Whereas local field operators play the crucial role in reconciling quantum mechanics and special relativity, they are not trivially compatible with the diffeomorphism invariance of gravity. In order to address this issue, we consider the background geometry which breaks the isometry spontaneously. Then the local gauge invariant operator can be constructed through the Stückelberg mechanism, where the fluctuation of the metric in the direction of the isometry breaking combines with that of matter whose classical solution breaks the isometry. This is equivalent to introducing the clock and the rod to promote the local field operators to the gauge invariant ones. A typical example is the curvature perturbation in quasi-de Sitter space arising from the spontaneous breaking of the timelike isometry. We also discuss the features of the local gauge invariant operator when the spacelike isometry is spontaneously broken. Meanwhile, even if the local gauge invariant operators exist, it does not guarantee the reliable construction of the gauge invariant operators on the local region like the island, which is regarded as an essential ingredient to resolve the black hole information paradox. This is because the fluctuation of the spacetime point is accumulated in time, which in fact also gives rise to eternal inflation in quasi-de Sitter space. In order to suppress the fluctuation at late time, the isometry must be strongly broken by the background. In the case of the evaporating black hole, it may be achieved by the transition to the higher dimensional black hole.

South Korea

Min-Seok Seo

Local gauge invariant operator on isometry breaking background

When states in a tower like the Kaluza-Klein or the string tower couple to another state through the irrelevant operators of the same type, their contributions to the loop corrections of the relevant or the marginal operators are not negligible, threatening the perturbativity. This can be avoided provided the cutoff scale is lower than the species scale associated with the irrelevant operator. We apply this to towers of states associated with the neutrino which couple to the Higgs through the Weinberg operator, the dimension-5 irrelevant operator generating the Majorana neutrino mass. Requiring the ‘Majorana species scale’, the species scale associated with the Weinberg operator, to be below the gravitational species scale, one finds the lower bound on the Majorana neutrino mass determined by the species number. The Festina-Lente bound also gives the lower bound on the Majorana neutrino mass, but it is not so stringent. Meanwhile, even if the neutrino mass is of the Dirac type at the renormalizable level, the Majorana mass term still can be written in the effective field theory action so far as the Weinberg operator is not forbidden. Even if the Majorana neutrino mass is larger than the Dirac one, so far as there are sufficient degrees of freedom with mass smaller than the scale of the cosmological constant, the observation of the Majorana nature of the neutrino may not contradict to quantum gravity constraints which rules out the neutrino mass purely given by the Majorana type.

Species scale associated with Weinberg operator and bound on Majorana neutrino mass

As a cutoff scale of quantum gravity, the species scale can be defined by the scale at which the perturbativity of the non-renormalizable gravitational interaction begins to break down. Since it is determined by the number of species in the effective field theory, we can find the close connection to the distance conjecture, which predicts the lowering of the cutoff at the asymptotic limit of the moduli space caused by the descent of a tower of states from UV. Meanwhile, the same kind of the cutoff scale can be obtained from any non-renormalizable interaction, in particular the interaction between the axion and the gauge field through the $\textit{θF}$ ∧ $\textit{F}$ term. Demanding this ‘axion species scale’ not to exceed the gravitational species scale, we obtain the bound (8$\textit{π}$ /$\textit{g}$ )$\textit{f}$ ≲ $\textit{M}$ . This is quite similar to the axion weak gravity conjecture bound, but can be applied to any gauge as well as the string interactions which are relevant to towers of states. We also investigate the implications of the (axion) species scale and the axion weak gravity conjecture-like bound by considering the Peccei-Quinn charge reduction of black hole through the interaction between black hole and the string or wormhole.

Axion species scale and axion weak gravity conjecture-like bound

We study the late time behavior of the scalar part of the volume modulus and the dilaton in stringy quintessence model, focusing on their contributions to the Hubble slow-roll parameter ϵ which directly measures the deviation of the spacetime geometry from de Sitter space. When only one of the moduli is allowed to move, ϵ converges to the stable fixed point at late time. The fixed point value is larger than 1, thus the slow-roll cannot be realized. Moreover, if the decay rate of the quintessence potential is larger than some critical value, the positivity of the potential imposes that the stable fixed point value is just given by 3, independent of the details of the moduli dynamics. Otherwise, the fixed point value coincides with the potential slow-roll parameter. When both the volume modulus and the dilaton roll down the potential simultaneously, we can find the relation between the contributions of two moduli to ϵ satisfied at the fixed point. In this case, the fixed point value is not in general the simple sum of fixed point values in the single field case and cannot be larger than 3.

