charge of our configuration is zero because the integral of the current on the first asymptotic boundary cancels the one from the other due to the opposite sign of the normal vector perpendicular to the two boundaries. If the topology was trivial (for a pure meron) we would instead have found half a unit of topological charge $q_T = 1/2$, due to the integral on a single three-sphere at infinity. A further analysis can be performed for each fixed $S^3$ in the radial gauge, if we use the Hopf fibration $S^{1}\hookrightarrow S^{3}{\xrightarrow {\ p\,}} \, S^{2} $ that describes the three sphere as an $S^1$ bundle over $S^2$, see appendix~\ref{MaurerHopf}. We can then compute the magnetic flux along various submanifolds of $S^3$ and in particular we find that the only non-zero result is \begin{equation}\label{gauge_i} \Phi = \int_{S^2} \Tr F = \frac{1}{4}\int_0^{\pi} dt_1 \int_0^{2 \pi} dt_2 \left(\sin t_1 + (\cos t_3 - \sin t_3) \cos t_1 \right) = \pi \, . \end{equation} This means that there is a constant magnetic flux piercing the $S^2$ pointing along the fiber coordinate $t_3$. Since this coordinate is also periodic we have closed magnetic lines on each $S^3$ following the fibers. This constant magnetic flux is responsible for supporting the throat. Using this result we also find that the meron has first Chern-number $c_1= \Phi/2 \pi = 1/2$, unlike the magnetic monopoles for which it is an integer. For an anti-meron the topological numbers and the flux pick a minus sign (it reverses orientation along the fiber). ]]>

0$, it has a single minimum at $u=0$ (above zero) and blows up near the ${\rm AdS}_2$ boundaries at $u = \pm \pi/2$ as \begin{equation} V(u) \sim \frac{m^2}{\left( u \pm \frac{\pi}{2} \right)^2} \, . \end{equation} The periodicity results in the presence of multiple ``sectors'', which for $m^2>0$ are completely decoupled since the potential is not penetrable. The energies are negative and below the minimum of the potential. Therefore, there are no bound state solutions to this equation and the boundary value problem has a unique solution, for more details see also the discussion in section~\ref{CorrelatorsMeronwormhole}. In case that $m^2 <0$, one has to distinguish two cases, since the potential admits no ground state if $m^2 < m^2_{\rm BF} = - 1/4$ which is in-line with the system being unstable below the BF-bound, where the conformal dimensions $\Delta_\pm$ become complex. When the mass is between $m^2_{\rm BF} < m^2 < 0$ perturbations naively seem to be able to pass between different ``universes'', but one also needs to impose a Hermiticity condition on the Schr\"{o}dinger operator~\eqref{3e}, discussed in appendix~\ref{Homogeneousstability}. This condition is a zero probability-flux condition across the boundaries, which is also consistent with the fact that the space ends there (the conformal factor blows up).\footnote{In a more general context the appropriate condition is that the Euclidean fluctuation operator be an elliptic operator~\cite{Witten-ml-2018lgb}. The Hermiticity condition for the scalar Schr\"{o}dinger ODE is a particular case of this more general property to be imposed on solutions.} One can therefore maintain stability in this regime and focus on a single period of the potential. According then to our previous discussion there is no normalisable bound state (for which the fluctuation operator is Hermitean) in the spectrum. In the opposite case (positive energies-normalisable bound solutions) one would have the freedom to add any normalisable solution that vanishes at the boundary, hence we would have a family of different solutions with the same boundary values. We therefore also conclude that the boundary value problem has a unique solution (in analogy with the Fefferman-Graham theorem for a single boundary AdS space). Later on we will see that similar conclusions can