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Area metrics and area metric backgrounds provide a unified framework for quantum gravity. They encode physical degrees of freedom beyond those of a metric. These non-metric degrees of freedom must be suppressed by a potential at sufficiently high energy scales to ensure that in the infrared regime classical gravity is recovered. On this basis, we first study necessary and sufficient algebraic conditions for an area metric to be induced by a metric. Second, we consider candidate potentials for the area metric and point out a possible connection between the reduction of area metric geometry to metric geometry on the one hand, and the smallness of the cosmological constant on the other. Finally, we consider modifications of the Nambu-Goto action for a string, from a metric background to an area metric background. We demonstrate that area metric perturbations introduce an interaction corresponding to a singular vertex operator in the classically equivalent Polyakov action. The implications of these types of vertex operators for the quantum theory remain to be understood., Comment: v1: 17 pages; v2: minor modifications, matches version published in PRD

We study the implications of the stringy space-time uncertainty relation (STUR) for inflationary cosmology. By demanding that no fluctuation modes that exit the Hubble radius are affected by the nonlocality resulting from the STUR, we find an upper bound on the number of e-foldings of inflation. The bound is a factor of 2 weaker than what results from the Trans-Planckian Censorship Criterion (TCC). By demanding that the inflationary phase is simultaneously consistent with STUR and sufficiently long for inflation to provide a causal explanation of structure on the scale of the current Hubble radius, we find an upper bound on the energy scale of inflation. The bound is less restrictive than what follows from the TCC, but it remains in conflict with canonical single-field inflation models., Comment: 7 pages, 1 figure

In string theories, interactions are exponentially suppressed for trans-Planckian space-like external momenta. We study a class of quantum field theories that exhibit this feature modeled after Witten's bosonic open string field theory, and discover a Lorentz-invariant UV/IR relation that leads to the spacetime uncertainty principle proposed by Yoneya. Application to a dynamical black hole background suggests that Hawking radiation is turned off around the scrambling time., Comment: 27 pages, 1 figure; references and comments added, updated to match the published version

The generalized uncertainty relation is expected to be an essential element in a theory of quantum gravity. In this work, we examine its effect on the Hawking radiation of a Schwarzschild black hole formed from collapse by incorporating a minimal uncertainty length scale into the radial coordinate of the background. This is implemented in both the ingoing Vaidya coordinates and a family of freely falling coordinates. We find that, regardless of the choice of the coordinate system, Hawking radiation is turned off at around the scrambling time. Interestingly, this phenomenon occurs while the Hawking temperature remains largely unmodified., Comment: 23 pages, 1 figure; v2: references added; v3: published version with more references and added clarifications

We revisit the connection between Hawking radiation and high-frequency dispersions for a Schwarzschild black hole following the work of Brout et al.. After confirming the robustness of Hawking radiation for monotonic dispersion relations, we consider non-monotonic dispersion relations that deviate from the standard relation only in the trans-Planckian domain. Contrary to the common belief that Hawking radiation is insensitive to UV physics, it turns out that Hawking radiation is subject to significant modifications after the scrambling time. Depending on the UV physics at the singularity, the amplitude of Hawking radiation could diminish after the scrambling time, while the Hawking temperature remains the same. Our finding is thus not contradictory to earlier works regarding the robustness of Hawking temperature., Comment: 24 pages, 8 figures; v2: minor changes, references added; v3: relation to Parikh-Wilczek tunneling clarified, factors of 2 corrected

We study the interior metric of 4D spherically symmetric static black holes by using the semi-classical Einstein equation and find a consistent class of geometries with large curvatures. We approximate the matter fields by conformal fields and consider the contribution of the 4D Weyl anomaly, giving a state-independent constraint. Combining this with an equation of state yields an equation that determines the interior geometry completely. We explore the solution space of the equation in a non-perturbative manner for $\hbar$. First, we find four types of asymptotic behaviors and examine the general features of the solutions. Then, by imposing physical conditions, we obtain approximately a general class of interior geometries: various combinations of dilute and dense structures without a horizon or singularity. This represents the diversity of the interior structure. Finally, we show that the number of possible patterns of such interior geometries corresponds to the Bekenstein-Hawking entropy., Comment: 32 pages, many figures

