--- title: Euclidean Wineglass Wormholes url: https://www.emergentmind.com/topics/euclidean-wineglass-wormholes type: topic --- # Euclidean Wineglass Wormholes Euclidean wineglass wormholes are smooth Euclidean saddles whose transverse size narrows to a throat and then flares toward one or more larger regions, producing the characteristic “wineglass” profile of the scale factor or radius function. The term is used explicitly in some recent papers and only geometrically in others, but across the literature it consistently refers to two-ended Euclidean geometries with a regular neck, or to O(4)-symmetric cosmological instantons with both a stem and a rim from which an expanding Lorentzian universe can be continued [2505.21118; 2605.10548]. In asymptotically flat axion gravity the canonical example is a smooth, \( \mathbb{Z}_2 \)-symmetric throat with two asymptotically flat ends, whereas in inflationary constructions the same label is applied to wormholes whose Euclidean scale factor has a local minimum and a local maximum, so that continuation across the maximum yields an expanding universe [2505.21118; 2603.11003]. ## 1. Geometric archetypes and terminology The simplest wineglass geometry is the two-ended Euclidean wormhole supported by axion flux. In four-dimensional Euclidean gravity with spherical symmetry, \[ ds^2 = N(r)^2 dr^2 + a(r)^2 d\Omega_3^2, \] timelike target-space geodesics \(c<0\) produce wormholes, and in conformal gauge \(N=a\) one finds \[ a^2(r) = \sqrt{\frac{\kappa_4^2 |c|}{6}} \cosh(2r). \] This gives a smooth minimum at \(r=0\) and symmetric growth toward both asymptotic ends, with \( \mathcal{H}=\dot a/a=\tanh r \) vanishing at the throat [2505.21118]. In this sense, the “wineglass” label denotes a regular neck together with two flaring asymptotic regions. A second, cosmological use of the term arises in O(4)-symmetric Euclidean instantons of the form \[ ds^2 = d\tau^2 + a(\tau)^2 d\Omega_3^2. \] In the minisuperspace constructions used for inflationary nucleation, the scale factor decreases from an asymptotic region, reaches a local minimum identified as the throat, then increases to a local maximum identified as the mouth or rim. Analytic continuation at that rim yields a Lorentzian universe with zero initial expansion rate and positive acceleration [2603.11003; 2605.10548]. In this class, the wineglass profile is not just a symmetric neck; it is a throat-plus-mouth geometry adapted to cosmological materialization. The same geometric language also appears in asymptotically AdS settings with non-spherical slices. Janus-type wormholes in Einstein–dilaton gravity have topology \[ \mathcal{M}_{d+1} \cong \Sigma_d \times I, \] with compact hyperbolic \( \Sigma_d \) and an interval joining two asymptotic AdS boundaries; the metric can be written as \[ ds^2 = \frac{du^2}{4u^2(1-u+bu^d)} + \frac{1}{u} dH_d^2, \] with a smooth neck at the smallest positive root of \(1-u+bu^d=0\) [2407.15630]. In AdS\(_3\), genus-two and torus-interval handlebodies provide higher-genus realizations of the same visual motif, for example \[ \frac{ds^2}{\ell^2}= d\rho^2+\cosh^2\rho\, d\Sigma_{g=2}^2 \] and the \(T\times I\) handlebody that computes the double-torus amplitude in pure AdS\(_3\) gravity [2508.14165; 2309.02533]. Several papers analyze such geometries without using the term itself. Smooth Euclidean axion wormholes in AdS are described as having a nonzero neck radius and two asymptotic Euclidean-AdS regions, which is geometrically the same wineglass shape [2004.08956]. Euclidean AdS\(_4\) wormholes in the STU model likewise have \[ a(\mu)=\frac{1}{gk}\cosh(g\mu), \] with a unique minimum at \( \mu=0 \) and two asymptotically EAdS\(_4\) ends [2312.13955]. This suggests that “wineglass” is primarily a geometric descriptor rather than a model-specific designation. ## 2. Dynamical realizations and supporting matter sectors The best-known support mechanism is axion flux. In the asymptotically flat four-dimensional model revisited in “Euclidean wormholes stability analysis revisited” [2505.21118], the Euclidean action may be written in a dual two-form formulation with \(F=dB\) and optional dilaton \( \phi \), or in the pseudoscalar axion formulation after integrating out \(F\). The fixed axion charge \[ Q = \frac{1}{2\pi^2}\int_\Sigma F \] holds the throat open. In the pure-axion case \(c=-Q^2\), while in the axion–dilaton system regularity requires \[ b^2 < \frac{8}{3}\kappa_4^2, \] so that the dilaton and axion remain finite across the throat [2505.21118]. In asymptotically AdS or flat inflationary wineglass wormholes, the matter sector is enlarged by a scalar field with potential \(V(\phi)\). A representative minisuperspace action is \[ \frac{S_E}{2\pi^2} = \int d\tau\left(-3a\dot a^2+\frac12 a^3\dot\phi^2-3a+a^3V+\frac{Q_a^2}{a^3}+\frac{Q_m^2}{a}\right) -3a^2\dot a\big|_{\tau=0}, \] where \(Q_a\) is an axionic charge and \(Q_m\) a magnetic charge [2603.11003]. The charge terms enter the equations of motion as repulsive Euclidean contributions, \(Q_a^2/a^6\) or \(Q_m^2/a^4\), and determine the throat size. A closely related construction replaces the axionic or magnetic support by an electric Maxwell sector. In “Prepare inflationary universe via the Euclidean charged wormhole” [2411.13844], the Euclidean Maxwell equation integrates to a conserved flux, with \(F_{\mu\nu}F^{\mu\nu}=Q/a^4\). The turning-point equation at the rim becomes \[ \tilde V_0 x^2 - x + \tilde Q = 0,\qquad x=a(0)^2, \] with discriminant \( \Delta = 1-4\tilde V_0\tilde Q>0 \), giving two positive roots identified with \(a_{\max}\) and \(a_{\min}\) [2411.13844]. The larger root gives the Euclidean local maximum needed for expansion after continuation. Other realizations are less flux-based and more sigma-model driven. In the Euclidean STU truncation of \( \mathrm{SO}(8) \) gauged supergravity, wormholes are supported by three pairs of independent complex scalars \( (z_i,\tilde z_i) \), with the bulk metric \[ ds_4^2 = d\mu^2 + g^{-2} e^{2A(\mu)} d\Sigma_{S^3}^2,\qquad e^{2A}=\frac{\cosh^2(g\mu)}{k^2}, \] and scalar profiles constrained by \( \sum_i k_i^2 = k^2-2 \) [2312.13955]. In Janus wormholes, a single massless scalar interpolates between different marginal couplings on the two AdS boundaries, with the asymptotic jump \( \Delta\phi = \phi_+ - \phi_- \) encoded by the deformation parameter \(b\) [2407.15630]. Purely gravitational AdS\(_3\) realizations also exist. The \(T\times I\) handlebody in pure AdS\(_3\) gravity is the simplest two-boundary wineglass wormhole in the spectral-form-factor literature, while genus-two wormholes in AdS\(_3\) furnish time-reflection-symmetric state-preparation geometries with a minimal neck and two asymptotic regions [2309.02533; 2508.14165]. ## 3. Boundary data, conserved quantities, and on-shell actions Boundary conditions are central because the same geometry can represent distinct semiclassical problems in different ensembles. In asymptotically flat axion wormholes, the axion sector is treated at fixed charge: one fixes \(Q\), equivalently Neumann boundary conditions for \( \chi \) or Dirichlet boundary conditions for \(F\), while the dilaton obeys Dirichlet boundary conditions at infinity [2505.21118]. The boundary term \[ -\int d(e^{b\phi}\chi \star d\chi) \] is crucial both for bounding the action after imposing constraints and for fixing the axion charge at infinity [2505.21118]. In asymptotically AdS inflationary wineglass wormholes, half-wormholes are constructed on \( \tau\in[0,\tau_{\mathrm{end}}] \) with Neumann conditions \[ \dot a(0)=0,\qquad \dot\phi(0)=0 \] at the mouth, so that the geometry can be analytically continued to Lorentzian time, and Dirichlet conditions on \(a(\tau_{\mathrm{end}})\) and \( \phi(\tau_{\mathrm{end}}) \) at the asymptotic boundary [2603.11003]. Because the Euclidean action diverges for EAdS asymptotics, the Hamilton–Jacobi method is used to identify counterterms \[ 2\pi^2(a^3W+a\Phi), \] and the physical weighting is obtained by subtracting the background AdS action \(S_{EAdS}=-12\pi^2/V(\phi_{\mathrm{ext}})\) [2603.11003]. Charged wineglass half-wormholes use a similar asymptotically EAdS setup, but the on-shell minisuperspace action takes the reduced form \[ S_E^{\mathrm{on\text{-}shell}} = 4\pi^2 \int d\tau \left[\frac{Q}{a^2}-a^3V(\phi)\right] + S_{GHY}+S_{ct}, \] up to a \(V_0\)-independent additive constant from the EAdS boundary terms [2411.13844]. The half-wormhole then prepares a semiclassical wavefunction \( \Psi \propto e^{-S_E^{wh}} \), so relative creation probabilities depend directly on the \(V_0\)-dependence of the on-shell action [2411.13844]. In Janus wormholes, the boundary data are the two constant marginal couplings \( \phi_\pm \), and the renormalized amplitude is expressed directly as \[ Z_{WH}[\phi_+,\phi_-;\Sigma_d] = \exp\!