be drawn for the higher dimensional examples of two boundary wormholes. After discussing these properties of the problem, the general solution after the transformation \begin{equation} z= \sin u\sp z \in [-1,1]\;, \end{equation} can be written as \begin{equation}\label{4e} \phi(z) = (1-z^2)^{\frac{1}{4}} \, \left( \, C_1 P^\mu_\nu(z) + C_2 Q^\mu_\nu (z) \, \right) \, , \end{equation} with $P^\mu_\nu(z) , \, Q^\mu_\nu(z)$ the associated Legendre functions of the first and second kind \linebreak and~where\footnote{For these indices the functions are called \emph{conical-Mehler} functions. They also appear as kernels in the Mehler-Fock transform.} \begin{equation}\label{41e} \m= \half \sqrt{1+4 m^2}~~~{\rm and}~~~ \n = - \half + i k \,. \end{equation} From this solution, we can derive the BtB propagator as well as the btB propagators and then the correlators analytically. In particular the un-normalised btB propagator with a source inserted on the boundary located at $z=-1$,\footnote{We call the boundary located at $z=-1$ boundary one or the first boundary. Equivalently boundary two or second boundary is the one located at $z=1$.} is \begin{align}\label{5e} C_1 K^1(z, k) & = C_1 (1-z^2)^{\frac{1}{4}} \left(\frac{\pi}{\sin \pi \mu} P^{\mu}_\n(z) - \frac{2}{\cos \pi \mu} Q^\m_\n(z) \right)\, \nn \\ & = - C_1 (1-z^2)^{\frac{1}{4}} \frac{2 \pi \, \Gamma(\n + \m + 1)}{\sin (2 \pi \mu) \Gamma(\n - \m +1)} P^{-\mu}_\nu (z) \end{align} This propagator has the leading asymptotic behaviour at the first boundary and the subleading at the second i.e.\ in case of an irrelevant deformation, this solution blows up at the first boundary and asymptotes to zero at the second. One may normalise it at some cutoff $\epsilon$, by forming the ratio \begin{equation} f_k^1 = {K^1(z, k)\over K^1(\epsilon, k)}\,. \end{equation} The second linearly independent solution is \begin{align}\label{6e} C_2 K^2(z, k) & = C_2 (1-z^2)^{\frac{1}{4}} \left(\frac{\pi}{\sin \pi \nu} P^{\mu}_\n(z) - \frac{2}{\cos \pi \nu} Q^\m_\n(z) \right)\, \nn \\ & = C_2 \frac{ \pi \sin 2 \pi \mu }{\sin 2 \pi \nu} K^1(-z, k) \end{align} having exactly the opposite behaviour with the first btB propagator (after normalising, we obtain $f^1_k(z) = f^2_k(-z)$). Notice also that the two solutions are related through the exchange $\m \leftrightarrow \n$ in the respective prefactors. To compute the correlator using the general equation~\eqref{Onshellactionmomentum}, we need to differentiate the btB propagator using the contiguous relation of Legendre functions \begin{equation}\label{7e} (1-z^2)\frac{d P_\n^\m(z)}{d z} = -\n z P_\n^\m (z) + (\n + \m ) P_{\n - 1}^\m (z)\,, \end{equation} with the same formula also holding for $Q_\n^\m$. We then find \begin{equation}\label{8e} \frac{d f^1_k}{d z} \Big|_{z=-1+\epsilon} \, = \, \left[\frac{-(\n + \half) z}{(1-z^2)} + \frac{(\n - \m ) P^{-\mu}_{\nu-1} (z)}{(1-z^2) P^{-\mu}_\nu (z)} \right]_{z=-1+\epsilon} \, \end{equation} The structure of this ratio upon expanding for small $\epsilon$ is of the form \begin{equation}\label{9e} \frac{d f^1_k}{d z} \Big|_{z=-1+\epsilon} = \epsilon^{-1} \frac{(A_1+ A_2 \epsilon +\ldots ) + \epsilon^{\m}(B_1 + B_2 \epsilon + \ldots )}{(C_1+ C_2 \epsilon + \ldots ) + \epsilon^\mu (D_1 + D_2 \epsilon+\ldots )}\, , \end{equation} where the various terms depend on $k, \,m$. To compute the correlator, one should keep only fractional powers in $\epsilon$, since the integer divergences contain analytic terms in $k$ which can be removed by local counterterms. The result after the expansion is \begin{equation}\label{10e} \frac{d f^1_k}{d z} \Big|_{z=-1+\epsilon} = \epsilon^{-1+ \mu} \, G_{11}(k) \, + \ldots \,\, , \qquad G_{11}(k) = \left( \frac{B_1}{C_1} - \frac{A_1 D_1}{C_1^2} \right) \, , \end{equation} where the term in parenthesis is the renormalised $1-1$ correlator that takes the form\footnote{This is the correlator between two points on the same boundary.