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Antipodal Correlation on the Meron Wormhole and a Bang-Crunch Universe

Antipodal Correlation on the Meron Wormhole and a Bang-Crunch Universe

PHYSICAL REVIEW D 97, 126006 (2018)

Antipodal correlation on the meron wormhole and a bang-crunch

Panagiotis Betzios* Crete Center for Theoretical , Institute for Theoretical and Computational Physics, Department of Physics, University of Crete, 71003 Heraklion, Greece † Nava Gaddam Institute for and Center for Extreme and Emergent Phenomena, Utrecht University, 3508 TD Utrecht, Netherlands ‡ Olga Papadoulaki Mathematical Sciences and STAG Research Centre, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom

(Received 13 January 2018; published 11 June 2018)

We present a covariant Euclidean wormhole solution to Einstein Yang-Mills system and study scalar perturbations analytically. The fluctuation operator has a positive definite spectrum. We compute the Euclidean Green’s function, which displays maximal antipodal correlation on the smallest three sphere at the center of the throat. Upon analytic continuation, it corresponds to the Feynman propagator on a compact bang-crunch universe. We present the connection matrix that relates past and future modes. We thoroughly discuss the physical implications of the antipodal map in both the Euclidean and Lorentzian geometries and give arguments on how to assign a physical probability to such solutions.

DOI: 10.1103/PhysRevD.97.126006

I. INTRODUCTION whose few proposed resolutions suffer from closed - like curves and other intricacies [10]. Euclidean wormholes [1–3] are extrema of the Euclidean Inspired by these questions, we study a Euclidean meron action whose interpretation is still partially shrouded in wormhole [11] in of the antipodal Z2 mapping mystery. Originally they were proposed as a resolution to proposed by ’t Hooft. We find a positive definite spectrum the cosmological constant problem [4], and as objects that for scalar perturbations, whose Euclidean Green’s function lead to a loss of quantum coherence causing an inherent exhibits large antipodal correlation localized near the uncertainty in the fundamental constants of nature [2,5,6]. smallest sphere at the center of the throat after performing An alternative interpretation was given in [7] within the the antipodal map. The analytic continuation of this context of the Wheeler-de Witt equation. Besides several solution results in a finite bang-crunch geometry [12], problems specific to these proposals, a basic general with temporal and spatial boundaries, opening up a question that remains unanswered is whether such solutions possible handle on the problem of observables. are stable (and thus minima of the Euclidean action), or if they should alternatively be thought of as bounces (or maxima) that might nevertheless contribute to the path II. THE MERON WORMHOLE integral in some nonperturbative fashion akin to [8]. Consider the Euclidean Einstein-Yang Mills system: On a parallel note, defining quantum gravitational Z   observables in closed is an acute problem [9] pffiffiffi 1 1 4 a 2 S ¼ d x g − R þ ðFμνÞ 16πGN 4

* [email protected] a † in units ℏ ¼ c ¼ 1, where R is the Ricci scalar and Fμν, the [email protected] ‡ field strength for the SUð2Þ (possibly embedded in SUðNÞ) [email protected] a a gauge field Aμ. The field strength is defined as Fμν ≔ ∂ a − ∂ a þ ϵabc b c Published by the American Physical Society under the terms of μAν νAμ gYM AμAν. In addition to the Yang- the Creative Commons Attribution 4.0 International license. μ a Mills equations of motion D Fμν ¼ 0 and the correspond- Further distribution of this must maintain attribution to the author(s) and the published article’s title, journal citation, ing Bianchi identities, the Einstein equations of motion are and DOI. Funded by SCOAP3. given by

2470-0010=2018=97(12)=126006(6) 126006-1 Published by the American Physical Society BETZIOS, GADDAM, and PAPADOULAKI PHYS. REV. D 97, 126006 (2018)