We study the time-dependence of Hawking radiation for a black hole in the Unruh vacuum, and find that it is not robust against certain UV and IR effects. If there is a UV cutoff at the Planck scale, Hawking radiation is turned off after the scrambling time. In the absence of a UV cutoff, Hawking radiation is sensitive to the IR cutoff through a UV/IR connection due to higher-derivative interactions in the effective theory. Furthermore, higher-derivative interactions with the background contribute to a large amplitude of particle creation that changes Hawking radiation. This unexpected large effect is related to a peculiar feature of the Hawking particle wave packets., Comment: 35 pages, 5 figures, references added, minor modification

In the background of a gravitational collapse, we compute the transition amplitudes for the creation of particles for distant observers due to higher-derivative interactions in addition to Hawking radiation. The amplitudes grow exponentially with time and become of order 1 when the collapsing matter is about a Planck length outside the horizon. As a result, the effective theory breaks down at the scrambling time, invalidating its prediction of Hawking radiation. Planckian physics comes into play to decide the fate of black-hole evaporation., Comment: 24 pages, no figure, reference added, minor modification

Unitarity demands that the black-hole final state (what remains inside the event horizon at complete evaporation) must be unique. Assuming a UV theory with infinitely many fields, we propose that the uniqueness of the final state can be achieved via a mechanism analogous to the quantum-mechanical description of dissipation., Comment: 12 pages, LaTeX, 1 figure

In the matrix model approaches of string/M theories, one starts from a generic symmetry $gl(\infty)$ to reproduce the space-time manifold. In this paper, we consider the generalization in which the space-time manifold emerges from a gauge symmetry algebra which is not necessarily $gl(\infty)$. We focus on the second nontrivial example after the toroidal compactification, the coset space $G/H$, and propose a specific infinite-dimensional symmetry which realizes the geometry. It consists of the gauge-algebra valued functions on the coset and Lorentzian generator pairs associated with the isometry. We show that the $0$-dimensional gauge theory with the mass and Chern-Simons terms gives the gauge theory on the coset with scalar fields associated with $H$., Comment: 29 pages

Assuming the standard effective-field-theoretic formulation of Hawking radiation, we show explicitly how a generic effective theory predicts a firewall from an initially uneventful horizon for a spherically symmetric, uncharged black hole in $D$ dimensions for $D \geq 4$. The firewall is created via higher-derivative interactions within the scrambling time after the collapsing matter enters the trapping horizon. This result manifests the trans-Planckian problem of Hawking radiation and demonstrates the incompatibility between Hawking radiation and the uneventful horizon., Comment: 14 pages, 1 figure

For an effective field theory in the background of an evaporating black hole with spherical symmetry, we consider non-renormalizable interactions and their relevance to physical effects. The background geometry is determined by the semi-classical Einstein equation for an uneventful horizon where the vacuum energy-momentum tensor is small for freely falling observers. Surprisingly, after Hawking radiation appears, the transition amplitude from the Unruh vacuum to certain multi-particle states grows exponentially with time for a class of higher-derivative operators after the collapsing matter enters the near-horizon region, despite the absence of large curvature invariants. Within the scrambling time, the uneventful horizon transitions towards a firewall, and eventually the effective field theory breaks down., Comment: 37 pages, 2 figures

Assuming that the vacuum energy-momentum tensor is not exceptionally large, we consider 4D evaporating black holes with spherical symmetry and evaluate the proper distance $\Delta L$ between the time-like apparent horizon and the surface of the collapsing matter after it has entered the apparent horizon. We show that $\Delta L$ can never be larger than $\mathcal{O}(n^{3/2}\ell_p)$ when the black hole is evaporated to $1/n$ of its initial mass, as long as $n \ll a^{2/3}/\ell_p^{2/3}$ (where $a$ is the Schwarzschild radius and $\ell_p$ is the Planck length). For example, the distance between the matter and the apparent horizon must be Planckian at the Page time., Comment: 16 pages

We study analytically the spacetime geometry of the black-hole formation and evaporation. As a simplest model of the collapse, we consider a spherical thin shell, and take the back-reaction from the negative energy of the quantum vacuum state. For definiteness, we will focus on quantum effects of s-waves. We obtain an analytic solution of the semi-classical Einstein equation for this model, that provides an overall description of the black hole geometry form the formation to evaporation. As an application of this result, we find its interesting implication that, after the collapsing shell enters the apparent horizon, the proper distance between the shell and the horizon remains as small as the Planck length even when the difference in their areal radii is of the same order as the Schwarzschild radius. The position of the shell would be regarded as the same place to the apparent horizon in the semi-classical regime of gravity., Comment: 42 pages, 8 figures, minor modification