\left[-\frac{d}{8\pi G_N}F(b)\,V(\Sigma_d)\right]. \] The function \(F(b)\) is analytic and strictly increasing on the regular branch \(0\le b0 \) in Lorentzian time, so the universe initially expands [2603.11003]. Smaller axionic or magnetic charges drive the scalar higher up the potential barrier and therefore produce initial data more favorable to a longer phase of slow-roll inflation [2603.11003]. The central structural result is the no-boundary limit. As \(Q_a\to 0\) or \(Q_m\to 0\), the throat radius shrinks to zero, the asymptotic AdS side disconnects, and the remaining compact geometry is a Hartle–Hawking no-boundary instanton with the scalar at the positive maximum \( \phi_{\max} \) [2603.11003]. The renormalized, background-subtracted actions of all wineglass wormholes converge to \[ -S_E^{\mathrm{no\text{-}boundary}}=\frac{12\pi^2}{V(\phi_{\max})}, \] and the no-boundary instanton has the highest weighting overall [2603.11003]. “Birth of Inflationary Universes via Wineglass Wormholes and their No-Boundary Relatives” develops the same transition in both asymptotically flat and asymptotically AdS settings [2605.10548]. There, the defining feature of a wineglass wormhole is the presence of a local maximum of \(a(\tau)\), the rim, in addition to a local minimum, the stem. The local maximum is what allows analytic continuation to a Lorentzian spacetime with \[ a(0)=a_0,\qquad \dot a(0)=0,\qquad \ddot a(0)>0, \] so that the offspring universe expands after materialization [2605.10548]. As the conserved charge tends to zero, the stem pinches off and the solution splits into the parent background plus a disconnected no-boundary instanton [2605.10548]. Charged wineglass half-wormholes refine this picture by identifying a parameter regime in which the Euclidean action decreases with the initial potential height \(V_0\). In the broad-plateau regime \(a_{\min}\simeq a_{\max}\), the action of the charged half-wormhole becomes a decreasing function of \(V_0\) in an allowed window compatible with the turning-point discriminant \( \Delta=1-4\tilde V_0\tilde Q>0 \); the resulting creation probability \[ P \propto e^{-2\,\mathrm{Re}\,S_E^{wh}} \] then favors larger \(V_0\) and a longer period of inflation [2411.13844]. Under equal dimensionless charge and equal initial \(V_0\), the charged wineglass half-wormhole is found to have smaller action, and hence larger probability, than the corresponding axion wineglass solution [2411.13844]. ## 6. Holography, factorization, and CFT interpretations Euclidean wineglass wormholes play several distinct roles in holography. In pure AdS\(_3\) gravity, the \(T\times I\) wormhole is the simplest two-boundary handlebody and computes the correlator of two torus partition functions. In the chaotic-CFT regime its translation sector reproduces the linear ramp of the spectral form factor, while the Kuznetsov trace formula fixes the modularly required subleading corrections through Kloosterman sums and Bessel kernels [2309.02533]. In this setting, the wineglass wormhole is the minimal bulk realization of spectral correlations consistent with modular invariance and quantum chaos [2309.02533]. In AdS/CFT with matter, two-boundary Euclidean wormholes also compute averaged products of partition functions. Janus wormholes with different marginal couplings on the two boundaries satisfy \[ \langle Z[\Sigma_d]^{\phi_+} Z[\Sigma_d]^{\phi_-}\rangle \propto \exp\!