1 throat. The IR divergence can be evaded by considering Rμν − gμνR ¼ 8πG Tμν 2 N similar solutions in the presence of a small cosmological constant that provides a natural regulator for such IR ¼ λ ¼ a aρ − 1 ð a Þ2 where Rμν Rμλν and Tμν FμρFν 4 gμν Fμν . The divergences. This is one possible and known way of a ¼ η −1 x−2 meron configuration Aμ aμνxνgYM yields the field making sense of similar configurations; however, a strength satisfactory solution for asymptotically flat -   is not known. Inspired by the work of Kosterlitz and 1 f1 f2 Thouless [21] (whose relevance to merons is sketched in a ¼ η þð η − η Þ Fμν aμν 2 xμ aργxγ xρ aμγxγ 2 ; [19]), we propose that the seemingly infinite classical gYM x x action must be compared with the of such configurations. A known issue with the pure meron back- with f1 ¼ f2 ¼ −1. Here, ηaμν are the ’t Hooft Eta symbols [13]—whose conventions we follow—that mix the gen- ground [19] is that one needs to regulate the meron core erators of space-time SOð4Þ with those of the SUð2Þ gauge with an ad-hoc procedure which results in a violation of the symmetry. The summation over all lower indices is carried equations of motion for the regulated configurations. This UV problem is evaded by the meron wormhole owing to its out by the flat metric δμν; however, owing to the conformal nature of the Yang-Mills equations, this is a solution in any finite-sized throat, a fact that will become even more transparent in our novel interpretation—using an antipodal conformally flat background. The meron configuration was — one of the first proposed mechanisms for confinement identification of the wormholes as holes of nothing. [14,15]. This meron-pair solution has a unit topological instanton charge split equally between the two points at A. Symmetries of the background x ¼ 0 and x ¼ ∞; the pair may be brought closer together Flat space has ten killing vectors (KVs) and five with a conformal transformation [16]. Unlike the Belavin- conformal KVs (CKVs). Since the wormhole is confor- Polyakov-Schwartz-Typkin (BPST) instanton [13,17,18], mally flat, its symmetries may be found by studying if the this solution is not self-dual. The meron’s magnetic field former remain KVs and if any of the latter get promoted to lines provide the required locally-negative density KVs. It may be checked that while rotations remain killing to support a wormhole throat and source the right hand side vectors of the wormhole background, translations become of the Einstein equations which are solved by the metric CKVs. Additionally, none of the five CKVs (dilatation and   special conformal transformations) of flat space are pro- 2 2 2 r1 μ ν 2 −2 moted to KVs. Finally, the wormhole background pos- ds ¼ 1 þ δμνdx dx r1 ¼ πG g ; μ 2 μ −2 x N YM sesses a discrete inversion symmetry under x → r1x x . In the r coordinates, it reads r → −r. This is an antipodal 2 ¼ 2 þð 2 þ 4 2Þ Ω2 ð Þ ds dr r r1 d 3 1 mapping when appended with appropriate maps on the pffiffiffiffiffiffiffiffiffi three-sphere, similar to the one considered in [22,23] in the x ¼ μ ≥ 0 ¼ x − 2x−1 with xμx and r r1 . This transforma- context of ’t Hooft’s black hole S-Matrix. tion maps the regions x ≶ r1 to the two sides of the ≶ 0 x ¼ wormhole r and the smallest sphere to r1. III. SCALAR PERTURBATIONS The geometry is that of a two-sided Euclidean space (with the two asymptotic regions at r ¼∞) connected by a The action governing the fluctuations of aR chargedpffiffiffi scalar ð2Þ ¼ 4 Lð2Þ ≡ 2r1 fieldR in this background, is given by Sϕ d x g ϕ wormhole throat of smallest size ; it has the curious 4 pffiffiffi ⋆ property that the regime of its semiclassical validity d x gϕ Mϕϕ with coincides with the perturbative regime of the YM theory. ð2Þ 2 2 2 The metric admits a natural analytic continuation r ¼ it Lϕ ¼½jDμϕj þðm þ ζRÞjϕj ; into a big bang-big crunch universe, with singularities at ¼2 ϕ ≕ ð∇ þ Þϕ r r1 [12]. It is an open problem to find general where Dμ μ igYMAμ;a . In order to explicitly 2 multimeron solutions specified by functions f1ðx Þ evaluate the spectrum of the operator Mϕ, it is first useful 2 and f2ðx Þ, constrained by both Einstein and Yang-Mills to define the following space-time operators, which re- equations. present rotations in the two invariant SU(2) subgroups of Let us mention here that the Euclidean action is the rotation group SO(4): logarithmically divergent. This is easy to see since the 1 1 Ricci scalar vanishes (R ¼ 0) and, from the Yang-Mills a a L ≔− iη μνxμ∂ν and L ≔− iη¯ μνxμ∂ν Lagrangian, one finds the usual log-divergence of the 1 2 a 2 2 a meron configuration ∼ logðL=aÞ where L is the system ½ a b¼ δ ϵabc c 2 ¼ 2 ¼ 2 ¼ −ð1 8Þ size (IR-cutoff) and a is a UV-cutoff [19]. This problem has with Lp;Lq i pq Lp and L1 L2 L = × 2 been discussed in the past, for example in [11] and later in ðxμ∂ν −xν∂μÞ . Owing to the SUð2Þ projection of these [20]. The UV problem is evaded by the finite size of the operators, the eigenvalues of L2 are given in terms of