Assuming spherical symmetry and the semi-classical Einstein equation, we prove that, for the observers on top of the trapping horizon, the vacuum energy-momentum tensor is always that of an ingoing negative energy flux at the speed of light with a universal energy density ${\cal E} \simeq - 1/(2\kappa a^2)$, (where $a$ is the areal radius of the trapping horizon), which is responsible for the decrease in the black hole's mass over time. This result is independent of the composition of the collapsing matter and the details of the vacuum energy-momentum tensor. The physics behind the universality of this quantity ${\cal E}$ and its surprisingly large magnitude will be discussed., Comment: 17 pages, 3 figures

For a black hole of Schwarzschild radius $a$, we argue that the back-reaction of the vacuum energy-momentum tensor is in general important at the apparent horizon when the time scale of a process is larger than order $a$. In particular, in a double-shell model, we show that the ignorance of the back-reaction leads to a divergence in the outgoing vacuum energy flux. The main result of this paper is that, once the back-reaction is included, the vacuum energy density for observers on top of the trapping horizon in vacuum is given by ${\cal E} \simeq -1/2\ell_p^2 a^2$, which is of the same order of magnitude as the classical matter but opposite in sign. Remarkably, this formula is independent of both the details about the collapsing matter and the vacuum energy-momentum tensor., Comment: latex,17 pages, 6 figures

Following the work of Kawai, Matsuo, and Yokokura, we study the dynamical collapsing process with spherical symmetry in the time-dependent space-time background including the back reaction of Hawking radiation. We show that in this model there are two classes of asymptotic solutions. One of the two classes is known previously. These states have the slope $\partial a/\partial r$ approximately equal to 1. The other class of asymptotic solutions is that of shells with a small thickness. We emphasize that these thin shells should be properly understood as configurations in the low-energy effective theory. They behave characteristically differently from the singular states of ideal thin shells of zero thickness., Comment: 34 pages, latex, minor modification

Static, spherically symmetric solutions to the semi-classical Einstein equation are studied, including the effect of the quantum energy-momentum tensor for conformal matters with 4D Weyl anomaly. Through both perturbative and non-perturbative methods, we show that the quantum effect can play a crucial role in shaping the near-horizon geometry, and that the existence of the horizon requires fine-tuning., Comment: 35 pages, minor modification

The near-horizon geometry of evaporation black holes is determined according to the semi-classical Einstein equation. We consider spherically symmetric configurations in which the collapsing star has already collapsed below the Schwarzschild radius. The back-reaction of the vacuum energy-momentum, including Hawking radiation, is taken into account. The vacuum energy-momentum plays a crucial role in a small neighborhood of the apparent horizon, as it appears at the leading order in the semi-classical Einstein equation. Our study is focused on the time-dependent geometry in this region., Comment: 23 pages, 3 figures

With the back reaction of the vacuum energy-momentum tensor consistently taken into account, we study static spherically symmetric black-hole-like solutions to the semi-classical Einstein equation. The vacuum energy is assumed to be given by that of 2-dimensional massless scalar fields, as a widely used model in the literature for black holes. The solutions have no horizon. Instead, there is a local minimum in the radius. We consider thin shells as well as incompressible fluid as the matter content of the black-hole-like geometry. The geometry has several interesting features due to the back reaction of vacuum energy. In particular, Buchdahl's inequality can be violated without divergence in pressure, even if the surface is below the Schwarzschild radius. At the same time, the surface of the star can not be far below the Schwarzschild radius for a density not much higher than the Planck scale, and the proper distance from its surface to the origin can be very short even for very large Schwarzschild radius. The results also imply that, contrary to the folklore, in principle the Boulware vacuum can be physical for black holes., Comment: 44 pages, 20 figures. v2: 49 pages, minor modifications, references added