} \begin{align} G_{11}(k) & = \mathcal{N}_{11}(m) \frac{ \Gamma(\half + \half \sqrt{1+4m^2} + i k) \Gamma(\half + \half \sqrt{1+4m^2} - i k)}{\Gamma(\half + i k) \Gamma(\half - i k)} \nn \\ \mathcal{N}_{11}(m) & = \frac{ \Gamma \left(1-\frac{1}{2} \sqrt{4 m^2+1}\right) }{2^{-1+\frac{1}{2} \sqrt{4 m^2+1}} \Gamma \left(\frac{1}{2} \sqrt{4 m^2+1}\right)} \, . \label{AdS11correlatormomentum} \end{align} \begin{figure} \centering \includegraphics[width=0.43\textwidth]{figures/AdS2correlatorgappedir} \hspace{4mm} \includegraphics[width=0.43\textwidth]{figures/AdS2correlatorgappedr} ]]>

0$ the minimum of the potential is at \begin{equation}\label{211e} V_{\rm min} = m^2 + q^2 - \frac{q^2 k_r^2}{m^2 + q^2} \, , \end{equation} and the allowed energies $E = - k_r^2 + q^2$ are always below $V_{\rm min}$, so that no bound states exist. In case $m^2 < 0$, one needs to take also into account the BF-bound of appendix~\ref{AdS2Stability}, that is $0 > m^2 \geq m^2_{qBF} = - 1/4 + q^2$, else the background is unstable. This also implies that in this case $q^2 < 1/4$. We then need to further consider the two subcases: \begin{itemize} \item In the first case the potential is bounded below $m^2 + q^2 >0$ so that $1/8 < q^2 < 1/4$. This then means that bound states can exist for \begin{equation}\label{222e} E^{\rm bound} = -k^2_{\rm min} + q^2 \geq V_{\rm min} \quad \Rightarrow \quad 0 \geq m^2 + \frac{m^2 k^2}{m^2 + q^2} \, , \end{equation} which can be satisfied for states with low enough momentum $k$ (the high momentum states have energies below the minimum of the potential). Such a case is plotted in the left side of figure~\ref{fig:Morse-Rosen}. \item In the second case the potential is unbounded below $0 > m^2 + q^2 \geq - 1/4 + 2 q^2$. This can only hold for $q^2 < 1/8$. Such a case is plotted on the right side of figure~\ref{fig:Morse-Rosen} and can in principle support a bound state if the potential is not too ``steep''. \end{itemize} This would then mean that the FG theorem fails in such cases since there exist normalisable bound states that violate the uniqueness of the boundary value problem. Nevertheless in addition one should check whether the fluctuation operator is Hermitean for these states, according to the discussion of appendix~\ref{Homogeneousstability}. We leave the details of such an analysis for the future. \begin{figure} \centering \includegraphics[width=0.43\textwidth]{figures/Morse-Rosen} \hspace{4mm} \includegraphics[width=0.43\textwidth]{figures/potentialchargedads2unbound} ]]>

0$, $b_{12}<0$) so that the 1-1 IR minimum corresponds to an 1-2 IR maximum. ]]>

0$). The reason is that, due to the negative energies of the Schr\"{o}dinger problem, there are no normalisable solutions to this equation. This is similar to the single boundary global Euclidean ${\rm AdS}_4$, (see appendix~\ref{AdSODE}) and the two-boundary global ${\rm AdS}_2$, see~\ref{AdS2correlators} where we do not find normalisable solutions to the homogeneous fluctuation equation either.\footnote{This is a general feature of the solutions we examine in this paper.} As previously discussed in the literature, this feature leads to uniqueness of the boundary value problem~\cite{Witten-ml-1998qj}. In the opposite case (positive energies-normalisable bound solutions) one would have the freedom to add any normalisable solution that vanishes at the boundary, hence we would have a family of different solutions with the same boundary values. In the limit where $B=\half$, (the limit where the two sides of the wormhole pinch-off and we have two disconnected ${\rm AdS}_{4}$ spaces) the equation~\eqref{WormE=0} reduces to equation~\eqref{AdSGlobalE=0}. In this limit the boundary value problem changes nature, and only one type of correlation function can be defined. \begin{figure} \centering \includegraphics[width=80mm]{figures/Potential1} ]]>