126006-2 ANTIPODAL CORRELATION ON THE MERON WORMHOLE … PHYS. REV. D 97, 126006 (2018) half-integers l via L2 ¼ lðl þ 1Þ. Furthermore, for scalars, breaks all the noncompact symmetries of flat space, there isospin rotations are generated by the operators Ta with are no zero modes either. The vacuum spectrum is eigenvalues T2 ¼ tðt þ 1Þ, with an arbitrary total isospin t. calculated most easily in the original r coordinates and The operator Mϕ (whose vacuum counterpart we call M0) the vacuum regulated one-loop determinant of scalar can now be written as fluctuations det Mϕ=det M0 can be calculated following the prescription in [13]. Instead, in what follows we turn to x2½ 2 þ 4ð aÞ a calculation of the Green’s function for the scalar M ¼ −∇2 þ T T · L1 þ 2 ϕ 2 2 2 m ; perturbations. ðx þ r1Þ where ∇2 is the Laplace-Beltrami operator of the wormhole IV. CORRELATION FUNCTION background. In order to calculate the spectrum (of The Euclidean Green’s function can be written in terms the vacuum and the wormhole background), it turns of the homogeneous solutions (λ ¼ 0)to(3) and μ μ μ 0μ out to be convenient to solve the rescaled operator ð ν νÞ¼ νð Þ νð Þ− −2 the Wronskian determinant W P ;Q P x Q x V ϕ ¼ λ ϕ V ≔ AM A A ¼ 0μ μ equation ϕ;0 r1 , with ϕ;0 ϕ;0 ; P νðxÞQνðxÞ, with a prime indicating a derivative with 2 2 2 2 2 ðx þ r1Þ =ð4r1x Þ [13]. This corresponds to solving the respect to the argument x. This two-point function Schrödinger equation with homogeneous boundary conditions for the field ψ     is given by 3 2 2 2 16 2ðλ − 2 2Þ ∂2 þ − r1 ∂ − J1 þ r1 r1m ϕ ¼ 0 x 2 2 x 2 2 2 2 ; ∞ þ þ1 μ−1 ˜ 2x xðx þ r1Þ x ðx þ r1Þ X kXt X Xl GEðx; Ω; y; Ω0Þ¼ ð2Þ k¼0 μ¼k−tþ1 l˜¼0 m˜ ¼−l˜  μ μ  Θð − Þ νð Þ νð Þþ ↔ where J1 ≔ T þ 2L1. In order to solve this equation, y x P x Q y x y × 2 μ μ we expand the field in three-sphere harmonics that ð1 − y ÞWðPν;QνÞðyÞ 2 2 0 satisfy −∇ 3 Y ˜ ˜ ðΩÞ¼kðk þ 2ÞY ˜ ˜ ðΩÞ with ðk þ 1Þ - ðΩÞ ðΩ Þ S kl m kl m × Yðμ−1Þl˜ m˜ Yðμ−1Þl˜ m˜ : fold degeneracy since m˜ ¼ −l;˜ …; l˜ and l˜ ¼ 0; 1; …;k. 2 ðΩÞ¼ ð þ 2Þ ðΩÞ ¼ − ϕ Similarly, J1Ykl˜ m˜ j1 j1 Ykl˜ m˜ with j1 k The corresponding correlator for is achieved by the t; …;kþ t provided j1 ≥ 0. The Schrödinger equation (2) rescaling that relates the fields. First we perform the sum then reduces to a one-dimensional problem whose over the indices l˜ and m˜ and then the sum over μ using solutions are