Asymptotic bound on slow-roll parameter in stringy quintessence model

We study the implication of the distance and the cobordism conjecture on the 6-dimensional Einstein–Maxwell theory compactified on  $$S^2$$  <math> <msup> <mi>S</mi> <mn>2</mn> </msup> </math>  . In this toy model, the radion potential is stabilized by the conspiracy of the curvature of  $$S^2$$  <math> <msup> <mi>S</mi> <mn>2</mn> </msup> </math>   and the flux through  $$S^2$$  <math> <msup> <mi>S</mi> <mn>2</mn> </msup> </math>   parametrized by f, and uplifted by the positive 6-dimensional cosmological constant parametrized by  $$\lambda $$  <math> <mi>λ</mi> </math>  . When  $$\lambda =0$$  <math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math>  , the radion is stabilized at the anti-de Sitter (AdS) vacuum, which cannot be interpolated to the Minkowski vacuum since the Kaluza–Klein (KK) tower descends from UV in the vanishing limit of the 4-dimensional cosmological constant. For nonzero  $$\lambda $$  <math> <mi>λ</mi> </math>   which realizes the metastable de Sitter (dS) vacuum, as well as the AdS and the Minkowski vacuum, such an obstruction can be found provided the combination  $$f^2\lambda $$  <math> <mrow> <msup> <mi>f</mi> <mn>2</mn> </msup> <mi>λ</mi> </mrow> </math>   is fixed and the limit  $$\lambda \rightarrow 0$$  <math> <mrow> <mi>λ</mi> <mo>→</mo> <mn>0</mn> </mrow> </math>   is taken. Moreover, the 6-dimensional Einstein–Maxwell theory allows the transition between vacua through the nucleation of the bubble. In this case, the values of the 4-dimensional cosmological constant inside and outside the bubble are different as f is changed at the bubble wall, while  $$\lambda $$  <math> <mi>λ</mi> </math>   remains unchanged. Regarding the AdS vacuum with the vanishing curvature radius as the ‘nothing’, we find that the transition from the metastable dS vacuum to the nothing is not prevented by the descent of the KK tower since  $$f^2\lambda $$  <math> <mrow> <msup> <mi>f</mi> <mn>2</mn> </msup> <mi>λ</mi> </mrow> </math>   is not fixed.

Kaluza–Klein tower and bubble nucleation in six dimensional Einstein–Maxwell theory

In Type IIB flux compactification realizing the metastable de Sitter (dS) vacuum, the uplift potential can be generated by <math><mover><mrow><mrow><mi>D3</mi></mrow></mrow><mo>‾</mo></mover></math>-branes at the tip of Klebanov-Strassler throat. Then the uplift potential obeys the scaling law with respect to the tower mass scale <math><msub><mrow><mi>m</mi></mrow><mrow><mi>sc</mi></mrow></msub></math>, which can be the Kaluza-Klein (KK) mass scale associated with the throat containing <math><mover><mrow><mrow><mi>D3</mi></mrow></mrow><mo>‾</mo></mover></math>-branes or the bulk tower mass scales, depending on the warping of the throat. On the other hand, in the presence of another throat of stronger warping, the KK mass scale associated with this throat is lower than <math><msub><mrow><mi>m</mi></mrow><mrow><mi>sc</mi></mrow></msub></math>. Nevertheless, the Higuchi bound and the condition that the tower mass scale is higher than the gravitino mass provide the upper bound on <math><msub><mrow><mi>m</mi></mrow><mrow><mi>sc</mi></mrow></msub></math> determined by the lowest tower mass scale (or gravitino mass). This bound also can be interpreted as the lower bound on the lowest tower mass scale determined by <math><msub><mrow><mi>m</mi></mrow><mrow><mi>sc</mi></mrow></msub></math>. We investigate this bound in detail when the throat containing <math><mover><mrow><mrow><mi>D3</mi></mrow></mrow><mo>‾</mo></mover></math>-branes is strongly and weakly warped, respectively.