d$ is a positive constant such that the cross-correlator $G_{12}^{\rm UV}$ asymptotes to zero for large momenta and there is no short-distance singularity, based on the results of section~\ref{HomogeneousODES}.\footnote{Notice that one could also take the cross-correlator to be exponentially suppressed $G_{12}^{\rm UV} \sim p^a e^{- b p}$, the qualitative results remain unaltered.} In the case where only $C_{11}\not =0$, we must take the operator $O_1$ to be such that $O_1^2$ is relevant, $2\Delta_1\leq d$. In that case the short distance structure of $G_{11}$ and $G_{12}$ remains unaltered, but the short distance structure of $G_{22}$ could in principle change. Indeed, from~\eqref{GIJ11}, if $\Delta_{12}<{d\over 2}-\Delta_2$, then the $G_{12}^2$ term would dominate the $G_{22}$ term. However, the constraint $\Delta_{12}>d$ forbids this case. We now examine the representative example of the $11$ correlator when only the coupling $C_{12}$ is turned on. In this case we obtain \begin{equation} \label{G1112} \tilde{G}_{11,12}=\frac{G_{11}}{\left(1 - C_{12} G_{12}\right)^2 - C_{12}^{2} G_{11} G_{22}} \, . \end{equation} In the case of theories where the spectra at the two boundaries are symmetric, $\Delta_1=\Delta_2$ and the UV structure of the deformed correlator $\tilde{G}_{11,12}$ is similar to $G_{11}$. However, if the dimensions on the two boundaries are different, and if $\Delta_1+\Delta_2-d>0$, then \begin{equation} \lim_{p\to\infty}\tilde{G}_{11,12}=-{1\over C_{12}G_{22}} \end{equation} However, the most radical change happens in the deformed $\tilde{G}_{12,12}$ correlator \begin{equation} \label{G1212} \tilde{G}_{12,12} = \frac{C_{12}(G_{11} G_{22}- G_{12}^2)+G_{12}}{(C_{12} G_{12}-1)^2-C_{12}^2 G_{11} G_{22}} \, . \end{equation} If we assume the canonical case $\Delta_1+\Delta_2-d<0$, then the denominator is near one, in the UV limit. However, in the numerator, the term $G_{11}G_{22}$ dominates in the UV\@. If \begin{equation} {d\over 2} ~~<~~\Delta_1+\Delta_2~~<~~d \end{equation} then the cross correlator now acquires a singularity at short distance. It is not clear whether the whole setup allows double-trace deformations with $C_{12}\not=0$. If it does however, this seems to be in agreement with qualitative properties of the putative field theory example discussed in the next section. ]]>

1~~~{\rm and}~~~ \Lambda^2-m^2\geq 2 \label{a35}\end{equation} the theory in~(\ref{a9}) is reflection positive, quadratic and satisfies all Euclidean QFT axioms including cluster decomposition. Interestingly, in this example, reflection positivity puts a lower bound on the amount of non-locality in the inter-theory coupling, controlled by $\Lambda$. \looseness=-1 We now shall attempt to rotate this theory to Minkowski space by the standard rescription $p^0\to ip^0$ or $x^0\to ix^0$. This will allow in particular $p^2$ to be positive, negative or zero. We shall find now the poles of the propagators of~\eqref{a11} or equivalently the zeros of the functions $D_{\pm}(q)$. This will be useful in order to understand if the analytically continued theory to Lorentzian signature is sensible. We start with $D_+$, \begin{equation}\label{a18} D_{+}(q)=0~~~\to~~~ (q^2+m^2)(q^2+\Lambda^2)+1=0 \end{equation} with two solutions, $- q_{\pm}^2$ with $q_{\pm}^2$ given in~(\ref{a21}). If $|m^2-\Lambda^2|>2$ both $q^2_\pm$ are real and positive and therefore they correspond to positive mass squared solutions. If $|m^2-\Lambda^2|<2$ the solutions are complex, and therefore the analogue of the mass square is complex. This is the case where the Euclidean theory violates reflection positivity. From~(\ref{a20}) we observe that the propagator has two poles, that correspond to positive masses squared (when $|m^2-\Lambda^2|>2$), but the respective residues have opposite signs and therefore