associated Legendre polynomials. To bring addition theorems [24]. The result for t ¼ 0 is them into canonical form, we redefine the field ϕ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϕ ¼ð1 − 2Þ1=2ψ ð2 Þ ð1 − 2Þð1 − 2Þ as y = r1 and change variables as E 0 x y 1 2 2 2 2 G ðx; Ω; y; Ω Þ¼− pffiffiffiffiffiffiffiffiffiffiffiffi QνðξÞ y ¼ðr1 − x Þ=ðr1 þ x Þ. In these coordinates, the two 4π2 1 − ξ2 boundaries x ¼ 0; ∞ are at y ¼1 while the smallest x ¼ ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sphere r1 lies at y . As it does on the r coordinate, where ξ ¼ xy þ ð1 − x2Þð1 − y2Þ cos ðδΩÞ and cos ðδΩÞ Z → − the discrete 2 map acts as a reflection y y on this is the angular difference, on the three-sphere, between Ω coordinate. The Schrödinger equation (2) then reduces to and Ω0. A nonvanishing isospin requires an extra finite the canonical associated Legendre form summation that can be performed but is not presented here.    2 μ2 Taking the massless limit m → 0, we find ð1 − y2Þ∂2 − 2y∂ þ νðν þ 1Þ − ψ ¼ 0 ð Þ z z 1 − 2 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ð1 − 2Þð1 − 2Þ E 0 x y G 2 ðx; Ω; y; Ω Þ¼ : ð4Þ μ m ¼0 4π2ð1 þ ξÞð1 − ξÞ whose two linearly independentpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi solutions are PνðyÞ μ 1 2 2 and QνðyÞ with ν ¼ 2 ð 1 þ 16λ − 16r1m − 1Þ and μ ¼ þ 1 μ As it must, the function displays maximal correlation when j1 . Since the order is an integer, the normalisable ð ΩÞ ð Ω0Þ solutions (in the domain y ∈ ½−1; 1) to this equation are x; coincides with y; . Rather beautifully, however, given by demanding that the degree ν also be an integer. we find maximal correlation in equal measure when ð Ω0Þ¼ð− Ω¯ Þ Therefore, for the spectrum, we find y; x; is the reflected-antipodal point of ðx; ΩÞ, allowing for a natural Z2 identification between ð ΩÞ ≡ ð− Ω¯ Þ 2 2 1 2 the two sides of the wormhole y; y; . This is λ − r m ¼ ðn þ nÞ;n≥ j1 þ 1: n;j1 1 4 further corroborated by the fact that modes on either side are related by a Unitary mapping through Legendre Of immediate note is the fact that the spectrum is positive function connection formulae [24]. After the identification, definite and the wormhole stable under (Gaussian) scalar the correlation between antipodal points, on the same side perturbations. Furthermore, since the meron wormhole of the wormhole, is a local effect near the smallest sphere.