We study spherically symmetric static solutions to the semi-classical Einstein equation sourced by the vacuum energy of quantum fields in the curved space-time of the same solution. We found solutions that are small deformations of the Schwarzschild metric for distant observers, but without horizon. Instead of being a robust feature of objects with high densities, the horizon is sensitive to the energy-momentum tensor in the near-horizon region., Comment: 40 pages, 14 figures. v2: 43 pages, minor modifications, references added

Following earlier works on the KMY model of black-hole formation and evaporation, we construct the metric for a matter sphere in gravitational collapse, with the back-reaction of pre-Hawking radiation taken into consideration. The mass distribution and collapsing velocity of the matter sphere are allowed to have an arbitrary radial dependence. We find that a generic gravitational collapse asymptote to a universal configuration which resembles a black hole but without horizon. This approach clarifies several misunderstandings about black-hole formation and evaporation, and provides a new model for black-hole-like objects in the universe., Comment: 34 pages, references added

Nambu proposed an extension of dynamical system through the introduction of a new bracket (Nambu bracket) in 1973. This article is a short review of the developments after his paper. Some emphasis are put on a viewpoint that the Nambu bracket naturally describes extended objects which appear in M-theory and the fluid dynamics. The latter part of the paper is devoted to a review of the studies on the Nambu bracket (Lie 3-algebra) in Bagger-Lambert-Gustavsson theory of multiple M2-branes. This paper is a contribution to the proceedings of Nambu memorial symposium (Osaka City University, September 29, 2015)., Comment: 25 pages, contribution to the proceedings of Nambu memorial symposium, Osaka City University, September 29, 2015, v2 minor modifications

With the back-reaction of Hawking radiation taken into consideration, the work of Kawai, Matsuo and Yokokura has shown that, under a few assumptions, the collapse of matter does not lead to event horizon nor apparent horizon. In this paper, we relax their assumptions and elaborate on the space-time geometry of a generic collapsing body with spherical symmetry. The geometry outside the collapsing sphere is found to be approximated by the geometry outside the white-hole horizon, hence the collapsing matter remains outside the Schwarzschild radius. As particles in Hawking radiation are created in the vicinity of the collapsing matter, the information loss paradox is alleviated. Assuming that the collapsing body evaporates within finite time, there is no event horizon., Comment: 28 pages, minor modification

To define a free string by the Nambu-Goto action, all we need is the notion of area, and mathematically the area can be defined directly in the absence of a metric. Motivated by the possibility that string theory admits backgrounds where the notion of length is not well defined but a definition of area is given, we study space-time geometries based on the generalization of metric to area metric. In analogy with Riemannian geometry, we define the analogues of connections, curvatures and Einstein tensor. We propose a formulation generalizing Einstein's theory that will be useful if at a certain stage or a certain scale the metric is ill-defined and the space-time is better characterized by the notion of area. Static spherical solutions are found for the generalized Einstein equation in vacuum, including the Schwarzschild solution as a special case., Comment: latex, 34 pages, minor modification

In an earlier work, Kawai et al proposed a model of black-hole formation and evaporation, in which the geometry of a collapsing shell of null dust is studied, including consistently the back reaction of its Hawking radiation. In this note, we illuminate the implications of their work, focusing on the resolution of the information loss paradox and the problem of the firewall., Comment: 16 pages, 6 figures, references added

We propose a generalization of Yang-Mills theory for which the symmetry algebra does not have to be factorized as mutually commuting algebras of a finite-dimensional Lie algebra and the algebra of functions on base space. The algebra of diffeomorphism can be constructed as an example, and a class of gravity theories can be interpreted as generalized Yang-Mills theories. These theories in general include a graviton, a dilaton and a rank-2 antisymmetric field, although Einstein gravity is also included as a special case. We present calculations suggesting that the connection in scattering amplitudes between Yang-Mills theory and gravity via BCJ duality can be made more manifest in this formulation., Comment: 25 pages, Latex, reference added, minor modification

A supersymmetric non-Abelian self-dual gauge theory with the explicit introduction of Kaluza-Klein modes is proposed to give a classical description of multiple M5-branes on $R^5 \times S^1$. The gauge symmetry is parametrized by Lie-algebra valued 1-forms with the redundancy of a 0-form, and the supersymmetry transformations without gauge-fixing are given. We study BPS configurations involving KK modes, including M-waves and M2-branes with non-trivial distributions around the circle. Finally, this supersymmetric gauge theory of two-forms can be equipped with more general non-Abelian gerbes in five dimensions., Comment: 34 pages, minor modification