\left[-\frac{d}{8\pi G_N}F(b)V(\Sigma_d)\right], \] and because \(F(b)\) is strictly increasing, the two boundary theories become increasingly decorrelated as \(|\Delta\phi|\) grows [2407.15630]. The same paper uses the wormhole amplitude as a generating functional for integrated dilaton correlators and finds crossed two-point functions that decay monotonically with the coupling difference [2407.15630]. A different use appears in the state-preparation literature. Slicing a genus-two Euclidean wormhole on a time-reflection-symmetric surface \( \Sigma \) with \(K_{ab}(\Sigma)=0\) prepares a semiclassical state whose initial data coincide with those of the spinless BTZ black hole in hyperbolic coordinates. Because half of the wormhole and half of Euclidean BTZ can be glued smoothly across \( \Sigma \), the normalized overlap between the wormhole-prepared state and the thermofield-double state is \( \mathcal{O}(1) \) when the initial data match [2508.14165]. The same analysis emphasizes an “infinite-family puzzle”: varying the handle moduli without changing the slice data produces infinitely many distinct Euclidean preparations with order-one overlap with BTZ/TFD [2508.14165]. Asymptotically AdS wineglass wormholes also illuminate factorization from the opposite direction. In the correlator analysis of [1903.05658], local operator correlators between distinct boundaries are finite at short distances, and cross-boundary Wilson-loop correlators are likewise finite. This is interpreted as evidence that the dual theory factorizes in the ultraviolet into two sectors coupled only by a soft nonlocal interaction in the infrared [1903.05658]. By contrast, in the string-theoretic AdS\(_3\times S^3\times T^4\) embedding of smooth Euclidean axion wormholes, AdS/CFT is used to argue that Coleman’s \( \alpha \)-parameter interpretation cannot be correct, because a fixed CFT Hilbert space factorizes whereas \( \alpha \)-averaging does not [0705.2768]. The holographic role of Euclidean wineglass wormholes is therefore double-edged: they furnish controlled bulk saddles for averaged observables and state overlaps, yet they sharpen the conceptual tension between connected bulk geometries and boundary factorization. ## 7. Analytical methods and unresolved problems Several recent papers emphasize that wineglass wormholes are as much a methodological problem as a geometric one. The revised flat-space stability analysis of [2505.21118] is notable for three techniques: choosing gauge-invariant variables that avoid \(1/\mathcal{H}\) singularities, treating unavoidable throat poles by a principal-value contour prescription in the complex plane, and testing positivity of the two-field quadratic form by the Sylvester criterion rather than heavy numerics. These methods are presented as reusable tools for other wormhole backgrounds [2505.21118]. Holographic renormalization is equally central. Inflationary AdS wineglass wormholes use Hamilton–Jacobi counterterms and explicit background subtraction to resolve the long-known puzzle that wineglass actions can become negative while remaining finite and bounded below by the no-boundary value [2603.11003]. In the STU-model EAdS\(_4\) solutions, the supersymmetric renormalization scheme yields the exact cancellation \(S_{\mathrm{on\text{-}shell}}=J_{\mathrm{on\text{-}shell}}=0\), but the field-theoretic explanation of that cancellation remains open [2312.13955]. Open problems recur across the literature. Stability is unresolved outside a few highly constrained sectors; one-loop determinants are not fixed by the local gluing arguments in semiclassical state preparation; AdS boundary conditions for axions can differ sharply from the asymptotically flat fixed-charge ensemble; and more exotic quasi-oscillatory wineglass families with multiple extrema of \(a(\tau)\) require both dynamical and contour analyses before their path-integral relevance can be assessed [2508.14165; 2602.23432]. The current state of the subject is therefore technically rich but not yet conceptually closed: Euclidean wineglass wormholes are established as explicit classical solutions in flat space, AdS, string compactifications, and inflationary minisuperspace, but their ultimate status as gravitational saddles still depends on boundary ensemble, matter content, and the treatment of fluctuations and contours. Source: https://www.emergentmind.com/topics/euclidean-wineglass-wormholes