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This may be seen from the following observation: With x; y ≥ 0, and Ω0 ¼ Ω¯ , the antipodal point of Ω, the correlation is maximal when x ¼ y ¼ 0. That is when they lie on the smallest sphere. As we move away from the center of the throat into the single exterior, the correlation on antipodal points of the three sphere diminishes, render- ing the maximal entanglement a local effect near the center of the throat. This is a concrete realisation of the antipodal identification suggested in [22,23]. As in that case, the spectrum is naturally halved owing to the following relation between three-sphere harmonics on ˜ ðΩ¯ Þ¼ð−1Þj1 ˜ ðΩÞ FIG. 1. Left: The bang-crunch universe in real time with Ω antipodal points: Yj1l m˜ Yj1l m˜ . i Ω¯ The x, y dependence of the massless correlator (4) can be being antipodal to i. Right: Antipodal identification results in half of the Einstein static universe in conformal time, with half entirely written in terms of a cross-ratio ½ðy3 − y1Þðy4 − y2Þ= ð − Þð − Þ1=2 ¼ð 1 1 Þ a three sphere (whose coordinates are collectively represented y3 y2 y4 y1 with y1;2;3;4 y; x; =x; =y . This by Ω) at the future boundary T ¼ 0. The solid line inside is null. cross-ratio is invariant under the entire projective linear group of Möbius transformations. Moreover, the Z2 map- 2 2 2 2 2 ping renders the manifold nonorientable. It is curious to ds ¼ −dt þð4r1 − t ÞdΩ3;t∈ ½−2r1; 2r1 note that this is reminiscent of string theory which admits 2 4r1 such vacua, in the presence of orientifolds, that contribute ¼ ð−dT2 þ dΩ2Þ;T∈ ð−∞; ∞Þ: 2T 3 to the path integral. cosh Having discussed the antipodal mapping, we now return 1 The end points of this closed universe represent bang and to the issue of the divergence of the Euclidean action. As crunch singularities. This geometry is depicted in Fig. 1. we saw above, after the said antipodal map, one is left The early universe begins to expand at t ¼ −2r1 and after with flat space from which a ball of radius r1 is excised. Let reaching maximum size at t ¼ 0, collapses into a crunch at us now consider an arbitrary Euclidean spacetime region of t ¼ 2r1. In conformal time, the geometry captures the 4 “ ” 4 volume L . This hole with a rough core volume of r1 complete Einstein static universe. could be centered at any spacetime point in this volume. 1 1 Defining μ˜ ¼ ν þ 2 and ν˜ ¼ μ − 2, a good basis of The probability of such a configuration appearing is μ˜ ð Þ μ˜ 4 −c log L=r1 2 solutions is Pν˜ tanh T . When is real, the solutions thus ðL=r1Þ e , (with c ∼ 1=g ) containing an YM blow up at T → ∞ and result in an instability. For “entropic” prefactor and of course, the action [25]. purely imaginary μ˜ (that is, for m2 > 1=16), the Therefore, when computing the free energy, this entropic solutions oscillate in conformal time as exp ðijμ˜jTÞ or term competes against the on-shell action. For appropriate exp ð∓ijμ˜jTÞ as T → ∞. Moreover, the modes are related g , the entropic term may dominate the classical action YM þμ˜ ð Þ¼½ −μ˜ ð Þ signalling a phase where the previous wormholes (now holes by complex conjugation Pν˜ tanh T Pν˜ tanh T . Therefore, the scalar field can consistently be quantized of nothing) proliferate. A transition between such a phase and P μ˜ ψðT; ΩÞ¼ ˜ a ˜ P ð TÞY ˜ ðΩÞþ the usual phase of would be analogous to the via j1;l;m˜ j1l m˜ j1þ1=2 tanh j1l m˜ † −μ˜ unbinding transition of vortices in the Kosterlitz-Thouless a P ð TÞY ˜ ðΩÞ ˜ ˜ j1þ1=2 tanh j1l m˜ , with the creation and anni- example [21]. Furthermore, since r2 ∼ G =g2 , the result j1l m 1 N YM hilation operators obeying canonical commutation rela- also depends on the running of the gravitational coupling tions. The vacuum is then defined by a ˜ j0i¼0. The constant. Of course this argument is incomplete, and one j1l m˜ μ˜ ð− Þ needs to perform a complete one loop computation of expanding modes Pν˜ tanh T may be obtained from the ∓μ˜ fluctuations on such a background because it directly affects contracting ones Pν˜ ðtanh TÞ via the connection matrix the entropic prefactor. Since this would be an extremely 0 1 ! interesting phase of gravity, we intend to address such a Γðν˜þμ˜þ1Þ sin πðμ˜þν˜Þ − sin ðπν˜Þ αβ B Γðν˜−μ˜þ1Þ sin ðπμ˜Þ sin ðπμ˜Þ C possibility in full detail in future work [26]. ¼ @ h i h i A β α − sin ðπν˜Þ Γðν˜þμ˜þ1Þ sin πðμ˜þν˜Þ sin ðπμ˜Þ Γðν˜−μ˜þ1Þ sin ðπμ˜Þ V. BIG-BANG BIG-CRUNCH UNIVERSE The euclidean meron-wormhole metric (1) yields a This is a canonical matrix with unit determinant, thereby closed big-bang big-crunch universe upon analytically gaining an interpretation of a Bogolyubov matrix in T continuing r → it ¼ 2ir1 tanh T: coordinates. Relative probability of creation in a given mode per unit volume can now be calculated to be 2 −2 1A similar argument might also work for the usual interpre- jβj jαj ¼ sechðπμ˜Þ. This is similar to the corresponding tation of meron wormholes but would be more involved. result in de-Sitter space [27].