We construct the low-energy effective field theory for a D3-brane in constant R-R 2-form potential background as the S-dual theory of a D3-brane in NS-NS B-field background. Despite the non-Abelian algebra of the noncommutative U(1) gauge symmetry, the electromagnetic duality transformation can be carried out to all orders, and the dual Lagrangian is given in a compact form. The gauge algebra is found to be a mixture of a deformed area-preserving diffeomorphism and the usual U(1) gauge symmetry., Comment: 29 pages, minor changes

We revisit the problem of quantizing a chiral boson on a torus. The conventional approach is to extract the partition function of a chiral boson from the path integral of a non-chiral boson. Instead we compute it directly from the chiral boson Lagrangian of Floreanini and Jackiw modified by topological terms involving auxiliary fields. A careful analysis of the gauge-fixing condition for the extra gauge symmetry reproduces the correct results for the free chiral boson, and has the advantage of being applicable to a wider class of interacting chiral boson theories., Comment: 31 pages, minor modification

We construct the low energy effective action for the bosonic sector on a Dp-brane in large constant RR (p-1)-form field background. The action is invariant under both U(1) gauge symmetry and the volume-preserving diffeomorphism characterizing the RR-field background. Scalar fields representing transverse coordinates of the Dp-brane are included. It also respects T-duality and is consistent with the action for M5-brane in C-field background., Comment: 16 pages, reference list corrected

Motivated by application to multiple M5 branes, we study some properties of non-Abelian two-form gauge theories. We note that the fake curvature condition which is commonly used in the literature would restrict the dynamics to be either a free theory or a topological system. We then propose a modification of transformation law which simplifies the gauge transformation of 3-form field strength and enables us to write down a gauge invariant action. We then argue that a generalization of Stueckelberg mechanism naturally gives mass to the two-form gauge field. For the application to multiple M5-branes, it should be identified with the KK modes., Comment: 21 pages

We extensively study BPS solutions of the low energy effective theory of M5-brane in large C-field background. This provides us an opportunity to explore the interactions turned on by C-field background through the Nambu-Poisson structure. The BPS states considered in this paper include the M-waves, the self-dual string (M2 ending on M5), tilted M5-brane, holomorphic embedding of M5-brane and the intersection of two M5-branes along a 3-brane., Comment: 25 pages, reference added

We construct a non-Abelian gauge theory of chiral 2-forms (self-dual gauge fields) in 6 dimensions with a spatial direction compactified on a circle of radius R. It has the following two properties. (1) It reduces to the Yang-Mills theory in 5 dimensions for small R. (2) It is equivalent to the Lorentz-invariant theory of Abelian chiral 2-forms when the gauge group is Abelian. Previous no-go theorems prohibiting non-Abelian deformations of the chiral 2-form gauge theory are circumvented by introducing nonlocality along the compactified dimension., Comment: 23 pages, reference added

The purpose of this paper is to understand the low energy effective theory of a Dp-brane in the background of a large constant R-R (p-1)-form field. We start with the M5-brane theory in large C-field background. The C-field background defines a 3-dimensional volume form on an M5-brane, and it is known that the low energy M5-brane theory can be described as a Nambu-Poisson gauge theory with the volume-preserving diffeomorphism symmetry (VPD). Via a double dimensional reduction we obtain a D4-brane in R-R 3-form field background. This theory has both the usual U(1) gauge symmetry and the new symmetry of VPD. We find that the gauge potential for VPD is electric-magnetic dual to the U(1) gauge potential, sharing the same physical degrees of freedom. The result can be generalized to Dp-branes., Comment: 26 pages, latex, minor modification in the introduction section

We consider D1-string in a constant R-R 3-form flux background and analyze its low energy limit. The leading order low energy theory has reparametrization symmetry and is a generalization of an earlier work by Takhtajan. We show that the dynamical evolution of the theory takes a generalized Hamiltonian form in terms of a Nambu bracket. This description is formulated in terms of reparametrization invariant quantities and requires no fixing of the reparametrization symmetry. We also show that a Nambu- Poisson (p + 2)-bracket arises naturally in the reparametrization invariant description of the low energy theory of a p-brane in a constant (p + 2)-form flux background. For example, our results apply for a fundamental string in a constant NS-NS 3-form flux H3 and an M2-brane in a constant 4-form flux F4., Comment: 30 pages, latex, references added