one of them is a ghost. On the other hand, in the minus sector the spectrum is healthy as can be checked from~(\ref{a22}). \looseness=-1 We conclude that the free theory in question, exists only as a healthy theory in Euclidean space, when~(\ref{a35}) is valid, where it exhibits the phenomena we have seen in the holographic model. It is not clear however, if analogous non-trivial interacting theories are possible in Lorentzian signature. Moreover, here we encounter an apparent puzzle: this theory seems to violate the Osterwalder-Schrader theorem,~\cite{Osterwalder-ml-1973dx} but we do not understand why. It is clear that this theory is a special case of a large class of possible theories that can be constructed by generalizing the ingredients used here. One clear generalization is to use a more general function $f(p^2)$ to couple the two operators. One such parametrization is \begin{equation} f(p^2)=\sum_{n=1}^{\infty}{c_n\over p^2+\Lambda_n^2} \end{equation} We can also add interactions in the two individual theories, or interaction vertices that include composite operators with more than two elementary fields. It is not a priori clear what are the properties of such theories and whether they are sensible at least as Euclidean theories, but we expect that a subset of them will be. It is not also clear whether a subset of them will have a healthy Minkowski continuation but this is an interesting question worthy of further study. ]]>

-1$ (always), while $\Psi_-$ is normalisable when $s<0$ and non-normalisable when $s>0$. The bulk field $\xi(u)$ has the two asymptotic behaviours \begin{equation} \xi_\pm(u) \sim u^{{3\over 2}\pm \left(s+{1\over 2}\right)} \end{equation} from which one finds the canonical conformal dimensions. The relation is $\Delta_\pm = {3\over 2}\pm \left(s+{1\over 2}\right)$. The conformal dimensions in AdS are then \begin{equation} \Delta_\pm = \frac{3}{2} \pm \frac{1}{2} \sqrt{9+ 4 m^2} \end{equation} In terms of these if $\Delta_\pm >1$ the corresponding solution is normalisable ($\Delta_+$ is always in this range), while if $\Delta_- <1$ it is not normalisable. In addition solving the equation in the IR $u \rightarrow - \infty$ one finds the solutions \begin{equation} \Psi^{\rm IR}_{\pm}\sim e^{\pm (\ell+1) u} \end{equation} Since one of the two solutions is divergent in the IR, one can also impose IR regularity to remove it. ]]>

0$ can be used to construct the propagator in the bulk of the space-time. All the equations we shall study in the rest, will reduce to ODE's due to the radial dependence of the background fields. It will be convenient then to bring the fluctuation equation into a Schr\"{o}dinger form with $E$ the energy of the Schr\"{o}dinger problem, so that stability can be readily tested. In particular for a metric of the form \begin{equation}\label{F2} d s^2 = e^{2 \Omega} \left(du^2 + ds_{d}^2 \right)\, , \qquad ds_{d}^2 = g_{i j}(u) dx^i dx^j \, , \end{equation} where $g_{ij}(u)= e^{2 f(u)} \tilde{g}_{ij}(x)$ one finds the scalar fluctuation equation \begin{equation}\label{Genericfluctuationeqn1} \left( \partial_u^2 + e^{- 2 f} \Box_{\tilde{g}} \right) \phi +\left[ (d-1) \Omega' + d f' \right]\phi' - m^2 e^{2 \Omega} \phi = 0 \end{equation} To remove the first derivative of $\phi$ and bring the equation into Schr\"{o}dinger form, we perform the following transformation $\phi = e^{-\beta}\Psi$ where: \begin{equation}\label{F3} 2 \beta' = (d-1) \Omega' + d f' \end{equation} and after an integration \begin{equation}\label{F4} \beta = \frac{d-1}{2} \Omega +\frac{d}{2} f \end{equation} then the differential equation takes its final form \begin{equation}\label{F5} -\Psi'' + \left(m^2 e^{2 \Omega} + \beta'' + {\beta '}^{2}- e^{- 2 f} \Box_{\tilde{g}} \right) \Psi = 0 \end{equation} in terms of the metric functions $\Omega(u), f(u)$. \pagebreak Let us now use as a warmup