VI. QUANTUM OBSERVABLES AND thoroughly study the stability [26] of this solution under HOLOGRAPHY generic metric and gauge field perturbations. Should there be negative modes as predicted in [1,12,20,28], it would be Big-bang big-crunch universes such as this pose a interesting to see if the antipodal identification projects serious problem of defining consistent quantum observ- them out. Returning to the logarithmic divergence of the ables in the theory. While AdS has a conformal boundary at spatial infinity where correlators may be defined, the future classical Euclidean action, we note that the solution we use and past null infinity of flat space allow for an S-matrix to is naturally UV regulated by the size of the hole of nothing be defined. While dS possesses a future boundary in (after antipodal identification). We can nevertheless make conformal time, the case at hand is more difficult for there sense of such solutions by comparing the Euclidean action is neither a boundary in space, nor in time. Nevertheless, to their entropy in a given volume of space. For a complete there is a possible resolution to this issue arising from the semiclassical treatment though, one needs to study all the antipodal correlation between modes at either side of field fluctuations on the background up to one-loop. zero time. Technically one computes a complete one loop determinant Since the antipodal mapping acts on the coordinates x, y with the modes split in terms of their spin and where ghost just as it acts on the coordinate r—as a reflection—the degrees of freedom are properly introduced to account for Green’s function (4) may easily be analytically continued gauge invariance. This calculation has not been performed by x → ix and y → iy. It may be checked that the maximal for a reason, in the literature. The Yang-Mills fluctuations antipodal correlation on either side of T ¼ 0 survives this are coupled to the gravitational ones on such a background. continuation, rendering the Z2 mapping legitimate. Upon This mixing results in an intricate calculation. In ongoing this identification, one obtains a big-bang universe natu- work [26], we find that this task is technically involved but rally possessing a future boundary in time, at t ¼ T ¼ 0; possible to achieve. For instance, we find that spin-2 this is a spatial section of the geometry that is an fluctuations have no negative modes and are stable. We antipodally identified three-sphere. Should a holographic hope to report on the complete 1-loop computation in the dual exist, we would expect it to have a finite Hilbert space near future [26]. of states, owing to the compactness of the geometry. This On another note, in AdS, finding large correlation boundary could then provide for a natural habitat for between the two exteriors (as we found in this work) definition of observables in the quantum theory in a similar could address several CFT puzzles raised in [12]. It would fashion to the dS proposed dualities. also be interesting to revisit instabilities of other euclidean Finally, it is worth mentioning that at t ¼ 0, half of a wormholes in this light. The antipodal map’s role in Euclidean meron wormhole may be glued to extend space- confinement is worth exploring in the light of center time. This may be done in two distinct ways. One is by symmetry in SU(N) YM theory. It is also of importance gluing the smallest three-sphere of the Euclidean wormhole to further investigate the “wrong-sign” problem of the to the largest one of the expanding universe. Another is conformal mode of the metric upon the background by gluing together antipodally identified three-spheres of coupling to the longitudinal mode of the gauge field, the euclidean and time-dependent geometries at r ¼ 0 and perhaps also in conjunction with the antipodal identifica- t ¼ 0 respectively. Of course, additional possibilities arise tion; this may provide interesting constraints on the when AdS and dS versions of the euclidean meron worm- swampland in view of the weak gravity . holes of [11,12] are considered.

VII. CONCLUSIONS ACKNOWLEDGMENTS In this article we have reinterpreted euclidean wormholes We take pleasure in thanking Bernard de Wit, Umut as “holes of nothing”; after the antipodal map, the single Gürsoy, Miguel Montero, Kostas Skenderis, Phil exterior is that of flat space from which a sphere of size 2r1 Szepietowski and especially Elias Kiritsis and Gerard ’ has been excised. We analytically computed the spectrum t Hooft for valuable discussions and suggestions. We also and the two-point function of scalar fluctuations on a wish to thank the anonymous referee for comments and Euclidean meron wormhole. The solution is found to be suggestions. P. B. is supported by the Advanced ERC stable under probe scalar perturbations. We showed that Grant SM-grav, No. 669288. O. P. is supported by the the Green’s function exhibits large antipodal correlation STFC Grants No. ST/K005391/1 and near the center of the throat. This gives good reason to No. ST/M004147/1.

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