In previous works, we constructed UV-finite and unitary scalar field theories with an infinite spectrum of propagating modes for arbitrary polynomial interactions. In this paper, we introduce infinitely many massive vector fields into a U(1) gauge theory to construct a theory with UV-finiteness and unitarity., Comment: 25 pages

We continue our investigation on the Nambu-Poisson description of M5-brane in a large constant C-field background (NP M5-brane theory) constructed in Refs.[1, 2]. In this paper, the low energy limit where the NP M5-brane theory is applicable is clarified. The background independence of the NP M5-brane theory is made manifest using the variables in the BLG model of multiple M2-branes. An all order solution to the Seiberg-Witten map is also constructed., Comment: expanded explanations, minor corrections and typos corrected

In this work, we study Lagrangian formulations for self-dual gauge theories, also known as chiral $n$-form gauge theories, for $n = 2p$ in $D = 4p+2$ dimensional spacetime. Motivated by a recent formulation of M5-branes derived from the BLG model, we generalize the earlier Lagrangian formulation based on a decomposition of spacetime into $(D-1)$ dimensions plus a special dimension, to construct Lagrangian formulations based on a generic decomposition of spacetime into $D'$ and $D" = D - D'$ dimensions. Although the Lorentz symmetry is not manifest, we prove that the action is invariant under modified Lorentz transformations., Comment: 27 pages, the modified Lorentz transformation rule revised, minor modifications

The BLG model for multiple M2-branes motivates an M5-brane theory with a novel gauge symmetry defined by the Nambu-Poisson structure. This Nambu-Poisson gauge symmetry for an M5-brane in large C-field background can be matched, on double dimension reduction, with the Poisson limit of the noncommutative gauge symmetry for a D4-brane in B-field background. Naively, one expects that there should exist a certain deformation of the Nambu-Poisson structure to match with the full noncommutative gauge symmetry including higher order terms. However, We prove the no-go theorem that there is no way to deform the Nambu-Poisson gauge symmetry, even without assuming the existence of a deformation of Nambu-Poisson bracket, to match with the noncommutative gauge symmetry in 4+1 dimensions to all order, regardless of how the double dimension reduction is implemented., Comment: v4: minor modifications.

We give a concise review of recent developments of the M5-brane theory derived from the Bagger-Lambert model. It is a 6 dimensional supersymmetric self-dual gauge theory describing an M5-brane in a large constant $C$-field background. The non-Abelian gauge symmetry corresponds to diffeomorphisms preserving an exact 3-form on the worldvolume, which defines a Nambu-Poisson bracket. Various interesting geometric and algebraic properties of the theory are discussed., Comment: 18 pages, minor modification, references added

In this article we give a concise review of recent progress in our understanding of the Lie 3-algebra and their application to the Bagger-Lambert-Gustavsson model describing multiple M2-branes in M theory., Comment: 13 pages, prepared for The Fourth International Symposium on Symmetries in Subatomic Physics (SSP 2009), minor modification, references added

Following a previous work (hep-th/0410248), where a scalar field theory with a modified propagator and phi^4 interaction in 4 dimensions is constructed to be UV-finite, unitary and Lorentz invariant, we discuss in this paper general phi^n theory in arbitrary even space-time dimensions. We show that the theory is still UV-finite, unitary and Lorentz invariant if the propagators are chosen to meet certain simple conditions depending on the space-time dimension but independent of n. We also comment that our model is reminiscent of string theory in the way UV divergence is avoided., Comment: 18 pages

We construct a class of Lie 3-algebras with an arbitrary number of pairs of generators with Lorentzian signature metric. Some examples are given and corresponding BLG models are studied. We show that such a system in general describes a supersymmetric massive vector multiplets after the ghost fields are Higgsed. Simple systems with nontrivial interaction are realized by infinite dimensional Lie 3-algebras associated with the loop algebras. The massive fields are then naturally identified with the Kaluza-Klein modes by the toroidal compactification triggered by the ghost fields. For example, Dp-brane with an (infinite dimensional) affine Lie algebra symmetry $\hat g$ can be identified with D(p+1)-brane with gauge symmetry $g$., Comment: 39 pages; v2: minor corrections, reference added