Bounds on tower mass scales in the presence of throats of different warping

We study the von Neumann algebra description of the inflationary quasi-de Sitter (dS) space. Unlike perfect dS space, quasi-dS space allows the nonzero energy flux across the horizon, which can be identified with the expectation value of the static time translation generator. Moreover, as a dS isometry associated with the static time translation is spontaneously broken, the fluctuation in time is accumulated, which induces the fluctuation in the energy flux. When the inflationary period is given by  $$(\epsilon _H H)^{-1}$$  <math> <msup> <mrow> <mo>(</mo> <msub> <mi>ϵ</mi> <mi>H</mi> </msub> <mi>H</mi> <mo>)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math>   where  $$\epsilon _H$$  <math> <msub> <mi>ϵ</mi> <mi>H</mi> </msub> </math>   is the slow-roll parameter measuring the increasing rate of the Hubble radius, both the energy flux and its fluctuation diverge in the  $$G \rightarrow 0$$  <math> <mrow> <mi>G</mi> <mo>→</mo> <mn>0</mn> </mrow> </math>   limit. Taking the fluctuation in the energy flux and that in the observer’s energy into account, we argue that the inflationary quasi-dS space is described by Type II $$_\infty $$  <math> <msub> <mrow></mrow> <mi>∞</mi> </msub> </math>   algebra. As the entropy is not bounded from above, this is different from Type II $$_1$$  <math> <msub> <mrow></mrow> <mn>1</mn> </msub> </math>   description of perfect dS space in which the entropy is maximized by the maximal entanglement. We also show that our result is consistent with the observation that the von Neumann entropy for the density matrix reflecting the fluctuations above is interpreted as the generalized entropy.

Von Neumann algebra description of inflationary cosmology

The distance conjecture claims that as the modulus traverses along the trans-Planckian geodesic distance, the effective field theory becomes invalid by a descent of a tower of states from UV. Moreover, according to the recent (strong version of) emergence proposal, the kinetic term of the modulus is entirely generated by the wavefunction renormalization in which a tower of states are integrated out. Assuming these two conjectures, we explore the role of a tower of states coupled to the modulus in (in)stability of the de Sitter (dS) vacuum by studying the one-loop effective potential generated by a tower of states. We find that a fermionic tower of states makes the effective potential more or less consistent with the dS swampland conjecture: either the slope or the curvature of the potential is sizeable. In contrast, the effective potential generated by a bosonic tower of states seems to allow the stable dS vacuum. Therefore, in order to argue the instability of the dS vacuum, the additional ingredient like supersymmetry breaking needs to be taken into account.

(In)stability of de Sitter vacuum in light of distance conjecture and emergence proposal

We investigate the connection between the distance conjecture and the uplift potential. For this purpose, we consider the concrete model, the warped deformed conifold embedded into Type IIB flux compactifications, with the uplift potential produced by   <math> <mover> <mrow> <mi>D</mi> <mn>3</mn> </mrow> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{D}3} $$ -branes at the tip of the throat. Whereas the various mass scales associated with towers of states can be found, it turns out that the lightest tower mass scale satisfies the scaling behavior with respect to the uplift potential, which is meaningful provided the number of   <math> <mover> <mrow> <mi>D</mi> <mn>3</mn> </mrow> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{D}3} $$ -branes is nonzero. This indicates that the effective theory becomes invalid in the vanishing limit of the uplift potential by the descent of an infinite tower of states from UV, as predicted in the distance conjecture. Since too large uplift potential is also problematic due to the runaway behavior of the moduli potential as well as the sizeable backreaction of   <math> <mover> <mrow> <mi>D</mi> <mn>3</mn> </mrow> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{D}3} $$ -branes, the uplift potential is bounded from both above and below. In the simple model like the KKLT or the large volume scenario in which non-perturbative effect is dominatd by the single term, this bound can be rewritten as the bound on the size of the superpotential.

Uplift and towers of states in warped throat

Whereas a static observer in de Sitter (dS) space detects thermal radiation emitted by the horizon, the dS isometries impose that the radiation is in equilibrium with the background. This implies that for the static observer to find the information paradox, the background must be deformed to quasi-dS space in which the dS isometries are spontaneously broken. We study the condition that the information paradox arises in quasi-dS space with the monotonically increasing horizon size which is used to describe the inflationary cosmology. For this purpose, the dimensional reduction of three-dimensional dS space with thermal radiation modelled by the JT gravity coupled to CFT is considered. We argue that when the central charge monotonically increases in time, the information paradox arises but the conditions for the existence of the island become more restrictive. As the central charge can be interpreted as the number of degrees of freedom, the absence of the island in quasi-dS space supports the entropy argument for the dS swampland conjecture.