example the case of pure AdS\@. We expand the modes in $S^3$ harmonics $\phi=\chi_{\ell}(r)Y_{\ell m p}(\Omega_3)$ with $-\Box_\Omega^2 Y_{\ell m p} = \ell(\ell + 2) Y_{\ell m p}$. Then, the radial fluctuation ODE becomes \begin{eqnarray}\label{F6} \chi_{\ell}''+3 \coth r \chi_{\ell}'+\left( E-m^2-\frac{\ell(\ell+2)}{\sinh^2 r} \right) \chi_{\ell}=0 \end{eqnarray} One can then rescale $\chi(r) = \sinh(r)^{-3/2} \psi(r)$ so that \begin{equation}\label{AdSglobalcoords} -\psi_{\ell}'' + \left[ \frac{\ell(\ell+2)}{\sinh^2 r} + \frac{3}{4} \frac{1}{\sinh^2 r} \right] \psi_{\ell} = \left(E-m^2 - \frac{9}{4} \right)\psi_{\ell} \end{equation} This potential has a divergence (blows up) at $r=0$, see figure~\ref{fig:Potentials}. This is a manifestation that pure AdS ends at $r=0$. The spectrum in these coordinates is continuous and bounded below. The constant shift $9/4$ corresponds to the fact that AdS can support negative $m^2$ as long as one stays above the BF bound $m^2_{\rm BF}=-9/4$. After this warmup, we will now perform a preliminary stability analysis of our solutions and make a comparison with results obtained previously in the literature for similar solutions (mainly in~\cite{Maldacena-ml-2004rf} and~\cite{Hertog-ml-2018kbz}). ]]>

0$, this potential has a quite interesting behaviour: it could either have a single maximum but there is also the possibility for it to exhibit two maxima and one minimum depending on the values of $s,B$. These two cases are depicted in~\ref{fig:Potentials2}. The physical meaning is that we can have both continuous (scattering) and bound states. The first type of states was also encountered in the meron wormhole. In this case fluctuations can penetrate and pass through the other side. Therefore, the two sides ``communicate'' and one finds both a reflection and a transmission amplitude. On the other hand, the normalisable bound states correspond to states that are localised near the throat and since they can have energies below zero, they lead to a perturbative instability for some regime of parameters. In order to remove the instability, we need to impose $B>1/2$, in line to what was found in the meron case, although in the latter case this condition was coming from the fact that the space pinches off, while now it is a stability criterion. The BF bound then is again given by the ${\rm AdS}_3$ condition $m^2_{\rm BF} = -1$. On the other hand, if we stay in the region $B<1/2$, even if the bound states have $E-m^2-1<0$, we find that for large positive $m^2$ these bound states will also have positive $E$, and therefore there can exist localised states that are still stable. To quantify this, we note that for $B<1/2$, the lowest energy supported states are for \begin{equation}\label{F10B} E_{\rm min}-m^2-1 \, = \, V_{\rm min} \, = \, \frac{s(1-s)}{B+\half} - \frac{\frac{1}{4}- B^2}{\left(B+\half \right)^2}\, , \end{equation} so that iff \begin{equation}\label{F10C} E_{\rm min} = m^2 + 1 + \frac{s(1-s)}{B+\half} - \frac{\frac{1}{4}- B^2}{\left(B+\half \right)^2} > 0 \, \end{equation} then the operator has both scattering and bound states but they are stable. This condition is most strict for $s=0$, so that iff \begin{equation}\label{F10D} m^2 > \frac{\frac{1}{4}- B^2}{\left(B+\half \right)^2} - 1 \equiv m^2_{\rm WBF} \, , \end{equation} then all states are stable. $m^2_{\rm WBF}$ then can be thought of as an analogue of the BF bound for this wormhole solution. \begin{figure} \centering \includegraphics[width=0.45\textwidth]{figures/HyperbolicfiniteE1} \hspace{4mm} \includegraphics[width=0.43\textwidth]{figures/HyperbolicfiniteE2} ]]>

1/2$. We notice that we can obtain the pure AdS quotients for $B= \half$, which is the limit when the potential starts to form a well and negative normalisable states can appear. Nevertheless one can extend the stability for $B<1/2$, provided that $m^2 > m^2_{\rm WBF}$, but one has to understand the role of the bound normalisable modes. It would be also interesting to complete our stability analysis using the tools of~\cite{Hertog-ml-2018kbz}, since in this more recent work another similar wormhole supported by an \emph{axion} field was found to have an infinite number of unstable modes. Nevertheless there is a significant difference of our solution compared to these single axion wormholes, since we have a positive definite kinetic term for our scalar field, while the afforementioned axionic solution comes from an action with the wrong sign for the axion kinetic term.\footnote{This is also in line with the statement in~\cite{Hertog-ml-2018kbz} that dilaton solutions do not suffer from these instabilities found for axions.} \boldmath ]]>

0 \, . \end{equation} Therefore is this condition holds, none of the bound or scattering states lead to an instability. This condition is most strict for $k_r=0$, so that iff \begin{equation}\label{F17} m^2 > q^2 - \frac{1}{4} \equiv m^2_{qBF} \, , \end{equation} all the states are stable. This is again the bulk analogue of the dual Euclidean field theory operators having real conformal dimensions. We conclude that simple probe scalar perturbations do not lead to an instability for the solutions studied in this paper, as long as one stays above the appropriate BF-bound, which is the bulk analogue of the dual theory conformal dimensions remaining real. We expect this dual field theory criterion to extend to other modes and correlation functions as well. Nevertheless our analysis is far from complete- one should study all possible gauge invariant combinations of perturbations around our specific wormhole backgrounds for a complete stability analysis that can settle the physical interpretation of such solutions in the semi-classical Euclidean path integral. ]]>

0$ there is no bound state. The question is what happens when the potential switches sign. In the case of ${\rm AdS}_2$ we find the exact solutions as a linear combination of Legendre functions $P_\nu^\mu(u), Q_\nu^\mu(u)$ with \mbox{$\mu =\half \sqrt{1+ 4 m^2}$}, $\nu = - \half + i k $ with $E = - k^2$, together with the connection formulae relating the two sides of the wormhole \begin{equation}\label{V5} \begin{pmatrix} P_\nu^\mu(-u) \\ Q_\nu^\mu(-u) \end{pmatrix} = \begin{pmatrix} \cos \pi (\nu+\mu) & - \sin \pi (\nu+\mu) \\ - \sin \pi (\nu+\mu) & - \cos \pi (\nu+\mu) \end{pmatrix} \begin{pmatrix} P_\nu^\mu(u) \\ Q_\nu^\mu(u) \end{pmatrix} \end{equation} It is then a tedious but straightforward exercise using the properties of Legendre functions and asymptotic expansions to show that there exist no normalisable states for which the operator is Hermitean for $\mu \in (0, \half)$. An easier method to check this is using~\eqref{Schroendinger} for $E_1 = E_2 = - k^2$, in conjunction with the Wronskian of the associated Legendre functions on the real axis \begin{equation}\label{V4} \mathcal{W}\left(P_\nu^\mu , Q_\nu^\mu \right)(u) = \frac{1}{1-u^2} \frac{\Gamma(\nu + \mu + 1)}{\Gamma(\nu - \mu + 1)}\, , \end{equation} which cannot vanish if $\mu$ is a positive real number and $\nu = - \half + i k $. This result is easy to understand since the wave-functions do not change nature as long as $\mu$ is a positive real number and this covers the whole range $m^2 > -1/4 $. Once $m^2 < - 1/4$, the parameter $\mu$ becomes complex. Then one finds that $P_\nu^\mu(u), P_\nu^\mu(-u)$ or $P_\nu^\mu(u), P_\nu^{-\mu}(u)$ form the correct complex pair of orthogonal states than can be used as a basis for the wave-functions~\cite{Zwillinger}, which are again found to oscillate rapidly near the two boundaries. This of course is in line with the bound derived in section~\ref{AdS2Stability}, where the analysis is much simpler, so we omit the rest of the examples that are treated there. ]]>